Swaps and Other Derivatives (With CD-ROM) (The Wiley Finance Series)

Swaps and Other Derivatives

Wiley Finance Series

Securities Operations: A Guide to Trade and Position Management Michael Simmons Monte Carlo Methods in Finance Peter Jackel Modeling, Measuring and Hedging Operational Risk Marcelo Cruz Building and Using Dynamic Interest Rate Models Ken Kortanek and Vladimir Medvedev Structured Equity Derivatives: The Definitive Guide to Exotic Options and Structured Notes Harry Kat Advanced Modelling in Finance using Excel and VBA Mary Jackson and Mike Staunton Operational Risk: Measurement and Modelling Jack King Advanced Credit Risk Analysis: Financial Approach and Mathematical Models to Assess, Price and Manage Credit Risk Didier Cossin and Hugues Pirotte Dictionary of Financial Engineering John F. Marshall Pricing Financial Derivatives: The Finite Difference Method Domingo A. Tavella and Curt Randall Interest Rate Modelling Jessica James and Nick Webber Handbook of Hybrid Instruments: Convertible Bonds, Preferred Shares, Lyons, ELKS, DECS and Other Mandatory Convertible Notes Izzy Nelken (ed.) Options on Foreign Exchange, Revised Edition David F. DeRosa The Handbook of Equity Derivatives, Revised Edition Jack Francis, William Toy and J. Gregg Whittaker Volatility and Correlation in the Pricing of Equity, FX and Interest-rate Options Riccardo Rebonato Risk Management and Analysis vol. 1: Measuring and Modelling Financial Risk Carol Alexander (ed.) Risk Management and Analysis vol. 2: New Markets and Products Carol Alexander (ed.) Implementing Value at Risk Philip Best Credit Derivatives: A Guide to Instruments and Applications Janet Tavakoli Implementing Derivatives Models Les Clewlow and Chris Strickland Interest-rate Option Models: Understanding, Analysing and Using Models for Exotic Interest-rate Options (second edition) Riccardo Rebonato

Swaps and Other Derivatives

Richard Flavell

JOHN WILEY & SONS, LTD

Copyright © 2002 John Wiley & Sons, Ltd, Baffins Lane, Chichester, West Sussex PO19 1UD, UK National 01243 779777 International (+ 44) 1243 779777 e-mail (for orders and customer service enquiries): cs-books @wiley.co.uk Visit our Home Page on http://www.wiley.co.uk All Rights Reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, scanning or otherwise, except under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency, 90 Tottenham Court Road, London W1P 9HE, UK, without the permission in writing of the publisher. Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158–0012, USA WILEY-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany John Wiley & Sons Australia Ltd, 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd, 2 Clementi Loop #02–01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons (Canada) Ltd, 22 Worcester Road, Rexdale, Ontario M9W IL1, Canada

British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 0 471 49589 1

Typeset in 10/12pt Times from the author's disks by Dobbie Typesetting Limited, Tavistock, Devon Printed and bound in Great Britain by Antony Rowe Ltd, Chippenham, Wilts. This book is printed on acid-free paper responsibly manufactured from sustainable forestry. in which at least two trees are planted for each one used for paper production.

Contents Preface and Acknowledgements

ix

1

Introduction 1.1 Introduction 1.2 Applications of swaps 1.3 An overview of the swap market 1.4 The evolution of a swap market 1.5 Conclusion

1 1 3 6 8 10

2

Short-term interest rate swaps Objective 2.1 Discounting, the time value of money and other matters 2.2 Forward rate agreements and interest rate futures 2.3 Short-term swaps 2.4 Future valuing a swap

11 11 11 16 20 31

3

Generic interest rate swaps Objective 3.1 Generic interest rate swaps 3.2 Pricing through comparative advantage 3.3 The relative pricing of generic IRS 3.4 The relationship between the bond and swap markets 3.5 Implying a discount function 3.6 Building a blended curve

35 35 35 38 41 43 50 58

4

The pricing and valuation of non-generic swaps Objective 4.1 The pricing of simple non-generic swaps 4.2 Rollercoasters 4.3 A more complex example 4.4 An alternative to discounting 4.5 Swap valuation

65 65 65 72 75 85 85

5

More complex swaps Objective 5.1 Asset packaging

95 95 95

Contents

5.2 Credit swaps 5.3 Credit-adjusted swap pricing 5.4 Simple mismatch swaps 5.5 Average rate swaps 5.6 Overnight indexed swaps 5.7 Basis swaps 5.8 Yield curve swaps 5.9 Convexity effects of swaps 5.10 Inflation swaps 5.11 Equity and commodity swaps 5.12 Volatility swaps Appendix Measuring the convexity effect

106 121 128 129 131 137 142 152 156 165 175 184

6

Cross-currency swaps Objective 6.1 Floating-floating cross-currency swaps 6.2 Pricing and hedging of CCBS 6.3 CCBS and discounting 6.4 Fixed-floating cross-currency swaps 6.5 Floating-floating swaps continued 6.6 Fixed-fixed cross-currency swaps 6.7 Cross-currency swaps valuation 6.8 Dual currency swaps 6.9 Cross-currency equity swaps 6.10 Conclusion Appendix Adjustments to the pricing of a quanto diff swap

205 205 205 207 211 224 229 234 241 247 259 262 262

7

Interest rate OTC options Objective 7.1 Introduction 7.2 The Black option pricing model 7.3 Interest rate volatility 7.4 Par and forward volatilities 7.5 Caps, floors and collars 7.6 Digital options 7.7 Embedded structures 7.8 More complex structures 7.9 Swaptions 7.10 Structures with embedded swaptions 7.11 FX options 7.12 Hedging FX options

267 267 267 268 271 277 288 299 300 307 309 316 320 326

8

Traditional market risk management Objective 8.1 Introduction 8.2 Interest rate risk management 8.3 Gridpoint risk management — market rates

333 333 333 336 337

mtents

vii

9

Index

8.4 Equivalent portfolios 8.5 Gridpoint risk management — forward rates 8.6 Gridpoint risk management — zero coupon rates 8.7 Yield curve risk management 8.8 Swap futures 8.9 Theta risk 8.10 Risk management of IR option portfolios 8.11 Hedging of inflation swaps Appendix Analysis of swap curves

337 340 344 347 355 360 362 373 375

Imperfect risk management Objective 9.1 Introduction 9.2 A very simple example 9.3 A very simple example extended 9.4 Multifactor delta VaR 9.5 Choice of risk factors and cashflow mapping 9.6 Estimation of volatility and correlations 9.7 A running example 9.8 Simulation methods 9.9 Shortcomings and extensions to simulation methods 9.10 Delta-gamma and other methods 9.11 Spread VaR 9.12 Equity VaR 9.13 Stress testing Appendix Extreme value theory

379 379 379 380 386 388 394 399 401 405 414 427 433 439 441 444 447

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Preface and Acknowledgements This book is designed for financial professionals to understand how the vast bulk of OTC derivatives are structured, priced and hedged, and ultimately how to use such derivatives themselves. A wide range of books already exist that describe in conceptual terms how and why such derivatives are used, and it is not the ambition of this book to supplant them. There are also a number of books which describe the pricing and hedging of derivatives, especially exotic ones, primarily in mathematical terms. Whilst exotics are an important and growing segment of the market, by far the majority of derivatives are still very much first generation, and as such relatively straightforward. For example, interest rate swaps constitute over half of the $100 trillion OTC derivative market, and yet there have been few books published in the last decade that describe how they are created and valued in practical detail. So how do many of the professionals gain their knowledge? One popular way is "learning on the job", reinforced by the odd training course. But swap structures can be quite complex, requiring more than just superficial knowledge, and probably every professional uses a computer-based system, certainly for the booking and regular valuation of trades, and most likely for their initial pricing and risk management. These systems are complex, having to deal with real-world situations, and their practical inner details bear little resemblance to the idealized world of most books. So often practitioners tend to treat the systems as black boxes, relying on some initial and frequently inadequate range of tests and hoping their intuition will guide them. The greatest sources of comfort are often the existing customer list of the system (they can't all be wrong!) and, if the system is replacing an old one, comparative valuations. The objective of this book is to describe how the pricing, valuation and risk management of generic OTC derivatives may be performed, in sufficient detail and with various alternatives, so that the approaches may be applied in practice. It is based upon some 15 years of varying experience as a financial engineer for ANZ Merchant Bank in London, as a trainer and consultant to banks worldwide, and as Director of Financial Engineering at Lombard Risk Systems responsible for all the mathematics in the various pricing and risk management systems. The audience for the book is firstly traders, sales people and front-line risk managers. But increasingly such knowledge needs to be more widely spread within financial institutions, such as internal audit, risk control and IT. Then there are the counterparties such as organizations using derivatives for risk management, who have frequently identified the need for transparent pricing. This need has been exacerbated in recent years as many developed countries now require that these organizations demonstrate the effectiveness of risk management, and also perform regular (usually annual) mark-to-

Preface and Acknowledgements

market. Similarly, organizations using complex funding structures want to understand how the structures are created and priced. Turning to the other side, many fund managers and in particular hedge funds are also using derivatives to manage their risk profile, and then to report using one of the value-at-risk techniques. This has been particularly true since the collapse of Long Term Capital Management, despite the fact that most implementations of VaR would not have recognized the risk. Other potential readers are the auditors, consultants and regulators of the banks and their client organizations. Institutions offer derivatives with a wide range of maturities, ranging from a few hours (used to provide risk management over the announcement of an economic figure) to perpetuals (i.e. no upfront maturity defined). There is however a golden rule when pricing derivatives, namely, always price them off the market that will be used to hedge them. This leads to the first separation in the interest rate swap market between the following. Chapter 2. The short end of the curve, which uses cash, futures and occasionally FRAs to hedge swaps. This chapter first discusses the derivation of discount factors from cash rates, and concentrates on the range of alternative approaches that may be used. It then looks at the derivation of forward interest rates, and how FRAs may be priced using cash and futures. The convexity effect is highlighted for future discussion. Finally an approach is introduced that does not require discounting, but permits the introduction of a funding cost. Chapter 3. The medium to long end of the curve. The highly liquid inter-bank market typically trades plain swaps (usually known as "generic" or "vanilla"), very often between market makers and intermediaries. These are hedged in other financial markets, typically futures for the shorter exposures and bonds for the longer ones. This chapter concentrates initially on the relationship between the bond and swap markets, and how generic swap prices may be implied. It concludes by developing various techniques for the estimation of discount factors from a generic swap curve. Chapter 4. The end-user market provides customers with tailored (i.e. non-generic) swaps designed to meet their specific requirements. Such swaps are not traded as such, but created as one-off structures. This chapter describes a range of simple non-generic swaps, and discusses various techniques for pricing them, including one that requires no discounting. Finally, two approaches to the ongoing valuation of an existing (seasoned) swap are demonstrated. Chapter 5. There are a wide variety of potential swap structures, and this chapter covers the pricing and hedging of some of the more complex and popular ones. These include asset packages, credit swaps, mismatch swaps of various types including yield curve and overnight average. It concludes with a discussion on two less common structures, inflationlinked (which are growing rapidly) and volatility swaps. Chapter 6. The earliest swap structures were cross-currency swaps, although this market has long been overtaken by interest rate swaps. Nevertheless, they possess some unique characteristics and structures. This chapter starts with the fundamental CCS building block, the cross-currency basis swap, and explores its characteristics, uses, pricing and hedging. This employs a novel approach: worst-case simulation. The role of CCBSs in the derivation of cross-currency discount factors is also explored. The other main types of swaps are then discussed: fixed-floating, floating-floating, diff and quantodiff. Fixed-fixed swaps occupy a special place because they are a general case of long-term FX forward contracts, so the pricing and hedging of these is considered in some detail. Finally, swap valuation is revisited because, in the CCS market, such swaps are frequently valued annually and the principals reset to the current exchange rate.

Preface and Acknowledgements

Chapter 7. There is an active market in many currencies in medium to long-term options on forward interest rates, usually known as the cap & floor market. Such structures are intimately linked to swaps for two reasons: first, because combinations of options can create swaps and second, swaps are generally used to hedge them. In many banks, they are actually traded and risk-managed together. This chapter reviews a range of different option structures, and touches albeit briefly on option pricing. Volatility plays a crucial role, and various techniques for estimation, including transformation from par to forward, are described in detail. These options are also frequently embedded in many swap structures, and the breakdown and pricing of a range of structures is discussed. There is also an active market in options on forward swaps (aka swaptions or swoptions) which, not unnaturally, is closely related to the swap market. The pricing and embedding of swaptions is described. The chapter concludes with two sections on FX options. These options are mainly traded OTC, although there is some activity on a few exchanges such as Philadelphia. The first section concentrates on the pricing of these options, and how it may be varied depending on the method of quoting the underlying currencies. The second section shows how traders would dynamically create a delta-neutral hedge for such an option, together with the hedging errors through time. Chapter 8. In the early days of the swap market, swap portfolios were risk-managed using either asset-liability methods such as gapping or the more advanced institutions used bond techniques such as duration. By the late 1980s a number of well-publicized losses had forced banks to develop more appropriate techniques such as gridpoint hedging. These (in today's eyes) traditional approaches have stood the banks in good stead for the next decade. This chapter describes the main techniques of both gridpoint and curve hedging, taking into account both first and second-order sensitivities. In passing, mapping cashflows to gridpoints is also discussed. The use of swap futures, as a relatively new hedging instrument, is also considered. The chapter then extends the risk management to interest rate options. Most texts discuss the "greeks" using short-dated options; unfortunately, the discussion often does not apply to long-term options, and so their different characteristics, especially as a function of time, are examined. The effectiveness of some optimization techniques to construct "robust" hedges is examined as an alternative to the more traditional deltagamma methods. Finally, the chapter shows how the same techniques can be used to create an inflation hedge for a portfolio of inflation swaps. Chapter 9. Risk management, however, is not a static subject, but has evolved rapidly during the latter half of the 1990s and beyond. Traditional risk management operates quite successfully, but there is a very sensible desire by senior management to be able to assess the riskiness of the entire trading operation and even wider. The traditional risk measures are not combinable in any fashion, and cannot be used. Value-at-risk was developed as a family of approaches designed very much to address this objective. It is now being developed further to encompass not only market risk but also credit and even operational risks into the same set of measures 1 .

1 See the proposed Basel Accord (for details, see BIS website: www.bis.org) for the regulatory requirements using VaR-style approaches.

xii

Preface and Acknowledgements

This chapter describes the major approaches used to estimate VaR: delta, historic and Monte-Carlo simulations, as well as second-order delta-gamma approaches. The advantages and disadvantages of each approach are discussed, along with various extensions such as extreme value theory and sampling strategies. The measurement of spread VaR and equity VaR using either individual stocks or a stock index are also considered. Finally, stress testing, or how to make significant moves in the properties of the underlying risk factors (especially correlation), is described. The book is supported by a full range of detailed spreadsheet models, which underpin all the tables, graphs and figures in the main text. Some of the models have not been described in detail in the text, but hopefully the instructions on the sheets should be adequate. Many of the models are designed so that the reader may implement them in practice without, hopefully, too much difficulty. Many of the ideas, techniques and models described here have been developed over the years with colleagues at both ANZ and Lombard Risk Systems, and through various consulting assignments with a wide range of banks across the world. Particular thanks go to Ronny Moller, Richard Szwagrzak and Sean Register for their careful review and insightful comments when the book was in a pretty unreadable shape. I also wish to thank the various editors and personnel at Wiley, especially Sam Whittaker who has remained cheerful and supportive despite the range of missed deadlines. Finally, for my wife Marilyn, who has shown much forbearance during all the hours that I have disappeared, and who hopes I will now regain a life.

— 1 — Introduction

1.1

INTRODUCTION

In the 1970s there was an active parallel loan market. This arose during a period of exchange controls in Europe. Imagine that there is a UK company that needs to provide its US subsidiary with $100 million. The subsidiary is not of sufficiently good credit standing to borrow the money from a US bank without paying a considerable margin. The parent however cannot borrow the dollars itself and then pass them on to its subsidiary, or provide a parent guarantee, without being subject to the exchange control regulations which may make the transaction impossible or merely extremely expensive. The parallel loan market requires a friendly US company prepared to provide the dollars, and at the same time requiring sterling in the UK, perhaps for its own subsidiary. Parallel loan

k

United States

United Kingdom

US Corporation

UK plc

interest & principal

$100m

,4

£60m

ir

^

US sub of UK plc

UK sub of US Corp

interest & principal

Two loans with identical maturities are created in the two countries as shown. Usually the two principals would be at the prevailing spot FX rate, and the interest levels at the market rates. Obviously credit is a major concern, which would be alleviated by a set-off clause. This clause allows each party to off-set unpaid receipts against payments due. As the spot and interest rates move, one party would find their loan "cheap", i.e. below the current market levels, whilst the other party would find their loan "expensive". If the parties marked the loans to market, in other words, valued the loans relative to the current market levels, then the former would have a positive value and the latter a negative one. A "topping-up" clause, similar in today's market to a regular mark-to-market and settlement, would often be used to call for adjustments in the principals if the rates moved by more than a trigger amount.

Swaps and Other Derivatives As exchange controls were abolished, the parallel loan became replaced with the backto-back loan market, whereby the two parent organizations would enter into the loans directly with each other. This simplified the transactions, and reduced the operational risks. Because these loans were deemed to be separate transactions, albeit with an offsetting clause, they appeared on both sides of the balance sheet, with a potential adverse effect on the debt/equity ratios. Back-to-back loan $100m loan

US Corp

UK pic

4 £60m loan $ interest

US Corp

*

UK plc £ interest

US Corp

4

n

$100m

»

UK pic

The economic driving force behind back-to-back loans is an extremely important concept called "comparative advantage". Suppose the UK company is little known in the US; it would be expensive to raise USD directly. Therefore borrowing sterling and doing a back-to-back loan with a US company (who may of course be in exactly the reverse position) is likely to be cheaper. In theory, comparative advantage cannot exist in efficient markets; in reality markets are not efficient but are racked by varieties of distortions. Consider the simple corporate tax system: if a company is profitable, it has to pay tax; if a company is unprofitable, it doesn't. The system is asymmetric; unprofitable companies do not receive "negative" tax (except possibly in the form of off-sets against future profits). Any asymmetry is a distortion, and it is frequently feasible to derive mechanisms to exploit it — such as the leasing industry. Cross-currency swaps were rapidly developed from back-to-back loans in the late 1970s. In appearance they are very similar, and for an outside observer only able to see the cashflows, identical. But they are subtly different in that all cashflows are described as contingent sales or purchases, i.e. each sale is contingent upon the counter-sale. These transactions, being forward conditional commitments, are off-balance sheet. We have the beginning of the OTC swap market! Cross-currency swap sale of $100m US Corp

UK plc sale of £60m

$sale US Corp

UK plc Esale resale of $100m

US Corp

UK plc resale of £60m

Introduction

The structure of a generic (or vanilla) cross-currency swap is therefore: • initial exchange of principal amounts; • periodic exchanges of interest payments1; • re-exchange of the principal amounts at maturity. Notice that, if the first exchange is done at the current spot exchange rate, then it possesses no economic value and can be omitted. Interest rate, or single currency swaps, followed soon afterwards. Obviously exchange of principals in the same currency makes no economic sense, and hence an interest swap only consists of the single stage: • periodic exchanges of interest payments where interest is calculated on different reference rates. The most common form is with one side using a variable (or floating) rate which is determined at regular intervals, and the other a fixed reference rate throughout the lifetime of the swap.

1.2 APPLICATIONS OF SWAPS As suggested by its origins, the earliest applications of the swap market were to assist in the raising of cheap funds through the comparative advantage concept. The EIB-TVA transaction in 1996 was a classic example of this, and is described in Box 1.1. Both parties benefited to the total of about $3 million over a 10 year period, and therefore were both willing to enter into the swap. It was quickly realized that swaps, especially being off-balance sheet instruments, could also be effective in the management of both currency and interest rate medium-term risk. The commonest example is of a company that is currently paying floating interest, and is concerned about interest rates rising in the future. By entering into an interest rate swap to pay a fixed rate and receive a floating rate, uncertainty has been removed: ».

Bank

Company 4

floating interest floating interest

To ensure that the risk management is effective, the floating interest receipts under the swap must exactly match the interest payments under the debt. Therefore the swap must mirror any structural complexities in the debt, such as principal repayment schedules, options to repay early, and so on. Usually a swap entered into between a bank and a customer is tailored specifically for that situation. This book will provide details of many of the techniques used to structure such swaps. 'Remember: legally these cashflows are not "interest" but contingent sales, but for clarity of exposition they will be called "interest" as they are calculated in exactly the same way.

Swaps and Other Derivatives

A well-known and very early example of the use of swaps is the one conducted between the World Bank and IBM in August 1981—described in Box 1.2. This swap has the reputation of kick-starting the swap market because it was performed by two extremely prestigious organizations, and received a lot of publicity which attracted many other endusers to come into the market. It was the first long-term swap done by the World Bank, which is now one of the biggest users of the swap market.

Box 1.1

Comparative Advantage: European Investment Bank-Tennessee Valley Authority Swap, September 1996

Both counterparties had the same objective: to raise cheap funds. The EIB, being a European lender, wanted Deutschmarks. The TV A, all of whose revenues and costs were in USD, wanted to borrow dollars. Their funding costs (expressed as a spread over the appropriate government bond market) are shown below:

USD

EIB TVA Spread

T+17 7+24

7bp

DEM B+13 5+17 4bp

Whilst both organizations were AAA, the EIB was deemed to be the slightly better credit. If both organizations borrowed directly in their required currency, the total funding cost would be (approximately — because strictly the spreads in different currencies are not additive) 37 bp over the two bond curves. However the relative spread is much closer in DEM than it is in USD. This was for two reasons: • the TVA had always borrowed USD, and hence was starting to pay the price of excess supply; • it had never borrowed DEM, hence there was a considerable demand from European investors at a lower rate. The total cost if the TVA borrowed DEM and the EIB borrowed USD would be only 34 bp, saving 3bp pa. The end result was: • EIB issued 10-year $1 billion bond; • TVA issued 10-year DM1.5 billion bond; • they swapped the proceeds to raise cheaper funding, saving roughly S3 million over the 10 years. This was a real exercise in comparative advantage; neither party wanted the currency of their bond issue, but it was cheaper to issue and then swap.

Introduction

Box 1.2 World Bank-IBM Swap, August 1981 This is a simplified version of the famous swap. The two counterparties had very different objectives. IBM had embarked upon a world-wide funding programme some years earlier, raising money inter alia in Deutschmarks and Swiss Francs. The money was remitted back to the US for general funding. This had created an FX exposure, because IBM had to convert USDs into DEMs and CHFs regularly to make the coupon payments. Over the years the USD had significantly strengthened, creating a gain for IBM. It now wished to lock in the gain and remove any future exposure. The World Bank had a policy of raising money in hard currency; namely DEM, CHF and Yen. It was a prolific borrower, and by 1981 was finding that its cost of funds in these currencies was rising simply through an excess supply of WB paper. Its objective, as always, was to raise cheap funds. Salomon Brothers suggested the following transactions. (a) The WB could still raise USD at relatively cheap rates, therefore it should issue two Eurodollar bonds: • one matched to the principal and maturity of IBM's DEM liabilities equivalent to $210 million; • the other matched to IBM's Swiss Franc liabilities equivalent to $80 million. Each bond had a short first period to enable the timing of all future cashflows to match. (b) There was a two-week settlement period, so WB entered into an FX forward contract to: • sell the total bond proceeds of $290 million; • buy the equivalent in DEM and Sw Fr. (c) IBM and WB entered into a two-stage swap whereby: USD coupons DM/Sw Fr coupons IBM

World Bank At maturity USD principals DM/Sw Fr principals

DM/Sw Fr coupons

DM/Sw Fr principals

USD coupons

USD principals

so that IBM converted its DEM and Sw Fr liabilities into USD, and the WB effectively raised hard currencies at a cheap rate. Both achieved their objectives!

Swaps and Other Derivatives

87 88 89 90 91 92 93 94 95 96 97 98 99 00 Source: ISDA. Reproduced by permission of the International Swaps and Derivatives Association. Figure 1.1 Size of the swap market (notional principal outstanding $tr)

1.3 AN OVERVIEW OF THE SWAP MARKET From these earliest beginnings, the swap market has grown exponentially. As Figure 1.1 shows, the volume of interest rate swap business now very much dominates cross-currency swaps2, suggesting that risk management using swaps is commonplace. The graph is shown in terms of notional principal outstanding, i.e. the principals of all swaps transacted but not yet matured. For the cross-currency swap described above, this would be recorded as [$100m + £60mxS]/2 where S is the current spot rate. The market has shown a remarkable and consistent growth in activity, although there has been a significant decline in the CCS market in 2000 due to the emergence of the single currency Euro. It is arguable whether this is a very appropriate way of describing the current size of the market, although it certainly attracts headlines. Many professionals would use "gross market value" or total replacement cost of all contracts as a more realistic measure. This measure has been in broad decline as banks improve their risk management, and are unwilling to take on greater risks due to the imposition of capital charges: Gross market value ($tr)

98 99 Jun-00 Dec-00

IRS 1.509 1.150 1.072 1.260

CCS 0.200 0.250 0.239 0.313

Source: BIS. Reproduced by permission of the Bank for International Settlement.

"The original source of these data was the International Swaps and Derivatives Association (ISDA) which for many years conducted a semiannual survey of its members. In 1995, the Bank for International Settlement (BIS) started a triennial survey of OTC derivative activity via the central banks. In 1997. ISDA stopped their original survey, whilst BIS expanded theirs to cover currently 48 central banks and monetary authorities. The jump in the graph from 1997 to 1998 is in part due to the shift from ISDA to BIS. The BIS also conducts a semiannual review of global derivative activity across a smaller range of participants to provide more regular indications. At the time of writing, the last available statistics were for the triennial survey of end-June 2001.

Introduction A brief overview of the OTC derivative market is shown in Box 1.3. Probably the most important statistic is that, despite all the publicity given to more exotic transactions, the overwhelming workhorse of this market is the relatively short-term interest rate swap. The derivative markets continue to grow at an astounding rate — why? There are two main sources of growth—breadth and depth. • Financial markets around the world have increasingly deregulated over the past 30 years, witness activities in Greece and Portugal, the Far East and Eastern Europe. As they do, cash and bond markets first develop followed rapidly by swap and option markets. • The original swaps were done in relatively large principal amounts with high credit counterparties. Banks have however been increasingly pushing derivatives down into the lower credit depths in the search for return. It is feasible to get quite small transactions, and some institutions even specialize in aggregating retail demand into a wholesale transaction. Box 1.3

A Brief Overview of the Current State of the Derivative Market

• The total OTC derivative market was estimated by the latest triennial survey to be $100 trillion, measured in terms of outstanding principal amount, broken up as shown below: FX contracts outfights and swaps CCS options IR contracts FRAs swaps options Credit Other* Other** Total Exchange-traded FX IR equity

1995 13.1 8.7 2.0 2.4 26.6 4.6 18.3 3.5

7.8 47.5 10.3 0.1 9.7 0.5

1998 22.1 14.7 2.3 5.0 48.1 6.6 32.9 8.5 0.1 2.0 11.4 80.3 13.9 0.1 12.8 1.0

2001 20.4 13.3 4.3 2.8 75.9 7.7 57.2 10.9 0.7 2.7 12.9 95.2 19.5 0.1 17.4 1.9

equity and commodity related ^estimated non-regular reportingI counterparties

• Currently growing at 11 .4% pa, slowing down from 15% in 1998. • The total gross value was $3 trillion, or 3% of the notional amount. • If netting between reporting institutions is taken into account, total market exposure drops to only about 1.3% of notional amount. • Removing the irregular reportees, IR products constitute some 76% and growing at 16.4% pa. (Continued)

Swaps and Other Derivatives Box 1.3 (Continued) • Whereas traditional FX forward contracts, which are predominantly short-term, are only 13% and have declined by 3% pa (due particularly to the advent of the Euro). • Although CCS have nearly doubled over the period. • All other products are very small in comparison, although they may attract considerably more publicity: credit derivatives have expanded very rapidly from virtually nothing. • The market is concentrated in the short-end: Maturity

FX

under 1 year 1-5 years over 5 years

77% 17% 6%

IRS 38% of total market 29% 23%

which is very much the easier part of the curve to hedge The percentage share of each currency: Currency

Percentage of market share of IR derivatives

USD Euro Yen GBP Sw Fr Can$ SwKr

June 98 28.5% 32.9 19.5 7.8 2.6 1.5 1.9

June 01 34.2% 33.2 16.7 7.3 2.5 0.1 1.5

Source: Extracted from BIS survey to end June 2001, published December 2001. Reproduced by permission of the Bank for International Settlement.

1.4 THE EVOLUTION OF A SWAP MARKET The discussion below refers to the evolution of the early swap market in the major currencies during the 1980s. It is however applicable to many other generic markets as they have developed. There are typically three phases of development of a swap market. 1. In the earliest days of a market, it is very much an arranged market whereby two swap end-users negotiate directly with each other, and an "advisory" bank may well extract an upfront fee for locating and assisting them. This is obviously a slow market, with documentation frequently tailored for each transaction. The main banks involved are investment or merchant banks, long on people but low on capital and technology as of course they are taking no risk. Typical counterparties would be highly rated, and therefore happy to deal directly with each other:

Introduction

A

B

d

The first swap markets in the major currencies were even slower, as there was considerable doubt about the efficacy of swaps. End-users were dubious about moving the activities off-balance sheet, and there was apprehension that the accounting rules would be changed to force them back on-BS. The World Bank-IBM swap (described previously) played a major role in persuading people that the swap market was acceptable. 2. In the second phase, originally early to mid 1980s, commercial banks started to take an increasing role providing traditional credit guarantees: T

Bank

T

The counterparties would now both negotiate directly with the bank, which would structure back-to-back swaps but take the credit risk, usually for an ongoing spread not an upfront fee. The normal lending departments of the bank would be responsible for negotiating the transaction and the credit spread. The documentation is now more standardized and provided by the bank. This role is often described as acting as an "intermediary", taking credit but not market risk. The role of intermediary may also be encouraged by external legislation. In the UK for example, if a swap is entered into by two non-bank counterparties, the cashflows are subject to withholding tax. This is not true if one counterparty is a bank. The concept of a market-making bank originally developed by the mid to late 1980s, whereby a bank would provide swap quotations upon request. This means they would be dealing with a range of counterparties simultaneously, and entering into a variety of non-matching swaps. With increased market risk, such banks required considerably more capital, pricing and risk management systems, and very standardized documentation. The swap market became dominated by the large commercial banks who saw it as a volume, commoditized business. These banks would typically be off-setting the market risk by hedging in another market, usually the equivalent government bond market as this is the most liquid. Therefore banks with an underlying activity in this market are likely to be at a competitive advantage. Local domestic banks usually have close links with the local government bond market, and hence they are frequently dominant in the domestic swap market. Probably the only market where this is not the case is the USD market, where the markets are so large that a number of foreign banks can also be highly active and competitive. It might be worth making the point here that banks frequently and misleadingly talk about "trading" swaps, as if a swap were equivalent to a spot FX transaction which is settled and forgotten about within two days. A swap is actually a transaction which has created a long-term credit exposure for the bank. The exposure is likely to remain on the bank's books long after the swap "trader" has been paid a bonus and left the

10

Swaps and Other Derivatives bank. From this perspective, swaps fit much more comfortably within the traditional lending departments with all the concomitant credit controlling processes and not within a treasury which is typically far more lax about credit.

This link with the bond market has meant that a bank may well adopt different roles in different markets. For example, a Scandinavian bank such as Nordea Bank would be a market-maker in the Scandinavian and possibly some of the Northern European currencies. On the other hand, it would act as an intermediary in other currencies. For example, if a customer wanted to do a South African Rand swap, it would enter into it taking on the credit risk, but immediately laying off the market risk with a Rand marketmaking bank. In this context, the 1996 EIB-TVA swap was interesting. The deal was brokered by Lehmann Brothers, but they played no role in the swap. At one point the swap had been out for tender from a bank but (rumour has it) the bid was a 1 bp spread. Why, asked the two counterparties, do we need to deal with a bank at all, especially given that we are both AAA which is better than virtually all banks? So they dealt directly! As the relative credit standing of banks declines, the market may well see more transactions of this nature — back full circle. One cannot really talk about a "global" swap market. There are obviously some global currencies, notably USD, Yen and the Euro, which are traded 24 hours a day, and then it would be feasible to get swaps. But most swap markets are tied into their domestic markets, and hence available only during trading hours. Swap brokers still play an important role in this market. Their traditional role has been to identify the cheapest suitable counterparty for a client, usually on the initial basis of anonymity. This activity creates liquidity and a uniformity of pricing, to the overall benefit of market participants. However, as the markets in the most liquid currencies continue to grow, the efficiency provided by a broker is less valued and their fees have been increasingly reduced to a fraction of a basis point. They are having to develop more electronic skills to survive.

1.5

CONCLUSION

The story of the swaps market has been one of remarkable growth from its beginnings only some 30 years ago. This growth has demonstrated that there is a real demand for the benefits swaps can bring, namely access to cheap funds and risk management, globally. Furthermore, the growth shows little sign of abating as swap markets continue to expand both geographically as countries deregulate and downwards into the economy. Hopefully this book will play some small role in the continued expansion, assisting the orderly development of the market by ensuring that people are well-trained in their understanding of the pricing, structuring and risk management of swaps and related derivatives.

Short-term Interest Rate Swaps OBJECTIVE The main objective of this chapter is to provide an introduction to the construction and pricing of short-term IRS using futures contracts. However, because a simple swap may be regarded as an exchange of two streams of cashflows which occur at different points in the future, extensive use is made of the concept of discounting. The chapter therefore begins with a brief discussion on the time value of money, and demonstrates how implied discount factors may be derived from the cash market. Because rates are only available at discrete maturities, interpolation is a necessary technique, and there are a number of different approaches which end up with different results. The chapter then discusses how to estimate forward rates, and how to price FRAs first off the cash market and then off the futures market. This leads naturally to the pricing and hedging of short-term IRS off a futures strip. Examination of the hedging reveals a convexity effect which is discussed in more detail in Chapter 5. Finally, an alternative approach to pricing swaps without discounting is briefly discussed.

2.1

DISCOUNTING, THE TIME VALUE OF MONEY AND OTHER MATTERS

Today's date is Tuesday 4 January 2000, and you have just been offered a choice of transactions: Deal 1: to lend $10 million and to receive 6.25% for 3 months Deal 2: to lend $10 million and to receive 6.70% for 12 months Which do you find more attractive? The current London rates at which you could normally deposit money are 61/32% pa and 19/32 6 % pa for 3 and 12 months respectively, and we will assume that the creditworthiness of the counterparty is beyond question. Comparing the transactions with these market rates, the 3 month deal is 22 bp above the market, whilst the 12 month deal is only 10bp. Intuitively you favour the first transaction, but wish to do some more analysis to be certain. These market rates suggest that the following transactions are currently available1: Dates 4-Jan-00 6-Jan-00 6-Apr-00 8-Jan-01

Days 91 368

3mo. Cash

12mo. Cash

- 10,000,000 10,152,457

-10,000,000 10,674,028

where nejjative signs indicate payments, positive or no sign receipts

Please note that the calculations for all the numbers are replicated on the accompanying CD.

12

Swaps and Other Derivatives

Note the following: (a) Whilst the rates are being quoted on 4 January, they are with effect from 6 January. In other words, there is a two-day settlement period between the agreement of the transaction and its start. This is the normal convention in the USD market, although it is feasible to organize a "same day" transaction. Conventions vary between markets; for example, the GBP convention is normally "same day". (b) Interest rates are invariably quoted on a "per annum" basis, even if they are going to be applied over a different period. It is therefore necessary to have a convention that translates the calendar time from, say, 6 April 2000 back to 6 January 2000 into years. The USD money market, in common with most money markets, uses an "Actual/360" daycount convention, i.e. calculates the actual number of days: 6 April 2000 - 6 January 2000 = 91 days and then divides by 360 to convert into 0.252778 years. The other common convention is "Actual/365", which is used in the sterling market and many of the old Commonwealth countries. The cashflow at the end of 3 months is given by: $10,000,000 x (1+6.03125% x 0.252778) = $10,152,456.60 (c) 6 January 2001 is a Saturday, and a non-business day in London. Payments can only be made on business days, and therefore a convention has to be applied to determine the appropriate date. The most popular is the "modified following day" convention, i.e. the operating date moves to the next business day unless this involves going across a month-end, in which case the operating date moves to the last business day in the month. Using this convention, the 12 month transaction ends on the next business day, i.e. Monday 8 January 2001, and interest is calculated accordingly: $10,000,000 x (1 + 6.59375% x 368/360) = $10,674,027.78 The concept of discounting will be used extensively throughout this book. The "time value of money" suggests that the value of money depends upon its time of receipt; for example, $1 million received today would usually be valued more highly than $1 million to be received in 1 year's time because it could be invested today to generate interest or profits in the future. If Ct represents a certain cashflow to be received at time r>0, then a discount factor df, relates this cashflow to its value today (or present value) Co by:

C0 = C, x dft Note that this does not presuppose any source or derivation of the discount factor. The present value of each of these two market-based transactions may easily be calculated as: -$10,000,000 + $10,152,456.60 x df3 -$10,000,000+ $10,674.027.78 x df12 where df3 and df12 are the 3 and 12 month discount factors respectively. The market rates are obviously freely negotiated, and we will assume that, at the moment of entering into the transactions, they represent no clear profit to either party. In other words, at inception the transactions would be deemed to be "fair" to both parties, and hence have a zero net value. This is of course ignoring market realities such as bid-offer or bid-ask spreads (or "doubles" as they are frequently called). In practice, most analysis uses mid-rates, i.e. the

Short-term Interest Rate Swaps

arithmetic average between bid and offer, simply to enable the statement of "fairness" to be made, and subsequently adjusted for various spreads. These issues will be discussed in more detail later; for the current discussion they will be ignored. If the present values are both zero, we can solve for the two discount factors, i.e. df3 = 0.984983 and df12 = 0.936853 respectively. A general formula for discount factors from the money markets is: J j . f l / / 1 i J\ df t = 1/1 + r1 x dt)

/ "^ 1 \ (2.1)

where dt is the length of time (in years) and rt is the rate (expressed as % pa). Turning back to the two original transactions, these will generate the following cashflows: Dates 4-Jan-00 6-Jan-OO 6-Apr-00 8-Jan-01

Deal 1

Deal 2

-10,000,000 10,157,986

- 10,000,000 10,684,889

These cashflows may be present valued using the discount factors derived from the market rates, giving: Deall

Deal 2 10,175,27

5,446.48

Thus we can see, perhaps against our intuition, that the second transaction would be the more profitable of the two. This is of course because the deal is longer: 22 bp over 3 months is roughly half of 10bp over a year. The current money market data readily available is: Today's date: 7 day 1 month 3 month 6 month 12 month

4-Jan-00 6-Jan-OO 13-Jan-OO 7-Feb-00 6-Apr-OO 6-Jul-00 8-Jan-01

Rates 517/32% 513/16% 61/32% 67/32 %

619/32%

Discount factors at each of the maturities can easily be calculated as above, i.e. Today's date: 7 day 1 month 3 month 6 month 12 month

4-Jan-OO 6-Jan-OO 13-Jan-OO 7-Feb-OO 6-Apr-OO 6-Jul-OO 8-Jan-01

Rates 17/32

5

%

5 732 /o

5

13/16%

61/32%

67/32% 6 19/32%

DFs

0.998926 0.994860 0.984983 0.969519 0.936853

14

Swaps and Other Derivatives

Table 2.1 Calculation of discount factor on 21-Sep-OO Interpolation of rates

Rates DFs

Interpolation of DFs

Linear

Cubic

Linear

Cubic

Log-linear

6.374% 0.956153

6.330% 0.956443

0.955996

0.956184

0.955860

You are now offered the opportunity to purchase a riskless $100 million on 21 September 2000. What value would you place on this transaction? To answer this question, the discount factor on 21 September is required — but how to calculate it? The obvious approach is "interpolation", but this raises two questions: • what is interpolated: cash rates or discount factors? • how is the interpolation calculated: linear, polynomial, exponential, etc.? with associated questions "do the answers change the valuation?" and "are there any 'right' answers?". The simple answers to the latter questions are "yes" and "no, but some are better than others"! The results from some popular methods are shown in Table 2.1 where: • "linear" is simply straight-line interpolation; • "cubic" implies fitting a cubic polynomial of the form a + bt + ct2 + dt3 through the four neighbouring points and solving for {a,b,c,d}; • "log-linear" is the straight-line interpolation of the natural logarithm of the discount factors (this last one is often suggested since a discount curve is similar to a negative exponential curve). The deal value fluctuates by some $50,000 or roughly 5 bp, which is perhaps not significant but worthwhile. It is more common practice to interpolate rates rather than discount factors at the short end of the curve. This is probably because it would be perfectly feasible to get a quote for a rate out to 21 September for depositing, and of course the two transactions should be arbitrage free. Cash rates are of course spot rates, i.e. they all start out of "today". The cash curve may be used to estimate forward rates, i.e. rates starting at some point in the future. For example, if we knew that we would receive $100 million on 6 April 2000 for, say, 3 months, we could lock in the investment rate today by calculating the 3/6 rate2. Forward rates are usually estimated using an arbitrage argument as follows: 1. we could borrow $100 million for 3 months at 61/32% which would cost: $100 m x (1 + 6.03125% x 0.252778) = 101,524,566 2. and then lend $100 million for 6 months at 6/32% to generate: $100 m x (1 + 6.21875% x 0.505556) = 103,143,924 2 Forward rates are conventionally quoted as "start/end" (or "start x end" or "start vs. end") rather than "start maturity" or "start/tenor".

15

Short-term Interest Rate Swaps Table 2.2 Calculation of discount factor on 6-Oct-00 Interpolation of DFs

Interpolation of rates

Rates DFs

Linear

Cubic

Linear

Cubic

Log-linear

6.404% 0.953522

6.355% 0.953865

0.953362

0.953556

0.953222

6.3100% 6.5505% 6.8280%

6.3100% 6.6901% 6.6913%

3mo. Forward rates 3/6 6/9 9/12

6.3100% 6.6317% 6.7485%

6.3100% 6.4216% 6.9544%

6.3100% 6.5648% 6.8140%

The break-even or implied 3/6 rate is therefore given by the equation: $100,524,566 x (1 + r3/6 x d3/6) = $103,143,924 => r3/6 = 6.3100% A general expression for a forward rate Ft/T, from t to T, is: Ft/T = ([1 + rT x d T }/(1 + rt x dt}] - 1}/(T - t)

(2.2)

However to use this expression, zero coupon spot rates are required with maturities t and T. This is acceptable for when T is under 1 year, but they are unlikely to be available for longer maturities. A more widely used expression for longer-dated forward rates is:

(2.3)

Fl/T={(df!/dfT)-1}/(T-t)

using discount factors estimated off the discount curve (which is of course synonymous for cash rates). Returning to the cash curve above, we want to estimate the 3-monthly forward rates, 3/6, 6/9 and 9/12. To do this, we need to estimate the 9 month discount factor df9. Table 2.2 shows it being estimated in a variety of ways, and the resulting forward rates. The impact of the different methods on the forward rates is quite dramatic, showing differences of up to 30 bp. See Figure 2.1. Contrast this with the difference in the discount factors, which in the previous example only reached 5 bp. To understand why, rewriting Equation (2.3) as: F t/T//r

=

{(dft-dfr)/dfr}/(T-t) Linear interpolation of rates Cubic interpolation of rates Linear interpolation of DFs Cubic interpolation of DFs Log-linear interpolation of DFs

6/9

9/12

Figure 2.1 3-Monthly forward rates

16

Swaps and Other Derivatives

Table 2.3

15-Day forward curve Interpolation of rates

6-Jan-00 21-Jan-00 5-Feb-00 20-Feb-00 6-Mar-00 21-Mar-00 5-Apr-00 20-Apr-OO 5-May-OO 20-May-00 4-Jun-OO 19-Jun-00 4-Jul-00 19-Jul-00 3-Aug-00 18-Aug-00 2-Sep-OO 17-Sep-00 2-Oct-OO 17-Oct-00 1-Nov-00 16-Nov-00 1-Dec-00 16-Dec-00 3 1-Dec-00

Interpolation of DFs

Linear

Cubic

Linear

Cubic

Log-linear

5.6213% 2.9724% 1.9911% 1.5097% 1.2268% 1.0380% 0.8804% 0.7747% 0.6936% 0.6287% 0.5755% 0.5312% 0.4931% 0.4607% 0.4327% 0.4082% 0.3866% 0.3673% 0.3501% 0.3345% 0.3205% 0.3076% 0.2959% 0.2852%

5.6381% 3.5995% 2.3344% 1.6358% 1.1502% 0.8110% 0.8236% 0.7064% 0.6963% 0.6279% 0.5717% 0.5247% 0.4851% 0.4515% 0.4228% 0.3982% 0.3772% 0.3594% 0.3442% 0.3315% 0.3211% 0.3126% 0.3059% 0.3010%

5.7145% 2.9415% 2.0159% 1.5216% 1.2203% 1.0195% 0.8885% 0.7803% 0.6954% 0.6275% 0.5719% 0.5257% 0.5006% 0.4682% 0.4382% 0.4119% 0.3887% 0.3682% 0.3497% 0.3332% 0.3182% 0.3046% 0.2922% 0.2808%

5.6902% 2.9497% 1.9953% 1.5154% 1.2254% 1.0302% 0.8826% 0.7763% 0.6940% 0.6282% 0.5744% 0.5297% 0.4920% 0.4596% 0.4317% 0.4073% 0.3858% 0.3667% 0.3497% 0.3345% 0.3207% 0.3082% 0.2969% 0.2865%

5.7189% 2.9401% 2.0223% 1.5239% 1.2191% 1.0159% 0.8938% 0.7836% 0.6965% 0.6268% 0.5699% 0.5224% 0.4956% 0.4621% 0.4313% 0.4044% 0.3806% 0.3594% 0.3405% 0.3235% 0.3081% 0.2941% 0.2813% 0.2696%

highlights the fact that a forward rate is related to the gradient of the discount curve and is therefore much more sensitive to small differences in the estimates. To demonstrate this more clearly, Table 2.3 calculates a 15-day forward rate curve using all the five different methods of interpolation. The average difference between the highest and lowest curves is 7.8 bp. In practice, whilst there is no "right" method, most people interpolate the cash rates using either linear if the cash curve is relatively flat, or polynomial if the curve is quite steep.

2.2 FORWARD RATE AGREEMENTS AND INTEREST RATE FUTURES An FRA is an agreement between two counterparties whereby: • • • •

seller of FRA agrees to pay a floating interest rate and receive a fixed interest rate; buyer of FRA agrees to pay the fixed interest and receive the floating interest; on an agreed notional principal amount; over an agreed forward period.

For example, a company is a payer of 3 month floating interest on $10 million of debt. The company is concerned about interest rates rising, and on 4 January 2000 it buys a S10 million 3/6 FRA at a fixed rate of 6.31% from a bank. The following operations occur:

Short-term Interest Rate Swaps

4 April 2000: 6 July 2000:

3mo. $ Libor is fixed out of 6 April 2000 net cash settlement (L-6.31%) x $10m x (6 July-6 April)/360 is paid. This is shown from the point of view of the company, and will be positive if L>6.31% or negative if L<6.31%

Hence, the company is locked into the fixed rate even if rates do rise over the period from 4 January to 4 April. In practice, the net amount is discounted back to 6 April 2000 using the recent Libor fixing, i.e.: {(L- 6.31 %) x $10m x (6 July - 6 April)/360}/[l + L x (6 July - 6 April)/360] and paid then. The usual reason given for this market convention is a reduction in the credit exposure between the two parties: (a) On 4 January, the current exposure is assumed to be zero, i.e. the FRA would have a zero valuation. (b) However, there is a "potential future exposure" over the period from 4 January to 4 April which would fluctuate as the estimate of the Libor fixing on 4 April varies. If the estimate rises, then the FRA has a negative value for the bank and hence the company has a credit exposure on the bank. Conversely, if the estimate falls, then the FRA has a positive value for the bank, and it has a credit exposure on the company. (c) On 4 April, the official Libor fixing is known, which then fixes the net settlement amount and crystallizes the residual credit exposure. (d) The two parties could wait until 6 July with one of them having this known residual exposure. By making the payment immediately on 6 April, this residual is removed. Discounting the net settlement amount appears to favour the bank, as it implies that, for a given credit limit, in the case above of the 3/6 FRA the bank could effectively do twice the total business. This impact of discounting on reducing the total credit exposure obviously declines as the time to the fixing date lengthens. The benefit to the company is less clear. Whilst the value of the net settlement remains constant whether discounted or not, most companies neither mark-to-market nor are overly concerned about credit exposures. The cashflows from the FRA and from the underlying debt are not on the same dates, therefore creating a mismatch which may cause accounting and tax problems. It is highly unlikely that the company could reproduce the undiscounted net settlement, as it would not be able to deposit or borrow at Libor flat for an odd cashflow, irrespective of its creditworthiness. It is perfectly feasible for banks to provide non-discounted FRAs 3 at a price, but this is seldom done. We saw in Section 2.1 how a forward rate may be created by spot money market transactions. However FRAs are off-balance sheet whereas cash trades are on-balance sheet, which is not a good mix. If a liquid interest rate (or deposit) futures market exists, then this is much more likely to be used to price and hedge FRAs. A brief reminder about futures contracts: • equivalent to standardized FRA contracts, traded through exchanges; • standardized notional principal amounts, maturity dates and underlying interest rates; • futures are deemed to be credit risk free as each contract is guaranteed by the -'Which could of course be thought of as a single period swap!

18

Swaps and Other Derivatives exchange — to achieve this, when entering into a contract, each party must place an initial margin with the exchange (sufficient to cover an extreme movement in the market) plus variation margin because each contract is valued and settled daily.

For example: the most liquid contract in the world is 3 month Eurodollar traded on Chicago Mercantile Exchange: • • • •

notional principal amount is $1 million; maturity dates: third Wednesday of a delivery month; delivery months: March, June, September and December; in theory, 40 contracts (i.e. spanning the next 10 years) are open at any time; in practice, there is good liquidity in the near 20 contracts; • underlying interest rate: 3 month USD Libor quoted on a "price" basis; on 4 January 2000, the quote for the March contract was 93.80, implying that the market was anticipating the 3 month Libor rate out of 15 March 2000 to be 6.20%; • variation margin: $1,000,000 x 1 bp x (90/360) = $25 per basis point movement has to be paid or received daily (notice this simplistically assumes 3 months is equal to 0.25 of a year, and does not use Actual/360). The current quotes for the Eurodollar futures contracts are:

March June

Maturity date 15 March 2000 21 June 2000

Price 93.80 93.50

Implied rate 6.20% 6.50%

Given these rates, we wish to price the FRA above by estimating the fair 3 month rate out of 6 April: this is usually done by simple linear interpolation between the neighbouring implied futures rates as shown in Figure 2.2. Since 6 April—15 March = 22 days and 21 June —6 April = 76 days, linear interpolation gives: (76/98) x 6.20% + (22/98) x 6.50% = 6.267% The reason for writing the interpolation in this fashion is that it provides a clear indication of the contribution of each futures contract, i.e. March provides 78%, June 22%, to the 6.6 6.5 6.4

6.1 15 March

6 April

6.0

Figure 2.2

21 June

19

Short-term Interest Rate Swaps 6.6 6.56.46 3

i - ' 6.26.16.0-

15 March 6 April

21 June

Figure 2.3

Figure 2.4

price estimate of the FRA. Leading on from that, it also provides a clear indication of the futures required to hedge the FRA; the bank has sold $10 million of a 3/6 FRA, therefore 10 futures contracts need to be sold to hedge it in the proportion of 7.8 March contracts, 2.2 June contracts. Why do we sell futures, especially as the bank has sold the FRA? The bank is paying the floating rate on the FRA, and is therefore concerned about rates rising. Futures are quoted on a price basis, i.e. the bank sells the March contract at 93.80. If rates then rise, the price will fall, and the bank can buy the contract back at a lower price. The profit gained — the variation margin — should offset the loss on the FRA. How good is this hedge? This will be discussed conceptually first, and then in detail later. Consider first of all a 10 bp parallel shift in the 3 month forward rate curve. See Figure 2.3. The bank would have to pay $10,000,000 x 10bp x (91/360) = $2,258 extra on the FRA and would receive $25x10 contracts x 10 bp = $2,500 from the futures. So the hedge is fairly effective, given the slight daycount mismatch. In theory, the size of the futures hedge could have been adjusted slightly, but this is obviously impractical. Next a rotational shift, pivoting around 1 April 2000. See Figure 2.4. This results in the following shifts: March contract: June contract: FRA contract: Net effect:

-6.8bp + 32.4bp + 2.0bp

value = -$1,326 value = + $1,782 value = -$505 -$55

20

Swaps and Other Derivatives

The hedge appears to be quite effective against both parallel and rotational shifts. However, if the rates move to increase their curvature, for example both futures rates decrease but the FRA rate remains constant, then the hedge will fail. As time passes, the hedge needs to be rebalanced as the proportions of the two contracts change. Eventually the March contract will expire, leaving the FRA hedged only with the June contract. This exposes the bank to rotational risk for the remainder of the contract. This may be reduced by selling a small amount of September contracts, but this is unlikely to be very effective given the short time to the FRA fixing. By this, we mean that the correlation between the remainder of the FRA contract and the September contract is likely to be quite small, and hence a large degree of curve risk has been introduced. The time of greatest risk therefore when hedging a FRA with futures is when one of the bracketing contracts has matured. The only way of removing this residual risk completely is to sell an IMM FRA, i.e. when the FRA fixing date falls on a futures maturity date, so it may be hedged with a single contract.

2.3 SHORT-TERM SWAPS There are some other issues that we need to discuss, and these will be done in the context of a more complex example. A money market swap is a short-dated swap typically priced and hedged using a futures strip. The swap will be: • • • •

notional principal amount of $10 million; 1 year maturity, starting on 4 January 2000; to receive fixed F annual Actual/360; to pay 3 month Libor quarterly.

In this context, the "fair price" of a swap is the fixed rate F such that the net present value of the swap is zero. The structure of the swap is:

4-Jan-00 6-Jan-00 4-Apr-00 6-Apr-00 4-Jul-00 6-Jul-00 4-Oct-00 6-Oct-OO 8-Jan-OO

First Libor fixing ( = current 3mo. cash rate 6.03125%) Start of swap (start of interest accruing on both sides) Second Libor fixing First floating payment = $10,000,000 x 6.0313% x (6 Apr-6 Jan)/360 = $152,457 Third Libor fixing Second floating payment = $ 10,000,000 x L x (6 Jul - 6 Apr)/360 Fourth Libor fixing Third floating payment Final cashflow: fourth floating payment & single fixed receipt = $10,000,000 x Fx (8 Jan 01 -6 Jan 00) 360

Note that, whilst described in detail above, the distinction between the fixing date of a floating reference rate and the start of the accruing period will generally be ignored unless it has some special significance. Future examples will tacitly assume that fixing takes place on the start date of each period. The current market information out of 4 January 2000 is:

21

Short-term Interest Rate Swaps

6-Jan-00 13-Jan-00 7-Feb-00 6-Apr-OO 6-Jul-00 8-Jan-01

Cash rates

Futures prices

5.5313% 5.8125% 6.0313% 6.2188% 6.5938%

March June September December March 01

93.80 93.50 93.27 93.05 93.03

We will look at various ways of determining F, and then will return to hedging. As with all swaps, there are two main issues: 1. What to do about the unknown forward floating rates? 2. As the cashflows occur at different points in time, they need to be made comparable in some fashion; usually by discounting all the cashflows back to the present. Whilst there are a variety of approaches that may be used to address the first issue, as we shall see later, using the futures to estimate the forward Libor rates as we did on the FRA, and subsequently to hedge them is a very natural choice. The first Libor is of course fixed today to be the current 3mo. cash rate. The second Libor is none other than the 3/6 rate: we estimated this using the March and June futures above to be 6.267%. Similarly, we can estimate the other two Libor fixings, which are the 6/9 and 9/12 rates respectively, as follows: 6/9: [(76 x 6.50%) + (15 x 6.73%)]/91 = 6.5379% 9/12: [(75 x 6.73%) + (16 x 6.95%)]/91 = 6.7687% These calculations are shown in column [1] of Worksheet 2.1. Hence the floating cashflows can be constructed; see column [2]. The second issue is more complex. We need to estimate appropriate discount factors. The most obvious choice is to derive them from the cash rates as above, interpolating the rates as required; this values the floating side to be —$628,655 as shown in columns [3] and [4]. To calculate the fixed rate, we could either calculate it analytically or numerically. • The value of the fixed side for F= 1 is simply: $10,000,000 x [(8 Jan 01-6 Jan 00)/360] x 0.936853 = $9,576,724 For the swap to be fairly priced, F= $628,655/$9,576,724 = 6.5644%. This will work in this situation as the present value of the fixed side is linear in terms of the fixed rate; this is true in many relatively simple structures. • Alternatively, we could guess the fixed rate, construct the cashflow in column [6], calculate the present value of the fixed side, and if it is not the same as the floating PV, adjust the guess. A good starting guess would be to use the average of the floating rates, i.e. 6.40% and adjust from quarterly to annual using the formula (l+r qu /4) 4 = (l + >rann), i-6. 6.5566%; this gives a net PV of —$747. But the starting point seldom matters as the iterations are well behaved. When pricing transactions in a spreadsheet, most people make extensive use of the goal seeking or solver functions to do this type of calculation. There are probably two reasons why this is so popular:

to

Worksheet 2.1 Pricing a money market swap using cash Today's date: 4-Jan-OO 1. Swap details: 10 million USD 1 year 6.5644% ann, act/360 3mo. Libor qu, act/360 Futures dates Cash rates 6-Jan-OO 13-Jan-00 5.53125% March 7-Feb-OO 5.81250% June 6-Apr-OO 6.03125% September 6-Jul-OO 6.21875% December 8-Jan-Ol 6.59375%

Futures prices 93.80 93.50 93.27 93.05

15-Mar-00 21-Jun-00 20-Sep-OO 20-Dec-OO

Implied Swap forward dates rates 6-Jan-OO 6.200% 6-Apr-OO 6.500% 6-Jul-OO 6.730% 6-Oct-OO 6.950% 8-Jan-Ol

Initial guess

6.5566%

Final value

6.5644%

Cash DFs

Fixed cashflows

[2]

Interpolated cash rates [3]

[4]

[5]

-152,457 -158,425 -167,080 -176,738

6.031% 6.219% 6.404% 6.594%

0.984983 0.969519 0.953522 0.936853

671,028

628,655

628,655

Net present value = Present value of fixed side for F = 1 9,576,724

0

Estimated Floating 3mo. Libor cashflows fixings [1] 6.03125% 6.26735% 6.53791% 6.76868%

Present value =

C/3

v> CO

a ex 0 cr

3. » rt

C/3

cr o

2. Hedge:

3. Hedge effectiveness: Shift (bp) in prices

Shift (bp) Futures in prices dates

[6]

[7] Y

[8] 0

March June September December

7.76 10.60 9.89 1.76

-25 -50 -100 -75

0 0 0 0

Total contracts

30.00

Implied forward rates

Swap dates

New Libor fixings [9]

15-Mar-OO 21-Jun-00 20-Sep-OO 20-Dec-00

6.450% 7.000% 7.730% 7.700%

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01Ol

Total change in value =

Hedge ratio

6.0313% 6.5735% 7.1203% 7.7247%

Change in cashflow on swap [10]

Discounted change Payments in or receipts cashflow from on swap futures [11] [12]

0 -7,738 -14,884 -24,963

-7,502 -14,192 -23,387

4,847 13,246 24,725 3,297

-47,585

-45,081

46,115

1.03

0.98

24

Swaps and Other Derivatives

(i) it directly generates the actual cashflows likely to happen under this swap, which is extremely useful for checking the structure; (ii) the method may easily be modified to enable the pricer to calculate a fixed rate that will generate a desired profit (non-zero net PV) for the transaction. As before, the hedges for the three unknown Libor fixings may be calculated: 6/9: (76/91) x 10 = 8.35 June and (15/91) x 10 = 1.65 September 9/12: (75/91) x 10 = 8.24 September and (16/91) x 10 = 1.76 December A total of 30 contracts are required, as shown in Box 2 of Worksheet 2.1. The effectiveness of this hedge is explored in Box 3. The futures prices are shifted, either individually in column [7] or in parallel in [8]. The new Libor estimates are calculated in [9], and the resulting change in the swap cashflows in [10]; obviously the cashflow corresponding to the first Libor fixing does not change. The margin cashflows from the futures hedge are calculated in column [12]; for example: March: 7.76 contracts x —25 bp shift in price x $25 per bp = $4,847 received We can see that the total changes in the swap cashflows shown in column [10], and the total receipts under the futures hedge in [12], are very similar. They should be equal if the hedge is completely effective; the reason why they are not is because of the differences in daycounts as discussed above [the resulting hedge ratio of 1.03 is roughly the ratio of the length of 3 months under the swap convention of Actual/360 and under the futures convention being equal to1/4of a year, which suggests that about 31 contracts are actually required]. However, column [10] ignores the timing of the cashflows and simply adds them up. The hedge is said to be a "cash hedge". In practice, the futures would pay the receipts on margin received today, whilst the additional payments under the swap would only occur on the payment dates. To make the results comparable, the changes in swap cashflows need to be discounted as in column [11]. In that case, the swap is overhedged, i.e. the changes in the value of the swap will always be smaller than the off-setting changes in the value of the futures receipts, so that the net effect is that we are short futures contracts, as shown in Figure 2.5. There is however a serious practical flaw in the model, and this refers back to the second issue. The model uses futures for estimating the future Libor fixings, and cash for deriving the discount factors. Both markets are providing information over the 12 month period; some of the information must therefore be redundant, and it may also be contradictory. 4,000 3,000

0-

- 1,000 - 2,000 -

)0

-150

-100

-50

100

150

Change in futures prices (bp) Figure 2.5

Hedge effectiveness on a discounted basis

210

Short-term Interest Rate Swaps

25

The hedge only protected against movements in the fixings, whereas shifts in the underlying interest rates should also affect the discounting process. These effects have been ignored. If we were to attempt to introduce this effect, we would have to link shifts in futures to shifts in the discounting. The discounting process is going to have to be rebuilt, this time using the following nonredundant or parsimonious set of market information:

Cash rates 6-Jan-00 15-Mar-00 6-Apr-00

5.99% 6.03125%

Futures prices March June September December March 01

93.80 93.50 93.27 93.05 93.03

Notice that there is a short cash rate from today to the maturity date of the first futures contract: this is often called the "cash stub" or "cash to first futures" (CTFF) and in this case would be roughly equivalent to a 2% month cash rate. Initially, let us assume that the implied futures rates apply from the maturity of one futures contract to the next one, e.g. the implied rate of 6.20% applies from 15 March until 21 June, the rate of 6.50% from 21 June until 20 September, etc. In this case we can build a discount curve as follows: • define D F ( t 1 , t 2 ) to be the discount factor at time t2 that will discount back to t1; • obviously DF(t 1 , t3) = DF(t1, t2) x DF(t2, t3): 1. DF(0, 15 March) = (1 + 69/360 x 5.99%)-1 = 0.988649 (the usual simple DF); 2. DF(15 March, 21 June) = (1+98/360x6.20%)-'=0.983402; DF(0, 21 June) = 0.988649x0.983402 = 0.972240; 3. DF(21 June, 20 September) = (1+91/360 x 6.50%)-1 = 0.983835; DF(0, 20 September) = 0.972240 x 0.983835 = 0.956524, etc. The resulting discount curve may be seen in column [1] of Worksheet 2.2. However the discount factors are required on the cashflow dates of the swap, so they need to be interpolated. In the worksheet, this is done by: • converting the DFs into zero coupon rates by zt= — ln(DFt)/t: see column [2]; • linearly interpolating the zero rates in [6]; • finally transforming back to discount factors using DFt = exp{ — ztt} in [7]. This method of interpolating the discount curve is widely used, often under the name of "continuously compounded interpolation". Its implications will be explored later. The fair rate for the swap can now be calculated as before to be 6.5635% —see column [8] — which is very slightly different to the earlier rate. In practice futures are not exactly 3 months apart; sometimes they gap, sometimes overlap. If it is deemed necessary, then one approach is as follows (see Worksheet 2.3 for details):

26

Swaps and Other Derivatives

Worksheet 2.2 Pricing a money market swap off a futures strip Today's date: 4-Jan-00 1. Swap details:

Cash rates

Futures prices

6-Jan-00 15-Mar-00 5.99% March 93.80 6-Apr-00 6.03125% June 93.50 September 93.27 December 93.05 March 01 93.03

Principal amount Maturity

10 million USD 1 year 6.5635% ann, act/360 3mo. Libor qu, act/360

From cash

6.5644%

Futures dates

6-Jan-OO 15-Mar-OO 21-Jun-00 20-Sep-00 20-Dec-00 21-Mar-01 21-Jun-01

Implied forward rates

6.200% 6.500% 6.730% 6.950% 6.970%

DFs

Z-c rates

[1]

[2]

1 0.988649 0.972240 0.956524 0.940524 0.924286 0.908110

5.9559% 6.0688% 6.2022% 6.3251% 6.4419%

27

Short-term Interest Rate Swaps

2. Hedge

Fixed InterEstimated Floating 3mo. cashflows cashflows polated Z-c rates Libor on swap fixings dates [4] [5] [6] [3] 6-Jan-00 6.03125% 5.9812% 6-Apr-00 6.2673% -152,457 6.0908% 6-Jul-00 6.5379% -158,425 6.2238% 6-Oct-00 6.7687% -167,080 -176,738 670,935 6.3495% 8-Jan-01 Swap dates

Swap DFs

Discounted net cashflows

PV01

Hedge

[7]

[8]

[9]

[10]

0.452 0.984995 -150,169 Cash stub -193.555 7.72 0.969677 -153,621 March -265.737 10.63 0.953734 -159,350 June -252.915 10.12 0.937156 463,139 Sep Dec -45.033 1.80

Present value =

0.00

30.27

28

Swaps and Other Derivatives Worksheet 2.3 The construction of a discount curve when futures are not evenly spaced

Today's date: 4-Jan-00 Swap details: Principal amount Maturity

10 million USD 1 year 6.5635% ann, act/360 3mo. Libor qu, act/360 1. Building a discount curve

Futures prices

Cash rates 6-Jan-OO 15-Mar-OO 6-Apr-OO

5.99% 6.0313%

March June September December March 01

93.80 93.50 93.27 93.05 93.03

Futures dates maturity end of rate 6-Jan-00 15-Mar-00 21-Jun-00 20-Sep-00 20-Dec-00 21-Mar-01

15-Mar-00 15-Jun-00 21-Sep-00 20-Dec-00 20-Mar-01 21-Jun-01

tenor 92 92 91 90 92

29

Short-term Interest Rate Swaps

Futures rates [1]

D-C rates [2]

6.200% 6.500% 6.730% 6.950% 6.970%

6.152% 6.447% 6.674% 6.891% 6.909%

Non-overlapping periods start end [3] [4] 6-Jan-OO 6-Jan-OO 15-Mar-OO 15-Mar-OO 15-Jun-OO 15-Jun-OO 21-Jun-OO 21-Jun-OO 20-Sep-OO 20-Sep-OO 21-Sep-OO 21-Sep-OO 20-Dec-OO 20-Dec-OO 20-Mar-Ol 20-Mar-Ol 21-Mar-Ol 21-Mar-Ol 21-Jun-01

Length of period 69 92 6 91 1 90 90 1 92

D-C rates [5] 6.152% 6.300% 6.447% 6.561% 6.674% 6.891% 6.900% 6.909%

simple IR

DFs

[6]

[7]

5.990% 6.200% 0.302% 6.499% 6.561% 6.729% 6.950% 6.900% 6.970%

1 0.988649 0.973229 0.972208 0.956494 0.956319 0.940497 0.924435 0.924258 0.908083

2. Pricing the swap Fixed rate Swap dates 6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01

Interpolated Z-c rates on swap dates

Swap DFs

5.9826% 6.0975% 6.2280% 6.3524%

0.984991 0.969644 0.953704 0.937128

Present value =

Estimated 3 mo. Libor fixings 6.0313% 6.2673% 6.5379% 6.7687%

6.5635% Floating cashflows

– – – –

152,457 158,425 167,080 176,738

Fixed cashflows

670,938

Discounted net cashflows – 150,168 – 153,615 – 159,345 463,129

0.0

Swaps and Other Derivatives

30

1. convert the simple interest rates in column [1] into daily compounded rates in [2] to place them all on the same basis; 2. create a set of non-overlapping contiguous periods as shown in [3] and [4]; 3. for periods that are uniquely defined, simply copy over the compound rates; 4. for periods that are undefined, such as 15 to 21 June, take the average of the two neighbouring periods, i.e. 0.5 x (6.152% + 6.447%) = 6.300%; 5. for periods that are defined twice, such as 20 to 21 September, again take the average; 6. once column [5] has been completed, then convert the rates back to simple interest and calculate the discount curve in the usual way; see columns [6] and [7]. The pricing differences, as may be seen by comparing the two worksheets, are extremely small. However such an approach can be invaluable for the estimation of short-dated forwards or FRAs over a gap or overlap. We will now return to the hedging of this money market swap. Because futures affect the swap through both the estimation and discounting processes, it is easiest to use numerical perturbation to estimate the impact of changing market conditions. In turn, each of the market rates, i.e. the cash stub and the futures rates, were perturbed upwards by 1 bp, and the change in the value of the swap noted; see column [9] headed PV01 (present value of 1 bp, also known as PVBP) in Worksheet 2.2. This is often called "blipping" a curve, i.e. take a curve of rates, perturb one rate, note the change in value, return the perturbed rate to its original value and move on to the next rate. The hedge amounts are now calculated by dividing the PV01 by, in this case, $25 for a Eurodollar future to give the futures contracts. Notice that the PV01 for the cash stub has been included in the PV01 for the first futures, as it tends to be relatively small and the objective is to create a fully offbalance sheet hedge. This is frequently described as a "tailed" hedge. The effectiveness of the hedge, in contrast with the previous one, is shown in Figure 2.6. The net effect is much smaller, suggesting that the hedge is much more effective. However, another phenomenon has arisen, namely that the net effect is always positive! The transaction plus hedge cannot lose irrespective of what happens to rates. This is an example of a "convexity (or gamma) effect". How it arises, how to measure the likely impact, and its implications will be discussed in the Appendix to Chapter 5. 4,000 3,000

Tailed

2,000

Untailed

1,000 0

1,000 2,000 3,000 4,000 -200

-150

-100

-50

0

50

Change in futures prices (bp)

Figure 2.6

Hedge effectiveness

100

150

200

Short-term Interest Rate Swaps

31

2.4 FUTURE VALUING A SWAP Before we finish with this structure, we are going to price the swap from a different point of view. Returning to Worksheet 2.2, we find the following information:

Swap dates

Estimated 3mo. Libor fixings

6-Jan-00 6-Apr-OO 6-Jul-00 6-Oct-00 8-Jan-01

6.0313% 6.2673% 6.5379% 6.7687%

Floating cash flows

Fixed cash flows

– 152,457 -158,425 – 167,080 -176,738

670,935

Three cash payments have to be made, and at the end of the year there is a cash in-flow. Where is the money raised for the payments? Let us assume we can borrow it at Libor flat; i.e. a payment of $152,457 has to be made on 6 April 2000 and we will assume that we borrow it for 3 months at a rate of 6.2673%. Therefore, the payment due on 6 July is now: $152,457 x (1 + 0.252 x 6.2673%) + $158,425 = $313,296 This amount itself may have been borrowed for 3 months at 6.5379%, and so on. At the end of the swap, there is a receipt of $670,935 on 8 January 2001. The net future value of the swap can therefore be calculated. A fair price for this swap would be when the NFV is zero, as we can see in column [4] of Worksheet 2.4. The price is in fact exactly the same as before, implying that discounting off a Libor curve is the same as reinvesting at the implied forward rates. An alternative view is that the swap cash payments are tantamount to lending the counterparty money. What would be a fair rate for this counterparty, bearing in mind its relative creditworthiness? For example, if we assume that we would only lend to this particular counterparty at Libor +100 bp, then we should use this rate as our effective borrowing cost. The fair price for the swap increases by about 1 bp to 6.58%. The detailed cashflows are shown in columns [5] and [6] — notice of course that the floating cashflows in [5] are unchanged — and the reinvested ones in [7]. The change in price may not be very significant for a short-term swap with a relatively small notional principal of only US$10 million; it would be much more significant for a longer-term structured swap on a larger notional principal, as we shall see later.

[WORKSHEET

2.4 OVERLEAF]

32

Swaps and Other Derivatives Worksheet 2.4 Pricing a money market swap off a futures strip using future valuing

Today's date: 4-Jan-00 Swap details: Principal amount Maturity

Cash rates

6-Jan-00 15-Mar-00 5.99% 6-Apr-00 6.0313%

10 million USD 1 year

•

March June September December March 01

Futures prices

93.80 93.50 93.27 93.05 93.03

Futures dates

Implied forward rates

15-Mar-00 21-Jun-00 20-Sep-00 20-Dec-00 21-Mar-01

6.200% 6.500% 6.730% 6.950% 6.970%

Swap dates

6-Jan-OO 6-Apr-00 6-Jul-00 6-Oct-00 8-Jan-01

0.253 0.253 0.256 0.261

Short-term Interest Rate Swaps

Borrowing margin 100 bp over Libor

Borrowing margin -bp over Libor

6.5759%

6.5635% Estimated 3mo. Libor fixings

Floating cashflows

Fixed cashflows

Accumulated cashflows at Libor flat

Floating cashflows

Fixed cashflows

[1]

[2]

[3]

[4]

[5]

[6]

670,931

– 152,457 – 313,296 – 485,611 0

6.0313% 6.2673% 6.5379% 6.7687%

– – – –

152,457 158,425 167,080 176,738

Net future value =

0.000

– – – –

152,457 158,425 167,080 176,738

672,199

Lending money at Libor + margin

[7] – 152,457 – 313,682 – 486,412 0 0.000

This page intentionally left blank

Generic Interest Rate Swaps

OBJECTIVE The previous chapter discussed short-term IRS, priced and hedged off a futures strip. Such a strip will not go out very far, and medium to long-term swaps are much more closely related to the bond market. This chapter first introduces a generic or "vanilla" swap, and shows how it may be regarded either as an exchange of cashflows, or as a link between two distinct markets. The pricing of a generic swap is then explored, first through the concept of comparative advantage, and then through the mechanism of hedging the two sides separately. During this latter process, we discuss a widely held belief in the swap market, namely the floating side of a generic swap including notional principals has no value. This leads us on to the identification of the fixed side as a par bond, and to a discussion of the relationship between the bond and swap market. Hedging swaps with bonds to protect against interest rates changing adversely is quite common, but we also explore what would happen to such a hedge if the rates do not move: namely, cost of carry issues. Finally the chapter concludes with the description of various ways, some bad but popular and some good, to imply discount factors for a given generic swap curve.

3.1 GENERIC INTEREST RATE SWAPS A generic or "plain vanilla" interest rate swap (a term probably first coined by the swap group at Salomon Brothers in the mid 1980s) is the simplest form of medium-term IRS. These constitute the vast bulk of inter-bank trading. Because of their maturity, they are associated far more with an underlying bond market than a deposit futures market for hedging. A generic swap is defined in Table 3.1. The important elements of the definition are as follows. • The minimum maturity typically reflects the length of the liquid futures market: this is obviously currency-specific. • The maximum maturity usually indicates the end of the very liquid swap market for which the bid-offer spread is tightest and constant: US dollars is currently on a 3 bp spread for 30 years whereas sterling, for example, is on a 5 bp spread for 10 years which widens rapidly after that to 12bp for 30 years. It is feasible to get longer swaps in the major currencies out to 50 years, but these lack liquidity. • The effective date depends on the convention in the floating rate reference market: so for US dollars this would be two business days after the trade date, for sterling same day, and so on. • A generic swap is a "spot" swap, therefore the fixed rate is the current market rate. • The frequency of the fixed side usually matches the frequency of the coupon in the hedging bond market, for example sterling swaps are semiannual reflecting the semiannual coupons in the gilts market: there are exceptions to this, such as USD where

36

•

•

•

•

Swaps and Other Derivatives

the swaps are usually quoted annually whilst the T-bond pays semiannually, and the South African market where the swaps are quoted quarterly whilst the bond pays semiannually. The daycount convention on the fixed side again often but not invariably matches the underlying bond conventions: US T-bonds are Act/Act, the fixed side of the swaps is usually quoted Act/360; swaps in continental Europe used to be quoted 30/360 to match the government bond markets, but the bond convention changed to Act/Act upon the introduction of the Euro single market. The floating side almost invariably follows the convention in the domestic money market: for US dollars, Libor is fixed two business days before the start of each floating or roll-over period, and paid at the end of the period using Act/360 daycount convention. Therefore the first fixing is the current Libor rate. It is important that the tenor of the floating rate, its frequency of reset fixings, and the frequency of payment all match: for example, if the floating rate is 6mo. Libor, then Libor is re-fixed at the beginning of each 6 month period and paid at the end of each 6 month period. It is perfectly feasible to get mismatch swaps, where these conditions are not true, such as using 6mo. Libor but paying every 3 months as we shall see later, but these are not generic swaps. Some of the generic swaps traded in the domestic US market, with reference rates such as the weekly T-bill fixings, violate these conditions; these will be discussed later. Finally there is no spread on the floating rate, nor any lump sum payments indicating that both counterparties deem the swap to be "fair", i.e. its value at mid-rates should be zero.

Interestingly, whilst the definition includes a statement on the range of possible maturity of a generic swap, it does not include any guidance as to the likely size of the underlying Table 3.1 Generic US dollar swap terms maturity trade date effective date settlement date

5–30 years date of agreeing to enter into the swap depends upon convention in floating rate market effective date

Fixed side fixed coupon frequency daycount pricing date

current market rate either annual or semiannual depends on the market trade date

Floating side floating index spread payment frequency daycount reset frequency first coupon

defined money market indices none tenor of the floating index depends on the market tenor of floating index (except T-bills) current market rate for index

premium/discount

none

Source: The Interest Rate Swap Market, Salomon Brothers. June 1985 (modified slightly).

Generic Interest Rate Swaps

37 Table 3.2

7-Year generic US dollar swap

notional principal maturity trade date effective date settlement date

100 million 7 years 4 January 2000 6 January 2000 6 January 2000

Fixed side fixed coupon frequency daycount pricing date

7.225% annual Act/360 4 January 2000

Floating side floating index spread frequency daycount reset frequency first coupon

6mo. Libor none 6 months Act/360 6 months 67/32% ( = 6.21875%)

premium/discount

none

principal to which the interest rates are applied. Market practice would probably imply $10–50m; that is not to say that larger swaps could not be obtained relatively routinely, it is just that the bid-offer spread on the pricing might be slightly wider. To make this more precise, a 7 year generic US dollar swap is defined in Table 3.2. The cashflows generated by this swap are shown in Table 3.3. Notice that, as in the earlier money market swap example, the periods are adjusted for non-business days, and the receipts on the fixed side of the swap reflect these adjustments. This is in contrast with the bond market, when interest will also only be paid on a business day but the amount will not vary. It is necessary to take these different conventions into account when structuring a bond-swap package, as we shall see later. The Libor values, other than the first fixing, are of course not known. A generic swap is usually considered as an agreement to exchange two streams of cashflows, one calculated with reference to a fixed rate of interest and the other with reference to a floating rate. We can however change the frame of reference if we pretend that the notional principal amounts (NPA) are also exchanged at the beginning and end of the swap, as shown in Table 3.4. The (pretend) exchange does not affect the economic reality of the swap, as the NPAs are assumed to be paid and received simultaneously at the start and end of the swap. However, the swap may now be thought of as: • buying a fixed rate bond (albeit with slightly strange interest payments); • either: issuing or selling a floating rate note at Libor flat or: borrowing money on the money markets. In either case, it may be considered as an instrument that links together two distinct markets.

Swaps and Other Derivatives

38

Table 3.3 Cashflows of a generic swap Trade date:

4-Jan-00

Settlement date: Notional principal: Maturity: To receive fixed rate: To pay floating rate: First Libor fixing:

6-Jan-00 100 million 7 years 7.225% ANN, Act/360 6mo. Libor 6.21875%

Days 6-Jan-OO 6-Jul-00 8-Jan-01 6-Jul-01 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06 6-Jul-06 8-Jan-07

182 186 179 185 182 182 182 183 182 184 181 184 181 186

Fixed cashflows

7,385,555.56 7,305,277.78 7,305,277.78 7,325,347.22 7,345,416.67 7,325,347.22 7,365,486.11

Floating cashflows -3,143,923.61 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor

Table 3.4 Cashflows of a generic swap with notional exchange Days 6-Jan-OO 6-Jul-OO 8-Jan-Ol 6-Jul-Ol 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06 6-Jul-06 8-Jan-07

182 186 179 185 182 182 182 183 182 184 181 184 181 186

Fixed cashflows

Floating cashflows

– 100,000,000

100,000,000 -3,143,923.61 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor – Libor – 100,000,000

7,385,555.56 7,305,277.78 7,305,277.78 7,325,347.22 7,345,416.67 7,325,347.22 107,365,486.11

3.2 PRICING THROUGH COMPARATIVE ADVANTAGE There are various ways of pricing financial instruments. Probably the most common is to price an instrument relative to similar instruments already in the marketplace. But this begs the question as to how the first instruments receive their price. A second approach is to estimate the cost of replicating the instrument using financial instruments drawn from

Generic Interest Rate Swaps

39

other, more liquid, financial markets. However the oldest method is to identify a price which will provide both the seller and the buyer with some perceived economic benefit, i.e. an arbitrage price. This last method was most common in the early days of the swaps market, but as the market has grown in size and increased in speed, the first and second approaches are far more prevalent. Nevertheless it is important to understand the last, as fundamentally it is this rationale that drives the market. Recall the EIB–TVA swap described in Chapter 1. Briefly, the EIB wished to borrow DEM, and the TVA USD. Their funding costs are shown below: USD

EIB TVA Spread

T+17 T + 24 7 bp

DEM B + 13 B+ 17 4 bp

We could use this information to price a swap. Remember, the upshot was that the EIB issued a USD bond at T + 17 bp, and the TVA issued a DEM bond at B + 17 bp as shown below. They entered into a swap, where m is the "price". What is a fair price? T +m

7+17

B + 17

If we look at their net costs of funds: Cost of bond Receipt on swap Payment under swap Net cost

EIB – (T+ 17)

TVA

– (B + 17) +B –(T+m)

-(B+17 – m)

To make the swap attractive to both parties, the net costs must be less than funding directly in the markets, i.e.: 17 - m < 13 bp for EIB, and 17 + m < 24 bp for TVA. The margin must therefore lie in the range: 4 < m < 7. If the advantage were divided equally, then m = 5.5 bp. The underlying principle is that the two counterparties will both perceive a benefit from entering into a swap at some agreed price. It is important to stress that the key word is perceive, namely it is feasible for two parties to do a swap if they have strongly held but diametrically opposite views as to the future movement of interest rates. More likely however is when the parties have asymmetric advantages, for example different access to markets. It is frequently argued that such arbitrages will disappear as markets become more efficient. However there are many sources that consistently distort markets, such as governments with asymmetric taxation and cheap subsidization, investors with arbitrary credit limits, capital regulations on banks with existing exposures, different perceptions of credit pricing, and so on, that suggest the arbitrages will continue. It is possible to create a matrix of comparative advantage, which may be used to identify opportunities. Consider the following (simplified) US market data:

Swaps and Other Derivatives

40

US government Governmental agencies Banks Financial institutions Prime corporates Lower-credit corporates

Bond market

Inter-bank money market

B B +3 B + 25

L – 12 bp L – 10 bp L L + 5 bp L + 10 bp L + 40 bp

B + 35 B + 50 B + 100

This table conveys two main messages, namely that the cost of borrowing increases as the creditworthiness of the borrower decreases, and secondly that the relative cost of borrowing is typically much lower in the floating rate market than in the fixed rate market. There are a variety of reasons for this: • because the floating rates are reset periodically back to the current market rates, the potential credit exposure for the lender on fixed interest payments is considerably greater than on floating payments; • margins are in part determined by supply and demand, with the floating rate loan market usually being far larger and more liquid than the fixed rate bond market. The second message can lead to potential swap opportunities. If we take an extreme (and unrealistic) case, suppose: • US government wished to borrow floating; • lower-credit corporate wished to borrow fixed. If each went directly to the relevant market, this would cost a total of (L — 12) + (B + 100) = B + L + 88 bp (assuming that the basis points in the two markets are "additive"). However the lower-credit corporate only pays 52 bp more than the government in the floating market, compared to at least 100 bp in the fixed market. Therefore a cheaper way of raising the money would be for: • government to borrow fixed rate at B, • lower-credit corporate to borrow floating rate at L + 40 bp, • and enter into a swap with each other, resulting in a total cost of current arrangement:

40 bp, i.e. saving 48 bp. The figure below shows the Libor - m

Government

Lower corporate

L + 40 bp

Using the same argument as above, namely "what margin m would make this transaction attractive to both counterparties?", we can calculate the net cost of funds for each party:

Generic Interest Rate Swaps

41

Table 3.5 Matrix of comparative advantage US Governmental Financial Prime Lower-credit government agencies Banks institutions corporates corporates US government Governmental agencies Banks Financial institutions Prime corporates

1

13 12 -

18 17 5 -

28 27

15 10 -

48 47 35 30 20

Note: figures are (difference in fixed funding) less (difference in floating funding). Government

Net cost of funds –(L – m)

Lower-credit corporate

–(B

+ 40 + m)

To make this transaction attractive to them both, 12 < m < 60. For example, assume that m = 40 bp. The net funding cost to the government is L — 40 bp; a saving of 28 bp. The corporate funds itself at B + 80 bp; a saving of 20 bp. Both parties achieve cheaper funding, with a total saving as expected of 48 bp divided between them. The source of the saving is that the government, as the stronger credit, is prepared to take a different credit view on the lower-credit corporate than the fixed rate market. This is tempered by the fact that the government is not taking a risk on the principal amount of the borrowing, merely on the difference between the fixed and floating swap payments. In practice, the pricing of the swap would be by negotiation, and obviously the strong credit has considerably more power. Also a bank would typically be acting as an intermediary and credit guarantor, and would require part of the savings. Such an example is extreme and unrealistic. A matrix of comparative advantage may be derived as in Table 3.5. This suggests the apparent savings that may be made between pairs of counterparties; for example, a bank entering into a swap with a prime corporate credit might achieve an overall saving of 15 bp.

3.3 THE RELATIVE PRICING OF GENERIC IRS In this section, we wish to explore the relationship between the swap market and other financial markets, so that we may be able to understand relative swap pricing. Just to remind ourselves, when a generic swap is first entered into, the two counterparties perceive themselves as being in equal positions. To be more precise, each of the two counterparties perceives that the total value of the anticipated receipts is not less than the total value of the anticipated payments. If we assume the two counterparties use the same valuation process, and that they have the same access to the market, then this reduces to: "value of receipts = value of payments" or alternatively "net value of swap = 0". If this equality were not true, then one party would deem itself to be disadvantaged and refuse to enter into this freely negotiated contract until the appropriate changes were made. As we have already seen, a generic swap may be broken down into streams of cashflows, some generated with reference to a fixed interest rate, others possibly with reference to a variable rate. When we discussed the valuation of cashflows in the money market swaps, we either discounted them back to the day of analysis, which is the commonest method, or future valued them to the end of the swap. We will use discounting as the main method of analysis, and therefore the above expression may be modified to:

42

Swaps and Other Derivatives "present value of receipts = present value of payments"

We also showed above how a generic swap may be represented as two synthetic instruments, a fixed rate bond (with slightly unusual coupon) and a money market transaction. A swap will first be discussed in this guise, and then later without principal amounts. Consider the floating side of the above generic swap only: (a) it is issued for $100 million, (b) it pays interest at Libor flat, and (c) at maturity, it is redeemed at $100 million as shown below: $100 million

$Libor

$100 million

Assume that we are a bank that can borrow or lend/deposit money at Libor ± margin within a marketplace, as shown below. The overall economic value of this floating transaction therefore depends solely upon the achievable margin, and not upon the particular levels of Libor. In particular, if the margin were zero, then the transaction would also have a zero value, which would lead in turn to the following statement: "value of a generic floating side of a swap, including the notional principals, is zero" $100 million

$100 million

$Libor ± margin

$Libor

$100 million

$100 million

Understanding this aspect of swaps is absolutely key to the pricing of swaps. Suppose, in the example above, that the margin for depositing is —10bp. The floating side would

Generic Interest Rate Swaps

43

therefore be effectively making a running loss of $100 million x 10 bp = $100,000 pa or roughly $700,000 over the lifetime of the swap. This money would have to be recovered from the fixed side, which would have to be increased by (roughly) 10 bp above the current market, thus making it uncompetitive. Alternatively, if the $100 million could have been lent out at a margin above Libor, then this value could have been used to subsidize the fixed side of the swap. So swap pricing should depend in part directly upon the abilities of banks to raise or to place money in their various local money markets; this in turn depends upon their creditworthiness. For example, one highly rated European bank was able to raise funds recently at an average cost of {Libor—6 bp}, whereas for most of the 1990s, Japanese banks have been unable to borrow USD at Libor, but have had to pay a premium which has ranged up to some 40 bp. However, in practice, many banks assume, implicitly or explicitly, that the funding margin for the purpose of pricing swaps is Libor flat. The European bank above, for example, has instructed its swap desk to assume a funding cost of Libor flat for pricing purposes, so that the prices are not being subsidized by the bank's credit rating. Many swap pricing systems do not permit a true funding cost to be entered, but tacitly assume Libor flat. It could be argued that if a swap portfolio is relatively flat, i.e. the payments and receipts approximately balance, then this assumption is unnecessary. Unfortunately portfolios are seldom flat, unless they have been constructed over an entire economic cycle. When interest rates are perceived high, then most end-users wish to pay floating and receive fixed. The reverse is true when rates are low. Therefore demand for new swaps is frequently one-way round, creating an imbalance for a market maker. A generic swap is a medium-term instrument; in this instance, the counterparties are committed to meeting their obligations for 7 years. How realistic is it to assume their creditworthinesses will remain constant, and hence their ability to raise or to deposit money, over the lifetime of the swap? The average funding cost for USD of one of the major US banks increased to {Libor + 2 bp} during the height of the S&L crisis in the late 1980s. All their swap pricing — plus P&L and bonuses — had been calculated on the basis of Libor flat, so suddenly their (extremely large) swap portfolio started to haemorrhage profits! Nevertheless, the assumption of zero margin is widely made, and we will (albeit with reservations) do the same for the remainder of this book. We will however show how to modify the pricing to include the funding cost for some structures later in the book 1 .

3.4 THE RELATIONSHIP BETWEEN THE BOND AND SWAP MARKETS Turning to the fixed side of the swap, we can immediately conclude that: "value of the fixed side of a generic swap, including the notional principals, is zero" because the net value must equal zero. However this synthetic bond has some further properties, namely it is a par bond. Such a bond satisfies three conditions: • its current price is par, i.e. $100 million; • it is redeemed at par at maturity; • the first period is a full period, or the accrued interest is zero. 1

As indeed we have already done for the money market swap using future valuing.

Swaps and Other Derivatives

44

In this situation, the yield-to-maturity equals the coupon of the bond. For the generic 7 year swap, the fixed rate is 7.225% ANN Act/360 whilst its yield is 7.221 %, as may be seen in the box of Worksheet 3.1. The slight difference is due to the impact of the non-business days, which causes the fixed cashflows to vary slightly. Let us now turn the argument on its head. We are a swap market maker who has just been asked to make a price for a generic 7 year swap. We could turn to the bond market to identify the current yield of a 7 year par bond. See Figure 3.1. Obviously such a bond will not be trading, but it can be estimated by interpolating the benchmark curve to give 6.445%.

Worksheet 3.1 Calculating the yield of a par bond Trade date:

4-Jan-00

Settlement date: Notional principal: Maturity: To receive fixed rate: To pay floating rate: First Libor fixing: Yield-to-maturity:

6-Jan-OO 100 million 7 years 7.225% ANN, Act/360 6mo. Libor 6.21875% 7.221%

Days 6-Jan-00 6-Jul-00 8-Jan-01 6-Jul-01 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06 6-Jul-06 8-Jan-07

Fixed cashflows – 100,000,000

182 186 179 185 182 182 182 183 182 184 181 184 181 186

7,385,555.56 7,305,277.78 7,305,277.78 7,325,347.22 7,345,416.67 7,325,347.22 107,365,486.11

Yield-based DFs Floating cashflows 1 100,000,000 -3,143,923.61 0.931209 — Libor — Libor 0.867822 — Libor — Libor 0.808749 — Libor — Libor 0.753552 — Libor — Libor 0.701986 — Libor — Libor 0.654076 — Libor — Libor 0.609199 – Libor–100,000,000 0.0000

Present value =

6.60

6.30

5

|

10

15

20

25

Maturity (yrs) Figure 3.1

US benchmark bond curve, 4 January 2000

30

Generic Interest Rate Swaps

45

Unfortunately, the most liquid bond market in most countries is the governmental market, which is by definition deemed to be of a higher credit rating than the average inter-bank swap market, generally considered to be somewhere between AA— and A+. This implies that the equivalent bond yield is likely to be lower than that quoted in the swap market. Nevertheless, it is a starting point, as shown in Table 3.6. The bond yields are interpolated from the benchmark bonds, the most liquid being 1, 5, 10 and 30 years maturity. The table shows mid-spreads and mid-swap rates. To convert from the bond yield, which is quoted on a semiannual Act/Act basis, to a swap basis requires the following calculations: (a) (b) (c) (d)

consider the 7 year bond yield of 6.445%; add the spread of 75 bp = 7.195%; convert the bond to annual by (1 + 0.5 x 7.195%)2 – 1 = 7.324%; convert the daycount by multiplying by (360/365) = 7.225%.

(There may be small differences due to rounding.) So the first and most important reason for the swap spread is the difference in credit between the underlying bond market and the inter-bank swap market. But there are other influences on the apparent spread. As we have already suggested, demand for paying or receiving swaps is seldom balanced, but depends Table 3.6 US generic swaps being quoted as a spread above the bonds

6-Jan-00 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05 6-Jan-06 8-Jan-07 7-Jan-08 6-Jan-09 6- Jan- 10 6-Jan-11 6-Jan-12 7-Jan-13 6-Jan-14 6-Jan-15 6-Jan-16 6-Jan-17 8-Jan-18 7-Jan-19 6-Jan-20 6-Jan-21 6-Jan-22 6-Jan-23 8-Jan-24 6-Jan-25 6-Jan-26 6-Jan-27 6-Jan-28 8-Jan-29 7-Jan-30

Interpolated yields

Spread (bp)

Swap rates ANN, Act/360

6.326% 6.352% 6.378% 6.403% 6.424% 6.445% 6.466% 6.487% 6.501% 6.503% 6.505% 6.507% 6.509% 6.511% 6.513% 6.515% 6.517% 6.519% 6.521% 6.523% 6.525% 6.527% 6.529% 6.531% 6.533% 6.535% 6.537% 6.539% 6.540%

55 65 69 70 72 75 77 78 80

6.895% 7.025% 7.085% 7.135% 7.175% 7.225% 7.265% 7.295% 7.335%

85

7.385%

89

7.435%

89

7.445%

88

7.445%

86

7.435%

46

Swaps and Other Derivatives

upon the perception of the economic cycle. If rates are perceived to be low and therefore likely to increase, most end-users want to pay fixed, receive floating. The market maker, observing the high demand for paying fixed, will in turn increase the fixed quote which is effectively equivalent to increasing the spread. In this fashion the demand will be managed. Conversely, when rates are perceived to be high and will only come down, then the spread is reduced. An extreme case of this occurred in the early 1990s in Germany, when interest rates were increased substantially to fund the reunification, driving the swap spread down until at times it became negative! Obviously such an arbitrage situation is seldom sustainable for long periods. The South African swap market for some considerable time had been unusual in that swaps are frequently quoted at very little spread off the government (and paristatal) bond curve. The reason would appear to be that, locally at least, the major financial institutions are deemed to be as creditworthy as the government. Indeed the long end of the swap is often significantly, i.e. 30 or 40 bp, below the bond curve. At the time of writing in 2001, swap spreads were becoming more positive in line with international expectations. There was a similar situation in Italy in the mid 1990s, when certain Italian organizations could raise money more cheaply in the international bond market than the Italian government. Another reason for the spread is the cost of hedging a swap portfolio. Consider a simple situation in which a bank has just entered into a swap to pay fixed, receive floating. The swap could be hedged by buying a specific bond, as follows: 1 Floating interest

Floating Money to buy the bond Fixed

Buy the bond

Fixed coupon

For reasons of liquidity, the bond is likely to be governmental. If interest rates fall, the swap loses in value, but the bond value increases. Hopefully, if the hedge is calculated correctly, one will off-set the other, as we shall see in Chapter 8. But suppose rates do not change at all. Do we make money, lose money, or remain flat? Over 1 day, we will:

Generic Interest Rate Swaps

47

pay accrued interest F on the swap receive accrued interest C on the bond receive accrued floating interest L on the swap pay accrued floating interest L on the borrowing Assuming the bond is trading close to par, in other words the nominal amount of the bond to be bought would be similar to the notional principal of the swap, then every day we would effectively lose {(F — C) x P x 1-day} where (F — C) represents the swap spread, as the Libor cashflows would cancel. This carry-cost must be included in the pricing. If the swap and bond were to be held to maturity, and this is part of the argument in South Africa that the market is so illiquid, then F could not exceed C. In normal circumstances, such a hedge would only be held for a short period, so the total carry-cost over this period would then have to be spread over the lifetime of the swap. In practice, the market maker is more likely to use the bond repurchase (or "repo") market than the cash market, as this allows practitioners to go long or short bonds more efficiently and cheaply. Briefly, the repo market operates as follows. Consider an investor who owns a bond. He can partially fund his bond position by borrowing money and providing the bond as collateral. Usually the interest charged on the borrowing, the repo rate, is lower than Libor, as the credit risks are lower due to the collateralization. He is said to have sold a repo transaction, or alternatively to have bought a "reverse repo" transaction (the two are mirror images of each other). The repo market is very closely related to the "sale and buy back" market. This latter involves the sale of a security to a counterparty, with an agreed repurchase price on an agreed date in the future. The repurchase price is usually a neutral forward price, and therefore higher than the sale proceeds due to the accrual of coupon. The major difference between the two markets is ownership. In a repo transaction, the ownership of the bond remains with the original buyer of the bond who is entitled to any coupon payments or other events. Ownership in a "sale and buy back" transfers during the lifetime of the transaction, and the counterparty receives the coupon payments. Repos are becoming more popular than sale-and-buy-back, principally because the transactions are cleaner as many participants are using them for hedging purposes and do not want the added complication of dealing with cashflows. To ensure that the bond provides adequate security, the lender usually defines a collateral margin or "haircut". Suppose the investor wishes to borrow $100m. He would be required to place bonds exceeding that value by say 2% as collateral. If interest rates start to rise, the value of the collateral bond decreases. The additional 2% acts as a safety margin for the lender so that the loan would be fully collateralized during this interest rate rise. The size of the haircut also depends on the liquidity of the bond; if it is relatively illiquid, then the lender will demand a higher haircut to cover the risks if the borrower defaults and the bond has to be sold into an adverse market. During the late 1990s and early 2000s, the US and Western European governments moved into budget surpluses, with a concomitant decline in the size of their bond markets. Repo transactions were extended to non-governmental bonds, but with increases in the haircut. Most repos are transacted on a general collateral (GC) basis, i.e. a general interest is applied irrespective of the security. If you wanted a specific bond as collateral, then this may be said to be "on special", whereby the interest rate may be higher or lower than the GC rate. For example, we describe below how inflation swaps may be hedged using index-linked bonds; these bonds are usually on special due to the limited supply and excess demand.

48

Swaps and Other Derivatives

The repo market is highly liquid in many countries, but repo transactions are generally very short-term — approximately 80% of USD repos are overnight, and most European repos are under 14 days. Repo rates can be quite volatile, reflecting changes in supply and demand for the bonds. Coming back to the swap, the bank could fund the bond position by buying a reverse repo transaction, i.e. borrowing the bulk of the money from the repo market and providing the bond as collateral. There is likely to be a small positive accrued gain on the floating transactions, because the collateralized repo rate is likely to be below Libor. Let us assume that the floating side of the swap is quarterly; 3 month Libor is therefore fixed at the beginning of the quarter, to be paid at the end. But the repo is short-term, and so would have to be rolled over if the hedge were to be held for the full quarter. Hence, even if repo rates are initially below the Libor fixing, it is possible for them to rise over the quarter and convert the gain into a further carry-cost. All of this adds to the risks of hedging, and hence to the cost. It would be feasible to obtain a "term" repo, i.e. one agreed for a fixed period of time such as 3 months to match the Libor tenor. However, the rate on such a repo is likely to be higher than the GC rate, and nearer to Libor. Another aspect to consider is flexibility; does the bank really wish to hedge this swap fully for 3 months, or will the hedge change as additional swap transactions are done? An additional source of risk is the "basis". The repo, bond and swap markets are all traded markets in their own right, and whilst linked by arbitrage constraints, also have their individual characteristics. Basis risk is the term used to describe the risk of one market moving, possibly due to some internal factors, relative to the other markets. If a bank enters into a swap, it initially possesses a position which is open to the movement of both short and long interest rates. By then entering into a counter-position in the bond market, the bank has attempted to reduce its long interest risk by substituting basis risk. In some circumstances, as we shall see later, basis risk may be greater than the initial interest rate risk, which suggests that the specific hedge is increasing overall risk not reducing it. The concept of "comparative advantage" drives many capital market transactions. This was discussed in Chapter 1 and in Section 3.2, but briefly re-stated in this context it proposes that a bond issuer will issue a bond into the market where there is the greatest demand, hence pay the lowest yield or conversely receive the highest price, and subsequently swap it into the funds actually required. A bond is designed very much to meet the specific requirements of the investor community, and derivatives are then used to transform the bond into the specific requirements of the issuer. Very commonly, to assist the investors, newly issued bonds are quoted as a spread over some appropriate governmental reference bond. As we have seen above, swaps are also frequently derived as a spread over the bond curve. We can therefore have the following situation: Libor

B +S B + S'

49

Generic Interest Rate Swaps

where S is the swap spread, S' the issuance spread. Concentrating just on the swap, and employing the following rather dubious manipulations (broadly correct but only exactly correct under certain circumstances): • deduct the swap spread S from both sides • add the issuance spread S' to both sides we end up with a swap: Libor–S + S'

B + S'

The objective of the issuer, as always, is to obtain cheap, i.e. sub-Libor, funding. This is true when S > S', namely when the swap spread is wide and/or the issuance spread is tight. Under these circumstances, there will be a number of swapped bond issues. Simple supply and demand arguments suggest that the increased issuance will drive S' up, and the swap counterparty will reduce S, hence closing the issuance window. Consider the plight of a potential bond issuer. The swap market is currently trading at a wide spread to the bond market, but for some reason the potential issuer will not be in a position to issue for another 3 months. However if he waits that long, it is likely that the window will be closed. Therefore he would like to do a swap today, which locks in today's spread S0, but which starts in 3 months' time: Libor Issuer

Notice that he does not care what happens to the bond yield B over the 3 month period, as that will be negated by the absolute level of the bond issue. To understand such a "spreadlock" swap, we will examine it from the point of view of a bank provider: Libor

The bank enters into an off-setting generic swap today, matching the maturities: Libor

Libor Bank

d B0+S0

and simultaneously does a (ideally 3 month) repo to lend money and to receive repo interest plus the reference bond as collateral. The bank then sells the bond into the market, with the intention of buying it back in 3 months' time.

50

Swaps and Other Derivatives

Consider what may happen at the end of the 3-month period, when the repo terminates. First assume rates do not move. From the bank's perspective, it will: • pay accrued Libor, but receive accrued repo; • receive accrued fixed B0 + S0, but pay accrued coupon on the bond when repurchased. Remember that currently S0 is quite wide, so it is likely that S0 will exceed {Libor — repo}, and that the bank has a negative cost-of-carry. Now assume rates do move over the period: • if rates increase such that B3 > B0, the bond price will have decreased so that the bank can repurchase it cheaply and make a profit that should off-set the running loss from the two swaps; • conversely, if rates decrease so that B3 < B0, the running profit should off-set the increase in bond price. The amount of bond to be repoed is determined by the need to match the bond gain or loss with the swap's losses or gains. The nominal amount of the bond may therefore not match the notional principal of the swap, which will result in some accrued mismatches which must also be included in the pricing, i.e. some margin usually deducted from S0 to compensate for the hedging costs. Spreadlocks are often quoted in terms of this margin. For example, see Table 3.7. Spreadlocks only really occur during issuance windows for the reasons explained. As we shall see later, they should also be supplied during periods of investment demand, but for some reason the market does not appear to respond in that fashion.

3.5 IMPLYING A DISCOUNT FUNCTION At this point, we have discussed generic swaps and their relationship with the money market and, in more detail, the bond market. Interest rate swaps may be thought of as the arbitrage hinge between the two markets. This idea will be explored in more detail when asset packaging is considered. We are now going to move on and assume that we can observe the various traded markets, and discover what information they imply. Table 3.7 Spreadlock swaps as spread over mid-swap rates Spreadlock swap rates Maturity 1 yr 2 yr 3 yr 5 yr 10 yr 30 yr

Treasuries (yield %)

Mid-swap (bp spread)

6.29 6.79 6.72 6.65 6.42 6.15

112 72 83 99 131 156

Source: Prebon Yamane, 9 May 2000.

Forward start period l yr 2 yr

3 yr 5 yr 10 yr

-9 16 0 7 6 8

-6 14 0 5 4 6

-8 15 0 6 5 7

-2 14 0 4 3 4

2 12 0 3 2 2

51

Generic Interest Rate Swaps Today's date: 4-Jan-00

Spot 7 day 1 mo 3 mo 6 mo 12 mo

Libor cash Act/360 6-Jan-00 13-Jan-00 7-Feb-00 6-Apr-00 6-Jul-00 8-Jan-01

2 3 4 5 6 7 8 9 10 12 15 20 25 30

5.53125% 5.81250% 6.03125% 6.21875% 6.59375%

Mid-swap rates ANN Act/360 against 3mo. Libor yr 7-Jan-02 6.8950% yr 6-Jan-03 7.0250% yr 6-Jan-04 7.0850% yr 6-Jan-05 7.1350% yr 6-Jan-06 7.1750% yr 8-Jan-07 7.2250% yr 7-Jan-08 7.2650% yr 6-Jan-09 7.2950% yr 6-Jan-10 7.3350% yr 6-Jan-12 7.3850% 6-Jan-15 7.4350% yr yr 6-Jan-20 7.4450% yr 6-Jan-25 7.4450% yr 7-Jan-30 7.4350%

Current USD market data We have already seen how to estimate discount factors from cash rates: 6-Jan-OO 13-Jan-OO 7-Feb-OO 6-Apr-00 6-Jul-00 8-Jan-Ol

0.019 0.089 0.253 0.506 1.022

5.53125% 5.81250% 6.03125% 6.21875% 6.59375%

1 0.998926 0.994860 0.984983 0.969519 0.936853

Cash-based discount factors Consider the 2 year swap. Its cashflows are: Notional principal = 100 million 2 year swap rate = 6.8950% Daycount Act/360 6-Jan-OO 6-Apr-OO 6-Jul-OO 6-Oct-OO 8-Jan-Ol 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253

Fixed

7,048,222.22

6,971,611.11

Floating — 3mo. — 3mo. — 3mo. — 3mo. — 3mo. — 3mo. — 3mo. — 3mo.

Libor Libor Libor Libor Libor Libor Libor Libor

The value of this swap is of course zero. We are now going to proceed in two different ways. First, we also know that if we add the notional principal amounts to both sides, then under the assumption that we can always fund or deposit at Libor flat, the value of each side is zero. Therefore, considering the fixed side alone, we can write:

52

Swaps and Other Derivatives

-100,000,000 * 1 + 7,048,222.22 * DF, + (100,000,000 + 6,971,611.11) * DF2 = 0 where DFt is the discount factor at time t (in years). Now comes another big assumption in the swap market: namely, that all cashflows that occur at the same time in the future are discounted at the same rate. This means that we already know DF1 = 0.936853 from the cash rates, and hence we can solve this equation for DF2 = 0.873099. The assumption implies that all cashflows, and hence all swap counterparties, are of equal credit standing, which is approximately "interbank", as discussed above. This is in clear contrast to the bond market, in which cashflows from a particular bond are discounted at the yield-to-maturity of that bond, reflecting the issuer's anticipated credit over the bond's lifetime. The assumption probably reflects the early days of the swap market, when generally speaking only good credits could access the market, so that little attention was paid to potential credit exposures. Credit limits were frequently set in terms of notional principal amounts, and there was one price for all counterparties. Use of this assumption is probably one reason why credit-adjusted swap pricing has been so slow to catch on amongst market practitioners, albeit exacerbated by the first Basle Accord. This process may be repeated sequentially along the swap curve calculating the annual discount factors. A general expression for estimating DFt is: DFt = (1 – Qt-1 × s1)/(1 + dt × st) where st is the swap rate of maturity t dt is the length of the period from (/ — 1) to (t) Q t-1 = d 1 × DF 1 + d 2 x DF 2 + ••• + d t-1 × DFt-1 = Q t-2 + d t-1 × DFt-1

This is frequently called a "zero coupon bootstrapping" process, and the phrase "bootstrapping a curve" is in common usage. It means: • the process is initiated using a zero coupon rate, usually a cash rate but it may be off the futures curve; • the process then progresses sequentially up the swap curve. One necessary condition for the process is that swap rates must be known at annual intervals. Imagine the situation: we have just calculated DF10 and the next known rate is s12 as shown above. When we generate the cashflows for the 12 year swap, we will get the following: Daycount

11 year 12 year

6-Jan-11 6-Jan-12

1.014 1.014

Cashflow

7,487,569.44 7,487,569.44

i.e. two cashflows each with an unknown discount factor, but only one valuation equation. It is common therefore to "complete" the swap curve, estimating the missing points on the swap curve by some means of interpolation, usually either linear or some polynomial such as cubic. Table 3.8 shows the discount factors resulting from using both methods (for details, see the appropriate worksheets on the CD). The differences appear to be negligible and arise in the fourth decimal place. On a cashflow of, say, USD10 million that causes a difference in valuation of only $1,000. Such

Generic Interest Rate Swaps

53

Table 3.8

6-Jan-00 13-Jan-00 7-Feb-00 6-Apr-00 6-Jul-00 8-Jan-01 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05 6-Jan-06 8-Jan-07 7-Jan-08 6-Jan-09 6-Jan-10 6-Jan-11 6-Jan-12 7-Jan-13 6-Jan-14 6-Jan-15 6-Jan-16 6-Jan-17 8-Jan-18 7-Jan-19 6-Jan-20 6-Jan-21 6-Jan-22 6-Jan-23 8-Jan-24 6-Jan-25 6-Jan-26 6-Jan-27 6-Jan-28 8-Jan-29 7-Jan-30

Linearly interpolated DFs

Cubically interpolated DFs

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913 0.449607 0.416548 0.386092 0.357872 0.331445 0.308038 0.286206 0.265847 0.247058 0.229574 0.213420 0.198441 0.184513 0.171497 0.159491 0.148509 0.138304 0.128819 0.119937 0.111771

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913 0.449336 0.416567 0.385844 0.357627 0.331497 0.307627 0.285639 0.265372 0.246851 0.229740 0.213495 0.198469 0.184538 0.171559 0.159632 0.148555 0.138302 0.128816 0.119977 0.111895

differences would increase if the market rates possess a very large amount of curvature, but are unlikely to be significant. As we saw earlier, we can think of the floating side of the swap as a series of forward rates, which may be estimated using the formula for the rate from t to T: F(t, T) = [(DFt/DFT) – 1] / (T - t) Suppose we wish to construct the curve of 3-monthly forward rates (as this is the tenor of the floating side of the generic USD swaps). We have annual discount factors, but need to estimate them every 3 months! Again, we need to interpolate in some fashion. There are of course many ways of interpolating, but three are popular:

Worksheet 3.2 Summarizing the results of building a 3-monthly forward curve Today's date: 4-Jan-00 linear interpolation of swap curve log-linear zero-coupon linear 6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-01 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05 6-Apr-05 6-Jul-05 6-Oct-05 6-Jan-06 6-Apr-06 6-Jul-06 6-Oct-06 8-Jan-07 6-Apr-07 6-Jul-07 8-Oct-07

6.031% 6.310% 6.632% 6.749% 6.843% 6.963% 7.092% 7.222% 6.932% 7.055% 7.183% 7.316% 6.910% 7.032% 7.160% 7.293% 6.981% 7.107% 7.238% 7.375% 7.026% 7.153% 7.287% 7.425% 7.192% 7.325% 7.465% 7.613% 7.222% 7.357% 7.501%

6.031% 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219% 7.054% 7.084% 7.114% 7.144% 7.137% 7.162% 7.189% 7.215% 7.191% 7.213% 7.235% 7.256% 7.358% 7.386% 7.414% 7.444% 7.397% 7.421% 7.447%

6.031% 6.310% 6.690% 6.691% 7.030% 7.032% 7.034% 7.032% 7.121% 7.121% 7.121% 7.121% 7.099% 7.099% 7.099% 7.100% 7.175% 7.175% 7.176% 7.176% 7.223% 7.224% 7.224% 7.224% 7.400% 7.401% 7.401% 7.403% 7.431% 7.434% 7.436%

cubic interpolation of swap curve log-linear zero-coupon linear 6.031% 6.310% 6.632% 6.749% 6.843% 6.963% 7.092% 7.222% 6.932% 7.055% 7.183% 7.316% 6.910% 7.032% 7.160% 7.293% 6.981% 7.107% 7.238% 7.375% 7.026% 7.153% 7.287% 7.425% 7.192% 7.325% 7.465% 7.613% 7.222% 7.357% 7.501%

6.031% 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219% 7.054% 7.084% 7.114% 7.144% 7.137% 7.162% 7.189% 7.215% 7.191% 7.213% 7.235% 7.256% 7.358% 7.386% 7.414% 7.444% 7.397% 7.421% 7.447%

6.031% 6.310% 6.690% 6.691% 7.030% 7.032% 7.034% 7.032% 7.121% 7.121% 7.121% 7.121% 7.099% 7.099% 7.099% 7.100% 7.175% 7.175% 7.176% 7.176% 7.223% 7.224% 7.224% 7.224% 7.400% 7.401% 7.401% 7.403% 7.431% 7.434% 7.436%

55

Generic Interest Rate Swaps

-20

150 200 250 Time (months)

300

350

400

Figure 3.2 Average differences (bp) between interpolating the swap curve

linear interpolation of the DFs i.e. DFt = DFi-1 + {(DFi - DF i-1 )/(T i - Ti-1)} × (t - Ti-1) for i - 1 < t < i or DFt = DFi-1 + {gradient} × (t - Ti-1) linear interpolation of ln(DF) — effectively assuming the discount curve follows a negative exponential; linear interpolation of the equivalent zero coupon rates. In turn these rates may be: (i) continuously compounded: DFt = exp(—z,t) (ii) discretely compounded: DFt = (1 +z t /n) We saw quite significant differences when we used different methods of interpolating the short cash rates: the impact now will be even more dramatic. The results are summarized in Worksheet 3.2; the individual worksheets (which are not printed in this book but are on the enclosed CD) show the details of the calculations. To summarize the differences between the interpolation methods, Figure 3.2 was constructed by interpolating the swap curve both linearly and cubically, calculating the 3monthly forwards by using the three methods above, and then averaging the forwards. The graph shows the difference between the averages produced using the alternative methods of interpolating the swap curve. There is obviously no effect under 10 years. The big "steps" of the graph beyond that are quite characteristic. The swap curve for that day was a typical positive curve, and therefore the linearly interpolated swap rates will always be lower than the cubically interpolated one. Hence the estimated forward rates start off also being lower for the linear method, but because the two methods must be exact at the original swap points of 12 yr, 15 yr, etc., these linearly interpolated forward rates have to "catch up" to ensure that each original swap would be correctly priced. The size of the big steps decreases as the swap curve flattens at the long end. The small steps are more of a combined function of the interpolation to produce the forwards. The maximum difference between forward rates can be seen to be up to 30 bp! If we compare the three different ways of interpolating the discount curve, the linear method produces very distinct and different forward rates, with a characteristic zig-zag pattern. The other methods are relatively similar, at least where the curvature of the curve is small. See Figure 3.3. To understand why linear is so unacceptable, consider the following exaggerated simple situation:

56

Swaps and Other Derivatives

DFA

time

where a discount curve is known at time tl and t3 but interpolated at t2. Two methods of interpolation are shown: linear and non-linear. We calculate the forward rates from t1 to t2 and from t2 to t3 using: F12 = [(DFt1/DFt2) - 1]/(t 2 - t1) and F23 = [(DF t2 /DF t3 ) Because DFt2 > DFt2 by construction, this means that:

and

F12 <

r-L . r-NL ^23 > ^23

If we accept that the non-linear approximation of the curve is more accurate, the forward rates from the linear interpolation will oscillate around the non-linear forwards, resulting in the zig-zagging. We can approach the bootstrapping method from a different angle, and one that provides some additional insight. Assume that the floating side of the generic swaps is annual, i.e. matching the tenor of the fixed side. In theory this should not affect the swap pricing, as receiving 3 month Libor quarterly and 12 month Libor annually should be

Linear Zero-coupon — Loo-linear 150

200

250

300

350

400

Time (months)

Figure 3.3 3-Monthly forward rates using alternative methods of interpolating a discount curve

Generic Interest Rate Swaps_

57

equivalent. In practice, this may not be exactly true as the longer tenor creates a larger credit exposure for which there needs to be compensation. Assume we already know DF1 , . . . , DFt-1, i.e. we are part way through the bootstrapping process. Consider the next swap of maturity t, with a rate st. The value of the two sides may be written as: value of fixed side: ^

.stdiDFi

and value of floating side:

i

i

which of course should be equal. We can write the forward Libor rate as:

Therefore, substituting, we get: value of floating side: £(DFi-1 - DFi) = 1 - DFt / This is an extremely useful result which, as we shall see later, can be generalized for a variety of swaps. If we now turn to the fixed side of the swap, it may be rewritten as:

t-i value of fixed side — s, ^ diDFi + s t d t DF t i=1

Equating the two sides: stQt-1 + stdtDFt = 1 - DFt or DFt = (1 - stQt-1)/(1 + dtst) as before. Approaching the bootstrapping in this fashion is important, because it shows the equivalence of treating the floating side of the swap as either a money account with principal flows or as a strip of forward rates, which in turn form the foundation for the pricing of most non-generic swaps. The market swap prices above went out 30 years. It is becoming increasingly feasible to obtain even longer-dated swaps, certainly in the major currencies, out for 50 years or more. However there is a problem. Using the bootstrapping formula, it is easy to show that: DFt = {l - (1 - DF t-1 ) × (st/St-1)}/(1 +dtst) For DFt > 0 as required, this means that (1 — DF t-1 ) × (s t /s t _ 1 ) < 1. As t increases, (1 - DF t-1 ) tends to 1. Therefore, if the curve is rising at the long end2, i.e. (s t /s t _ 1 ) > 1, it is feasible for (1 — DFt-1) × (S t /s t - 1 ) > 1 for some t less than the longest maturity, and hence DFt < 0! Unfortunately there is nothing inherent in the bootstrapping process that will guarantee that the discount curve will be asymptotic to the time axis. For the USD market data above, as it is declining at the long end, this phenomenon does not occur. On the other hand it has been observed in some Euro swap curves beyond 35 years. When it does occur, it is a serious problem as most systems cannot cope and break down. The real difficulty is that the market provides information only at a small set of maturities. Unfortunately the bootstrapping algorithm requires all rates at the intervening 2 Suppose there were the following quotes: S20 = 6.0%, S30 = 6.2% and S50 = 6.3%. Naive interpolation would almost certainly guarantee s t /s t-1 > 1, with subsequent failure.

58

Swaps and Other Derivatives

maturities to be implied, which are then treated as if they had exactly the same validity as the original rates. Most practitioners look at the forward curve as a measure of appropriateness. This is because the forward curve is effectively the gradient of the discount curve, so any small misalignment in the latter is magnified in the former. Looking at the forward curves above, the one using linear interpolation was rejected because it fluctuated so much. But the other two curves were little better, as they both had significant discontinuities. A measure of a "good" curve is often taken to be its overall smoothness, defined in some fashion. These observations lead to an alternative approach to the derivation of discount and forward curves. Using the expressions above, we can write: net value of a swap = value of fixed side — value of floating side which is of course a linear function in DFs. We know that: NVt = 0 for all t e {original maturities} We can also define a link between discount factors and a forward rate of some tenor: F(t, T) = f(DFt, DFT) Finally, we could create some definition of a "good" forward curve; for example, one with minimum roughness defined by £]/{F(ti+1, ti) — F(t i, ti-1)}2. Constraints on the discount factors such as DFt > DFt+1 > DFt+2 • • • > 0 could also be included, but in practice these should be unnecessary, indeed worrying if required3. Worksheet 3.3 demonstrates one model for this approach. 3-monthly forward rates are treated as the unknown variables (except for the first two which are fixed off the cash curve); see column [1]. A smoothing function is created as described in column [2]. The discount factors are calculated from the forward rates in column [3] and finally the net value of each of the generic swaps in column [5]. The objective is to ensure that all entries in column [5] are zero and that the total smoothing function is minimized. The final forward curve is shown in Figure 3.4, compared with a traditional bootstrapping curve.

3.6 BUILDING A BLENDED CURVE So far in this book we have seen a number of different financial instruments — cash, futures, swaps — being used to build discount curves. It is conventional in some countries, typically ones that do not possess a liquid futures market, to incorporate FRAs as well. Furthermore it is feasible to use bonds and bond futures, although less likely due to the disparate implied creditworthiness. In practice a group of traders and risk managers would build a curve from a mixture of instruments in segments — this is usually known as "blending", for example: • cash for the first 12 months, • interest rate futures for the first 5 years, • interest swaps from 2 years onwards. 3

Using this definition of "good" implies that the forward curve would extrapolate flat. This is a common assumption, although one disputed by some historical evidence; see S. M. Schaefer et al., "Why do long term forward interest rates (almost) always slope downwards?". IFA working paper 299. 2000.

o

Worksheet 3.3 Building a curve by smoothing the forwards

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05 6-Apr-05 6-Jul-05 6-Oct-05 6-Jan-06

Estimated forward rates [1] 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256

6.0313% 6.3100% 6.5885% 6.7908% 6.9154% 7.0118% 7.0793% 7.1167% 7.1250% 7.1270% 7.1224% 7.1114% 7.0940% 7.0891% 7.0967% 7.1169% 7.1497% 7.1733% 7.1876% 7.1924% 7.1877% 7.2001% 7.2298% 7.2770%

Smoothness 19.87 [2]

0.000008 0.000004 0.000002 0.000001 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

DFs [3]

par rates

days Act/360

[4]

1

0.984983 0.969519 0.953465 0.936853 0.921280 0.905235 0.888806 0.873099 0.857653 0.842475 0.827576 0.812962 0.798640 0.784581 0.770755 0.756987 0.743549 0.730306 0.717134 0.704190 0.691760 0.679395 0.667070 0.654891

constraints

Q-factor

6.59375%

1.022

0.96

6.89500%

1.011

1.84

7.02500%

1.011

2.66

7.08500%

1.014

3.43

7.13500%

1.017

4.15

7.17500%

1.014

4.81

1 yr 2 yr 3 yr 4 yr 5 yr 6 yr 7 yr 8 yr 9 yr 10 yr 12 yr 15 yr 20 yr 25 yr 30 yr

[5] 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

Swaps and Other Derivatives

60

— Optimization — Traditional bootstrapping

0

50

100

150 200 250 Time (months)

300

350

400

Figure 3.4 Implication of DFs by optimization — forward curves

The location of the breakpoints will depend upon liquidity and knowledge of when which instruments will be used for hedging. There are two different ways of tackling this problem of building a curve. First, the segments may overlap — see the cash and futures above, and the diagram below—and in that case some weighting information usually has to be provided about the relative importance of each market. The result is likely to be relatively smooth (especially if the weighting is applied gradually) but, and this is a big but, will not be arbitrage free in the overlap portions. For example, any generic swap that matured in the overlap period between futures and swaps, and was used in the construction of the curve, would not be priced back to zero!

1. Overlap, weight and average

cash

• futures

2. No overlap

swaps

The alternative is to remove the overlaps as shown in part 2 of the diagram above. Arbitrage-freeness is maintained, which from a market practitioner's perspective is highly desirable. But the problem now is how to achieve the hand-over between the segments in the smoothest possible fashion. Developing a satisfactory balance between arbitrage-freeness and smoothness whilst using bootstrapping is extremely difficult, and many arcane multilayer algorithms have been developed. The optimization model described above however may be used to tackle the problem much more directly. For example, Worksheet 3.4 constructs an arbitrage free curve using short cash, a 5 year strip of futures, and swaps. The starting point of the worksheet is the estimated 3 month forward curve shown in column [1]. We are only going to use one cash rate, namely the spot 3 month one, and this is shown as not being a variable in the optimization. We can therefore estimate the smoothness of the curve in column [2]. The forward curve will be linearly interpolated to

61

Generic Interest Rate Swaps

3

4

5

6

10

Figure 3.5 3-Monthly forwards

estimate forward rates on the futures maturity dates, so we need the forward gradient in column [3], and finally the discount curve in [4]. Box 2 of Worksheet 3.4 calculates the arbitrage errors in the futures — as shown in columns [5]-[7] — and in the swap curve for 5 years and beyond—see columns [8]–[12]. All the errors have been calculated on a rate basis to ensure comparability. The optimization proceeds in two stages, first estimating the forward curve so that all the generic instruments are error free, and second the curve smoothing. The result is shown in Figure 3.5. This approach removes the need for complex blending difficulties, and can make the trade-off between arbitrage-freeness and smoothness quite explicit using a form of regression with multiple objectives. Whilst not easy to demonstrate within a spreadsheet formulation, probably the best overall approach is to model instantaneous forward rates rather than discrete tenor ones as above.

Worksheet 3.4 Market data for building a blended curve

Today's date Cash

26-Sep-Ol

1m 3m 6m 12m

26-Oct-01 26-Dec-Ol 27-Mar-02 26-Sep-02

24-Sep-01

Libor 2.64375% 2.60000% 2.57625% 2.75875%

Futures Maturity Settlement Maturity date price 19-Dec-01 97.5300 Dec-01 97.4100 20-Mar-02 Mar-02 19-Jun-02 96.9850 Jun-02 18-Sep-02 96.5000 Sep-02 18-Dec-02 96.0000 Dec-02 19-Mar-03 95.6800 Mar-03 18-Jun-03 95.3750 Jun-03 17-Sep-03 95.1450 Sep-03 94.9150 17-Dec-03 Dec-03 17-Mar-04 94.8350 Mar-04 16-Jun-04 94.6950 Jun-04 15-Sep-04 94.5750 Sep-04 15-Dec-04 94.4100 Dec-04

Mid

Swap

1y

2y 3y

4y

5y 7y 10 y

26-Scp-02 26-Sep-03 27-Sep-04 26-Sep-05 26-Sep-06 26-Sep-08 26-Scp-11

1. Constructing the 3 monthly forward curve Length of 3 mo. Smoothing time forwards 11970%

Ann Act/360 2.7450% 3.4750% 4.3850% 4.4150% 4.6650% 5.0750% 5.4350%

Implied Convexity? N forward 0.01 2.4700% 0.06 2.5900% 3.0150% 0.16 0.36 3.5000% 0.70 4.0000% 1.15 4.3200% 1.72 4.6250% 4.8550% 2.25 2.97 5.0850% 5.1650% 3.89 5.12 5.3050% 6.60 5.4250% 8.17 5.5900%

0.000 0.253 0.503 0.758 1.014 1.267 1.517 1.772 2.028 2.281 2.533 2.794 3.047 3.300 3.553 3.806 4.058 4.311 4.564 4.817 5.072 5.325 5.575 5.831 6.086 6.339 6.592 6.847 7.103 7.356 7.606 7.861 8.122 8.375 8.619 8.881 9.133 9.386 9.639 9.892 10.144

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120

26-Sep-01 26-Dec-01 26-Mar-02 26-Jun-02 26-Sep-02 26-Dec-02 26-Mar-03 26-Jun-03 26-Sep-03 26-Dec-03 26-Mar-04 28-Jun-04 27-Sep-04 27-Dec-04 28-Mar-05 27-Jun-05 26-Sep-05 26-Dec-05 27-Mar-06 26-Jun-06 26-Sep-06 26-Dec-06 26-Mar-07 26-Jun-07 26-Sep-07 26-Dec-07 26-Mar-08 26-Jun-08 26-Sep-08 26-Dec-08 26-Mar-09 26-Jun-09 28-Sep-09 28-Dec-09 26-Mar-10 28-Jun-10 27-Sep-10) 27-Dcc-10 28-Mar-11 27-Jun-11 26-Sep-11

[1]

2.60000% 2.45917% 2.59935% 3.04923% 3.54293% 4.04405% 4.34327% 4.65183% 4.87703% 5.10783% 5.17128% 5.32457% 5.44026% 5.61275% 5.76023% 5.78560% 5.68917% 5.65067% 5.66990% 5.74678% 5.88200% 6.00111% 6.10442% 6.19142% 6.26228% 6.31820% 6.35864% 6.38338% 6.39235% 6.40027% 6.40736% 6.41414% 6.42009% 6.42536% 6.42994% 6.43360% 6.43666% 6.43882% 6.44015% 6.44061%

[2] 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0%

0.253 0.250 0.256 0.256 0.253 0.250 0.256 0.256 0.253 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.250 0.256 0.256 0.253 0.253 0.256 0.256 0.253 0.250 0.256 0.261 0.253 0.244 0.261 0.253 0.253 0.253 0.253 0.253

Forward gradient

DFs

[3]

[4]

-0.00155% 0.00156% 0.00489% 0.00537% 0.00551% 0.00332% 0.00335% 0.00245% 0.00254% 0.00070% 0.00163% 0.00127% 0.00190% 0.00162% 0.00028% -0.00106% -0.00042% 0.00021% 0,00084% 0.00147% 0.00131% 0.00115% 0.00095% 0.00077% 0.00061% 0.00044% 0.00027% 0.00010% 0.00009% 0.00008% 0.00007% 0.00006% 0.00006% 0.00005% 0.00004% 0.00003% 0.00002% 0.00001% 0.00001%

1

0.993471 0.987400 0.980884 0.973300 0.964661 0.955006 0.944522 0.933425 0.922058 0.910305 0.898177 0.886249 0.874226 0.861997 0.849626 0.837379 0.825508 0.813882 0.802382 0.790769 0.779184 0.767667 0.755875 0.744101 0.732506 0.720991 0.709462 0.698075 0.686974 0.676155 0.665262 0.654304 0.643855 0.633899 0.623432 0.613455 0.603634 0.593966 0.584452 0.575089

2. Ensuring arbitrage-freeness Total error =>> Futures Start Market Interpolated rate rate [5] [6] 19-Dec-01 2.4700% 2.4700% 20-Mar-02 2.5900% 2.5900% 19-Jun-02 3.0150% 3.0150% 18-Sep-02 3.5000% 3.5000% 18-Dec-02 4.0000% 4.0000% 19-Mar-03 4.3200% 4.3200% 18-Jun-03 4.6250% 4.6250% 17-Sep-03 4.8550% 4.8550% 17-Dec-03 5.0850% 5.0850% 17-Mar-04 5.1650% 5.1650% 16-Jun-04 5.3050% 5.3050% 15-Sep-04 5.4250% 5.4250% 15-Dec-04 5.5900% 5.5900%

0.0000% Error ×10e6 [7] 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000%

Swaps

26-Sep-01 26-Sep-02 26-Sep-03 27-Sep-04 26-Sep-05 26-Sep-06 26-Sep-07 26-Sep-08 28-Sep-09 27-Sep-10 26-Sep-11

1.014 1.014 1.019 1.011 1.014 .014 .017 .019 .011 .011

DFs

Q

Market rate

[8] 1 0.973300 0.933425 0.886249 0.837379 0.790769 0.744101 0.698075 0.654304 0.613455 0.575089

[9] 0 0.9868 1.9332 2.8367 3.6834 4.4851 5.2396 5.9493 6.6163 7.2366 7.8180

[10]

Estimated generic rate [11]

Error ×10e6

[12]

4.4150% 4.41500% 0.0000% 4.6650% 4.66500% 0.0000% 5.0750% 5.07500% 0.0000% 5.4350% 5.43500% 0.0000%

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4 The Pricing and Valuation of Non-generic Swaps

OBJECTIVE Given discount and forward curves, we can now start to price and value swaps that are structured for end-users. These are commonly known as non-generic, as they frequently possess aspects tailored to the user's requirements. Two common structures, namely forward start and amortizing, plus a more complex one, are analysed in some detail and three alternative approaches are described. In passing, a very real practical problem is observed and an alternative known as the "reference rate" method. Future valuing as an alternative to discounting is then reintroduced. Finally swap valuation is discussed using two alternative approaches.

4.1 THE PRICING OF SIMPLE NON-GENERIC SWAPS Whilst the vast bulk of swaps traded between banks, or at least between market makers, are generic, most swaps conducted with non-banking counterparties are non-generic. Such swaps are usually structured to meet their specific requirements. In this chapter, we will discuss how to price such swaps. We start with some relatively simple structures, known as "par non-generic swaps" because, as we shall see, they can be cash hedged with par generic swaps. For example, suppose a company is currently paying quarterly floating interest on $100 million of debt maturing in 5 years' time. The treasurer believes that interest rates will continue to stay low for at least another year, but will continue to rise after that. Instead of entering into an ordinary 5 year swap to pay fixed annually, receive floating, she is considering a 1/5 forward starting swap. This means that the fixed rate would be agreed today, unlike the spreadlock swap, but the swap would only start in 1 year's time with a length of 4 years. Note that the usual convention for forward swaps is the same as for FRAs, namely {start/end}; an alternative is to use a phrase such as "1 into a 4 year swap" — if in any doubt, spell it out! The cashflows from the swap would be as follows (from a bank's point of view):

66

Swaps and Other Derivatives Today's date: 04-Jan-OO Fixed rate: 7.298% ann Dates 6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-00 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 6-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

Notional principal: 100 million USD Floating rate: 3 month $ Libor

Act/360

Floating side

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

— Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor

Fixed

7,378,664

7,378,664

7,398,935

7,419,206

The rate quoted, and remember this is a fair mid-rate so that the bank would be likely to add 2 basis points onto the fixed rate, is some 16 bp higher than the current 5 year swap rate of 7.135%. Why is this, and how did the bank arrive at its quote? There are a number of ways to approach this. First, let us consider how a bank might hedge such a transaction using a generic instrument. Obviously its main concern is that Libor might rise over the lifetime of the swap. Libor could therefore be hedged by entering into two generic swaps: 3mo. L

3mo. L

5 year generic 5

1/5 Forward start swap

Bank

3mo. L 1 year generic Fixed rate

The 5 year generic swap off-sets the Libor payments of the forward start, but also generates Libor receipts in the first year. The 1 year swap is required to off-set these receipts. The bank will be paying 7.135% ann over 5 years, and receiving 6.59375% in the first year. Thus there is a shortfall in the first year of 54 bp which will have to be recovered over the next 4 years. Therefore we would expect the forward start swap rate to be approximately:

o

B' era

Worksheet 4.1

Pricing of non-generic swap

Today's date:

100 USD

to receive to pay

7.298% ann 0 quarterly

Current 5 year swap rate: Current 1 year swap rate: Interpolated dates

using hedging swaps

04-Jan-OO

Notional principal: Fixed rate: Floating rate:

1/5 forward starting

7.135% 6.594%

DFs

Act/360

Floating side margin

Fixed

[1] 6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

Replace floating Pay 5 yr Receive 1 yr [2] [3]

Net cashflow [4]

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

-7,293,556 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor

0 0 0 0

7,378,664

-7,214,278

7,378,664

-7,214,278

164,386

0 0 0 7,398,935

-7,234,097

0 0 0 0

164,386

0 0 0

0 0 0 0

-553,278

0 0 0

0 0 0 0

6,740,278

164,838

0 0 0 7,419,206 Discounted value =

-7,253,917

165,289

0

68

Swaps and Other Derivatives

7.135% + 54 bp/4 = 7.27% crudely spreading the 54 bp over the 4 years. Worksheet 4.1 calculates this more accurately by taking the time value of money into account. Column [1] shows the fixed cashflows from the forward start, with the fixed cashflows from the two generic swaps in columns [2] and [3]. As we require the swap to have a zero value, we find that this is achieved by a forward rate of 7.30%1. We can also approach the pricing slightly differently. We know that the value of the fixed side of a unitary generic swap is Fn * Qn = 1 — DFn from above. By extension, the value of the fixed side of a forward start must be Fs/m * [Qm — Qs] = [DFs — DFm]. Therefore: Fl/5 = [DF5 - DF1]/[Q5 -Q1]= [0.936853 - 0.704190]/[4.14599 - 0.9577] = 7.298% Pricing such non-generic swaps always revolves around what to do with the floating side. We saw in the discussion on bootstrapping that we would regard a money account that paid the floating reference flat as having no economic value. This applies equally to a forward starting money transaction as it does to a spot one. Assume that the bank exchanges the principal amount of $100 million with the swap counterparty twice, once at the start of the first floating period, and reverses the exchange on the last payment date, as shown: Money account 6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-00 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

100,000,000 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor -Libor –100,000,000

Counter-entries

- 100,000,000

100,000,000

Clearly these have no economic impact on the value of the swap. However, the floating side has now become equivalent to the money account (with zero value) plus the two remaining principal cashflows. Worksheet 4.2 shows the swap consisting of the money account in column [1] and the counter-entries in column [3]. The fair price of the swap is of course the same as before. Finally we can always imply the Libor rates directly off the discount curve in the usual fashion. This is most straightforward, as shown in Worksheet 4.3. The implied rates are shown in column [1], and the resulting cashflows in column [2]. 1

By using goal-seek or the solver.

(TQ

Worksheet 4.2

Pricing of non-generic swap — 1/5 forward starting- using NPA

Today's date:

04-Jan-OO

Notional principal: Fixed rate: Floating rate: Interpolated dates

100 USD

to receive to pay

7.298% ann 0 quarterly

DFs

Act/360

Floating side

Fixed margin [2]

[1] 6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-00 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

Replace floating [3]

Net cashflow

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

100,000,000 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor - Libor - 100,000,000

-100,000,000

0

7,378,664

7,378,664

0 0 0

0 0 0 0

7,378,664

7,378,664

0 0 0

0 0 0 0

7,398,935

7,398,935

0 0 0 0

-100,000,000

0 0 0

0 0 0

0 0 0 7,419,206 Discounted value=

100,000,000

107,419,206

0

Worksheet 4.3

Pricing of non-generic swap — 1/5 forward starting —-using IF

Today's date:

04-Jan-00

Notional principal: Fixed rate: Floating rate:

100 USD

to receive to pay

7.298% ann 0 quarterly

Interpolated dates

DFs

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

Floating side margin

Act/360

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

— Libor — Libor — Libor - Libor — Libor — Libor — Libor - Libor — Libor — Libor - Libor — Libor — Libor - Libor — Libor — Libor

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Fixed Implied forwards

Cashflow

[1]

[2]

6.0312% 6.3100% 6.6317% 6.7485% 6.8429% 6.9634% 7.0923% 7.2218% 6.9315% 7.0551% 7.1832% 7.3161% 6.9096% 7.0324% 7.1597% 7.2931% 6.9813% 7.1067% 7.2381% 7.3745% Discounted value -

Net cashflow

0 0 0 0 -

,672,715 ,760,186 ,851,886 ,825,510 ,752,132 ,783,379 ,815,761 ,849,341 ,746,592 ,777,641 ,809,813 ,863,803 ,764,710 ,796,411 ,849,746 ,884,607

7,378,664

7,378,664

7,398,935

7,419,206

- ,672,715 - ,760,186 - ,851,886 ,553,153 - ,752,132 - ,783,379 - ,815,761 ,529,323 - ,746,592 - ,777,641 - ,809,813 ,535,132 - ,764,710 - ,796,411 - ,849,746 5,534,599

0

II

71

Pricing and Valuation of Non-generic Swaps

For a straightforward fixed-floating swap, the fixed rate may be thought of as some (albeit complex) average of the floating rates. For example, using implied 12-monthly forward rates, their simple average is close to the 5 year swap rate as shown below:

year year year year year

0/1 1/2 2/3 3/4 4/5

Average over all 5 years Actual 5 year rate

Implied 12mo. forward rates 6.594% 7.222% 7.316% 7.293% 7.375% 7.1599% 7.135%

The forward starting swap rate must be approximated by the average over years 1 to 5 only, i.e. 7.3014%. As the forward curve is rising, omitting the first rate will increase the average and hence the 1/5 forward swap rate will be higher than the 5 year spot rate. We have seen three approaches to the pricing of this forward start swap: • using hedging swaps to cancel the unknown Libors; • converting the floating side into a zero value money account by adding notional principal amounts to both sides; • implying the Libor rates off the discount curve. Each one is removing, in some fashion, the unknown floating rates. However all three methods are consistent with each other. Swap pricing systems such as those used by market makers are likely to use the last two methods, fair pricing swaps at mid-rates for subsequent adjustment. Whilst the notional principal amount (NPA) method is more traditional, harking back to the relationship with the bond market, the implied forward (IF) method, which had its foundations in the futures market, has probably overtaken it in popularity. IF is certainly more flexible and is also safer in the sense that it estimates what the actual cashflows would be if the curves remained valid. This may become more evident when we look at some more complex structures. Many banks however act as an intermediary, particularly in foreign illiquid currencies. This involves doing a non-generic swap with a customer and therefore taking on the credit exposure, but immediately passing on most of the market risk by entering into hedging swaps with a market maker. In this case, the hedging swaps would include a bid-offer spread which should then be reflected in the pricing of the non-generic swap. There is however a very real practical problem. The following is an extract from the swap dates (the two-day settlement period is not mentioned as it does not affect this argument):

Swap dates 6-Oct-00 8-Jan-01 6-Apr-01 6-Jul-01

3 Months later 6-Jan-01 8-Apr-01 6-Jul-01 6-Oct-01

Adjusted for business days 8-Jan-Ol 9-Apr-01 6-Jul-01 8-Oct-01

72

Swaps and Other Derivatives

For example, consider what happens on 8 January 2001. The actual Libor rate fixed at that time in the cash market would be based on the period from the swap date to the date "adjusted for business days", i.e. specifically from 8 January to 9 April 2001. The swap cashflow however would be calculated from the swap dates, i.e. from 8 January to 6 April 2001. The value of this cashflow would be: PV = P × F(8 Jan, 9 Apr) x (t6Apr - t8 Jan ) × DF6 Apr

However the NPA method implicitly assumes: PV = P × F(8 Jan, 6 Apr) x (t6Apr - t8 Jan ) × DF6 Apr

which simplifies to P × {DF8 Jan - DF6 Apr}. Therefore, despite its wide popularity, the NPA approach is not entirely consistent with reality. Is this effect significant? Worksheet 4.4 has incorporated this, estimating the forward rates to match the cash market and then applying them over the swap dates. Columns [1] and [2] show the swap dates and the discount factors out of those dates, columns [3] and [4] the adjusted end dates for each period and associated discount factors. Column [5] calculates the tenor of each forward rate, and [6] the level of the forward rate. Finally the cashflows are calculated using the dates in [1] and the length of time shown in [7]. The new price for the forward swap is about 4 bp lower, highly significant given a 2bp bid-offer spread. A more theoretical statement of this problem is as follows: • calculate a set of dates S1, .S2,. . . 3 months apart out of the start date; • adjust these dates onto business dates, giving S1, S2, . . . . The length of time between these dates will sometimes be greater than 3 months, and sometimes shorter; • estimate the end of a 3mo. rate out of the adjusted dates, i.e. Ei = Di + 3 months; • adjust these dates to give E1, E2,• • •; • implied 3-monthly forward rates would then be calculated from Di to Ei — this estimate would be consistent with the physical cash market; • but in the multiperiod instrument, it would be applied from Di to Di+1 which is inconsistent. For practical purposes, this approach is termed the "reference rate" methodology, as the forward rates follow the physical reference market. In contrast, the earlier method is to imply and then to apply forward rates over the period Di to Di+1; for obvious reasons this has been termed the "period date" approach. The reference rate method is theoretically correct; however the period date method would appear to be the approach widely used in practice.

4.2

ROLLERCOASTERS

Another common structure is the "rollercoaster" swap. Consider again a company that is currently paying floating interest on some debt, and wishes to swap into fixed. Instead of the debt being a "bullet", i.e. being drawn down and subsequently repaid as a single lump sum, it is very common for the debt to have agreed drawdown and repayment schedules. Obviously the swap must have the same underlying principal structure. Common names for such structures are "step-up" — when the notional principal increases in steps.

Worksheet 4.4

Pricing of non-generic swap 1/5 forward starting

using IF with reference rate method

Today's date: 04-Jan-00 Notional principal: 100 USD Fixed rate: to receive Floating rate: to pay Swap dates

DFs

[1]

[2]

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

7.260% ann 0 quarterly 3 month period dates

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

[3] 6-Apr-OO 6-Jul-OO 6-Oct-OO 8-Jan-01 9-Apr-01 6-Jul-Ol 8-Oct-Ol 8-Jan-02 8-Apr-02 8-Jul-02 8-Oct-02 7-Jan-03 7-Apr-03 7-Jul-03 7-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

DFs

Act/360 Floating side Fixed for forward marggin Implied fixing Length of time Cashflows rate of forwards for cashflow

[4]

[5]

0.984983 0.969519 0.953362 0.936853 0.920915 0.905502 0.889038 0.872934 0.858065 0.843031 0.827831 0.812808 0.799006 0.785051 0.770942 0.756987 0.743860 0.730733 0.717462 0.704190

0.253 0.253 0.256 0.261 0.253 0.253 0.261 0.256 0.253 0.253 0.256 0.256 0.253 0.253 0.256 0.256 0.253 0.253 0.256 0.256

[6]

— Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 ()

6.0312% 6.3100% 6.6317% 6.7485% 6.8468% 6.9634% 7.0923% 7.2187% 6.9315% 7.0551% 7.1847% 7.3117% 6.9096% 7.0324% 7.1611% 7.2931% 6.9813% 7.1067% 7.2381% 7.3745%

Net cashflow

[8]

[7] 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

0 0 0 0 –1,673,670 –1,760,186 - ,851,886 - ,824,730 - ,752,132 - ,783,379 – 1 ,816,124 - ,848,247 - ,746,592 - ,777,641 – 1,810,173 -1,863,803 – 1,764,710 – 1,796,411 – 1,849,746 - ,884,607

Discounted value =

7,421,720

7,381,385

7,381,385

7.381,385

– 1,673,670 – 1,760,186 -1,851,886 5,596,990 -1,752,132 – 1,783,379 -1,816,124 5,533,138 – 1,746,592 – 1,777,641 -1,810,173 5,517,582 – 1,764,710 – 1,796,411 – 1,849,746 5,496,778

0

Swaps and Other Derivatives

74

"step-down or amortizing"—when the principal decreases. Rollercoaster is the general name suggesting the principal rising and falling. It is important to note that the changes in the principal amount are defined in advance, and are not altered by subsequent events. There is a class of swaps, one example of which is "index amortizing", where the principal amount changes as a function of some external events such as increases or decreases in the floating rate fixings. These will be considered later. To be specific, we will look at the pricing of a 5 year amortizing swap, whereby the principal amount starts at $100 million and declines at the end of each year by $20 million as shown: 100

80 60 40 20

The bank will be receiving the fixed rate, paying Libor. It is shown like this because this also gives an idea how to hedge the Libor payments, namely by doing five swaps: 1. 2. 3. 4. 5.

20m 20m 20m 20m 20m

5 year 4 year 3 year 2 year 1 year

swap swap swap swap swap

to to to to to

pay pay pay pay pay

fixed fixed fixed fixed fixed

@ @ @ @ @

7.13500%, 7.08500%, 7.02500%, 6.89500%, 6.59375%,

to to to to to

receive Libor receive Libor receive Libor receive Libor receive Libor

We can quickly produce a crude estimate of the fair amortizing rate by averaging these generic swaps, i.e.: (5 × 7.13500%) + (4 x 7.08500%) + (3 x 7.02500%) + (2 x 6.89500%) + (1 + 6.59375%) (5 = 7.0316% Notice that a weighted average was calculated, reflecting the total contribution of each hedging swap. A more precise calculation is shown in Worksheet 4.5: the amortizing principal is shown in column [1], the cashflows using the estimated amortizing swap rate in column [2], and the fixed cashflows from the hedging swaps in columns [3] to [7]. As usual, the estimated rate of 7.026% is such that the net cashflows have zero value. To employ the NPA method, we have to do some more work. Under this approach, a stream of Libor payments {—L, — L, — L,. . .} can be replaced by {—P, . . . , + P}, signifying a payment of the principal on the first fixing date and receiving the principal on the last payment date. We have the following structure on the floating side of the swap, as shown in the box on the left:

75

Pricing and Valuation of Non-generic Swaps Original swap cashflows 6-Jan-OO 6-Apr-00 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

Adding the NPAs

Counter-entry

+100 -L

100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 40 20 20 20 20

-100

-L -L

-L-100 +80 -L -L -L -L-80

+100 - 80

+60 -L -L -L -L-60

+80 - 60

+40 -L -L -L

-L-40

+60 - 40

+20 -L -L -L -L-20

+40 - 20

+20

The first four cashflows are based on a principal of $100 million. Add the principal amount on the first fixing day and subtract it on the last payment day as shown in the small box. A counter-entry would have to be made to ensure that the swap value has not been changed. Under our assumptions the cashflows in the small box have zero value, so we are left only with the counter-entries. This process is then repeated throughout the lifetime of the swap, with the result that the floating leg has been completely replaced with a simple fixed cashflow, namely {-100, + 20, + 20, + 20, + 20, + 20}. Worksheet 4.6 shows the swap reduced to two columns: column [1] is the cashflows on the fixed side of the swap, and column [2] the strip of principal amounts, with a total net value of zero. The third approach is to use the implied forwards. These have been calculated in column [1] of Worksheet 4.7 in the usual way. The cashflows are then generated using the amortizing principals and finally the net cashflows discounted and shown to have a value of zero.

4.3 A MORE COMPLEX EXAMPLE Finally, to complete this section, we will apply these approaches to a slightly more complex swap. A company has some debt on which it is paying 6mo. Libor + 70 bp. The debt has the following principal structure: $40 million in year 1 $85 million in year 2 $120 million in year 3 $80 million in year 4 $50 million in year 5

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Worksheet 4.6 Pricing of non-generic swap— 5 year amortizing Today's date: Notional principal:

04-Jan-00

100 USD

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-00 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

stepping down by

20m pa

7.026% ann 0 quarterly

Fixed rate: to receive Floating rate: to pay Interpolated dates

using NPA

DFs

Act/360

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

NPA (m)

100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 40 20 20 20 20

Floating side margin

Fixed

[1] -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

Replace floating

Net cashflow

[2] –100,000,000

- 100,000,000

0 0 0 7,181,863

20,000,000

27,181,863

0 0 0 5,683,039

20,000,000

25,683,039

0 0 0 4,262,280

20,000,000

24,262,280

0 0 0 2,849,326

20,000,000

22,849,326

0 0 0 1,428,566

Discounted value =

20,000,000

21,428,566

0

Worksheet 4.7

Pricing of non-generic swap — 5 year amortizing — using IF

Today's date: Notional principal:

04-Jan-OO 100 USD

Fixed rate: to receive Floating rate: to pay Interpolated dates 6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

stepping down by

20m pa

7.026% ann 0 quarterly

DFs

Act/360

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

NPA (m)

100 100 100 100 80 80 80 80 60 60 60 60 40 40 40 40 20 20 20 20

Floating side margin

Fixed Implied forwards

[1] -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl -Fl

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6.0312% 6.3100% 6.6317% 6.7485% 6.8429% 6.9634% 7.0923% 7.2218% 6.9315% 7.0551% 7.1832% 7.3161% 6.9096% 7.0324% 7.1597% 7.2931% 6.9813% 7.1067% 7.2381% 7.3745%

Cashflows [2] -

,524,566 ,595,040 ,694,758 ,762,113 ,338,172 ,408,149 ,481,509 ,460,408 ,051,279 ,070,028 ,089,457 ,109,605 -698,637 –711,056 -723,925 -745,521 -352,942 -359,282 -369,949 -376,921

Discounted value =

Net cashflow

[3]

7,181,863

5,683,039

4,262,280

2,849,326

1,428,566

- ,524,566 - ,595,040 - ,694,758 5,419,750 - ,338,172 - ,408,149 - ,481,509 4,222,631 - ,051,279 - ,070,028 - ,089,457 3,152,675 -698,637 -711,056 -723,925 2,103,805 -352,942 -359,282 - 369,949 1,051,645

0

s 1'

79

Pricing and Valuation of Non-generic Swaps

i.e. a rollercoaster structure. The company proposes to restructure its debt so that it will pay 6mo. Libor on a constant principal amount of $65 million spread over a 6 year period. What margin should be applied to the Libor to make this a fair swap? First we will consider the hedging swap approach. The rollercoaster side is shown below: 120m 85m

80m

50m 40m

To decide how to hedge this, the trick is to always start at the far end and work backwards. From the bank's point of view, it has to pay Libor + 70 bp on this structure, therefore it could start to hedge the Libor payments by entering into: 5 year $50 million swap to pay fixed, receive Libor That will hedge the Libor payment it has to make in the final year. Working back, it will do: 4 year $30 million swap to pay fixed, receive Libor

3 year $40 million swap to pay fixed, receive Libor We now have the following situation: 120m 85m

80m

50m 40m

80

Swaps and Other Derivatives

Worksheet 4.8 Pricing a non-generic swap using hedging swaps Today's date:

4-Jan-00

Pay side of swap Maturity Notional principal

Generic swap rates 5 years

40 million 85 million 120 million 80 million 50 million

year 1 year 2 year 3 year 4 year 5 70 bp 6mo. Libor +

Rate

Receive side of swap Maturity Notional principal Rate 6mo. Libor + Interpolated dates

6-Jan-00 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

6 years 65 million 64.43 bp NPAon pay side

DFs

[1]

1 0.969519 0.936853 0.905502 0.873099 0.843031 0.812962 0.785051 0.756987 0.730733 0.704190 0.679743 0.654891

Margin cashflow on pay side

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511

40 40 85 85 120 120 80 80 50 50

-0.1416 -0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

1 yr 6.5938% 2 yr 6.8950% 3 yr 7.0250% 4 yr 7.0850% 5 yr 7. 1350% 6 yr 7.1750%

Pricing and Valuation of Non-generic Swaps

Hedging swaps to replace pay side of swap 4 yr 2yr 5yr maturity 1 yr 3yr rate 6.5938% 6.8950% 7 0250% 7.0850% 7.1350% NPA 45 35 40 50 30 Pay /Receive? Receive Receive Pay Pay Pay [2] [4] [6] [3] [5] 3.0331

2.4669

- 2.8724 -2.1727 -3.6468

2.4401

- 2.8412 -2.1491 -3.6071

— 2.8412 -2.1491 -3.6071 -2.1550 -3.6170 -3.6270

Hedging swap to replace receive side Margin of swap cashflow 6 yr on receive 7.1750% side 65 Receive [8] [7] 0.2117 0.2164 4.7674 0.2082 0.2152 4.7156 0.2117 0.2117 4.7156 0.2117 0.2129 4.7285 0.2117 0.2140 4.7415 0.2106 0.2140 4.7285

Discounted value =

Net cashflows [9] 0.0702 1.6471 -0.0876 –1.5324 -0.2129 -4.0949 -0.0714 -1.1153 0.0348 1.1497 0.2106 4.9426 0.0000

82

Swaps and Other Derivatives

in which the Libor payments to be made in the last 3 years are matched, but the first 2 years are now overhedged. To rectify this, we will do: 2 year $35 million swap to receive fixed, pay Libor

and 1 year $45 million swap to receive fixed, pay Libor so that now all the Libor payments to be made by the bank will be matched by Libor receipts. Turning to Worksheet 4.8, column [1] contains the cashflows for the 70 bp margin that has to be paid, whilst columns [2] to [6] are the fixed cashflows from the portfolio of five hedging swaps which have replaced the rollercoaster Libor payments. The other side of the swap is more easily dealt with. The Libor receipts may be matched by a single 6 year swap to receive fixed, as shown in column [8], and the cashflows from the calculated balancing margin of 64.4 bp are shown in column [7]. The overall swap may be seen to be fair as its total discounted value is zero. The NPA approach may be applied as follows. The rollercoaster side may be represented as shown below: • the floating cashflows are broken up into strips, each based upon a constant principal amount; • the NPAs are then added on the first fixing date of each strip, and subtracted on the last payment date, as shown in the boxes; • counter-entries have to be made to ensure that the value of the swap remains constant; • we can then argue that the value of cashflows in each box is zero, so we are left just with the counter-entries which are shown in column [1] of the worksheet. Original swa| cashflows 6-Jan-OO 6-Jul-0 8-Jan-01 6-Jul-01 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05

NPA 40 40 85 85 120 120 80 80 50 50

-L -L -L -L _ 1

-L -L -L -L -L

Counter-entry Adding the NPAs +40 -L -L-40

-40 +85 -L -L-B5

+40–85 +120 -L -L-120

+85–120

+ 80 -L -L-80

+120–80

+ 50 -L -L-50

+80 – 50 +50

The Libor leg based upon the constant principal may be replaced simply by principals at the start and end of the swap: see column [3] of Worksheet 4.9. The IF approach is, as usual, straightforward. Once the implied forwards are calculated, the two Libor cashflow streams may be constructed: see columns [1] and [3] of Worksheet 4.10.

Worksheet 4.9 Pricing a non-generic swap using NPA Today's date:

4-Jan-OO

Pay side of swap Maturity Notional principal

Rate

5 years year 1 year 2 year 3 year 4 year 5 6mo. Libor + 70 bp

Receive side of swap Maturity Notional principal Rate 6mo. Libor + Interpolated dates 6-Jan-OO 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-M-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

6 years 65 million 64.43 bp

DFs 1 0.969519 0.936853 0.905502 0.873099 0.843031 0.812962 0.785051 0.756987 0.730733 0.704190 0.679743 0.654890996

40 million 85 million 120 million 80 million 50 million

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511

Implied forward rates

NPA on pay side

6.2188% 6.7485% 6.9634% 7.2218% 7.0551% 7.3161% 7.0324% 7.2931% 7.1067% 7.3745% 7.1533% 7.4248%

40 40 85 85 120 120 80 80 50 50

NPA to replace Libor on pay side -40 -45 –35

40 30 50

Margin NPA to cashflow replace Libor on pay side on receive side 65 -0.1416 -0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

-65 Discounted value =

Margin cashflow on receive side 0.2117 0.2164 0.2082 0.2152 0.2117 0.2117 0.2117 0.2129 0.2117 0.2140 0.2106 0.2140

Net cashflows 25.0000 0.0702 -44.9283 -0.0876 -35.0906 -0.2130 39.7870 -0.0714 29.9282 0.0348 50.0352 0.2106 –64.7860 0.0000

Worksheet 4.10 Pricing a non-generic swap using IF Today's date:

4-Jan-OO

Pay side of swap Maturity Notional principal

5 years year 1 year 2 year 3 year 4 year 5 6mo. Libor+ 70 bp

Rate

Receive side of swap Maturity Notional principal Rate 6mo. Libor + Interpolated dates 6-Jan-OO 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

40 million 85 million 120 million 80 million 50 million

6 years 65 million 64.43 bp Implied forward rates

DFs

NPA on pay side

Libor cashflow on pay side [1]

40 40 85 85 120 120 80 80 50 50

-1.2576 -1.3947 -2.9430 -3.1545 -4.2801 -4.4384 -2.8442 -2.9659 -1.7964 –1.8846

1 0.969519 0.936853 0.905502 0.873099 0.843031 0.812962 0.785051 0.756987 0.730733 0.704190 0.679743 0.654890996

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511

6.2188% 6.7485% 6.9634% 7.2218% 7.0551% 7.3161% 7.0324% 7.2931% 7.1067% 7.3745% 7.1533% 7.4248%

Margin Libor cashflow cashflow on pay side on receive side [2] [3] -0.1416 -0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

2.0436 2.2664 2.2505 2.4123 2.3184 2.4041 2.3109 2.4098 2.3353 2.4500 2.3377 2.4667

Discounted value =

Margin cashflow on receive side [4]

Net cashflows

0.2117 0.2164 0.2082 0.2152 0.2117 0.2117 0.2117 0.2129 0.2117 0.2140 0.2106 0.2140

0.8561 0.9434 -0.7801 -0.8328 -2.1747 -2.2472 -0.6047 -0.6279 0.5737 0.6005 2.5483 2.6807 0.0000

Pricing and Valuation of Non-generic Swaps 9 . -x*"***>**^lr

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Figure 4.1 Accumulated cash

4.4 AN ALTERNATIVE TO DISCOUNTING All the swap valuations so far have been calculated using a single bootstrapped swap curve. The creditworthiness of the swap counterparty has not been a factor in the pricing, despite the fact that the swap may involve a considerable credit exposure. For example, consider the last swap. The best estimate of the future cashflows comes from the IF approach. The periodic cashflows are shown in column [1] of Worksheet 4.11, and the accumulative cashflows in column [2] are plotted in Figure 4.1. This shows that initially the counterparty is expected to be a net depositor of money, reaching a maximum of $2 million, but rapidly becomes a net borrower of some $6 million. Discounting off the swap curve is exactly equivalent to assuming that all money deposited or lent is at Libor flat. This is demonstrated in column [3]; each cashflow is rolled over at the appropriate Libor rate for the period, and the accumulated balance at the end of the swap is zero. However, if we regarded the swap as effectively a financing vehicle for receiving money from and lending money to the counterparty, then we would be likely to inflict margins on these activities. Let us assume that: • we would only pay Libor — 25 bp on deposits from this particular counterparty; « we would expect to receive Libor + 50 bp on loans to this counterparty. Therefore, when the accumulated cashflow is positive, indicating a net deposit, this will be rolled at L — 25 bp for 6 months. When the balance turns negative, this will be rolled over at L + 50 bp. The balance at the end of the swap is now negative, indicating that when receiving a margin of 64.43 bp on the swap the swap at maturity will have cost some $91,000 as shown in column [4]. Obviously the margin needs to be increased to compensate, and we find a fair margin is now 66.28 bp as shown in column [5]. This approach obviously has wide application, but in particular when there is a considerable disparity of credit rating between the two counterparties, and when there is a significant embedded loan or deposit.

4.5 SWAP VALUATION The swap has been agreed, traded between the counterparties, and recorded in a bank trading book. Banks are generally required by their regulators to mark their trading books to market every day. The main purpose of this is to establish the current value of a

86

Swaps and Other Derivatives

Worksheet 4.11 Pricing a non-generic swap using IF and future valuing Today's date:

4-Jan-00

Pay side of swap Maturity Notional principal

Rate

5 years year 1 40 million year 2 85 million year 3 120 million year 4 80 million year 5 50 million 6mo. Libor+ 70 bp

Receive side of swap Maturity 6 years Notional principal 65 million Rate 6mo. Libor+ 64.43 bp

Interpolated dates 6-Jan-OO 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

DFs

Implied forward rates

NPAon pay side

Libor cashflow on pay side

Margin cashflow on pay side

40 40 85 85 120 120 80 80 50 50

–1.2576 –1.3947 -2.9430 -3.1545 -4.2801 -4.4384 -2.8442 -2.9659 –1.7964 –1.8846

-0.1416 -0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

1 0.969519 0.936853 0.905502 0.873099 0.843031 0.812962 0.785051 0.756987 0.730733 0.704190 0.679743 0.654890996

0.506 6.2188% 0.517 6.7485% 0.497 6.9634% 0.514 7.2218% 0.506 7.0551% 0.506 7.3161% 0.506 7.0324% 0.508 7.2931% 0.506 7.1067% 0.511 7.3745% 0.503 7.1533% 0.511 7.4248%

87

Pricing and Valuation of Non-generic Swaps

–25bp 50 bp

Deposit margin: 6mo. Libor Borrowing margin: 6mo. Libor

New margins Libor cashflow on receive side

Margin cashflow on receive side

2.0436 2.2664 2.2505 2.4123 2.3184 2.4041 2.3109 2.4098 2.3353 2.4500 2.3377 2.4667

0.2117 0.2164 0.2082 0.2152 0.2117 0.2117 0.2117 0.2129 0.2117 0.2140 0.2106 0.2140

66.28 bp

Net Accumulative Accumulative Accumulative Accumulative cashflows cashflows cashflows cashflows cashflows reinvested reinvested reinvested at Libor at margin at margin [4] [2] [3] [5] [1] 0.8561 0.9434 -0.7801 -0.8328 -2.1747 -2.2472 -0.6047 -0.6279 0.5737 0.6005 2.5483 2.6807

0.8561 1.7995 1.0194 0.1866 –1.9880 -4.2353 -4.8399 -5.4678 -4.8941 -4.2936 –1.7453 0.9354

0.8561 1.8294 1.1126 0.3211 -1.8421 -4.1575 -4.9100 -5.7199 -5.3517 -4.9529 -2.5827 0.0000

0.8561 1.8283 1.1092 0.3162 –1.8476 -4.1679 -4.9313 -5.7545 -5.4021 -5.0190 -2.6638 -0.0910

0.8622 1.8408 1.1281 0.3419 –1.8149 -4.1277 -4.8835 -5.6988 -5.3381 -4.9463 -2.5823 0.0000

Swaps and Other Derivatives portfolio, so that the bank management has a clear idea of the trading assets and liabilities. Mark-to-market is not without controversy. It was introduced in recognition that, unlike a traditional bank loan, it is feasible to trade in and out of derivative (and other traded securities) positions. Therefore it would seem sensible to establish the current value of a traded position, just in case the bank wished to liquidate it. The traditional alternative was "accrual accounting"; consider a bank buying a bond at 95 with an annual coupon of 7% and a full redemption at 100 on maturity. If an accrued approach were applied, the capital gain of (100 — 95) would be spread evenly over the remaining lifetime of the bond, on the assumption that the bond was to be held until maturity. Mark-to-market on the other hand would try to establish the current price for that bond at the end of each time period, and use the change in value as the simple P&L on the bond. Accrual accounting is gradually but surely disappearing as it is becoming increasingly feasible for banks to sell away all kinds of past exposures. Traditional lending required the bank to hold the loan to maturity; now there is a burgeoning secondary loan market, banks are securitizing entire loan portfolios, credit derivatives allow the transfer of the credit exposures for a price, and so on. The original concept of taking on a static risk for a given return has disappeared, and hence mark-to-market is becoming increasingly common across all bank activities, actively encouraged by the banking regulators. However the market in "old" or "seasoned" OTC derivatives is hardly active. It would be virtually impossible to obtain a price for a swap traded some time ago. although obviously for unwinding purposes it is still feasible albeit seldom efficient. Therefore what passes for mark-to-market is usually "mark-to-model". This operates for a seasoned swap as follows: 1. using the current market levels of generic instruments, build a discount curve; 2. interpolate the discount curve to obtain discount factors on the relevant swap dates; 3. value the swap. This process raises a number of issues, such as: • what are the relevant market levels for this particular swap, and where are they? • as we have already seen, the process of estimating the relevant discount factors is not unique • this process will produce a mid-valuation, which is unlikely to be achievable in the event of an unwind which suggests that different banks may well produce different daily valuations for the same transaction. There have been a number of well publicized instances where P&L controllers, i.e. back office people responsible for the daily P&L, have had to rely upon the traders of the original transaction to advise them as to the current levels of the relevant rates, with disastrous consequences. Hardly the outcome originally envisaged by the advocates of mark-to-market! There are a number of market initiatives trying to overcome this, such as banks valuing each other's books, and closed or public clubs circulating market information. As an example of swap valuation, we will consider the complex rollercoaster swap we have just priced above. Its details were, from the bank's point of view: To pay 6mo. Libor-H 70 bp on the following structure: $40 million in year 1 $85 million in year 2

Pricing and Valuation of Non-generic Swaps

$120 million in year 3 $80 million in year 4 $50 million in year 5 To receive 6mo. Libor + 65.4 bp on $65 million for 6 years. The swap was traded on 4 January 2000, and the first Libor fixing was 6.21875%. Today's date is 31 January 2000. Over the 3 week period, rates have moved up slightly but because the swap is floating-floating, it is difficult to predict whether its value will be positive or negative. As with swap structuring, the key to valuation is what to do about the unknown Libor fixings. Using generic swaps to cancel these is not feasible because of the mismatch in the dates, therefore there are really only two common approaches: 1. inserting the notional principal amounts to create a par FRN with zero value; 2. implying the forward rates directly off the discount curve. The first stage is to construct the new discount curve. This is shown in the New Market Data Worksheet (not reproduced in the book) using linear interpolation on the swap curve. The discount factors on the relevant swap dates are then estimated using zero coupon interpolation, see columns [1] to [4] of Worksheet 4.12. Let us consider the two sides of the swap separately. The first Libor cashflow on the rollercoaster side is shown in column [6]; remember that this cashflow is to cover the interest over a full 6 month period, i.e. from 6 January to 6 July 2000, and the Libor fixing is known as it occurred at the beginning of the period. The remaining Libor fixings are unknown, but these may be replaced by notional principals in the usual fashion. For example, the Libor payment due on 8 January 2001 would have been fixed, ignoring the two-day settlement, on 6 July 2000, i.e. the cashflow date of the previous fixing: 6 July 2000 8 January 2000

Libor cashflows L fixing -40m x L x 0.517

Adding NPAs +40m -40m x (1 + L x 0.517)

Counter-entries -40m +40m

By adding the NPAs, we have created a single period money account with zero value, and with the above counter-entries. This may be repeated throughout the lifetime of the swap, resulting in column [7]. Finally the margin cashflows in column [8] must also be paid. A similar analysis on the straight side of the swap produces columns [10], [11] and [12]. Finally all the cashflows are netted and then discounted, using the DFs in column [4] of course, to produce a small negative valuation of $5,033. Using IF is equally straightforward. The unknown Libors are estimated in the usual way: see column [5] of Worksheet 4.13. Columns [8] and [12] contain the implied Libor cashflows. All the cashflows may then be netted and discounted to produce the same valuation. Both methods produce mid-valuations and are therefore interchangeable. However the NPA approach is the one embedded in regulatory capital calculations such as the BIS Accord and the EU Capital Adequacy Directive. This allows swaps, and similar derivatives, to be treated in a consistent fashion to bonds and other physical instruments. There is also another advantage to the NPA approach: consider the two net cashflow columns. As time progresses, and market rates change, the IF column changes daily, whilst the NPA column remains constant until the known Libor cashflows are actually made and a new fixing declared. Thinking about a large swap portfolio, it is possible to represent it by a relatively static "cash ladder"; this possesses numerous computational advantages for risk calculations.

90

Swaps and Other Derivatives

Worksheet 4.12

Valuing a non-generic swap using NPA

Today's date:

31-Jan-00

Pay side of swap Maturity Notional principal

Rate

6mo. Libor+

Last Libor fixing

Dates

DFs

1 0.998906 0.995276 0.984976 0.969210 0.936640 0.871948 0.811550 0.755417 0.702556 0.652837

40 million 85 million 120 million 80 million 50 million

6.21875%

Z-c Z-c rates daycount

Interpolated Z-c dates daycount

0.019 0.081 0.250 0.506 1.017 2.036 3.047 4.058 5.075 6.089

3.6282% 5.8786% 6.0552% 6.1860% 6.4384% 6.7298% 6.8524% 6.9113% 6.9563% 7.0034%

6-Jan-OO 2-Feb-OO 6-Jul-00 8-Jan-0l 4-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

0.431 0.947 1.444 1.958 2.464 2.969 3.475 3.983 4.489 5.000 5.503 6.014

DFs Daycount

Z-c rates [3]

[2]

[1]

2-Feb-OO 9-Feb-OO 2-Mar-OO 2-May-OO 2-Aug-OO 2-Feb-0l 4-Feb-02 3-Feb-03 2-Feb-04 2-Feb-05 2-Feb-06

5 years year 1 year 2 year 3 year 4 year 5 70 bp

6.1476% 6.4041% 6.5606% 6.7075% 6.7817% 6.8430% 6.8774% 6.9070% 6.9304% 6.9530% 6.9761% 6.9999%

[4] 1 0.973878 0.941142 0.909587 0.876906 0.846121 0.816116 0.787424 0.759475 0.732643 0.706348 0.681212 0.656413

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511

91

Pricing and Valuation of Non-generic Swaps

Receive side of swap Maturity Notional principal Rate 6mo. Libor +

NPA [5]

6 years 65 million 64.43 bp

Pay side Libor cashflow Replacing from Libor with first fix NPAs [7] [6]

40 -1.2576 40 — Libor 85 — Libor 85 — Libor 120 — Libor 120 — Libor 80 — Libor 80 — Libor 50 — Libor 50 — Libor

-40 -45 -35

40 30 50

Margin cashflow [8]

-0.1416 -0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

Receive side Libor NPA cashflow Replacing from Libor with first fix NPAs [10] [11] [9] 65 65 65 65 65 65 65 65 65 65 65 65

2.0436 + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor

65

-65

Margin Net cashflow cashflows

[12] 0.2117 25.8561 0.2164 -44.9283 0.2082 -0.0876 0.2152 -35.0906 0.2117 -0.2130 0.2117 39.7870 0.2117 -0.0714 0.2119 29.9282 0.2117 0.0348 0.2140 50.0352 0.2106 0.2106 0.2140 -64.7860 Value =

-5,033.49

92

Swaps and Other Derivatives

Worksheet 4.13

Valuing a non-generic swap using IF

Today's date:

31-Jan-OO

Pay side of swap Maturity Notional principal

Rate

6mo. Libor+

Last Libor fixing

Dates

DFs

1 0.998906 0.995276 0.984976 0.969210 0.936640 0.871948 0.811550 0.755417 0.702556 0.652837

40 million 85 million 120 million 80 million 50 million

6.21875%

Z-c Z-c rates daycount

Z-c Interpolated dates daycount

[2]

[1]

2-Feb-OO 9-Feb-OO 2-Mar-OO 2-May-OO 2-Aug-OO 2-Feb-0l 4-Feb-02 3-Feb-03 2-Feb-04 2-Feb-05 2-Feb-06

5 years year 1 year 2 year 3 year 4 year 5 70 bp

0.019 0.081 0.250 0.506 1.017 2.036 3.047 4.058 5.075 6.089

5.6282% 5.8786% 6.0552% 6.1860% 6.4384% 6.7298% 6.8524% 6.9113% 6.9563% 7.0034%

[3]

6-Jan-OO 2-Feb-OO 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06

DFs Daycount

Z-c rates [4]

1

0.431 0.947 1.444 1.958 2.464 2.969 3.475 3.983 4.489 5.000 5.503 6.014

6.1476% 6.4041% 6.5606% 6.7075% 6.7817% 6.8430% 6.8774% 6.9070% 6.9304% 6.9530% 6.9761% 6.9999%

0.973878 0.941142 0.909587 0.876906 0.846121 0.816116 0.787424 0.759475 0.732643 0.706348 0.681212 0.656413

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511

93

Pricing and Valuation of Non-generic Swaps

Receive side of swap Maturity Notional principal Rate 6mo. Libor+

Implied forward rates

5]

6.7322% 6.9772% 7.2523% 7.1966% 7.2724% 7.2074% 7.2394% 7.2442% 7.2836% 7.3389% 7.3918%

6 years 65 million 64.43 bp

Pay side Libor Implied Margin NPA cashflow from Libor cashflow first fix cashflows

[6]

40 40 85 85 120 120 80 80 50 50

[7] -1.2576 — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor — Libor

[8]

[9]

–1.3913 -2.9488 -3.1679 -4.3660 -4.4119 -2.9150 -2.9440 -1.8312 -1.8614

-0.1416 –0.1447 -0.2958 -0.3058 -0.4247 -0.4247 -0.2831 -0.2847 -0.1769 -0.1789

NPA [10]

65 65 65 65 65 65 65 65 65 65 65 65

Receive side Libor cashflow Implied Margin from Libor cashflow first fix cashflows [12] [13] [11] 2.0436 + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor + Libor

2.2609 2.2550 2.4225 2.3649 2.3898 2.3684 2.3920 2.3805 2.4198 2.3984 2.4557

0.2117 0.2164 0.2082 0.2152 0.2117 0.2117 0.2117 0.2129 0.2117 0.2140 0.2106 0.2140 Value =

Net cashflows

0.8561 0.9413 -0.7815 -0.8359 -2.2140 -2.2351 -0.6180 -0.6238 0.5841 0.5936 2.6089 2.6697 -5.033.49

This page intentionally left blank

More Complex Swaps

OBJECTIVE The previous chapter discussed so-called "par non-generic swaps", i.e. ones that can be created from generic swaps. This chapter looks at more complex swaps which cannot be created in that way. It starts by looking at the asset packaging of bonds, i.e. swapping the coupon from fixed into a spread over a floating reference. Various structures are described, but it is revealed that the whole mechanism can work due to the different ways in which the bond and swap markets value credit. This leads into credit swaps, and both total return and credit default swaps are discussed and priced in different ways. Default probabilities are implied, and compared to those observed historically. A related topic is the price adjustment of swaps due to the creditworthiness of the counterparty, and an approach is described. The next major segment of the chapter is mismatch swaps, namely ones where the usual conventions on the floating side are violated. A range of examples are discussed in some detail: in-arrears, average rate, overnight-indexed, different types of basis swap, and finally yield curve swaps. The concept of convexity is re-introduced (see Chapter 2), and various approaches are described in the Appendix. The chapter concludes by discussing some other types of swaps that are increasing in popularity: inflation-linked, equity, commodity and volatility.

5.1 ASSET PACKAGING Issuing a bond cheaply and swapping it into a desired currency and structure has been one of the main driving forces behind the swap market from its inception; estimates of the percentage of new bond issues swapped range up to 70%, particularly with the growth of disintermediation. The "international" or eurobond bond market is the unregulated offshore market, based mainly in London, which acts as the main channel for international issuers with good crediting ratings. In the early days, once a bond was issued, it would be sold through the primary market to investors. Secondary market trading, except for sovereign and supranational bonds, would typically only last for a short time and the bonds would then be held to maturity. The unregulated nature of the market meant that it was possible for two bonds of very similar characteristics to be issued at virtually the same time. The one with the higher yield would prevail, and the underwriters were usually left with the "dog" which had become unsaleable. Asset packaging was the market's response to this situation. The basic concept is extremely simple, and occurs in three basic steps: 1. an "investor" borrows some money at a floating reference rate plus margin; 2. the money is used to purchase a physical security; 3. the unwanted future receipts from the security are paid away under a swap, to receive floating rate plus (hopefully, a larger) margin. The structure is shown below for a fixed coupon bond:

96

Swaps and Other Derivatives Libor + margin1

Libor + margin2

coupons redemption—bond price

When the bond matures, it pays the redemption amount (implicitly assumed to be 100 in the discussion below). This is used first to repay the borrowing, and the balance (which we call the "capital gain") will be paid away under the swap. This capital gain may be positive, for a bond trading below par, or negative. Asset packaging has been increasing substantially over the past decade, providing much wanted liquidity to the secondary market. As a bond loses liquidity, its yield rises and price falls. Therefore the capital gain described above rises, and the margin achievable under the swap also increases. Investors usually have margin targets to cover their funding costs, and these targets place an upper limit on the bond yield before it becomes packageable. Consider the swap cashflows from the investor's point of view; see diagram on left. If we add the notional principal amount of D(irty) P(rice) of the receive side of the swap, we get: -DP

+DP -C -C

-C

+L +

-C

-C

-C

-C

-C

-C

-C -C- (100

–C–(100–DP)

+L

-DP)

-DP

+L +m: +m2 +L +L +m2 +m2 +L +L +m2 +m2 +L +m2 +L +m2 +L +m2 +L +m2 +L +m2: +L +m2 +L +L + DP +m2

where the par money account has, under our usual assumptions, zero value. The left-hand side of this second swap is simply the bond cashflows, so we have reduced the swap to: (a) the value of the bond; (b) the value of the margin cashflows. Worksheet 5.1 creates an asset swap for the following bond:

More Complex Swaps Today's date: issuer credit rating maturity coupon redemption current yield clean price

= = = = =

4 January 2000 Charter Communications B+ 15-Apr-09 8.63% ANN 100 10.1660% 91.0000

First the dirty price of the bond has to be calculated; this may be done either by using the yield-based DFs in column [1] to value the future bond cashflows or simply using the clean price plus accrued interest. The swap is then represented twice. First using the actual cashflows occurring under the swap; column [4] shows the coupons and capital gain being paid away, column [6] the implied Libor rates and column [7] the resulting cashflow including the fair margin required to make the net value equal to zero. Notice that in this model there is a short first floating receipt, from 6 January to 17 April which then lines up the subsequent cashflows and ensures that both sides of the swap finish on the same date. Second the swap may be represented by: (a) the value of the bond in column [8], and (b) the value of the margin cashflows in column [9] giving the same supra-Libor margin of 258 bp. DP packages are not common, because the amount to be borrowed is non-market size. Par packages are much more common, when the amount to be borrowed is 100. –(100–DP)

–(100–DP) +100

+L + m2 —C

—c -c -C –C

–C

+ L + m2

-C

+L + m2

+L + m2 +L + m2 +L + m2 +L + m2 +L + m2 +L + m2 + L + m2 +L + m2 +L + m2 +L+m2

-C

—C –C

–C

-C- 100

-100 +L +L

+ m2 + m2 + m2 + m2

+L +L +L + m2 +L + m2 +L + m2 +L + m2 +L + m2 +L + m2 +L + m2 +L + m2 +L + 100 + m2

Notice now that there is an immediate cashflow representing the difference between the par amount and the DP used to buy the bond. Obviously at maturity, the full redemption value is used to repay the borrowing. Adding the notional principals of 100 into the swap, we get the swap on the right where again the par money account has zero value. Worksheet 5.2 constructs the above swaps in a similar fashion as before. First the swap is shown using the actual cash payments in column [1], and the floating receipts including the margin in column [3]. Second, the bond cashflows are valued in column [4], and the margin alone in column [5]. Both approaches give a margin of 250 bp. The left-hand side of the two swaps which include the NPAs are identical, representing the bond cashflows including the bond price. The right-hand side is simply the margin,

Worksheet 5.1 Creating a simple DP asset swap with a bond trading below par Bond details: Today's date = 04-Jan-00 Maturity = l5-Apr-09 coupon = 8.63% ANN redemption = 100 Yield= 10.1660% dirty price = 97.2088 accrued interest = 6.2088 clean price- 91.0000

06-Jan-00 13-Jan-00 07-Feb-00 06-Apr-OO 06-Jul-00 08-Jan-0l 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 06-Jan-06 ()8-Jan-07 07-Jan-08 06-Jan-09 06-Jan-10

USD DFs

Bond dates

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913

06-Jan-OO 17-Apr-00 16-Apr-0l 15-Apr-02 15-Apr-03 15-Apr-04 15-Apr-05 17-Apr-06 16-Apr-07 l5-Apr-08 15-Apr-09

Daycount (30/360)

0.281 1.278 2.275 3.275 4.275 5.275 6.281 7.278 8.275 9.275

Yieldbased DFs

Bond cashflows

Swap dates

[1] [2] 1 -97.2088 06-Jan-OO 0.973203 8.63 17-Apr-00 0.883634 8.63 16-Oct-00 0.802309 8.63 16-Apr-Ol 0.728273 8.63 15-Oct-0l 0.661069 8.63 !5-Apr-02 0.600066 8.63 15-Oct-02 0.544400 8.63 15-Apr-03 0.494296 8.63 15-Oct-03 0.448804 8.63 15-Apr-04 0.407389 108.63 15-Oct-04 15-Apr-05 17-Oct-05 17-Apr-06 17-Oct-06 16-Apr-07 16-Oct-07 15-Apr-08 15-Oct-08 15-Apr-09

IBOR DFs

[3] 1 0.983137 0.951781 0.919781 0.887882 0.856701 0.826487 0.797476 0.769447 0.742256 0.715895 0.690520 0.665573 0.641628 0.618145 0.595715 0.573820 0.552859 0.532572 0.512853

Value of Leg

Implied forward rate method Swap Swap cashflows cashflows receive pay side side

NPA method Swap Swap Implied cashflows Bond cashflows cashflows receive forward receive rates side pay side side NPA = DP NPA = DP margin => 257.67 257.67 [4] [5] [6] [7] [8] [9] 97.2088 -8.63 Libor+margin 6.054% 2.3770 -8.63 0.7097 Libor+margin 6.516% 4.4688 .2663 -8.63 Libor+margin 6.882% 4.6484 -8.63 .2663 Libor+margin 7.106% 4.7586 .2663 -8.63 Libor+margin 7.199% 4.8044 -8.63 .2663 Libor+margin 7.192% 4.8270 .2733 -8.63 Libor+margin 7.196% 4.8025 -8.63 .2663 Libor+margin 7.166% 4.8143 .2733 -8.63 Libor+margin 7.207% 4.8343 -8.63 .2733 Libor+margin 7.244% 4.8527 .2733 -8.63 Libor+margin 7.269% 4.8385 -8.63 .2663 Libor+margin 7.294% 4.9309 .2872 -8.63 Libor+margin 7.382% 4.8940 -8.63 .2663 Libor+margin 7.473% 4.9662 .2733 -8.63 Libor+margin 7.489% 4.9196 -8.63 .2593 Libor+margin 7.506% 4.9824 .2733 -8.63 Libor+margin 7.499% 4.9518 -8.63 .2663 Libor+margin 7.494% 4.9762 .2733 –11.42119444 Libor+margin 7.605% 5.0039 –108.63 .2663 -64.3694 Net PV =

64.3694

–17.0144

17.0144

0.0000

Net PV =

0.0000

C/3

1

g.

BS

2

Worksheet 5.2 Creating a simple par asset swap with a bond trading below par Today's date =

06-Jan-00 13-Jan-00 07-Feb-00 06-Apr-00 06-Jul-00 08-Jan-0l 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06-Jan-09 06-Jan-10

Bond details: 04-Jan-00 Maturity = 15-Apr-09 coupon = 8.63% ANN redemption = 100 Yield = 10.1660% dirty price = 97.2088 accrued interest = 6.2088 clean price = 91.0000

USD DFs

Bond dates

Daycount (30/360)

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913

06-Jan-OO 17-Apr-00 16-Apr-Ol |l5-Apr-02 15-Apr-03 15-Apr-04 15-Apr-05 17-Apr-06 16-Apr-07 15-Apr-08 15-Apr-09

0.281 1.278 2.275 3.275 4.275 5.275 6.281 7.278 8.275 9.275

Yieldbased DFs

Bond cashflows

Swap dates

1 -97.2088 06-Jan-OO 0.973203 8.63 17-Apr-00 0.883634 8.63 16-Oct-00 0.802309 8.63 16-Apr-0l 0.728273 8.63 15-Oct-0l 0.661069 8.63 15-Apr-02 0.600066 8.63 l5-Oct-02 0.544400 8.63 15-Apr-03 0.494296 8.63 15-Oct-03 0.448804 8.63 15-Apr-04 0.407389 108.63 15-Oct-04 15-Apr-05 17-Oct-05 17-Apr-06 17-Oct-06 16-Apr-07 16-Oct-07 15-Apr-08 15-Oct-08 15-Apr-09

IBOR DFs

1 0.983137 0.951781 0.919781 0.887882 0.856701 0.826487 0.797476 0.769447 0.742256 0.715895 0.690520 0.665573 0.641628 0.618145 0.595715 0.573820 0.552859 0.532572 0.512853

Value of Leg

NPA method Implied forward rate method Swap Swap Swap Bond cashflows Swap cashflows Implied cashflows cashflows receive cashflows receive forward receive pay side side pay side side rates side NPA =100 NPA=l00 margin => 250.48 250.48 [4] [5] [1] [2] [3] 97.2088 -2.791 -8.63 0.7097 -8.63 Libor+margin 6.054% 2.4249 1.2663 Libor+margin 6.516% 4.5607 -8.63 1.2663 -8.63 Libor+margin 6.882% 4.7455 1.2663 Libor+margin 7.106% 4.8589 -8.63 1.2663 -8.63 Libor+margin 7.199% 4.9060 1.2733 Libor+margin 7.192% 4.9290 -8.63 Libor+margin 7.196% 4.9041 -8.63 1.2663 1.2733 Libor+margin 7.166% 4.9160 -8.63 1.2733 –8.63 Libor+margin 7.207% 4.9366 Libor+margin 7.244% 4.9555 1.2733 -8.63 1.2663 -8.63 Libor+margin 7.269% 4.9410 1.2872 Libor+margin 7.294% 5.0355 -8.63 1.2663 -8.63 Libor+margin 7.382% 4.9982 1.2733 Libor+margin 7.473% 5.0722 -8.63 1.2593 -8.63 Libor+margin 7.489% 5.0246 1.2733 Libor+margin 7.506% 5.0889 -8.63 Libor+margin 7.499% 5.0577 -8.63 1.2663 Libor+margin 7.494% 5.0826 1.2733 -108.63 1.2663 -8.63 Libor+margin 7.605% 5.1112 -65.7291 Net PV =

65.7291

-17.0144

17.0144

0.0000

Net PV =

0.0000

100

Swaps and Other Derivatives

albeit on different principals. If we were to wear a bond hat, i.e. discounting all cashflows using the yield to maturity as shown in column [2] of the previous worksheet, the left-hand side would be valued to zero, and hence the margin in each case would also be zero. Asset packaging wouldn't work! But we have valued the bond cashflows off the Ibor curve. which is lower and hence values the future bond cashflows more highly. The bond YTM reflects the creditworthiness of the issuer, whilst the swap curve assumes a general credit rating similar to the Ibor market, namely about A+. This discussion summarizes the typical traditional approaches to credit within the two markets: • the bond market has a well developed sense of credit-adjusted returns as represented by the existence of multiple spread curves; • when asked to do a transaction, a dealer in the swap market is much more likely to ascertain whether there is any spare capacity in the overall dealing limit, and if so make a price irrespective of the creditworthiness of the counterparty. Some banks have tried to move away from this simplistic approach and to introduce credit-adjusted swap pricing. Initially this met with little success as obviously they become uncompetitive, but following the economic crises in the Far East and Eastern Europe in 1997-9, pricing has become more credit sensitive. Therefore asset packaging is effectively an arbitrage between the two valuation approaches in the two markets. The margin being earned by the investor is not "profit", but a recompense for the risk that, if the bond defaults, he will still have to make the swap payments. Whether the margin is adequate or not is obviously a judgement which provides stimulus to the market. As the left-hand sides of the two swaps are identical, they obviously have the same value. Consider a stream of margin receipts. Its value may be written as: PV = m x P x

di

x DFi, = m x P x Q1,

For the two swaps, the values of the right-hand sides can be written as: mDP x DP x Q1,— m100 x 100 x Q, or

WDP x DP = w IOO x 100

As DP < 100,wDP > m100 as calculated above. The package as constructed matches the maturity of the bond, some 9.27 years, which is usual for single bond packages. However some investors require packages of regular maturities. It is also feasible to package a portfolio of bonds, particularly when there is a limited supply of each issue, and in this case a package of a regular maturity would be more likely. Suppose for example, the investor wished for a 10 year par package, but based upon the above bond. The underlying swap structure has to be modified as shown below. The bond matures and pays the redemption amount on 15 April 2009; however the investor does not receive the money back until 6 January 2010. Therefore the swap must actually contain the principal cashflows at the end as shown. Effectively the bond principal is lent to the swap counterparty for about 9 months: in return the investor is receiving an additional full Libor payment plus of course the balancing margin of 238 bp. See Worksheet 5.3.

More Complex Swaps 06 –Jan–00

–(100 –DP)

1 7– Apr–00

—C

16–Apr–0l

—C

1 5– Apr–02

—C

16-Apr-07

-C

1 5–Apr–08

-C

1 5– Apr–09

-C- 100

06–Jul–00 ()8-Jan-01 06–Jul–0l 07-Jan-02 ()8–Jul–02 06–Jan–03

+L +L +L +L +L +L

+m +m +m +m +m +m

07–Jan–08 ()7-Jul-08 06–Jan–09 06-Jul-09 06–Jan–10

+L +L +L +L

+m +m +m +m

+L + 100 + w2

A third type of asset package, popular in the Far East, is based on a discount amount. An example based upon a principal of 90 is shown in Worksheet 5.4. The swap structure is more complex, as shown below: +( DP –90) + 90

+(DP – 90) —C' —C

—C —C —C– (100 –90)

+L + +L+ +L + +1 + +L + +L+

m2 m2 m2 m2 m2

m2

-C -C -C

+L + m2

—C

+L + m2 +L + m2 +L + m2 +L + m2

—C– (100 –90) -90

-90 +L + m2 +L + m2 +L + m2 + m2 +L +L + m2 +L +m2 + m2 +L +L + m2 + m2 +L +L +m2 +L + m2: +L + 90 + m2

By adding the NPA however, it may be reduced to the more familiar form shown on the right. The margin of 278 bp is of course larger, being based upon a smaller principal than the other two examples. The first asset packaging diagram above showed the investor at the heart of the action. Whilst this is the normal final outcome, the starting point is usually different. Initially a potential investor will have been contacted by an asset packager, and a target such as: a minimum return of 6mo. Libor+35 bp for 5 years, with a credit rating of BBB or better agreed. Some other examples, at the time of writing: Libid to Libor flat for sovereigns and AAA depending on supply; Limean+4 bp for AA; Libor+l0 bp for A. currently tight due to lack of supply. As we can see, in theory at any rate it is feasible to hit any funding target simply by reducing the principal amount. But notice that under the swap structure, the first actual cashflow is positive, being the balance of the bond price received from the swap counterparty. This is effectively a loan, which of course is being implicitly charged at Libor flat. As the investor is looking to make a substantial margin over Libor, this is a cheap loan providing a subsidization to the package. To estimate a fair price for this type of discount swap, it

102

Swaps and Other Derivatives

Worksheet 5.3

Creating a 10 year par asset swap with a bond trading below par Bond details:

Today's date: 04–Jan–00 Maturity = 15–Apr–09 coupon = 8.63% ANN redemption = 100 Yield = 10.1660% dirty price = 97.2088 accrued interest = 6.2088 clean price = 91.0000

06˜Jan–00 13–Jan–00 07–Feb–00 06–Apr–oo 06–Jul–00 08–Jan–0l 07-Jan-02 06-Jan-03 06–Jan–04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06–Jan–09 06–Jan–10

USD DFs

Bond dates

Daycount (30/360)

Yieldbased DFs

IBOR DFs

Bond cashflows

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913

06-Jan-OO 17–Apr–00 16–Apr–0l 15–Apr–02 15-Apr-03 15–Apr–04 15-Apr-05 17-Apr-06 16-Apr-07 15–Apr–08 15–Apr–09

0.281 1.278 2.275 3.275 4.275 5.275 6.281 7.278 8.275 9.275

1 0.973203 0.883634 0.802309 0.728273 0.661069 0.600066 0.544400 0.494296 0.448804 0.407389

1 0.983137 0.919781 0.856701 0.797476 0.742256 0.690520 0.641628 0.595715 0.552859 0.512853

-97.2088 8.63 8.63 8.63 8.63 8.63 8.63 8.63 8.63 8.63 108.63

103

More Complex Swaps

Swap dates

IBOR DFs

Implied forward rate method Swap cashflows Swap receive cashflows side pay side

NPA method Implied forward rates

margin => 06–Jan–00 06–Jul–00 08–Jan–0l 06–Jul–0l 07–Jan–02 08–Jul–02 06–Jan–03 07–Jul–03 06–Jan–04 06–Jul–04 06–Jan–05 06–Jul–05 06–Jan–06 06–Jul–06 08–Jan–07 06–Jul–07 07–Jan–08 07–Jul–08 06–Jan–09 06–Jul–09 06–Jan–10

1 0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564 114 0.543575 0.523588 0.504100 0.484913 Leg

[1] -2.791 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -108.63

[2]

Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin

–117.0144 Net PV =

6.219% 6.749% 6.938% 7.246% 7.119% 7.252% 7.132% 7.194% 7.214% 7.268% 7.267%, 7.312% 7.440% 7.501% 7.478% 7.529% 7.474% 7.510% 7.689% 7.742%

Swap cashflows receive side NPA =100 237.90 [3] 4.3466 4.7159 4.6328 4.9462 4.8020 4.8688 4.8085 4.8661 4.8500 4.9306 4.8500 4.9534 4.9369 5.1045 4.9009 5.0914 4.9812 5.0267 5.0619 105, 1727

Bond cashflows pay side

[4] 97.2088 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 –108.63

Swap cashflows receive side NPA =100 237.90 [5] 1 .2027 1.2291 1.1829 1 .2225 1.2027 1 .2027 1 .2027 1 .2093 1.2027 1.2159 1.1961 1.2159 1.1961 1.2291 1.1829 1.2225 1.2027 1.2093 1.1961 1.2159

117.0144

-17.0144

17.0144

0.0000

Net PV =

0.0000

104

Swaps and Other Derivatives

Worksheet 5.4 Today's date =

Creating a simple discount asset swap with a bond trading below par Bond details: 04–Jan–00 Maturity = 15–Apr–09 coupon = 8.63% ANN redemption = 100 Yield = 10.1660% dirty price = 97.2088 accrued interest = 6.2088 clean price = 91.0000 Size of package = 90

06–Jan–00 13–Jan–00 07–Feb–00 06–Apr–00 06–Jul–00 08–Jan–0l 07-Jan-02 06-Jan-03 06–Jan–04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06–Jan–09 06–Jan–10

USD DFs

Bond dates

Daycount (30 360)

Yieldbased DFs

Bond cashflows

Swap dates

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913

06-Jan-OO 17–Apr–00 16–Apr–0l 15-Apr-02 15-Apr-03 15-Apr-04 15-Apr-05 17-Apr-06 16-Apr-07 15-Apr-08 15-Apr-09

0.281 1.278 2.275 3.275 4.275 5.275 6.281 7.278 8.275 9.275

1 0.973203 0.883634 0.802309 0.728273 0.661069 0.600066 0.544400 0.494296 0.448804 0.407389

-97.2088 8.63 8.63 8.63 8.63 8.63 8.63 8.63 8.63 8.63 108.63

06–Jan–00 17-Apr-OO 16–Oct–00 16–Apr–0l 15–Oct–0l 15-Apr-02 15–Oct–02 15-Apr-03 15–Oct–03 15-Apr-04 15–Oct–04 15–Apr–05 17–Oct–05 17-Apr-06 17–Oct–06 16–Apr–07 16–Oct–07 15-Apr-08 15–Oct–08 15–Apr–09

0.283 0.789 1.294 1.800 2.306 2.814 3.319 3.828 4.336 4.844 5.350 5.864 6.369 6.878 7.381 7.889 8.394 8.903 9.408

105

More Complex Swaps

IBOR

IBOR z-c rates

DFs

Implied forward rate method Swap cashflows Swap receive cashflows side pay side

NPA method Implied forward rates

margin = >

[2]

[1] 1 6.002% 6.265% 6.460% 6.606% 6.708% 6.773% 6.818% 6.847% 6.874% 6.899% 6.922% 6.943% 6.967% 6.994% 7.018% 7.041% 7.060% 7.077% 7.098%

0.983137 0.951781 0.919781 0.887882 0.856701 0.826487 0.797476 0.769447 0.742256 0.715895 0.690520 0.665573 0.641628 0.618145 0.595715 0.573820 0.552859 0.532572 0.512853 Leg

7.209 -8.63

-8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -18.63

Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin Libor+margin

-60.8577 Net PV =

6.054% 6.516% 6.882% 7.106% 7.199% 7.192% 7.196% 7.166% 7.207% 7.244% 7.269% 7.294% 7.382% 7.473% 7.489% 7.506% 7.499% 7.494% 7.605%

Swap cashflows receive side NPA = 90 278.31

[3] 2.2534 4.2313 4.3976 4.4996 4.5420 4.5635 4.5403 4.5517 4.5703 4.5872 4.5735 4.6606 4.6250 4.6923 4.6481 4.7073 4.6785 4.7016 4.7268

Bond cashflows pay side

[4] 97.2088 -8.63

-8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -8.63 -108.63

Swap cashflows receive side NPA = 90 278.31

[5] 0.7097 1.2663 1 .2663 1 .2663 1 .2663 1 .2733 1.2663 1.2733 1 .2733 1.2733 1.2663 1.2872 1 .2663 ! .2733 1.2593 1 .2733 1 .2663 1.2733 1 .2663

60.8577

-17.0144

17.0144

0.0000

Net PV =

0.0000

106

Swaps and Other Derivatives

would seem sensible for the counterparty to incorporate a realistic funding cost for at least the loan element, if not the entire balance of the swap. By doing this, the margin over Libor that may be achieved is effectively limited. See Worksheet 5.5. At the end of the packaging process, the investor will receive: • details of the bond • identification of the swap counterparty • details of the swap plus resulting margin over Libor plus full supporting documentation on the bond purchase from say EuroClear, and an ISDA Master Agreement for the swap. The investor becomes the owner of the two separate components, which would enable one to be sold off later if required. Obviously the investor is responsible for the swap obligations if the bond defaults. It is feasible to find a third party such as a bank to take on the credit risks of the bond, but there would be a compensating charge.

5.2 CREDIT SWAPS A crude summary of the previous section is that asset packaging involves the purchase of a bond trading on a high yield and hence low price, and then swapping the bond into a margin over Libor. This margin is very much a reflection of the demand for the bond and the creditworthiness of the issue. In the discussion below, we will assume that the swap counterparty is a much higher credit than the bond issuer. Consider a simple par package as shown on the left below:

L+m2 100

L + m1 100

L+m2 L+m

100 at maturity Investor

Investor

100–DP upfront Coupon 100 at maturity

100–DP upfront Coupon

Coupon 100

Par package

Coupon 100

Total return swap

The investor is taking the credit risk on the bond, and would have to make up any shortfall to the swap counterparty. On the other hand he is receiving the full margin. Suppose now the deal is modified, as shown on the right. The investor makes the normal upfront payment on the swap, and then pays all the bond cashflows only if he receives them from

Worksheet 5.5 Creating a simple discount asset swap with a bond trading belowpar (this worksheet incorporates a borrowing margin and creates the package by setting the FV = 0) Bond details: Today's date: 04-Jan-OO Maturity = 15–Apr–09 coupon = 8.63% ANN redemption = 100 Yield = 10.1660% dirty price = 97.2088 accrued interest = 6.2088 clean price = 91.0000 Size of package = 90 Implicit borrowing margin =

06-Jan-OO 13–Jan–00 07-Feb-OO 06–Apr–00 06–Jul–00 08–Jan–0l 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06-Jan-09 06–Jan–10

USD DFs

Bond dates

Daycount 30 / 360

1 0.998926 0.994860 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654891 0.607724 0.564114 0.523588 0.484913

06-Jan-OO 17-Apr-OO 16-Apr-Ol 15-Apr-02 15-Apr-03 15-Apr-04 15-Apr-05 17–Apr–06 16–Apr–07 15-Apr-08 15-Apr-09

0.281 1.278 2.275 3.275 4.275 5.275 6.281 7.278 8.275 9.275

Yieldbased DFs

250 bp

Bond cashflows

Swap dates

1 -97.2088 06-Jan-OO 0.973203 8.63 17-Apr-OO 0.883634 8.63 16–Oct–00 0.802309 8.63 16-Apr-Ol 0.728273 8.63 15–Oct–0l 0.661069 8.63 15–Apr–02 0.600066 8.63 15–Oct–02 0.544400 8.63 15-Apr-03 0.494296 8.63 15-Oct-03 0.448804 8.63 15–Apr–04 0.407389 108.63 15-Oct-04 15-Apr-05 17-Oct-05 17–Apr–06 17-Oct-06 16–Apr–07 16-Oct-07 15-Apr-08 15-Oct-08 15–Apr–09

Daycount (Act/360)

IBOR DFs

Implied forward rates

0.283 0.506 0.506 0.506 0.506 0.508 0.506 0.508 0.508 0.508 0.506 0.514 0.506 0.508 0.503 0.508 0.506 0.508 0.506

1 0.983137 0.951781 0.919781 0.887882 0.856701 0.826487 0.797476 0.769447 0.742256 0.715895 0.690520 0.665573 0.641628 0.618145 0.595715 0.573820 0.552859 0.532572 0.512853

6.054% 6.516% 6.882% 7.106% 7.199% 7.192% 7.196% 7.166% 7.207% 7.244% 7.269% 7.294% 7.382% 7.473% 7.489% 7.506% 7.499% 7.494% 7.605%

Implied forward rate method Swap Swap cashflows cashflows Swap receive receive cashflows side side pay side NPA = 90 margin — > 262.53 [3] [1] 7.209 Libor+margin 2.2131 -8.63 Libor+margin 4.1595 -8.63 Libor+margin 4.3258 Libor+margin 4.4278 Libor+margin 4.4702 -8.63 Libor+margin 4.4913 -8.63 Libor+margin 4.4685 Libor+margin 4.4796 Libor+margin -8.63 4.4981 Libor+margin 4.5151 Libor+margin 4.5018 -8.63 Libor+margin 4.5877 -8.63 Libor+margin 4.5532 Libor+margin 4.6201 Libor+margin 4.5767 -8.63 Libor+margin 4.6351 -8.63 Libor+margin 4.6067 Libor+margin 4.6294 -18.63 Libor+margin 4.6550

Money Account

7.209 0.967 5.170 1.111 5.593 1.707 6.283 2.429 7.028 3.243 7.919 4.182 8.980 5.352 10.243 6.704 1 1 .680 8.247 13.296 0.000

108

Swaps and Other Derivatives

the issuer. The swap counterparty pays Libor plus a margin at regular intervals plus 100 at the end. If there is no bond default, then the principal cashflows match and we are back with the par package. However the credit risk now rests on the counterparty, and obviously she will demand compensation in the form of a considerably lower margin. We have created a total return swap (TRS)! This is simply one source of such a swap. Probably the wider use is to manage the credit risk on loan portfolios. For example, suppose that a bank has an unacceptably high level of exposure to a company and indeed to that particular industrial sector. The physical solution would be to sell off some of this exposure in the secondary loan market, but this potentially could damage the relationship with the company and is of course subject to the liquidity and vagaries of documentation in the secondary market. A simple alternative would be to enter into a total return swap, whereby it pays away all cashflows received from the loan and receives Libor (say) plus principal at maturity. The loan assets still remain on the balance sheet of the bank, whilst at the same time reducing its exposure to the industrial sector and increasing its exposure to the counterparty 1 . One might ask why a TRS is viewed as a derivative at all, as it involves the exchange of the full cashflows from a physical security. Probably the best reason is that they are treated commercially, in terms of pricing, liquidity and documentation, alongside other OTC derivatives. As we shall see later, pure "credit risk" is isolated for pricing by extracting out market risks, so we could argue that the value of this swap is derived from the credit risk. There is however a danger in this attitude. OTC derivatives are widely treated as if they are liquid, tradeable instruments whereas in reality they are illiquid credit-based transactions. Treating credit derivatives in the former fashion may exacerbate the tension between competitive dealing requirements and an institution's customary credit standards. The other main type of credit swap is the "credit default swap" (CDS). This is in essence an insurance policy whereby the above investor would pay a premium to a third party, and in return the third party would make good any losses incurred if the issuer suffered some form of adverse credit event. Again there has been a transfer of credit, this time from the issuer to the third party. In many situations, either form may be effectively interchangeable. However consider the following dynamic situation which demonstrates the power of the credit default arrangements. Assume Bank A has entered into a long-term crosscurrency swap with Bank B. Unfortunately Bank B is severely downgraded after a few years. This may result in a higher capital charge and hence a lowering of return on capital to Bank A, together with an increased utilization or even a breaching of its counterparty limits. Correcting the exposure by unwinding or assigning the swap is likely to be difficult, and would impair the relationship with Bank B. Suppose however Bank A buys a credit default swap from Bank C:

Payment

$ Libor Bank A

Bank B Fixed Y

Bank C Premium

1 For considerably more details on the application of credit derivatives, and credit structures in general, see Credit Derivatives and Credit Linked Notes edited by Satyajit Das. Wiley. 2000.

More Complex Swaps

109

where a premium is paid, either a single upfront payment or on a periodic basis, in return for a payment in the event of a default event. The payment could be: max {0 (100% — recovery rate) x mark-to-market valuation before credit event} Note that the payment is dynamic and linked to movements in the financial markets, although very often the recovery rate is set upfront. Such deals can also be structured so that the payment: • is capped; • is only due if the MTM valuation exceeds a predefined level. Measuring the size of the exposure is generally straightforward, especially if it involves mark-to-market valuations. But this of course is linked to the recovery rate after the credit event, and there are various ways in which this may be determined. The main difficulty is that to determine the real recovery rate we may have to wait until any form of work-out has been completed, which may take a considerable time. So the credit derivative market is generally more pragmatic, for example: • the rate could be agreed upfront as a constant percentage of the nominal amount; • physical delivery of the defaulted obligation in exchange for a pre-agreed amount, usually the value of the obligation at the time of entry into the derivative; • if it is a traded security, then the market could be polled. Moody's, for example, suggest the change in valuation from before the credit event to 30 days after the event gives the market a reasonable time to absorb the event and settle down 2 . Some banks will actually use the recovery rate as a trading opportunity, selling derivatives with different recovery rates than expected. The third major type of credit derivative is written on the credit spread between two securities. For example, a company could buy a 1 year option from a bank for the ability to borrow money for (say) 3 years at Libor plus a fixed spread. If, at the end of the year, the company's borrowing spread has increased either because of a systemic shift in the market or because of some specific reason, the company would exercise the option. Like all derivatives, credit derivatives are entered into for one of two reasons: • return enhancement, in which somebody is prepared to adopt some credit risk in order to improve the return; • credit risk management, either by a bank itself or by an end-user. Financial institutions in particular may use them to manage concentration risk. From these early beginnings, a wide range of potentially complex derivative structures have been created, usually to meet the needs of a specific situation. For example, the default event above may be linked not to a single obligor3, but perhaps to the first-todefault within a basket of obligors. -"Moody's Special Report, "Understanding the risks in CDS", March 2001, p. 14. 3 The generic name commonly used to describe anybody who has a future liability to meet, and is therefore the source of a credit exposure.

110

Swaps and Other Derivatives

There are three particular factors that determine the value of a credit derivative over a simple interest rate or FX derivative, namely: • the probability of some credit event occurring; • the size of the exposure at the time of the credit event; • the likely recovery rate from the obligor. The credit event can be quite flexible, for example4: • • • •

payment default that remains unpaid after an agreed grace period; bankruptcy, restructuring or administration; credit rating downgrade; non-systemic material change in a defined credit spread.

The event may be structured depending upon the bank's expectation of the obligor's treatment of different classes of obligation. For example, a company may default on its bank loans but not on its public securities, or it may default on its domestic obligations but not on its international ones. Events in Russia during August 1998 focused the industry's attention on the importance of credit derivative documentation. Many Western banks, anxious to do business in Russia but unenthusiastic about the credit risks, entered into CDSs with Russian banks who would provide payment if a credit event occurred. On 17 August, the Russian government announced that it was rescheduling some short-term debt payments and, at the same time, imposed a moratorium on payments by Russian banks on their obligations under various FX-related contracts. A range of disputes broke out with the credit default swap counterparties as to whether this rescheduling was a credit event. Other disputed issues included: • the standard (ISDA) documentation didn't cover credit derivatives involving sovereign states; • did the rescheduling of one class of debt constitute a credit event for other classes? • there was no mention as to what would happen during the period of rescheduling, which could be many months; • what would happen to swaps that matured before the final rescheduling? This had a major impact on the burgeoning credit derivative market, with considerable attention now being paid to the precise definitions in the documentations. One key development was the revised ISDA credit derivative documentation issued in 1999 after the Russian experiences, which has now become very much the de facto standard. But there are still some interesting problems. The documentation very much reflects practice in the marketplace, for example the range of potential credit events. Unfortunately the range is wider than the range recognized by, say, Moody's rating databases5, so its historic default probabilities are inappropriate. But generally concerns over documentation and regulation are declining, to be replaced by the more usual 4

See ISDA's credit event support confirmation document for a fuller list of potential events. For example, ISDA recognizes a range of restructurings as default events, whereas Moody's (2001) only recognizes one. 5

More Complex Swaps

concerns over liquidity and client knowledge. Products are diversifying rapidly, especially embedded transactions such as credit-linked obligations. The first global survey of the credit derivative market was released by the BIS in December 20016. It showed that the total size of the market (in terms of notional principals) was $0.7 trillion at the end of June 2001, a rise of 86% pa since June 1998. The British Bankers Association has been doing annual surveys in London from 1997, and suggested7 that the size of the global market (excluding asset swaps) was $586 billion by the end of 1999, and would rise by about 40% pa to $1581 billion by the end of 2002. London is currently the major market, with just under half this volume. Bankers are currently still the major sellers (47% in 1999) and buyers (63%) of credit protection, although insurance companies were becoming increasing sellers (23% in 1999, up from 10% in 1998). The market is switching rapidly from writing protection against sovereigns (down to 20% in 1999 from 35% in 1998) to corporate protection (up from 35% to 55%). It was anticipated that these proportions will remain relatively stable for the next three years. The rapid development of the credit derivative market has been the stimulus, at least in part, for increased attention to the modelling of credit risk. Another major driving force is the regulatory proposal8 for the allocation of capital against credit exposure and the permitted use of some forms of internal modelling. There are a wide number of approaches to credit modelling, and more are being developed as the nature of credit risk becomes better understood. Crudely the models can be divided into two classes: • risk-neutral pricing models, based upon fundamental arbitrage relationships in the current markets; • credit risk management models, based upon either credit ratings or the actual capital structure of the obligor, which in turn are based upon average historical information. Usually credit derivatives are priced using the former approach, on the basis that they are tradeable instruments requiring regular mark-to-market. The risk management models are generally used in assessing the credit exposure inherent in a long-term stable portfolio such as a banking book. There are two simple approaches that are widely used. Assume that we possess two curves, one risk free and the other risky. In the worksheet, a risk free par yield curve was built using a small number of on-the-run T-bonds and a Nelson-Seigel curve9. Given also the current spread curve for BBB counterparties, a risky yield curve was also built. We can now simply bootstrap each curve to get risk free DFs and credit-adjusted (CA) DFs as shown in columns [4] and [5] of Worksheet 5.6. For TRSs, the approach can be very simple. From the swap provider point of view, he is receiving a risky cashflow and paying a risk free one. He would value the former using the CADFs and the latter using the risk free DFs. Consider the par asset package built from the Charter Communications bond in the previous section. This generated a margin of about 250 bp for the investor, but he was taking on the credit risk. If we now price this (rather crudely as we are treating it as a 9 year par deal) as a TRS, as shown in the box on the See Chapter 1 for more details. See the BBA website: www.bba.org.uk/html/1596.html 8 Basel II, likely to be implemented in 2004. 9 C. R. Nelson et al., Parsimonious modeling of yield curves, Journal of Business, 60(4). 1987, 473-489. 7

112

Swaps and Other Derivatives

Worksheet 5.6

Example spreadsheet to calculate credit-adjusted dfs

Market data Today's date:

15–Aug–02 15–May–05 15–Aug–10 15–Mav–30

30-Oct-OO Market yield

1.79 4.54 9.80 29.56

6.00% 5.80% 5.73% 5.75%

Estimated yields

Error

6.00% 5.80% 5.73% 5.75%

3.98E–15 4.65E–14 6.27E–14 9.58E–15

error => Building a Nelson-Siegel curve beta0 betal beta2 tau

0.0577 0.0056 -0.0078 2.5863

1.23E-07

Yields of par bonds Risk free Maturity [1] 0 1 6.12% 2 5.98% 3 5.88% 5.82% 4 5 5.78% 6 5.76% 7 5.75% 8 5.74% 9 5.73% 10 5.73% 11 5.73% 5.73% 12 13 5.73%

Risky spread [2]

Risky yield '[3]

4.64% 4.34% 4.14% 3.93% 3.74% 3.61% 3.45% 3.34% 3.20% 3.10% 2.99% 2.88% 2.78%

10.76% 10.32% 10.02% 9.75% 9.52% 9.37% 9.20% 9.08% 8.93% 8.83% 8.72% 8.61% 8.51%

Spread is roughly equivalent to a B counterparty

USD Bond data - 30 October 2000

12 11 10 9 8 7 6 5 10

12

14

113

More Complex Swaps

Risk free bootstrapped curve

[4] [ 0.942366 0.890480 0.842601 0.797754 0.755398 0.715228 0.677064 0.640794 0.606334 0.573613 0.542566 0.513126 0.485227

Risky bootstrapped curve

[5] 1 0.902887 0.822065 0.751757 0.691047 0.637564 0.588310 0.545805 0.505763 0.471526 0.438821 0.410088 0.384401 0.360770

Cashflows from Charter Communications par asset package Swap Swap payments receipts margin (bp) => 14.65 -97.208 8.63 -0.15 8.63 –0. 1 5 –0. 1 5 8.63 –0.15 8.63 8.63 -0.15 8.63 -0.15 8.63 –0.15 8.63 –0. 1 5 108.63 –0.15

PV = 1.01 Net PV = Price of the credit risk transfer = (upfront as % of NPA)

-1.01 0.0000

Old margin 250.48

-2.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50 -2.50

–17.20 16.20%

114

Swaps and Other Derivatives

worksheet, the bond cashflow receipts are discounted using the CADFs, the counterparty's payments using the DFs, and the fair margin is now only 15 bp. The price for transferring the risk, and ensuring the continued payments under the swap, is 16.2% of the notional principal payable upfront. The asset package and the TRS imply that the maturity of the swap matches reasonably closely the maturity of the asset. But there is no reason for this, and the TRS could have had a maturity considerably less than 9 years. In this case: • investor supplies 100 upfront, of which 97.21 is used to buy the bond and 2.79 is paid to the swap counterparty; • assume the swap matures at time T « 9 years; • let PT be the dirty price of the bond at that time; • there is therefore a cash settlement (PT — 100) x NPA at the end which may be positive or negative. Hence the counterparty is also guaranteeing the price of the asset. Short maturities, typically 6–12 months, are very common. Longer maturities may have an each-way option to terminate every 12 months, or even to terminate if there is a credit event on the bond. We can use the same structure to construct "credit spread" swaps: • to pay L + m1 based on credit of Security 1, or • to receive L + m2 based on credit of Security 2 and even extend the approach to include options on credit spreads. Credit default swaps are often more structured, and priced using default probabilities extracted from the market information. We will use the following notation: • a forward df from j -> i is defined as DF,; = DF, / DFj • let the cashflow from a bond of maturity i at time k be Cki • let Vki be the value of a risky bond with maturity i at time k: this is estimated by taking the cashflows Cti at time t > k, and discounting back to time k using the CADFs. Expected cashflows are no longer risky cashflows, and hence are discounted using the risk free DFs. Assume we have a set of risky par bonds, i.e. all current prices are equal to 100. Consider the one period bond. It either matures at the end of the period with the full payment C11 or it defaults. If the latter, then a percentage R1 is recovered:

1–P1

If p1 is the probability of not defaulting in the first period, then the expected payment is:

115

More Complex Swaps

E11 =P1 x C11 +(1 –p1) x R1 x C11

But E01 = E11 x DF0,1 = 100, therefore: pl = [(100 / DF0,1) — R1x C11]/[(l – R1) x C11]. Notice that this assumes a default will only occur on the date of an anticipated cashflow, but is realistic as that is the only real time a default can be observed. If the credit event were something different such as a downgrading, then we would have to consider the event happening on a more continuous basis. Next consider the two period risky bond with cashflows C12 and C22. This has the following possible structure:

The expected value may be calculated in three stages: 1 . E22 = p2 x C22 + (1 – p2) x R2 x C22 2. E12=p1 x (E22 x DF 22 + C1.2) + ( l — p 1 ) x R1 x Vl2 3. E02 x DF0.1 = 100

Thus, given p1, we can therefore solve for p2, etc. A general recursive relationship is: E k-1,i =p k – 1 x (E k,i x DFk–1,k + Ck–1,i) + (1 —pk–1) x R k _ 1 x

for

k = 1 to i– 1

and at maturity: EH E k,i = (E

k–1i

—Pk–1 x C

= P i xC k–l,i

i i+

( 1p

(1 —pk–1) x R k–1 x Vk–l,i

i

)x R i x (5.1)

where E0,i — 100, p0 — 1 and: Pi

= {Eii - Ri x Cii}/{(1 –Ri) x C]}

(5.2)

Thus, we can estimate Eii knowing pk for k = 1 to i — 1, and hence pi. Given the same risk free bond curve as before, we are now going to use the spread curve for a BBB issuer; we will also assume an average recovery rate of 50% 10. The first step is to calculate the terminal value of the risky bonds Vki using the CADFs. This is easily calculated using the relationships: Vii = Cii Vk,i = Ck,i + V k+l,i x CADF

k,,k+1

for k = 1 to i - 1

as shown in Box 1 of Worksheet 5.7. "This is the average for senior unsecured debt, which is the typical bond status — see Using Default Rates to Model the Term Structure of Credit Risk by Jerome Fons, in Das (2000) pp. 331-348.

116

Swaps and Other Derivatives

Worksheet 5.7

Box 1. Calculate the terminal value of the risky bond

Maturity

i => 0 Risky yields V(i.i) Risky 1 -period dfs

Risky 1 -period dfs

1 2 3 6.206% 6.115% 6.043% 106.2058 106.1154 106.0431 0.941568 0.943223 0.944417

4 6.013% 106.013 0.944182

V(k.i)- > k

1 2 3 4 5 6 7 8 9 10 11 12 13

0.941568 0.943223 0.944417 0.944182 0.944211 0.944333 0.943191 0.943414 0.942974 0.943717 0.942224 0.941828 0.943129

1 2 3 4 5 6 7 8 9 10 11 12 13

Box 2. Calculate the default probabilities 1 -period recovery risk free probs rates 1 –period dfs 1

0.998305 0.996364 0.996165 0.994521 0.994308 0.994741 0.992710 0.993625 0.993134 0.995100 0.992282 0.991729 0.994710

50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50%

1 36.21

1 -period probs- > 1 recovery rates- > risk free 1 -period dfs— >

1

k

0.942366 0.944941 0.946232 0.946776 0.946906 0.946822 0.946642 0.946430 0.946223 0.946035 0.945874 0.945740 0.945630

1 2 3 4 5 6 7 8 9 10 11 12 13

106.21 106.12

106.21 106.19 106.04

106.21 106.22 106.11 106.01

0.998305 0.996364 0.996165 50% 50% 50% 0.942366 0.944941 0.946232 [1] [2] [3] E(k.i)=> 100 106.116

100

100

106.116 105.922

106.116 105.999 105.840

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More Complex Swaps

11 7 8 9 10 12 6 5 13 5.990% 5.999% 6.010% 6.011% 5.994% 5.980% 5 985% 5.987% 5.992% 105.9945 105.9802 105.9853 105.9866 105.9919 105.9898 105.9992 106.0097 106.0108 0.944211 0.944333 0.943191 0.943414 0.942974 0.943717 0.942224 0.941828 0.943129

106.21 106.24 106.15 106.08 105.99

These are values of the risky 106.21 106.21 106.25 106.25 106.17 106.17 106.10 106.11 106.03 106.04 105.94 105.95 105.98 105.99 105.99

106.21 106.26 106.18 106.12 106.06 105.98

bonds during their lives, for estimation of recovery 106.21 106.21 106.21 106.21 106.21 106.23 106.24 106.25 106.25 106.23 106.11 106.12 106.14 106.15 106.16 106.02 106.02 106.06 106.09 106.08 105.92 105.97 105.92 106.02 106.01 105.87 105.92 105.91 105.80 105.80 105.94 105.80 105.80 105.88 105.95 105.77 105.87 105.78 105.96 105.94 105.79 105.80 105.91 106.01 105.99 105.73 105.75 105.87 105.99 105.84 105.85 106.00 105.99 106.01 106.01

0.994521 0.994308 0.994741 0.992710 0.993625 0.993134 0.995100 0.992282 0.991729 0.994710 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 0.946776 0.946906 0.946822 0.946642 0.946430 0.946223 0.946035 0.945874 0.945740 0.945630 [12] [10] [13] [5] [6] [4] [7] [8] [9] [11] 100

100

100

100

100

100

106.116 106.031 105.905 105.723

106 116 106.050 105.946 105 785 105.693

106.116 106.066 105.977 105.833 105.759 105.701

106.116 106.060 105.966 105.816 105.735 105.671 105.599

106.116 106.059 105.963 105.812 105.729 105.663 105.589 105.649

106.116 106.053 105.951 105.794 105.705 105.632 105.550 105.602 105.628

100 106.116 106.055 105.956 105.801 105.715 105.644 105.566 105.621 105.650 105.730

100

100

100

106.116 106.045 105.935 105.769 105.671 105.588 105.496 105.536 105.551 105.615 105.590

106.116 106.034 105.912 105.734 105.622 105.526 105.419 105.444 105.441 105.488 105.444 105.571

106.116 106.033 105.910 105.730 105.617 105.519 105.411 105.434 105.430 105.475 105.429 105.554 105.730

118

Swaps and Other Derivatives

Table 5.1 Market data M. IWlVliD

Maturity

1 2 3 4 5 6 7 8 9 10 11 12 13

VSA

|_TU..L

isw*..ivftL»

Risk free

Risky spread

Risky yield

Recovery rate

Non-default one period probability

Probability of default in each period

Probability of survival

6.12% 5.98% 5.88% 5.82% 5.78% 5.76% 5.75% 5.74% 5.73% 5.73% 5.73% 5.73% 5.73%

0.09% 0.14% 0.16% 0.19% 0.21% 0.22% 0.24% 0.25% 0.26% 0.26% 0.27% 0.28% 0.28%

6.21% 6.12% 6.04% 6.01% 5.99% 5.98% 5.99% 5.99% 5.99% 5.99% 6.00% 6.01% 6.01%

50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50% 50%

99.83% 99.64% 99.62% 99.45% 99.43% 99.47% 99.27% 99.36% 99.31% 99.51% 99.23% 99.17% 99.47%

0.17% 0.36% 0.38% 0.54% 0.56% 0.52% 0.71% 0.62% 0.66% 0.47% 0.73% 0.78% 0.49%

99.83% 99.47% 99.09% 98.54% 97.98% 97.47% 96.76% 96.14% 95.48% 95.01% 94.28% 93.50% 93.00%

Spread is roughly equivalent to a BBB counterparty (see Fons).

Table 5.2 Average cumulative default rate (1—survival) Maturity Rating

AAA AA A BBB BB B Inv grade Spec grade

0 0 0 0 0 0 0 0

0 0 0 0.2 1.7 7.9 0 4.3

0 0 0.1 0.5 4.1 14.2 0.2 8.1

0 0 0.1 0.2 0.3 0.4 1.4 0.9 8.9 6.5 19.3 23.3 0.4 0.6 11.4 14.3

0.1 0.2 0.3 0.4 0.5 0.3 0.4 0.4 0.5 0.6 1.5 0.6 0.8 1.2 1.0 1.9 2.3 2.9 3.6 4.2 11.1 12.9 14.4 15.8 17.1 26.5 29.7 31.8 33.8 35.3 0.8 1.1 1.4 1.7 2.1 16.8 19.0 20.7 22.2 23.5

10

11

12

0.7 0.8 1.8 4.7 18.4 36.7 2.4 24.8

0.8 0.9 2.1 5.3 19.6 37.7 2.8 26.0

1.0 1.1 2.4 5.8 21.0 38.3 3.2 27.1

Showing the probability of default for a rated counterparty within N years. Extracted from Moody's Corporate Bond Default Report, January 1995 reproduced with permission.

We now calculate the probabilities. Starting with column [1], we know E11 = 100/DF0.1 = 106.116. We can therefore calculate p1 using Equation (5.2) above. In column [2], again E12 = 106.116; we can calculate E22 using Equation (5.1) and p1, and hence p2. This is repeated across the E matrix to give the vector of non-default probabilities. Hence, the probability of survival 5, is IIPk from k = 1 to i. Finally the probability of surviving up to period i and then defaulting is Si_1 x (1 — p,) as shown in Table 5.1. Notice that, because the spread curve is not strictly monotonic increasing, the non-default probabilities are not monotonic declining. We have calculated risk-neutral default probabilities from the two bond curves. If we compare these with observed probabilities, as shown in Table 5.2, the estimated default probabilities are higher than observed. There may be a number of practical reasons for this

More Complex Swaps

119

as already discussed, but a likely one is that we assumed the entire spread was due to credit and for no other reasons. In practice, Treasuries are likely to be much more liquid than BBB corporate bonds and therefore the spread would contain a liquidity premium. Another argument refers to the Capital Asset Pricing Model (CAPM): ui = uf + (um — uf) x Bi. for all stocks where: Hi — return on stock i uf = return on risk free investment um = average return on market portfolio Bi. = beta coefficient = cov(ri,rm)/var(rm). In words, the return on a stock should be equal to a risk free return plus a risky premium measured purely in terms of risk relative to the "market". This latter point is very important: a stock is deemed to be riskier than the market if its beta is greater than 1, and hence the return demanded by investors should be greater than the average market return. The beta of corporate debt is quite low as it does not generally fluctuate with the market but simply gets repaid: for example, the beta of investment grade debt with under 3 years to maturity is about 0.1. However, as the credit rating on a bond becomes lower, its risk profile becomes similar to equities, therefore its beta increases, and investors demand an increasing excess over the risk free rate. So the observed credit spread may have an element of systemic (i.e. the whole equity market) risk as well as non-systemic (i.e. related to the corporate specifically) risk. Which probabilities should be used? Pricing of derivatives implicitly uses risk-neutral valuation, so it would be consistent to use the implied riskneutral probabilities to make the pricing arbitrage free. If, on the other hand, you wished to do scenario analyses on a portfolio, then it would make more sense to use the observed probabilities. As an example of using the probabilities, we will price the following 3 year swap: • • • • •

underlying reference: GECC 7% ANN bond which matures in 2012 current bond price: 108.3 credit rating: BBB for senior unsecured debt credit event: 5 day default on senior unsecured debt payment ={1 – R} x Pt x NPA

where R is the recovery rate (assumed to be a constant 50%) and Pt the estimated dirty price of the bond before default. In other words, the bond is valued at Pt before the credit event, the credit event then occurs, and its value drops effectively to R x Pt; i.e. the investor has lost {1 — R} x P, x NPA. There are two alternatives for the fee, either a lump sum is paid upfront, or an annual payment is made at the beginning of each year provided that the bond has not defaulted. See Worksheet 5.8. The first step is to value the bond using the CADFs, either by forward valuing the current price Pi =(pi–1 — C / _ 1 ) /CADF ii _ 1 (see column [1]) or backward valuing from the known value at maturity Pi–1 = Pi x CADFi,i-1 + Ci_1 (see column [2]). Note that this assumes if the bond is going to default, it will do so before paying a coupon — hence the price at the end of period / has full accrued. The expected payout EP, under the swap can

Worksheet 5.8 Pricing a credit default swap

NPA= Upfront fee = Margin (bp pa) =

Underlying asset: 12 year GECC bond paying 7.00% coupon ann Assume recovery rate is 50%

100 45.9 16.2

Pricing the bond off the risky dfs bond cashflows

1 2 3 4 5 6 7 8 9 10 11 12

7 7 7 7 7 7 7 7 7 7 7 107

CADF 1 0.941568 0.888108 0.838745 0.791928 0.747747 0.706122 0.666008 0.628321 0.592490 0.559144 0.526838 0.496191

1 -period DF

0.941568 0.943223 0.944417 0.944182 0.944211 0.944333 0.943191 0.943414 0.942974 0.943717 0.942224 0.941828

forward bond price [1] 108.302 115.023 114.525 113.853 113.170 112.443 111.659 110.963 110.198 109.439 108.549 107.776 107.000

forward bond price [2] 108.302 115.023 114.525 113.853 113.170 112.443 1 1 1 .659 110.963 110.198 109.439 108.549 107.776 107.000

prob of survival

prob of default in each period

expected default cashflows

risk free DF

[3]

100% 99.83% 99.47% 99.09% 98.54% 97.98% 97.47% 96.76% 96.14% 95.48% 95.01% 94.28% 93.50%

0.17% 0.36% 0.38% 0.54% 0.56% 0.52% 0.71% 0.62% 0.66% 0.47% 0.73% 0.78% PV =

-0.0975 -0.2075 -0.2160

1 0.942366 0.890480 0.842601 0.797754 0.755398 0.715228 0.677064 0.640794 0.606334 0.573613 0.542566 0.513126

-0.4586

expected fee cashflows [4] 0.1623 0.1620 0.1614

0.4586 Net value =

0.0000

00

g.

More Complex Swaps

121

be calculated for each period as Si_1 x (1 —pi) x (1 — R) x Pi as shown in column [3]. If the swap fee is going to be a single upfront payment, as a percentage of NPA: fee = 10,000 x j]T EPi x DFi /NPA (bp) = 46 bp Alternatively, if it is paid as a margin at the beginning of each period, conditional upon the bond not having yet defaulted, then it is given by: margin = 10,000 x | ] E P i x DFi \ , $ i - 1 x DFi–1 /NPA (bp) = 16.2bp This section has only lightly touched upon the topic of credit derivatives. It is a field which has only started relatively recently, and therefore, whilst growing rapidly, is very small compared to the interest rate and FX derivative markets. However, spurred on by the new Basel proposals which will encourage banks to manage their credit exposures much more actively, it is likely to continue to grow at a substantial rate into the foreseeable future. It is also changing a lot, as the concepts and techniques become more widely accepted, and people look to manage their credit exposures more dynamically.

5.3 CREDIT-ADJUSTED SWAP PRICING Associated with credit derivatives is the use of credit-adjusted pricing, i.e. adjusting the price of a derivative to incorporate some margin that reflects the potential loss if the counterparty defaulted at some stage during the derivative's lifetime. In the early days of the derivatives market, when counterparties all tended to be of good credit, such adjustments were never made. Even today, they are relatively unusual despite numerous attempts by selected banks to introduce them. This is because the competitive pressures are sufficiently large that it is possible for a poor-credit counterparty to flat unadjusted prices. Nevertheless, with the introduction of the proposed second Basel Accord, with its riskbased capital approach, there will be increasing pressures to impose credit-adjusted pricing. We will consider one simple approach to introduce this based upon an interest rate swap. When a generic swap is first entered into, there is no immediate credit risk. As time passes, and interest rates move, the swap value becomes non-zero; the party that has the positive valuation has a credit exposure on the other party. In the jargon of the Basel regulations, the current exposure of a generic swap is zero, but there is potential future exposure. The scale of this exposure can be estimated by constructing an interest rate envelope. Assume we have a current curve of forward interest rates F0 = {F0,0, F 0,1 ,F 0,2 , • • •} where F0,T is the estimate today of the forward rate that fixes at time T. We also assume a known forward rate volatility—we will assume a single constant volatility throughout to make the discussion easier. If we assume that the forward rates follow a lognormal process11, we can write:

See the discussion on Monte Carlo simulation in Section 9.8 for more details.

Worksheet 5.9 Modelling the IR envelope

Today's date 04–Jan–00 Envelope multiplier 1 .645 1 year's time Forward dates

Swap curve ANN, Act/360

[1] 06–Jan–00 08–Jan–0l 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06-Jan-09 06–Jan–10 06–Jan–ll 06–Jan–12 07–Jan–13 06–Jan–14 06–Jan–15 06–Jan–16 06–Jan–17 08–Jan–18 07-Jan-19 06-Jan-20

6.59375% 6.89500% 7.02500% 7.08500% 7.13500% 7.17500% 7.22500% 7.26500% 7.29500% 7.33500% 7.36000% 7.38500% 7.40174% 7.41835% 7.43500% 7.43700% 7.43900% 7.44101% 7.44301% 7.44500%

12mo. Forward rates

Volatility

down

mid curve

up

[4]

[5]

[6]

7.1410% 7.2342% 7.2116% 7.2920% 7.3417% 7.5280% 7.5602% 7.5487% 7.7783% 7.6585% 7.7401% 7.6512% 7.7117% 7.7759% 7.4107% 7.4192% 7.4281% 7.4376% 7.4476%

9.1395% 9.2588% 9.2298% 9.3328% 9.3963% 9.6348% 9.6760% 9.6612% 9.9552% 9.8018% 9.9062% 9.7925% 9.8699% 9.9521% 9.4847% 9.4955% 9.5070% 9.5191% 9.5319%

"iMUftT^' ""

[2] 6.59375% 7.22180% 7.31607% 7.29314% 7.37455% 7.42476% 7.61318% 7.64576% 7.63406% 7.86633% 7.74512% 7.82767% 7.73781% 7.79898% 7.86391% 7.49455% 7.50310% 7.51217% 7.52174% 7.53187%

[3] 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15% 15%

0 5.5795% 5.6524% 5.6346% 5.6975% 5.7363% 5.8819% 5.9071% 5.8980% 6.0775% 5.9838% 6.0476% 5.9782% 6.0255% 6.0756% 5.7903% 5.7969% 5.8039% 5.8113% 5.8191%

0

123

More Complex Swaps 18 16 -

___

14 -

1 yr forward 5 yr forward 12 yr forward

12 • 10 8 • 6 4 2 0

0

1 2

3 4

5 6

7 8 9 10 11 12 13 14 15 16 17 18 19 20

Figure 5.1 Forward curve with 95% envelope

This describes the possible evolution of the 7th forward rate through time, where e is a unit normal random variable. The drift uT is small in practice for interest rates, and will be ignored. A probability bound can be constructed: Fupr = F 0.T x exp{-|a21 + 1.645<7 V7} i.e. there is a 5% chance that the estimate of FTT at time t will lie above FupTT. We can also estimate the down bound by: Fdownt,T= FO,T x exp{-i<j2t – 1.645<7 Vr} Worksheet 5.9 has taken the current 12 month forward curve out for 20 years, and evolved it through time using a volatility of 15%. The printout shows the evolution over the first year: column [5] shows the expected evolution of the curve, and columns [4] and [6] show what is called the 95% envelope based upon the multiplier of 1.645. Of course, the current 12 month rate is already fixed, and does not evolve, so the new forward curve is only 19 years long. The evolution is then continued as shown in Figure 5.1. This shows not only the envelope in 1 year's time, but also after 5 and 12 years. The next step is to calculate how the valuation of a swap would change. Consider a vanilla swap to pay fixed which matures at T; its value at time t is {(1 — DFt,T) - S0 _T x Qt.T}So, given the current swap curve, we can take the forward curve which has been evolved out for 1 year, calculate the new discount curve DF1 T and values of Q1 T, and hence calculate the value of the swaps. This is demonstrated in the extract from Worksheet 5.10. The valuation is done after net payment of the cashflows at the end of the first year, hence the 1 year swap has zero value. It is probably more valid to include the payment in the valuation on the grounds that, if a default were to occur, it is likely to occur before payment; the model would be simple to modify.

124

Swaps and Other Derivatives

Worksheet 5.10

Modelling the potential swap exposure Evolution of the DFs

06–Jan–00 08–Jan–0l 07-Jan-02 06-Jan-03 06–Jan–04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06–Jan–09 06–Jan–10 06–Jan–ll 06–Jan–12

.022 .011 .011 .014 .017 .014 .019 .011 .014 .014 .014 .014

1 0.946597 0.895423 0.847032 0.800655 0.756648 0.713844 0.673611 0.635602 0.598710 0.564465 0.531853

1 0.932659 0.869088 0.809873 0.753976 0.701741 0.651725 0.605443 0.562400 0.521289 0.483729 0.448530

1 0.915407 0.837045 0.765418 0.699086 0.638278 0.581193 0.529399 0.482169 0.437963 0.398373 0.362013

Evolution of the Qs 06-Jan-OO 08-Jan-Ol 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06-Jan-09 06–Jan–10 06–Jan–ll 06–Jan–12

.022 .011 .011 .014 .017 .014 .019 .011 .014 .014 .014 .014

0 0.9571 1.8625 2.7213 3.5353 4.3024 5.0302 5.7113 6.3557 6.9627 7.5350 8.0743

Original rates 06-Jan-OO 08-Jan-Ol 07-Jan-02 06-Jan-03 06–Jan–04 06-Jan-05 06-Jan-06 08-Jan-07 07-Jan-08 06-Jan-09 06–Jan–10 06–Jan–ll 06–Jan–12

.022 .011 .011 .014 .017 .014 .019 .011 .014 .014 .014 .014

6.593750% 6.895000% 7.025000% 7.085000% 7.135000% 7.175000% 7.225000% 7.265000% 7.295000% 7.335000% 7.360000% 7.385000%

0 0.9430 1.8218 2.6429 3.4094 4.1209 4.7853 5.3975 5.9677 6.4962 6.9867 7.4414

0 0.9256 1.7719 2.5480 3.2587 3.9059 4.4983 5.0336 5.5225 5.9665 6.3704 6.7375

Evolution of the swap value

0.0% -1.3% -2.6% -4.0% -5.3% -6.5% -7.7% -8.9% -9.9% -10.9% –11.9% -12.8%

0.0% 0.2% 0.3% 0.3% 0.3% 0.3% 0.3% 0.2% 0.2% 0.2% 0.2% 0.2%

0.0% 2.1% 3.8% 5.4% 6.8% 8.1% 9.4% 10.5% 11.5% 12.4% 13.3% 14.0%

125

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Maturity Figure 5.2 95% exposure envelopes

On the basis that I am only concerned with my exposure, i.e. if rates rise, the evolution of the swap value through time can be tracked, producing the classical path rising and then dropping back to zero by maturity (see Figure 5.2). This is deemed to be a 95% potential future exposure (PFE) envelope, i.e. there is a 5% chance that my exposure may exceed this. From the graph, the maximum exposure for the 12 year swap is about 24% of the notional principal amount: this is often called the peak exposure limit (PEL). The worksheet contains an extract from a Bank of England report12, estimating the PELs for an IRS and a CCS under a range of maturities and probability envelopes. These results formed the foundation for the capital setting in the first Basel Accord published in 1987. The 95% anticipated loss in the tth period under this swap, following the envelope, is: St—1 x ( l – p t ) x PFEt where S t _ 1 and pt can be estimated either from the spread as above or using historic data: Worksheet 5.11 uses the historic cumulative default data in the matrix above. The required adjustment in the price — and note that this assumes that only the counterparty is likely to default: I will not adjust my prices to take into account the probability that I might default — is estimated in three steps: 1. the expected PV of the losses is £), St_i x (I –pt) x PFEt x DFt; 2. the expected PV of 1 bp received through the lifetime of the swap is ]Tf St x dt x DFt/10,000 expressed as a percentage of NPA; 3. the required adjustment is the ratio of these two results — see Figure 5.3. This pricing was done on the basis that the margin had to be sufficient to cover a 95% loss. This is quite pessimistic; the first Basel Accord operated on the basis of a 50% loss, i.e. the adjustment should cover the average credit loss. This also ignores any possibility that the swap might have a mark-to-market loss, in which case the counterparty has a credit exposure on you. But this follows the regulations, which state that the credit exposure has to be calculated as: max{0, current exposure} + PFE Again, it would be easy for the adjustment to reflect {current exposure + PFE}. 12

Report 1361d: "Potential Credit Exposure on IR and FX related instruments", Bank of England, 1987.

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Worksheet 5.11

3. PV Cumulative loss matrix 2

3

4

Aaa Aa A Baa Ba B Inv grade Spec grade

maturity

>

0.000% 0.000% 0.000% 0.004% 0.031% 0.143% 0.000% 0.078%

0.000% 0.000% 0.002% 0.014% 0.113% 0.412% 0.005% 0.233%

0.000% 0.003% 0.009% 0.033% 0.246% 0.776% 0.014% 0.452%

4. Credit-adjusted value of 1 bp Aaa Aa A Baa Ba B Inv grade Spec grade

0.018% 0.018% 0.018% 0.018% 0.018% 0.016% 0.018% 0.017%

0.027% 0.027% 0.027% 0.026% 0.026% 0.023% 0.027% 0.025%

0.034% 0.034% 0.034% 0.034% 0.033% 0.029% 0.034% 0.031%

5. Basis point adjustments Aaa Aa A Baa Ba B Inv grade Spec grade

0 0 0 0 2 9 0 5

0 0 0 1 4 18 0 9

0 0 0 1 8 27 0 15

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5

6

7

8

9

10

11

12

0.000% 0.008% 0.019% 0.064% 0.434% 1.221% 0.028% 0.734%

0.003% 0.015% 0.033% 0.107% 0.669% 1.718% 0.046% 1 .062%

0.009% 0.025% 0.053% 0.160% 0.944% 2.281% 0.073% 1.436%

0.017% 0.034% 0.076% 0.225% 1.236% 2.850% 0.106% 1.824%

0.028% 0.045% 0.103% 0.305% 1.537% 3.415% 0.144% 2.212%

0.040% 0.059% 0.138% 0.397% 1.857% 3.987% 0.192% 2.614%

0.062% 0.082% 0.190% 0.528% 2.334% 4.859% 0.261% 3.223%

0.081% 0.102% 0.236% 0.638% 2.674% 5.407% 0.322% 3.632%

0.041% 0.041% 0.041% 0.041% 0.039% 0.034% 0.041% 0.037%

0.048% 0.048% 0.048% 0.048% 0.045% 0.039% 0.048% 0.042%

0.054% 0.054% 0.054% 0.054% 0.050% 0.043% 0.054% 0.047%

0.060% 0.060% 0.060% 0.059% 0.055% 0.047% 0.060% 0.052%

0.065% 0.065% 0.065% 0.064% 0.059% 0.050% 0.065% 0.056%

0.070% 0.070% 0.070% 0.069% 0.063% 0.053% 0.070% 0.060%

0.075% 0.075% 0.074% 0.073% 0.067% 0.056% 0.074% 0.063%

0.079% 0.079% 0.078% 0.077% 0.070% 0.059% 0.078% 0.066%

0 0 0 2 11 36 1 20

0 0 1 2 15 44 1 25

0 0 1 3 19 53 1 30

0 1 1 4 23 61 2 35

0 1 2 5 26 68 2 40

1 1 2 6 29 75 3 44

1 1 3 7 35 86 4 51

1 1 3 8 38 92 4 55

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10

Figure 5.3 95% equivalent exposure (bp)

5.4 SIMPLE MISMATCH SWAPS Consider the process of determining the cashflow corresponding to a generic floating rate fixing: 1. the rate L is fixed at the start of the period, at time say /; 2. the cashflow is paid at the end of the period, say T; 3. the cashflow equals P x L x (T — t). Finally the tenor of the rate is also equal to (T— t). This determination contains a number of implicit assumptions, so let us make them explicit. We would write as a more general statement: 1. 2. 3. 4.

the rate L fixes at time t, payment happens at time T>>t, the cashflow equals P x L x q, the rate has a tenor T.

So the generic case has q = T = T — t. Consider probably the simplest form of a mismatch swap, namely an "in-arrears" swap, where the floating rate is fixed and paid effectively at the same time, T = t. These swaps arise when a user wishes to take a view on the movement of interest rates. Receiving conventional floating rates means that the first cashflow is fixed from the outset; hardly desirable if you wish to take an open position on the rates moving. Receiving in arrears means that the first fixing is not until the end of the first period, therefore giving some opportunity for the rate to move. Pricing such a swap is straightforward using implied forwards. For example, consider the following swap: maturity: principal: to receive: to pay:

5 years 100 million USD 3mo. Libor in arrears 3mo. Libor + 6.50 bp in advance

First the implied forward rates are estimated; see column [1] of Worksheet 5.12. For a single period, say from t1, to t2, the in-arrears cashflow is constructed by: Px F23 x (t 2 -t 1 )

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i.e. the forward rate for the next period applied backwards as shown in column [2]. Finally the conventional cashflow, including the margin, is calculated; see [3]. The swap is shown to have a zero fair value; see [4]. The margin can be anticipated as follows: • the spot 3mo. rate is 6.03125%, and the final rate out of 6 January 2005 is 7.19133%; • you are paying the former and receiving the latter, as all the other rates effectively cancel; • hence the swap counterparty needs to receive some compensation; • the difference is 116bp which divided by 20 periods gives just under 6bp per period. Obviously this crudely ignores the timing of the cashflows.

5.5 AVERAGE RATE SWAPS In-arrears swaps are a very simple, and seldom used, example of mismatch swaps. A more common application, because it has a firmer practical foundation, is the class of average rate swaps. We will consider two different types, arithmetic average rates used by endusers, and overnight average rates used mainly by banks. Consider a company that has the following debt structure: 1. 2. 3. 4. 5.

3mo. 3mo. 3mo. 3mo. 3mo.

Libor + 25bp Libor + 35 bp Libor + 30 bp Libor + 25 bp Libor + 50 bp

on on on on on

$100 million, out of 20 January; $50 million, out of 7 February; $75 million, out of 21 February; $100 million, out of 3 March; $25 million, out of 15 March.

All the debt has a long time to maturity. The company would like to swap the debt from floating to fixed for the next 3 years. Obviously one way to do this is to enter into a series of five individual swaps. However, it had recently constructed a 3 year budget with funding assumptions based upon an average value of 3mo. Libor. Therefore an alternative structure would be: to pay a fixed rate, and to receive Libor, based upon a weighted average, for 3 years. This is constructed in Worksheet 5.13. First the appropriate dates for each of the debts are constructed; see columns [1], [4], [7], [10] and [13]. This includes the date of the last fixing as shown. From the discount curve, the last fixing plus the implied forward rates have been constructed in columns [3], [6], [9], [12] and [15]. The weights to be applied to each debt are calculated, i.e. for debt 1, we get 100/(100 + 50 + 75+ 100 + 25)^28.57%, etc. The 3 year swap is priced out of 4 January 2000 from a bank's point of view. The cashflows at the end of each quarter are: to receive ]T wi, x Li based upon the implied Libor fixings: columns [18] and [19], and to pay fixed rate 6.7943% qu. Act/360: column [20]. The implied principal amount shown on the worksheet is $100 million, but obviously to provide a suitable hedge, it would need to be scaled up to $350 million. Compound swaps are another family of swaps related to average rate swaps. Consider the following swap:

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Worksheet 5.12 Creating an in-arrears swap Today's date = 04-Jan-00 Swap structure Maturity 5 years Principal 100m USD To receive 3mo. Libor in arrears To pay 3mo. Libor — 6.50 in advance

Swap dates 06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05

Daycount (Act/360)

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250

IBOR DFs 1 0.98498328 0.96951906 0.95353969 0.93685347 0.92152646 0.90561068 0.88911296 0.87309939 0.85786767 0.8427658 0.82779632 0.81296166 0.79871918 0.78466829 0.77080766 0.75698677 0.74357214 0.73034907 0.71717364 0.7041904 0.69175383

Implied forward rates [1] 6.03125% 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219% 7.054% 7.084% 7.114% 7.144% 7.137% 7.162% 7.189% 7.215% 7.19133%

Swap cashflows receive side [2] .5950 .6576 .7432 .7766 .6995 .7963 .8946 .7755 .7919 .8084 .8248 .7832 .7907 .7982 .8059 .8239 .8105 .8172 .8437 .8378 Present value =

Swap cashflows pay side [3] - .5410 - .6115 - .6924 - .7981 - .6791 - .7739 - .8725 - .8505 - .7920 - .8084 - .8248 - .8412 - .7996 - .8071 - .8146 - .8424 - .8205 - .8269 - .8537 - .8603

Net cashflows [4]

0.054 0.046 0.051 -0.021 0.020 0.022 0.022 -0.075 0.000 0.000 0.000 -0.058 -0.009 -0.009 -0.009 -0.018 -0.010 -0.010 -0.010 -0.023 0.0000

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131

Table 5.3 Time (months)

0 3 6 9 12

Fixing

L0/3 L 3/6 L6/9 L 9/12

Implied cashflow

Compounded cashflow

CF3 = P x (1 + d1 x L0/3) x LXN L3/6) 3 x — Jp x xs (1 V^i +d n^ 2 W") i-v^/^y CF6 — CF9 = P x (1 + d3 x L6/39)

CF3.acc = CF3 CF6-acc = CF6 + CF3-acc x (1 + d2 x L3/6)

CF12 = P x (1 + d4 x Z-9/n)

CF12-acc = CF12 + CF9-acc x (1 +d4x L9/u)

Cr9-acc = CF9 + CF6-acc X (1 + d3 X L6/9)

to receive F annually, and to pay Libor quarterly. Even if we assume rates do not change from those implied at the beginning, obviously the fixed rate receiver has an accumulating credit exposure which is effectively reset annually. We could however agree to pay the floating side on a compound basis. The simplest form is as given in Table 5.3. The cashflows are reinvested at the new Libor rates flat, and a single cash payment made at the end, thereby reducing the credit exposure substantially. The main complication with compound swaps is the existence of spreads. For example, there may be a spread on: • the original Libor fixing that determines the implied cashflows, • the Libor rates used for reinvestment, « the final (overall) implied compounded Libor rate, or any combination thereof. Obviously the specification of the contract needs to be carefully defined to avoid confusion and error.

5.6 OVERNIGHT INDEXED SWAPS Average rate swaps are being increasingly used by banks themselves for hedging. During a normal business day, a bank will make a large number of cash payments and receipts. These will ultimately flow down to the cash desk within the Treasury, who will have the final responsibility to fund the net payments or lend out the net receipts. At the end of each working day, the desk will ensure that its books, either in separate currencies or all netted back to a single home currency, are square within limits. Knowing the estimated future cash requirements of the bank, part of the expertise of the desk is to decide how much money will be borrowed or lent, and for what period of time. The remaining balances are invariably sourced into the bank via the overnight market. The overnight rates available depend upon the net positions of all the contributing banks, and can fluctuate violently from day to day. In order for the banks to perform some limited risk control over the fluctuations, overnight indexed swaps (OIS) have been developed. This is the generic name for a class of swaps that: pay some average of the overnight rate, and receive a fixed rate

132

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Worksheet 5.13

Creating a customized average rate swap

Today's date =

04-Jan-OO

Structure of Debt Next payment date Principal 20-Jan-00 100 07-Feb-00 50 21-Feb-00 75 03-Mar-00 100 15-Mar-00 25

Margin (bp) 25 35 30 25 50

Last fixing date 20-Oct-99 08-Nov-99 22-Nov-99 03-Dec-99 15-Dec-99

DFs

Implied 3mo. forward rates

Debt 1

Dates 20-Oct-99 [1] 06-Jan-00 20-Jan-OO 20-Apr-00 20-Jul-00 20-Oct-00 22-Jan-01 20-Apr-01 20-Jul-01 22-Oct-01 21-Jan-02 22-Apr-02 22-Jul-02 21-Oct-02

Last fixing 6.08750% 6.09375% 6.10000% 6.10325% 6.10500%

Weights 28.57% 14.29% 21.43% 28.57% 7.14%

Debt 3

Debt 2

Daycount (Act/360)

[3] 1 0.997823 6.08750% 6.116% 0.982633 6.349% 0.967111 6.597% 0.951077 6.827% 0.934420 6.827% 0.919082 6.976% 0.903157 7.129% 0.886652 7.226% 0.870748 7.034% 0.855536 7.099% 0.840454 7.164% 0.825505

[2]

0.256 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253

Dates 08-Nov-99 [4] 06-Jan-00 07-Feb-00 08-May-00 07-Aug-00 07-Nov-00 07-Feb-01 07-May-01 07-Aug-01 07-Nov-01 07-Feb-02 07-May-02 07-Aug-02 07-Nov-02

Daycount (Act/360)

DFs [5]

0.253 0.253 0.253 0.256 0.256 0.247 0.256 0.256 0.256 0.247 0.256 0.256

Implied 3mo. forward rates

[6] 1 0.994860 6.09375% 0.979595 6.165% 0.964002 6.399% 0.947899 6.648% 0.931636 6.831% 0.916112 6.855% 7.005% 0.900000 0.883838 7.156% 7.188% 0.867896 0.853041 7.044% 7.111% 0.837816 0.822727 7.176%

Dates 22-Nov-99 [7] 06-Jan-00 21-Feb-00 22-May-00 21-Aug-00 21-Nov-00 21-Feb-01 21-May-01 21-Aug-01 21-Nov-01 21-Feb-02 21-May-02 21-Aug-02 21-Nov-02

Daycount (Act 360)

0.253 0.253 0.253 0.256 0.256 0.247 0.256 0.256 0.256 0.247 0.256 0.256

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DebtS

Debt 4

DFs [8]

Implied 3mo. forward rates [9]

1 0.992563 0.977220 0.961575 0.945418 0.929199 0.913664 0.897544 0.881375 0.865551 0.850715 0.835511 0.820443

6.10000% 6.211% 6.437% 6.687% 6.830% 6.878% 7.028% 7.179% 7.154% 7.054% 7.121% 7.186%

Swap dates

[16] 06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-01l 06-Apr-01l 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 ()6-Jan-03

Dates 03-Dec-99 [10] 06-Jan-OO 03-Mar-OO 05-Jun-OO 04-Sep-OO 04-Dec-OO 05-Mar-01l 04-Jun-01l 03-Sep-01l 03-Dec-01l 04-Mar-02 03-Jun-02 03-Sep-02 03-Dec-02

Daycount (Act/360)

Implied 3mo. forward rates

DFs

[11]

[12]

1 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253

0.990737 6.10325% 0.974836 6.247% 0.959139 6.474% 0.943107 6.725% 0.927108 6.827% 0.911214 6.900% 0.895262 7.049% 0.879263 7.198% 0.863711 7.123% 0.848559 7.064% 0.833374 7.130% 0.818488 7.195%

Daycount (Act/360)

DFs

[17]

Dates 15-Dec-99 [13] 06-Jan-OO 15-Mar-00 15-Jun-00 15-Sep-00 15-Dec-00 15-Mar-01 15-Jun-01 17-Sep-01 17-Dec-01 15-Mar-02 17-Jun-02 16-Sep-02 16-Dec-02

Daycount (Act/360)

Implied 3mo. forward rates

DFs [14]

[15]

1 0.253 0.256 0.256 0.253 0.250 0.256 0.261 0.253 0.244 0.261 0.253 0.253

0.988726 6. 10500% 6.273% 0.973126 6.503% 0.957219 6.756% 0.941147 6.822% 0.925965 6.918% 0.909289 7.072% 0.892803 7.221% 0.876798 7.084% 0.861872 0.846239 7.075% 7.139% 0.831239 7.204% 0.816373

Weighted average forward rates

Average rate cashflows

Fixed rate cashflows

[18]

[19]

6.7943% [20]

6.097% 6.192% 6.422% 6.671% 6.828% 6.869% 7.019% 7.170% 7.165% 7.051% 7.117% 7.182%

1.5411 1.5652 1.6411 1.7420 1.6690 1.7364 1.8327 1.8124 1.8112 1.7824 1.7990 1.8155

-1.7175 -1.7175 -1.7363 -1.7741 -1.6608 -1.7175 - .7741 - .7175 - .7175 - .7175 - .7175 -1.7175

18.5759

-18.5759

1 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962 Present value = Net difference

0.0000

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for some notional principal amount at the end of a pre-specified period. Most OIS have a maturity of less than 1 year, although longer OIS are starting to appear, and with a constant principal, but these represent market practice, not theoretical constraints. Most wholesale bank lending takes place during the business day. For example, the Libor rates are officially determined by taking an arithmetic average of the inter-bank deposit rates between a number of contributor banks at 1 lam each London business day. Obviously these rates are inappropriate for overnight transactions, and most financial centres produce an official overnight rate late in the business day, or retrospectively the next day. One of the earliest overnight rates to be published is Taux Moyen Pondere (TMP) which is an official governmental rate reported by the Banque de France at 11.30am the next day. Sterling Over Night (SON) is the average interest rate, weighted by volume, of all unsecured overnight Sterling deposits arranged by specific brokers, and has been available since March 1997. It is published by 5pm on the day of calculation. The European OverNight rate (EON) is published by the European Central Bank by 7pm (Central European Time) each day. The precise structure of an OIS varies from centre to centre, and generally reflects the particular market operations. For example, Table 5.4 shows an extract from the average rate side of a Frankfurt Indexed OverNight Average (FIONA) swap (EONIA swaps follow exactly the same convention). The first rate, 3.41667% applies from 11 February 1998 overnight to 12 February. Based on a principal amount of 1, the principal & interest (P&I) on 12 February is given by: 1 x (1 + 3.41667% x 1/360) = 1.00009491 Carrying on, the P&I is then accrued from 12 to 13 February: 1.00009491 x (1 + 3.41600% x 1/360) = 1.00018981 Table 5.4

11/02/98 12/02/98 13/02/98 16/02/98 17/02/98 18/02/98 19/02/98 20/02/98 23/02/98 24/02/98 25/02/98 26/02/98 27/02/98 02/03/98

Day interval

Cumulative days

1 1 3 1 1 1 1 3 1 1 1 1 3

1 2 5 6 7 8 9 12 13 14 15 16 19

O/N rate 3.41667 3.41600 3.41200 3.41867 3.41067 3.39667 3.39800 3.40000 3.42533 3.44933 3.48933 4.45000 4.45333

Compounding P&I

Accrued interest

Implied accrual rate

1.00009491 1.00018981 1.00047419 1.00056920 .00066400 .00075841 .00085287 .00113645 .00123170 .00132763 .00142469 .00154848 .00192016

9,490.75 18,980.54 47,419.27 56,920.08 66,399.55 75,841.01 85,287.06 113.644.56 123.170.18 132,763.45 142,468.90 154,847.62 192.016.17

3.416670 3.416497 3.414187 3.415205 3.414834 3.412846 3.411482 3.409337 3.410866 3.413917 3.419254 3.484072 3.638201

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135

13 February 1998 was a Friday, so the next rate of 3.41200% applies over the weekend. FIONAs use simple interest over this period, i.e. the accrued P&I is given by: 1.00018981 x (1 + 3.41200% x 3/360) = 1.00047419 and so on to the end of the swap. The total accrued interest is given by P&I-P = 0.00192016 or $192,016.17 on a principal of $100 million. This can then be converted into an average rate: 0.00192016 x 360/19 = 3.638201% as the extracts span 19 calendar days. Finally there would be a net cash settlement at the end of the swap, taking the difference between the average rate and the constant fixed rate: P x {3.638201% - F} x 19/360 = P x {0.00192016 - F x 19/360} calculated on a simple basis, where P is the actual principal amount. One practical point to note is that invariably the daily compounding calculations are done to a pre-specified number of decimal places, usually rounded to 6. This can be significant over the full lifetime of the swap. When an OIS is first entered into, obviously the overnight fixings are unknown and need to be implied off a forward curve. Not knowing where the overnight rates are likely to be, banks typically use the IBOR curve as a reference, recognizing that this is not ideal but possessing little else. Use of a governmental curve is likely to be worse, and repo rates are of course collateralized. The unknown average rate for a FIONA (and SONIA, EONIA, etc.) swap may easily be estimated from the following: P&I = HO + ri x di) where ri< = [DFs(i)/DFe(i) - l]/di- = niDFs((i)/DFe(i) = l/DFend Thus for a FIONA that was entered into some time previously, P&I would consist of real compounding using the rates already determined in the past plus the estimated forward rates off the curve. The OIS described above are relatively straightforward, applying the same compounding approach throughout the lifetime of the swap. Some OIS are more complex: for example, the French Taux Moyen Mensual du Marche Monetaire (TMMMM or T4M) or South African Rand Overnight Deposit (ROD) swaps take an arithmetic average of the overnight rates during each month, and then compound up that average for each month. For example, Worksheet 5.14 shows an example of a ROD swap. The swap is just over 2 months long, and spans three calendar months. The overnight rates corresponding to each business day are shown in column [2]. The arithmetic average of the rates for the first month, weighted by the days, is calculated to be 11.09247% in column [5]. This is for the period from 11 September and finishing on 1 October; the last rate is 12.3905% from 30 September to 1 October. The arithmetic average is then calculated for the month of October, starting on 1 October and 31 October irrespective whether or not these are business days. Finally the average is calculated for the remaining part of November. These three averages are then compounded using Act/365 convention (columns [6] and [7]) and the annualized rate calculated over the total 70 calendar days. OIS were originally conceived as single fixed-average floating cash-settled swaps under 1 year's maturity. However more complex structures are now easily available, such as

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Worksheet 5.14 Example RODS spreadsheet, month beginning in red, bold end of compounding period Business days

[1]

10-Sep-99 13-Sep-99 14-Sep-99 15-Sep-99 16-Sep-99 17-Sep-99 20-Sep-99 21-Sep-99 22-Sep-99 23-Sep-99 24-Sep-99 27-Sep-99 28-Sep-99 29-Sep-99 30-Sep-99 l-Oct-99 4-Oct-99 5-Oct-99 6-Oct-99 7-Oct-99 8-Oct-99 ll-Oct-99 12-Oct-99 13-Oct-99 14-Oct-99 15-Oct-99 18-Oct-99 19-Oct-99 20-Oct-99 21-Oct-99 22-Oct-99 25-Oct-99 26-Oct-99 27-Oct-99 28-Oct-99 29-Oct-99 l-Nov-99 2-Nov-99 3-Nov-99 4-Nov-99 5-Nov-99 8-Nov-99 9-Nov-99 10-Nov-99 ll-Nov-99 12-Nov-99 15-Nov-99 16-Nov-99 17-Nov-89 18-Nov-99 19-Nov-99

O/N rates

[2] 9.9751% 10.4367% 10.7107% 10.3559% 10.1313% 10.5022% 10.8649% 11.1844% 11.4647% 11.2372% 11.6594% 12.1439% 12.6174% 12.9942% 12.3905% 12.9431% 12.8269% 13.4430% 13.9281% 13.4256% 13.1199% 12.9976% 12.9336% 13.1273% 12.8848% 12.3544% 12.2853% 12.6257% 12.6701% 12.6847% 12.0992% 11.9472% 11.9884% 12.2190% 12.5062% 12.5338% 13.0599% 12.7173% 12.7580% 12.7664% 12.7487% 12.5346% 12.3602% 12.8772% 13.3751% 13.2664% 13.0306% 12.8841% 13.1910% 12.9277%

No. of days

[3]

Interest accruing [4]

Average rate [5]

3 0.08199% 1 0.11058% 1 0.13993% 1 0.16830% 1 0.19605% 3 0.28237% 1 0.31214% 1 0.34278% 1 0.37419% 1 0.40498% 3 0.50081% 1 0.53408% 1 0.56865% 1 0.60425% 1 0.63820% 11.09247% 3 0.10638% 1 0.14152% 1 0.17835% 1 0.21651% 1 0.25330% 3 0.36113% 1 0.39674% 1 0.43217% 1 0.46814% 1 0.50344% 3 0.60498% 1 0.63864% 1 0.67323% 1 0.70795% 1 0.74270% 3 0.84214% 1 0.87488% 0.90772% 0.94120% 0.97546% 3 1.07848% 12.69822% 0.03578% 0.07062% 0.10558% 0.14055% 3 0.24534% 0.27968% 0.31354% 0.34882% 0.38547% 3 0.49450% 0.53020% 0.56550% 0.60164% 0.63706% 12.91819% 70 RODS interest payable = or converted to annualized rate

P&I over month [6]

P&I compounded over swap

1.00638

1.00638

1.01078

1:01724

1.00637

[7]

'.

;:/f«fP?2

2.3716% of principal amount 12.3662%

More Complex Swaps

137

forward starts in particular, rollercoasters, basis swaps (such as 3 month average on one side, 6 month average on the other) and longer maturities with multiple cash payments.

5.7 BASIS SWAPS Interest rate basis or floating-floating swaps are intrinsically straightforward. However their pricing can be complex, which is why they are being discussed in this section. There are two classes of IR basis swaps. The first class use the same reference index but of different tenors. For example: to pay to receive

Imo. Libor 12mo. Libor

In theory the fair mid-price of such a swap should be zero. In practice, there is usually a small margin on one side representing the relative supply and demand in the two cash markets, their liquidities, and the inherent credit exposure. The latter point is very evident in the above swap, namely the bank will be making 12 monthly payments before the annual receipt. These swaps are generally used only by market professionals to risk manage the floating sides of their portfolios. Related to these swaps are yield curve swaps, which typically use short-term Libor on one side and a very long-term rate on the other. These are an extremely important class of swap, and the next section is devoted to their discussion. The other major class of basis swap uses two reference rates, one typically Libor and the other a floating rate from a completely different market. There are a variety of such swaps available, especially in the US due to the large number of possible reference rates. Some example reference rates are: Commercial Paper

Municipal Prime

Fed Funds

T-Bills

rate of interest paid on short-term securities issued by corporates to fund their working capital requirements (obviously the level depends upon their credit rating) rate of interest paid on securities issued by state and local government agencies rate of interest at which a commercial bank in the US would lend to its most creditworthy domestic customers (Note: this is effectively a regulated domestic rate, and should not be confused with Libor which is effectively an unregulated international rate. Base rate would be the UK equivalent) Federal funds are non-interest bearing reserves deposited by member banks at the Federal Reserve. The Fed Funds rate is the rate of interest charged by banks trading these reserves, and is closely monitored by the Fed rate of interest paid on short-term government securities

See Table 5.5. Forward curves can be constructed from each of the basis swap curves, using the Libor forward curve and Libor discount factors on each side. For example, consider a CP swap of maturity n:

138

Swaps and Other Derivatives

Table 5.5

CP

1 year 2 year 3 year 4 year 5 year 7 year 10 year

Muni

Prime

Fed Funds

3mo. T-Bills

Pay

Rec

Pay

Rec

Pay

Rec

Pay

Rec

Pay

Rec

13.0 10.3 9.0 8.0 7.5 6.8 6.3

17.0 14.3 13.0 12.0 11.5 10.8 10.3

63.00 64.25 65.00 66.00 67.00 68.50 70.50

64.00 65.25 66.00 67.00 68.00 69.50 71.50

-286.5 -284.0 -283.5 -283.0 -282.5 -282.0 -282.0

-282.5 -280.0 -279.5 -279.0 -278.5 -278.0 -278.0

14.8 15.5 16.5 18.0 18.5 19.3 20.5

18.8 19.5 20.5 22.0 22.5 23.3 24.5

60.0 63.0 68.0 71.0 71.0 71.0 71.0

70.0 73.0 78.0 81.0 81.0 81.0 81.0

All quotes against 3 month Libor. All quotes are in basis points on non-Libor side, except muni which are a % of Libor. Source: Tullett & Tokyo, except muni from Prebon Yamane.

~L -L -L -L -L -L -L -L -L -L -L

+CP + mn + CP + 7mn

+ CP + mn

-L

If we put the principals in on both sides, the left-hand side will have a zero value and so we could write: -1 + £,(CPt+ mn) x dt x DFt + DFn = 0 We could either adopt a bootstrapping approach, i.e. estimate mj, for j= 1,2,... by interpolation from the market quotes, and then calculate CPj, sequentially, or use an optimization approach. This latter is shown in Worksheet 5.15. In the worksheet, we are going to estimate 1 month CP rates, which are probably the most common tenor; the same approach could easily be adapted for CP rates of other tenors. Given that one can get estimates for CP rates out for a year, it would be quite easy to estimate the first few forward rates off this physical market, but this has not been done in the worksheet. Instead we have adopted a slightly different tack. The CP curve is effectively a spread curve off Libor, so if we define CPt = L, — s, we can work in terms of the spreads {s}. The first worksheet (not printed) estimates a 1 month Libor curve using the optimization approach; the end result may be seen in columns [1] and [2]. The spread curve {5} is estimated in column [3], and the final values of the mid-rate CP-Libor swaps in column [8], using the above equation. The optimization approach has been used again, this time to smooth the spread curve (see Figure 5.4).

139

More Complex Swaps 8.0 7.5 7.0 6.5

™ CP forwards — Libor forwards

6.0 5.5 5.0

0

12

24

36

Figure 5.4

48

60

72

84

96

108

120

1 -Month CP-Libor curves

We can construct a muni curve in a similar way. These are actually average rate swap swaps, for example: to receive 1 3 week arithmetic average of a muni index, such as JJ Kenny or PSA; to pay OCT x 3 month Libor, where aT is a constant depending upon maturity (see Worksheet 5.16). We can estimate the implied muni rates by first calculating 1 month implied Libor rates at weekly intervals; see columns [1] to [8]. The next step is to calculate the 13 week average Libor rate; see column [9] and summarized in column [10]. We now guess the forward multipliers to be applied to these average rates to arrive at the average 3 month muni rate13, see column [11]. The swap is now as follows, where {a1 , a2, . . .} are the guesses, for a given maturity T. — a.T x L — oer x L —txT x L OJ j1 X

L

— ctT x L -XTX

L

+#i +a2 +a3 +a4

x Liboraverage x Liboraverage x Liboraverage x Liboraverage

+aT__2 x Liboraverage +aT-1 x Liboraverage +aT x Liboraverage

If we add ar x principal at the beginning and end as usual, the left-hand side is zero, and the right-hand side becomes: -<*r +

DFt + ocr x DFn = 0

note that typically dt is on 30/360 basis. Using the same optimization approach, we can value each of the par instruments based upon the initial guess, and then adjust the forward multipliers accordingly. See Figure 5.5. This time, the worksheet applies the smoothing to the forward multipliers. 13 An alternative construction would be to apply the multipliers to the 1 month Libor rates before averaging, and then proceed as before.

140

Swaps and Other Derivatives

Worksheet 5.15

CP-Libor swaps

Market data

Mid-rate margins

1 2 3 4 5 7 10

15 12.3 11 10 9.5 8.8 8.3

Daycount

Monthly Libor DF

[2]

[1] 0 1 2 3 4 5 6 7 8 9

10 11 12 13 14 15 16 17

18 19 20 21

22 23

06-Jan-00 07-Feb-00 06-Mar-00 06-Apr-00 08-May-00 06-Jun-00 06-Jul-00 07-Aug-00 06-Sep-00 06-Oct-00 06-Nov-OO 06-Dec-00 1 08-Jan-01l 06-Feb-01 06-Mar-01 06-Apr-01 07-May-01 06-Jun-01 06-Jul-01 06-Aug-01 06-Sep-01 08-0ct-01 06-Nov-01 06-Dec-01

Monthly Libor

1 0.089 0.078 0.086 0.089 0.081 0.083 0.089 0.083 0.083 0.086 0.083 0.092 0.081 0.078 0.086 0.086 0.083 0.083 0.086 0.086 0.089 0.081 0.083

0.994860 0.990174 0.984983 0.979591 0.974663 0.969519 0.963978 0.958741 0.953472 0.948005 0.942697 0.936853 0.931718 0.926762 0.921279 0.915806 0.910515 0.905237 0.899799 0.894379 0.888808 0.883778 0.878598

5.813% 6.084% 6.120% 6.193% 6.276% 6.366% 6.467% 6.555% 6.630% 6.698% 6.757% 6.804% 6.842% 6.876% 6.910% 6.941% 6.972% 6.997% 7.018% 7.037% 7.052% 7.066% 7.075%

141

More Complex Swaps

Spread (bp)

1 Month Smoothing CP rates conditions

[3]

[4]

16.42 16.36 16.24 16.06 15.82 15.52 15.16 14.74 14.26 13.72 13.13 12.47 11.76 11.10 10.51 9.98 9.52 9.14 8.82 8.57 8.39 8.28 8.23

5.6483% 5.9209% 5.9574% 6.0326% 6.1182% 6.2113% 6.3154% 6.4074% 6.4879% 6.5605% 6.6258% 6.6791% 6.7248% 6.7649% 6.8052% 6.8415% 6.8770% 6.9060% 6.9300% 6.9513% 6.9677% 6.9827% 6.9930%

[5] 4.4738 0.0040 0.0136 0.0324 0.0602 0.0901 0.1286 0.1777 0.2312 0.2865 0.3526 0.4274 0.5095 0.4303 0.3577 0.2804 0.2079 0.1469 0.1021 0.0625 0.0315 0.0124 0.0025 0.0002

CP*d*DF

[6] 0.004995 0.009555 0.014608 0.019861 0.024664 0.029682 0.035094 0.040213 0.045368 0.050724 0.055929 0.061665 0.066712 0.071588 0.076987 0.082383 0.087601 0.092810 0.098180 0.103533 0.109038 0.114009 0.119129

Q Maturity

[7] 0.088432 0.165446 0.250264 0.337338 0.415853 0.496646 0.582333 0.662228 0.741684 0.823318 0.901876 0.987754 1.062809 1.134891 1.214223 1.293084 1.368960 1.444397 1.521879 1.598895 1.677900 1.749094 1.822310

Net value

Margin

[8] 1 2 3 4 5 7 10

08-Jan-01 07-Jan-02 06-Jan-03 06-Jan-04 06-Jan-05 08-Jan-07 06-Jan-10

15 12.3 11 10 9.5 8.8 8.3

0.000000 0.000000 0.000000 0.000000 0.000000 0.000000 0.000000

142

Swaps and Other Derivatives

Muni forwards Libor forwards

12

24

36

48

60

72

84

96

108

120

Figure 5.5 3-Month muni-Libor swap

This section has described the existence of various basis swaps, and how the implied forward rates may be estimated based upon the market quotes. Most other basis swaps are priced as either a spread or a multiplier. Because they are effectively a link between two markets, the conventions may however appear unusual for people only familiar with Libor markets, but of course they have to be consistent with the practice in the non-Libor market. Prime (and base) swaps are another good example. Because these are quite regulated, the rates do not change from day to day, but typically only after a signal by the government. A time series of Prime therefore consists of a series of quite large step changes. Most people model Prime as a function of forward Libor, but based upon confidence bands. For example, Prime is held constant whilst Libor remains within a constant band; when Libor moves outside the band, then both Prime and the band are adjusted. So Prime typically lags Libor changes, although there can be asymmetry going up and down depending upon government policy.

5.8 YIELD CURVE SWAPS In 1993, short-term USD interest rates were at a historic low, had been for most of 1993, and despite the steep positive implied forward rate curve nobody was forecasting any increase. Money market investors were therefore actively searching for ways to enhance their returns. Three strategies were popular: 1. liquidation and reinvestment in foreign assets: DEM were extremely popular as the curve was inverted at the time due to the substantial borrowing to fund reunification; 2. liquidation and reinvestment further up the USD curve; 3. structured speculation based upon interest rate expectations. Strategies 1 and 3 will be discussed under cross-currency and embedded option structures respectively. To implement #2, this would require a full liquidation of the short-term assets followed by the purchase of long-term assets such as bonds, obviously involving substantial cashflows plus fairly wide dealing spreads. Alternatively investors could enter into a yield curve swap as shown below:

143

More Complex Swaps Worksheet 5.16 Muni-Libor swaps Monthly data from market data Market data Mid-rate margins (% of Libor)

Monthly rests

Zero

Libor DF

[1] 1 2 3 4 5 7 10

63.50 64.75 65.50 66.50 67.50 69.00 71.00

1

2

3

4

5

06-Jan-OO 13-Jan-00 07-Feb-OO 06-Mar-OO 06-Apr-OO 08-May-OO 06-Jun-OO 06-Jul-00 07-Aug-OO 06-Sep-OO 06-Oct-OO 06-Nov-OO 06-Dec-OO 08-Jan-0l 06-Feb-0l 06-Mar-0l 06-Apr-0l 07-May-0l 06-Jun-0l 06-Jul-0l 06-Aug-0l 06-Sep-0l 08-Oct-0l 06-Nov-0l 06-Dec-0l 07-Jan-02 06-Feb-02 06-Mar-02

1 0.998926 0.994860 0.990174 0.984983 0.979591 0.974663 0.969519 0.963978 0.958741 0.953472 0.948005 0.942697 0.936853 0.931718 0.926762 0.921279 0.915806 0.910515 0.905237 0.899799 0.894379 0.888808 0.883778 0.878598 0.873099 0.867974 0.863214

0.019 0.089 0.167 0.253 0.342 0.422 0.506 0.594 0.678 0.761 0.847 0.931 1.022 1.103 1.181 1.267 1.353 1.436 1.519 1.606 1.692 1.781 1.861 1.944 2.033 2.117 2.194

Zero gradient

[2]

[3]

5.5283% 5.7975% 5.9247% 5.9857% 6.0353% 6.0783% 6.1230% 6.1717% 6.2166% 6.2599% 6.3024% 6.3415% 6.3810% 6.4134% 6.4426% 6.4730% 6.5015% 6.5277% 6.5523% 6.5762% 6.5985% 6.6201% 6.6385% 6.6563% 6.6741% 6.6895% 6.7030%

0.00011 0.00005 0.00002 0.00002 0.00001 0.00001 0.00002 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00001 0.00000 0.00000

Continued

144

Swaps and Other Derivatives

Calculating 13 week average of monthly Libor Weekly rests start

Interpolated zero

[4] 06-Jan-00 13-Jan-00 20-Jan-00 27-Jan-00 03-Feb-OO 10-Feb-00 17-Feb-00 24-Feb-OO 02-Mar-OO 09-Mar-OO 16-Mar-00 23-Mar-OO 30-Mar-OO 06-Apr-OO 13-Apr-00 20-Apr-00 27-Apr-OO 04-May-OO 11-May-00 18-May-00 25-May-00 01-Jun-00 08-Jun-OO 15-Jun-00 22-Jun-OO 29-Jun-00 06-Jul-00 13-Jul-00 20-Jul-00 27-Jul-00 03-Aug-0l 10-Aug-0l 17-Aug-0l 24-Aug-OO 31-Aug-0l 07-Sep-0l 14-Sep-0l 21-Sep-0l 28-Sep-OO 05-Oct-OO 12-Oct-0l 19-Oct-0l 26-Oct-OO 02-Nov-OO 09-Nov-OO 16-Nov-0l 23-Nov-OO 30-Nov-OO 07-Dec-OO 14-Dec-0l 21-Dec-0l 28-Dec-0l 04-Jan-01

Monthly Libor DF

[6]

[5] 1 0.019 0.039 0.058 0.078 0.097 0.117 0.136 0.156 0.175 0.194 0.214 0.233 0.253 0.272 0.292 0.311 0.331 0.350 0.369 0.389 0.408 0.428 0.447 0.467 0.486 0.506 0.525 0.544 0.564 0.583 0.603 0.622 0.642 0.661 0.681 0.700 0.719 0.739 0.758 0.778 0.797 0.817 0.836 0.856 0.875 0.894 0.914 0.933 0.953 0.972 0.992 1.011

5.5283% 5.6037% 5.6791% 5.7545% 5.8112% 5.8430% 5.8748% 5.9066% 5.9306% 5.9444% 5.9582% 5.9720% 5.9857% 5.9966% 6.0074% 6.0183% 6.0291% 6.0397% 6.0501% 6.0605% 6.0709% 6.0813% 6.0917% 6.1021% 6.1126% 6.1230% 6.1336% 6.1443% 6.1549% 6.1656% 6.1762% 6.1866% 6.1971% 6.2076% 6.2180% 6.2281% 6.2382% 6.2484% 6.2585% 6.2681% 6.2777% 6.2874% 6.2970% 6.3063% 6.3155% 6.3246% 6.3337% 6.3427% 6.3511% 6.3595% 6.3678% 6.3762%

Weekly rests end

0.998926 0.997823 0.996693 0.995534 0.994366 0.993206 0.992036 0.990854 0.989675 0.988508 0.987337 0.986162 0.984983 0.983808 0.982631 0.981451 0.980268 0.979083 0.977896 0.976707 0.975515 0.974321 0.973124 0.971925 0.970723 0.969519 0.968311 0.967101 0.965888 0.964673 0.963456 0.962237 0.961015 0.959792 0.958566 0.957340 0.956112 0.954881 0.953649 0.952417 0.951184 0.949949 0.948712 0.947475 0.946239 0.945001 0.943761 0.942520 0.941283 0.940045 0.938805 0.937563

06-Jan-OO 14-Feb-00 21-Feb-00 28-Feb-00 03-Mar-00 10-Mar-00 17-Mar-00 24-Mar-00 03-Apr-00 10-Apr-00 17-Apr-00 24-Apr-00 01-May-00 08-May-00 15-May-00 22-May-OO 29-May-00 05-Jun-00 12-Jun-00 19-Jun-00 26-Jun-00 03-Jul-00 10-Jul-00 17-Jul-00 24-Jul-00 31-Jul-00 07-Aug-00 14-Aug-00 21-Aug-00 28-Aug-00 04-Sep-00 11-Sep-00 18-Sep-00 25-Sep-00 02-Oct-00 09-Oct-OO 16-Oct-00 23-Oct-00 30-Oct-00 06-Nov-00 13-Nov-00 20-Nov-00 27-Nov-00 04-Dec-00 11-Dec-00 18-Dec-00 25-Dec-00 Ol-Jan-01 08-Jan-0l 15-Jan-0l 22-Jan-0l 29-Jan-0l 05-Feb-0l

145

More Complex Swaps

Interpolated zero

0.108 0.128 0.147 0.158 0.178 0.197 0.217 0.244 0.264 0.283 0.303 0.322 0.342 0.361 0.381 0.400 0.419 0.439 0.458 0.478 0.497 0.517 0.536 0.556 0.575 0.594 0.614 0.633 0.653 0.672 0.692 0.711 0.731 0.750 0.769 0.789 0.808 0.828 0.847 0.867 0.886 0.906 0.925 0.944 0.964 0.983 1.003 1.022 1.042 1.061 1.081 1.100

5.8293% 5.8611% 5.8929% 5.9111% 5.9326% 5.9464% 5.9602% 5.9798% 5.9919% 6.0028% 6.0136% 6.0245% 6.0353% 6.0457% 6.0560% 6.0664% 6.0768% 6.0872% 6.0977% 6.1081% 6.1185% 6.1291% 6.1397% 6.1504% 6.1610% 6.1717% 6.1822% 6.1926% 6.2031% 6.2136% 6.2238% 6.2339% 6.2440% 6.2541% 6.2640% 6.2736% 6.2832% 6.2928% 6.3024% 6.3116% 6.3207% 6.3298% 6.3389% 6.3475% 6.3559% 6.3643% 6.3726% 6.3810% 6.3888% 6.3966% 6.4045% 6.4123%

Monthly Libor DF [7] 1 0.993705 0.992539 0.991362 0.990684 0.989509 0.988341 0.987169 0.985489 0.984312 0.983136 0.981957 0.980775 0.979591 0.978405 0.977217 0.976026 0.974833 0.973638 0.972439 0.971239 0.970035 0.968829 0.967620 0.966408 0.965194 0.963978 0.962760 0.961539 0.960316 0.959091 0.957865 0.956638 0.955409 0.954177 0.952945 0.951713 0.950479 0.949243 0.948005 0.946769 0.945532 0.944292 0.943051 0.941813 0.940575 0.939336 0.938096 0.936853 0.935616 0.934377 0.933137 0.931895

1 -month Libor [8] 5.9107% 5.9897% 6.0495% 6.0772% 6.0941% 6.1112% 6.1195% 6.1247% 6.1292% 6.1472% 6.1638% 6.1791% 6.1932% 6.2131% 6.2328% 6.2523% 6.2715% 6.2917% 6.3129% 6.3341% 6.3554% 6.3773% 6.3996% 6.4220% 6.4444% 6.4670% 6.4873% 6.5075% 6.5276% 6.5476% 6.5660% 6.5840% 6.6019% 6.6195% 6.6356% 6.6514% 6.6670% 6.6824% 6.6976% 6.7114% 6.7256% 6.7395% 6.7533% 6.7634% 6.7739% 6.7840% 6.7938% 6.8044% 6.8144% 6.8241% 6.8335% 6.8426%

13- week average

[9]

6.0992% 6.1224% 6.1411% 6.1567% 6.1717% 6.1869% 6.2024% 6.2189% 6.2366% 6.2557% 6.2751% 6.2950% 6.3154% 6.3365% 6.3576% 6.3787% 6.3999% 6.4211% 6.4422% 6.4630% 6.4836% 6.5040% 6.5238% 6.5432% 6.5621% 6.5804% 6.5981% 6.6153% 6.6321% 6.6484% 6.6642% 6.6794% 6.6940% 6.7081% 6.7215% 6.7344% 6.7470% 6.7591% 6.7707% 6.7818% Continued

146

Swaps and Other Derivatives

Muni swap: Libor average vs. muni rate 3-Monthly rests

Average Libor

3mo. Libor

Libor DF

[10] 06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-OO 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05 06-Jul-05 06-Oct-05 06-Jan-06 06-Apr-06 06-Jul-06 06-Oct-06 08-Jan-07 06-Apr-07 06-Jul-07 08-Oct-07 07-Jan-08 07-Apr-08 07-Jul-08 06-Oct-08 06-Jan-09 06-Apr-09 06-Jul-09 06-Oct-09 06- Jan- 10

1 6.0992% 6.3365% 6.5981% 6.7818% 6.8960% 6.9870% 7.0455% 7.0801% 7.0906% 7.0880% 7.0751% 7.0619% 7.0573% 7.0590% 7.0531% 7.0664% 7.1013% 7.1352% 7.1529% 7.1543% 7.1493% 7.1603% 7.1939% 7.2454% 7.3058% 7.3548% 7.3864% 7.4012% 7.4028% 7.3968% 7.3830% 7.3625% 7.3439% 7.3513% 7.3860% 7.4490% 7.5282% 7.5909% 7.6308% 7.5755%

0.984983 0.969519 0.953472 0.936853 0.921279 0.905237 0.888808 0.873099 0.857639 0.842453 0.827561 0.812962 0.798632 0.784549 0.770729 0.756987 0.743572 0.730330 0.717144 0.704190 0.691755 0.679391 0.667069 0.654891 0.643093 0.631299 0.619541 0.607724 0.596859 0.585836 0.574682 0.564114 0.553770 0.543612 0.533600 0.523588 0.513872 0.504149 0.494451 0.484913

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256 0.250 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.256 0.250 0.253 0.256 0.256

6.0313% 6.3100% 6.5855% 6.7937% 6.9156% 7.0110% 7.0789% 7.1177% 7.1315% 7.1310% 7.1188% 7.1045% 7.0984% 7.1010% 7.0940% 7.1035% 7.1374% 7.1729% 7.1945% 7.1982% 7.1909% 7.1995% 7.2279% 7.2764% 7.3384% 7.3904% 7.4266% 7.4468% 7.4472% 7.4437% 7.4328% 7.4113% 7.3901% 7.3922% 7.4224% 7.4826% 7.5630% 7.6296% 7.6746% 7.6969%

147

More Complex Swaps

Forward multiplier

3mo. Muni rates

[11]

[12]

63.67% 63.92% 64.43% 65.20% 66.25% 67.04% 67.57% 67.84% 67.87% 68.10% 68.55% 69.19% 70.04% 70.85% 71.63% 72.38% 73.09% 73.62% 73.98% 74.16% 74.17% 74.26% 74.42% 74.66% 74.98% 75.37% 75.84% 76.38% 76.99% 77.55% 78.05% 78.50% 78.90% 79.24% 79.53% 79.78% 79.97% 80.11% 80.21% 80.25%

3.8831% 4.0501% 4.2510% 4.4218% 4.5684% 4.6839% 4.7608% 4.8035% 4.8121% 4.8271% 4.8498% 4.8864% 4.9430% 5.0014% 5.0522% 5.1144% 5.1904% 5.2532% 5.2917% 5.3056% 5.3025% 5.3170% 5.3538% 5.4097% 5.4780% 5.5436% 5.6019% 5.6531% 5.6997% 5.7364% 5.7627% 5.7797% 5.7941% 5.8252% 5.8742% 5.9425% 6.0201% 6.0812% 6.1204% 6.0797%

Smoothing condition

Par multiplier 30/360

Q

Net value of par instruments

[13] 964.9441 0.0000 0.0000 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001 0.0001 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

63.50%

64.75%

65.50%

66.50%

67.50%

69.00%

71.00%

0.250 0.250 0.250 0.256 0.244 0.250 0.256 0.247 0.253 0.250 0.247 0.247 0.253 0.250 0.247 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.250 0.256 0.244 0.250 0.256 0.247 0.250 0.250 0.247 0.250 0.250 0.250 0.250 0.250

0.009562 0.019379 0.029511 0.040098 0.050386 0.060986 0.071800 0.082168 0.092600 0.102767 0.112689 0.122510 0.132489 0.142298 0.151925 0.161604 0.171252 0.180844 0.190331 0.199671 0.208842 0.217872 0.226801 0.235658 0.244465 0.253214 0.261891 0.270670 0.278986 0.287387 0.295851 0.303911 0.311933 0.319849 0.327599 0.335377 0.343111 0.350776 0.358341 0.365712

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

0.000000

148

Swaps and Other Derivatives 20 year bond yield-margin Investor Libor

Libor

where the long-term rate merely acts as a floating reference rate, traditionally refixed in advance at regular intervals and the cashflow paid at the end of each period. The margin is determined so that the swap is deemed to be fair. This is of course simply another form of basis swap. In theory any reference rate could be used, but in practice yield curve swaps typically use either the Treasury or the Swap curve. The former is probably more common in the US, the latter in Europe. For example, we will price (i.e. find the fair margin for) the following: trade date maturity principal to pay to receive

4 January 2000 4 year USD swap $100 million 6mo. Libor sa {5 year sa Act/360 swap rate—margin} sa

This means that the long-term reference rate is the 5 year semiannual Act/360 swap rate, and that the cashflows for the actual swap are themselves semiannual on both sides. Obviously there is no relationship between the frequency of the underlying reference swap and the frequency of the actual cashflows. This is an example of a constant maturity swap (CMS), where the structure of the underlying swap reference rate does not change over the lifetime of the swap. These are the most common type. VM (variable maturity) structures do exist; for example consider an investor who already possesses a portfolio with an average maturity of, say, 10 years, and wishes to swap it into a constant 5 year yield. As time passes, the reduction in the maturity of the physical portfolio needs to be reflected on the pay side of the swap. The actual sequence of events on the two sides is shown below:

0 l

/2

1 Ifc

2 21/2

3 31/2

4

06-Jan-00 06-Jul-00 08-Jan-01 06-Jul-01 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04

fixing fixing fixing fixing fixing fixing fixing fixing

payment payment payment payment payment payment payment payment

More Complex Swaps

149

Using FRA notation, we can represent the first yield curve fixing as FS(0,5), the second as FS(1/2,51/2), the third as FS(1,6), and so on. We know from the previous chapter how to estimate forward swap rates, i.e.:

where it is important that the Q represents the correct frequency of the underlying reference swap rate. See Worksheet 5.17. If we look at the worksheet, the Libor side may be estimated in the usual fashion using the implied forward rate formula:

as shown in column [3]. The Q column [2] is calculated on a semiannual basis, reflecting the underlying frequency, and hence the FS rates in column [5]: for example, consider the fixing on 8 January 2001 which results in a cashflow on 6 July 2001:

8 January 2001 6 January 2006

DF 0.936853 0.654891

Q 0.974187 4.897154

which gives FS(start,end) = (0.936853-0.654891)/(4.897154-0.974187) = 7.1875%. It is conventional that all the swap cashflows are then discounted using the usual Libor DFs. The margin in column [7] could be estimated in the usual fashion, namely what margin would set the net present value of the swap to zero? However in this instance, a more dynamic approach may be better by using the following property:

NPV = PV(Libor) + PV(CMS) + PV(m) = 0 PV(m) = P x m x Qend = -[PV(Libor) + PV(CMS)] which enables the margin m to be estimated directly. If the underlying reference frequency switches to quarterly, then the compounding of Q must also switch to quarterly, as shown in column [2] of Worksheet 5.18. The CMS rates are estimated semiannually based upon the quarterly Qs, and then the rest of the worksheet is as before.

Worksheet 5.17 Pricing a CMS using sa reference rate Date: Maturity: Principal: To pay: To receive: Swap dates

04-Jan-OO 4 year swap 100 million USD 6mo. Libor 5 year CMS less Daycount (Act/360)

Libor DFs interpolated [1]

06-Jan-OO 06-Jul-00 08-Jan-01 06-Jul-01 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05 06-Jul-05 06-Jan-06 06-Jul-06 08-Jan-07 06-Jul-07 07-Jan-08 07-Jul-08 06-Jan-09 06-Jul-09 06-Jan-10

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511 0.503 0.517 0.497 0.514 0.506 0.508 0.503 0.511

25.55bp margin

1 0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564114 0.543575 0.523588 0.504100 0.484913

Q [2] 0 0.490146 0.974187 1.424476 1.873153 2.299217 2.710215 3.106908 3.491710 3.860941 4.220861 4.562432 4.897154 5.214546 5.528537 5.819878 6.109770 6.384578 6.650735 6.904186 7.152030 |

Implied Libor

Libor cashflows discounted [4]

[3]

6.219% 6.749% 6.938% 7.246% 7.119% 7.252% 7.132% 7.194%

-3,143,923.61 -3,486,733.92 -3,449,914.84 -3,723,663.72 -3,599,290.91 -3,666,118.81 -3,605,774.39 -3,656,803.79

PV =

-24,301,323.10

CMS yield [5]

7.008% 7.125% 7.1875% 7.238% 7.2598% 7.295% 7.320% 7.356%

Fair value =

CMS cashflow [6]

Less margin on CMS 25.55 bp [7]

3,543,072.72 3,681,268.59 3,573,774.50 3,719,655.94 3,670,254.51 3,687,952.49 3,700,666.75 3,739,109.74

-129,182.46 -132,021.64 -127,053.08 -131,311.84 -129,182.46 -129,182.46 -129,182.46 -129,892.26

25,193,544.86

-892,221.76 0.0000

Worksheet 5.18 Date: Maturity: Principal: Swap dates 06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05 06-Jul-05 06-Oct-05 06-Jan-06 06-Apr-06 06-Jul-06 06-Oct-06 08-Jan-07 06-Apr-07 06-Jul-07 08-Oct-07 07-Jan-08 07-Apr-08 07-Jul-08

PricingI a CMS using qu reference rate paying sa 04-Jan-00 4 year swap To pay: 100 million USD To receive: Libor Daycount DFs interImplied (Act/360) polated Q Libor [1]

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256 0.250 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962 0.798719 0.784668 0.770808 0.756987 0.743572 0.730349 0.717174 0.704190 0.691754 0.679367 0.667035 0.654891 0.643062 0.631276 0.619537 0.607724 0.596931 0.585939 0.574763 0.564114 0.553761 0.543575

[2]

0 0.248982 0.494055 0.737737 0.982360 1.207622 1.436540 1.668698 1.889398 2.106248 2.319280 2.528529 2.734027 2.935926 3.134272 3.329115 3.522568 3.710526 3.895142 4.078420 4.258380 4.431318 4.603047 4.773511 4.940872 5.101638 5.261210 5.419537 5.578220 5.724137 5.872249 6.022326 6.164971 6.304900 6.442304

6mo. Libor 5 year CMS less Libor cashflows discounted

[3]

[4]

6.219%

-3,143,923.61

6.749%

-3,486,733.92

6.938%

-3,449,914.84

7.246%

-3,723,663.72

7.119%

-3,599,290.91

7.252%

-3,666,118.81

7.132%

-3,605,774.39

7.194%

-3,656,803.79

19.04bp margin CMS cashflow

Less margin on CMS 19.04bp [7]

6.947%

3,511,856.68

-96,269.53

7.061%

3,648,383.61

-98,385.34

7.123%

3,541,684.25

-94,682.67

7.173%

3,686,005.03

-97,856.39

7.194%

3,636,984.38

-96,269.53

7.229%

3,654,417.06

-96,269.53

7.253%

3,666,864.00

-96,269.53

7.288%

3,704,791.62

-96,798.48

24,966,225.81

-664,902.71

CMS yield i i I | | [ ! I | i i i ! 1 ! i

[5]

[6]

I i I i I [ i | | j

PV =

1 | -24,301,323.10 i \ i

Fair value —

0.0000

152

Swaps and Other Derivatives Proof of example 5 year forward swap rate (Worksheet 5.18) Rate = 1 123% -100,000,000 08-Jan-Ol ,741,163 06-Apr-Ol ,800,521 06-Jul-Ol ,859,879 08-Oct-Ol 07-Jan-02 ,800,521 ,800,521 08-Apr-02 ,800,521 08-Jul-02 ,800,521 07-Oct-02 ,800,521 06-Jan-03 ,800,521 07-Apr-03 ,800,521 07-Jul-03 ,800,521 06-Oct-03 ,820,307 06-Jan-04 .800,521 06-Apr-04 ,800,521 06-Jul-04 .820,307 06-Oct-04 ,820,307 06-Jan-05 ,780,735 06-Apr-05 ,800,521 06-Jul-05 ,820,307 06-Oct-05 101,820,307 06-Jan-06 PV =

0

If the underlying reference frequency is retained as semiannual, but the payment frequency is quarterly, then Q has to be calculated more carefully. The first CMS estimate is calculated from a semiannual Q starting on 6 January 2000. The second CMS estimate uses semiannual Q starting on 6 April 2000. Effectively therefore, there are two series of semiannual Qs which do not overlap but are a quarter apart, as shown in column [2]. Worksheet 5.19 is similar to before, but obviously with quarterly cashflows. The market in yield curve swaps has become quite liquid, with bid—offer spreads two to three times wider than generic IRS, as shown in Table 5.6. An alternative formulation is a "participation" yield curve swap, i.e. instead of quoting a spread off the long reference rate, a multiplier is applied to the reference rate instead. See Worksheet 5.20.

5.9 CONVEXITY EFFECTS OF SWAPS Consider the implied forward method for pricing some form of new swap. We can write the discount factors as functions of forward rates Fw, therefore our pricing approach has been to determine (say) the fixed rate F such that: , Fw] - PVnoating[Fw] = 0

(5-3)

Obviously in reality the forward rates are unknown (except for the initial fixing), and therefore could be regarded as random variables subject to some generating process. Hence the PVs are themselves random variables, and of course the above expression.

Worksheet 5.19 Pricing a CMS using sa reference rate paying qu Date: 04-Jan-00 Maturity: 4 year swap To pay: 6mo. Libor Principal: 100 million USD To receive: 5 year CMS less Libor Libor Swap Daycount DFs intercashflows Implied dates (Act/360) Libor discounted polated Q 06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05 06-Jul-05 06-Oct-05 06-Jan-06 06-Apr-06 06-Jul-06 06-Oct-06 08-Jan-07 06-Apr-07 06-Jul-07 08-Oct-07 07-Jan-08 07-Apr-08 07-Jul-08 06-Oct-08

[1]

0.000 0.506 0.508 0.517 0.506 0.497 0.514 0.514 0.506 0.506 0.506 0.506 0.506 0.506 0.506 0.508 0.508 0.506 0.508 0.511 0.506 0.503 0.508 0.511 0.506 0.503 0.508 0.517 0.506 0.497 0.514 0.514 0.506 0.506 0.506

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962 0.798719 0.784668 0.770808 0.756987 0.743572 0.730349 0.717174 0.704190 0.691754 0.679367 0.667035 0.654891 0.643062 0.631276 0.619537 0.607724 0.596931 0.585939 0.574763 0.564114 0.553761 0.543575 0.533554

[2]

0 0 0.490146 0.484716 0.974187 0.950599 1.424476 1.407504 1.873153 1.841204 2.299217 2.259701 2.710215 2.663498 3.106908 3.053184 3.491710 3.431166 3.860942 3,795730 4.220861 4.145450 4.562432 4.484526 4.897154 4.809629 5.214546 5.124560 5.528537 5.426342 5.819878 5.721706 6.109770 6.001663 6.384578 6.271405

[3]

0.2528 0.2528 0.2556 0.2611 0.2444 0.2528 0.2611 0.2528 0.2528 0.2528 0.2528 0.2528 0.2528 0.2528 0.2528 0.2556

PV =

6.031% 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219% 7.054% 7.084% 7.114% 7.144%

[4]

- ,524,565.97 - ,595,040.20 - ,675,795.61 - ,781,090.86 - ,663,220.50 -1,757,463.84 -1,855,525.21 -1,834,106.20 -1,775,532.75 -1,791,941.65 -1,808,353.19 - ,824,767.38 -1,783,164.06 -1,790,679.58 -1,798,195.65 -1,825,777.09

33.03 bp margin CMS cashflow

CMS yield

Less margin on CMS 33.03bp [7]

[5]

[6]

7.008% 7.074% 7.125% 7.163% 7.187% 7.216% 7.238% 7.252% 7.260% 7.278% 7.295% 7.309% 7.320% 7.338% 7.356% 7.372%

1,771,536.36 1,788,030.11 1,820,842.53 1,870,330.66 1,756,939.42 1,824,023.12 1,889,987.34 1,833,247.14 1,835,127.25 1,839,789.53 1,843,976.24 1,847,520.33 1,850,333.37 1,854,909.25 1,859,338.72 1,884,006.55

-83,492.58 -83,492.58 -84,410.08 -86,245.08 -80,740.08 -83,492.58 -86,245.08 -83,492.58 -83,492.58 -83,492.58 -83,492.58 -83,492.58 -83,492.58 -83,492.58 -83,492.58 -84,410.08

25,464,828.28

-1,163,505.18

-24,301,323.10 Fair value =

0.0000

Worksheet 5.20 Pricing a participation constant maturity swap - using sa reference rate Date: Maturity Principal: To pay: To receive: Swap dates

04-Jan-OO 4 year swap 100 million USD 6mo. Libor 5 year CMS times 96.459% participation Daycount (Act/360)

Libor DFs interpolated

[1] 06-Jan-OO 06-Jul-00 08-Jan-0l 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05 06-Jul-05 06-Jan-06 06-Jul-06 08-Jan-07 06-Jul-07 07-Jan-08 07-Jul-08 06-Jan-09 06-Jul-09 06-Jan-10

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511 0.503 0.517 0.497 0.514 0.506 0.508 0.503 0.511

1

0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564114 0.543575 0.523588 0.504100 0.484913

Q [2]

Implied Libor

Libor cashflows discounted

[3]

[4]

0 0.49014575 0.97418671 1.42447646 1.87315254 2.29921747 2.71021475 3.10690817 3.49170977 3.8609418 4.22086134 4.56243207 4.89715413 5.21454569 5.52853653 5.81987834 6.10977035 6.3845778 6.65073503 | 6.90419 7.15203013

1

6.219% 6.749% 6.938% 7.246% 7.119% 7.252% 7.132% 7.194%

PV =

-3,143,923.61 -3,486,733.92 -3,449,914.84 -3,723,663.72 -3,599,290.91 -3,666,118.81 -3,605,774.39 -3,656,803.79

Unmultiplied

Multiplied

CMS

CMS

CMS

yield

cashflow

cashflow 96.459%

[5] 7.008% 7.125% 7.1875% 7.238% 7.2598% 7.295% 7.320% 7.356%

-24,301,323,10 Fair value =

[6]

[7]

3,543,072.72 3,681,268.59 3,573,774.50 3,719,655.94 3,670,254.51 3,687,952.49 3,700,666.75 3,739,109.74

3,417,595.87 3,550,897.57 3,447,210.36 3,587,925.45 3,540,273.55 3,557,344.77 3,569,608.76 3,606,690.30

25,193,544.86

24,301,323.10 0.0000

155

More Complex Swaps Table 5.6 CMT swap rates against 6mo. Libor CMT reference Maturity l yr 2 yr 3 yr 4 yr 5 yr 7 yr 10 yr

5 yr

10 yr

-46/-41 -32/-26

-59/-S4 _44/_39 -36/-31 -34/-29 -31/-26 -2S/-23 -24/-19

2 yr

-22/-16 -13/-7 -7/-1 -5/+1 -3/ + 3 -1/ + 5 + 1/ + 7

-25 /-19 -22/-17 -20/-15 -17/-12 -13/-8

CMT is sa Act/Act yield. Source: Prebon Yamane, May 1997.

equating a random variable to a constant, is not appropriate. A more correct expression would be to take expectations with respect to some distribution, i.e.: - E{PVfloating[Fw]} = 0

(5.4)

The implied forward method effectively estimates the expected forward rates, and then calculates: =0

(5.5)

But we know, for a non-linear function/(•), E{f(x)} ^f(E{x}} for some random variable x under any distribution. Hence the estimate for F from Equations (5.4) and (5.5) will be different. For many classes of swaps, there is a simple way around this. A discount bond is a zero coupon bond that pays 1 at maturity T. Its price today is given by the discounted value of the cashflow, i.e. pT = 1 x DFT = DFT.We can therefore think of DFs as being tradeable discount bond prices, with their own market behaviours, volatilities, correlations, etc. Consider a generic swap, we know we can write: PVflx = P x F x Q

and

PV floating = 1 - DFend

i.e. the net value of the swap is a linear function of DFs. Under these circumstances, taking expectations of DFs not forward rates, Equations (5.4) and (5.5) are equivalent. However for more complex swaps such as mismatch, this linear relationship is no longer true. For example, if the CMS above were to be repriced after a parallel shift in the current Libor curve, the changes in value are small (as CMSs are very insensitive to parallel shifts anyway) but not symmetric as shown below: Change in value of CMS for shift in Libor curve Shift (bp) Libor up Libor down 100 6.31 -6.39 200 12.54 -12.85 300 18.66 - 19.34 400 24.67 -25.87 500 30.55 -32.40

156

Swaps and Other Derivatives

This suggests that, if movements in the curve are equally likely up or down, the receiver of the CMS would be slightly penalized and require some small compensation such as a reduction in the margin to enter into the swap. See Worksheet 5.21. More generally, when estimating the fixed rate F that makes Equation (5.4) true, higher order terms need to be considered. The impact of these terms on the pricing is termed the "convexity effect". In many cases the impact is negligible, but for some structures it should be taken into account. The Appendix describes various approaches that may be used to estimate the impact, and applies them to a range of different swaps.

5.10 INFLATION SWAPS Another form of swap is the inflation swap, where the cashflows on one or both sides are calculated with reference to an inflation rate. As we will see, there are two main types, either fixed-floating inflation or Libor—inflation basis swap. Before however we can discuss inflation swaps — known as RPI swaps in the UK as the Retail Price Index is usually used as the reference rate — we need to understand the inflation-linked bond market. Unlike traditional fixed coupon bonds, the coupons and/or the principal amount at redemption of an inflation bond is linked to an inflation index. For example, the box below shows an indexed gilt bond. At the time of issue, the RPI

Today's date: Issue date: Maturity date: Coupon: Current price:

6 October 1999 26 August 1982 26 August 2001 2.5% sa 202.81 plus accrued

RPI: 78.3

The principal is index-linked, defined as PT = 100 x {I T /I 0 }- The actual bond cashflows are: coupon at time T, CT = 0.5 x 2.5% x PT, principal payment at maturity, P'maturity. Inflation-linked securities have been issued by a number of organizations, including governments. There are typically two reasons for their issue. 1. When investors are concerned about extremely high future inflation (or hyperinflation), and are therefore unwilling to bear this risk. Governments such as Brazil, Mexico, Israel and Argentina have all found it necessary to issue short-dated inflationlinked securities regularly as the only way they could attract investors. 2. With low and apparently stable inflation across much of the developed world (at the time of writing), an increasing number of governments have been issuing index-linked bonds. There are three reasons for this. • In such circumstances, traditional fixed coupon bonds are unattractive to investors. Because of the low coupon, a small increase in inflation will result in a large decrease in bond price, i.e. the bond's price elasticity with respect to inflation is high. 14

The RPI was re-based to 100 in February 1987.

More Complex Swaps

157

• Governments frequently set themselves inflation targets; by issuing these bonds, they can act as a form of penalty if inflation is allowed to get out of control. • They broaden the range of potential investors in the government securities, which is likely to reduce the overall cost of debt. The first major market of the latter form was started in the UK, whose government has issued indexed-linked bonds from 1981. There are currently 10 issues outstanding, which represent about 25% of total gilt market (measured in terms of total redemption). The government has an ongoing commitment to issue a minimum of GBP 2.5 billion pa. to provide market liquidity. It has in fact usually exceeded those levels: 1996-7 5.9bn 1997-8 4.8bn 1998-9 2.6bn 1999-2000 3.5bn Other major markets include Australia (started issuance in 1985), Canada (1991), France (1998), Sweden (1994, including some zero-coupon bonds) and US TIIPs (1997). Current market value (USD billion) No. of issues outstanding

Canada 10.9

France 17.6

Sweden 11.2

UK 102.9

US 131.1

3

3

6

10

9

Reprinted by permission of Barclays Capital—last updated 31 January 2002

In October 2001, France announced its intention to issue bonds linked to a European-wide inflation rate. All of these government bonds use some version of a Consumer Price Index as a reference.There are also a number of non-governmental securities issued. For example: • there are 47 sterling corporate and supranational issues which raised a total of just over 5 billion • many of the Australian states have issued commodity price-linked bonds to fund specific projects Let us consider the mechanics of the above bond in more detail. Inflation is of course backward-looking, i.e. it is reported after the event. In the UK, the RPI for a month is published two weeks after the end of that month, i.e. January's RPI will appear in midFebruary. Most people will wait until (say) the end of February before using the figure in case of any subsequent adjustments. Inflation figures in the US are usually quoted with a 3 month lag. Therefore, to calculate the next coupon payable on this bond, which is on 26 February 2000, we cannot use the February index because it has not yet been published, but we will have to use an earlier one. More generally, to index a cashflow at time T, we would have to use the index at (T— T) where i is referred to as the reference lag. To be consistent, the lagged index I 0 _ T is used as the original reference; the index of 78.3 quoted above was actually the index for December 1981. To be sure about the reference index, i would have to be set to a minimum of 2 months. In actual fact, the lag in the indexed gilts market is 8 months! There is a sensible and pragmatic reason for this, as follows: • the last coupon on this bond was paid on 26 August 1999;

158

Swaps and Other Derivatives

Worksheet 5.21

Pricing a constant maturity swap: how does it change in value for a shift in the forward curve

Today's date:

04-Jan-00

Example: To pay: To receive:

4 year swap 6mo. Libor 5 year CMS less

25.55 bp margin

New value after shift (per bp)

Shift in Libor = Swap dates 06-Jan-00 06-Jul-00 08-Jan-0l 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05 06-Jul-05 06-Jan-06 06-Jul-06 08-Jan-07 06-Jul-07 07-Jan-08 07-Jul-08 06-Jan-09 06-Jul-09 06-Jan-10

Libor Daycount DFs interpolated (Act/360)

Implied FRA

1 0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564114 0.543575 0.523588 0.504100 0.484913

6.219% 6.749% 6.938% 7.246% 7.119% 7.252% 7.132% 7.194% 7.214% 7.268% 7.267% 7.312% 7.440% 7.501% 7.478% 7.529% 7.474% 7.510% 7.689% 7.742%

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511 0.503 0.517 0.497 0.514 0.506 0.508 0.503 0.511

Libor up

1 | | | | | | | | | | | | | | | ! i | ! i i

7.219% 7.749% 7.938% 8.246% 8.119% 8.252% 8.132% 8.194% 8.214% 8.268% 8.267% 8.312% 8.440% 8.501% 8.478% 8.529% 8.474% 8.510% 8.689% 8.742% PV =

new DF 1 0.964790 0.927653 0.892427 0.856148 0.822390 0.789456 0.758281 0.727960 0.698935 0.670596 0.643835 0.617596 0.592454 0.567528 0.544573 0.521708 0.500276 0.479533 0.459461 0.439810

6.3149

lOObp Libor cashflows Q discounted

new

0.487755 0.9670422 1.4107768 1.8507415 2.2665052 2.6656191 3.0489721 3.4190186 3.7723689 4.1151182 4.4388239 4.7544839 5.0523567 5.3455795 5.6163532 5.884453 6.1373702 6.3811326 6.6121392 6.8369312

-36,494.79 -40,034.01 -39,471.37 -42,375.53 -41,048.46 -41,716.74 -41,113.30 -41,651.37

-272,039.82

CMS CMS yield cashflow 25.55 8.005% 8.123% 8.186% 8.237% 8.258% 8.293% 8.318% 8.354%

39,176.46 40,649.97 39.434.30 41,018.18 40.459.25 40,636.01 40.761.19 41.167.28

272,046.14

More Complex Swaps

159

-6.3918

Libor down

new DF

5.219% 5.749% 5.938% 6.246% 6.119% 6.252% 6.132% 6.194% 6.214% 6.268% 6.267% 6.312% 6.440% 6.501% 6.478% 6.529% 6.474% 6.510% 6.689% 6.742%

1 0.974295 0.946192 0.919055 0.890473 0.863751 0.837288 0.812111 0.787322 0.763340 0.739645 0.717050 0.694639 0.672851 0.650986 0.630673 0.610202 0.590863 0.571937 0.553329 0.534898

new

Q 0.49256 0.9814259 1.4384005 1.8960046 2.3326786 2.7559741 3.1665411 3.5667631 3.952674 4.3307146 4.6912314 5.0462689 5.3845635 5.7209064 6.0344913 6.3480671 6.6467811 6.937516 7.2157174 7.4891097

Libor cashflows discounted

CMS yield

CMS cashflow 25.55

-26,383.68 -29,700.67 -29,526.93 -32,097.75 -30,937.35 -31,605.63 -31.002.19 -31,484.70

6.012% 6.127% 6.189% 6.239% 6.261% 6.296% 6.322% 6.357%

29,101.31 30,334.97 29,500.15 30,748.77 30,362.27 30,539.48 30,668.57 31,017.11

-212,678.04

212,671.65

160

Swaps and Other Derivatives

• after that date, the actual price for the bond is the quoted clean price plus accrued interest; • the accrued can only be calculated with certainty if the next coupon, which is to be paid on 26 February 2000, is known; • this coupon can only be known if the reference index is fixed by 26 August 1999. The disadvantage to this scheme is that the bond lags 8 months behind the actual inflation, and is therefore a less than perfect hedge. All other countries use a much shorter lag which overcomes the latter problem, but means that the accrued is susceptible to estimation adjustments. The published RPI for June 1999 was 164.90. The coupon is therefore: 0.5 x 2.5% x 100 x (164.90/78.3) = 2.63% of 100 and hence the accrued (ignoring the settlement period) is: 2.63% x (26 Feb 2000 - 6 Oct 1999)/182.5 = 0.591% and the dirty price for the bond is 202.81 + 0.591 = 203.401. Notice that the index is quoted for the month, rather than the inflation on a specific date. This is another peculiarity of the UK systems, in that a cashflow occurring at any time in the month of February would be inflated by the June index. In the US and Sweden for example, the inflation is calculated as of a particular date, and the reference index for a specific cashflow is calculated by (linearly) interpolating the bracketing published indices — again providing a better hedge. The full cashflows for this bond are shown below: Today's date: 6 October 1999 cashflow indexing inflation months index dates 26-Aug-82 78.3 06-Oct-99 Jun-99 26-Feb-00 164.90 26-Aug-00 Dec-99 ^Dec-99 26-Feb-0l Jun-00 IJune-00 Dec-00 26-Aug-0l IDec-00

non-indexed cashflows

1.25 1.25 1.25 101.25

indexed cashflows 203.401 2.63 1. 25 X {/oee^/78.3}

1.25x1/^.00/78.3} 101.25 xl/o^.oo/78.3}

If we wish to value this bond, there are two unknown components: 1. discount factors — if we assume that the risk premium between the conventional gilt market and the indexed market is zero, then we can use the conventional gilt discount curve; 2. future indices — define the average index growth rate g,T from / < T by:

One common assumption is to make g constant over the remaining lifetime of this bond — we can therefore calculate the value of g that enables the discounted value of the indexed cashflows to equal the current dirty price. If we turn to the spreadsheet model, Worksheet 5.22 constructs the gilt discount curve using the conventional repo and bond data.

More Complex Swaps Worksheet 5.22

161

Pricing an inflation swap: Step 1. Building a gilt DF

Today's date

06-Oct-99

Market data 1: Repos Risk free DF

Z-c rates

0.003 0.019 0.038 0.085 0.252 0.501 1.003

0.999884 0.999090 0.998124 0.995729 0.986449 0.972216 0.944143

4.25% 4.75% 4.90% 5.04% 5.41% 5.62% 5.73%

market prices 105.09 111.11 120.10 129.53 140.41

last coupon 26-Aug-99 07-Jun-99 13-Apr-99 06-Aug-99 25-Aug-99

dirty prices 106.21 113.93 124.44 131.03 141.42

GC repo rates 06-Oct-99 07-Oct-99 13-Oct-99 20-Oct-99 06-Nov-99 06-Jan-OO 06-Apr-OO 06-Oct-00

4.25% 4.75% 4.90% 5.05% 5.45% 5.70% 5.90%

Market data 2: Conventional Gilts coupon 10% 8.50% 9% 9% 8.75%

i 2 3 4 5

maturity 26-Feb-0l 07-Dec-05 13-Oct-08 06-Aug-12 25-Aug-17

estimated prices 106.21 113.93 124.44 131.03 141.42

difference 0.00 -0.00 -0.00 -0.00 -0.00

Implying a discount curve from gilts

1.0 Discount curve Zero-coupon curve

0.5

0.4

0.3

4.0

162

Swaps and Other Derivatives

A general repo curve is used to calculate the discount factors at the short end of the curve — see Box 1. When building bond discount curves, unlike swap curves, it is always a trade-off whether one uses a relatively small number of liquid gilts with a reasonable spread of maturities to construct a curve that prices all bonds exactly, or uses more bonds but recognizes that they may be mutually contradictory and therefore construct a regression curve which approximately prices the bonds. In this example, five liquid gilts have been selected which will be priced exactly — see Box 2. There are a variety of methods that could be used to estimate the discount curve; polynomial splining is a common approach. This worksheet uses the optimization approach as described in Chapter 2 to build the curve, i.e. it starts by guessing a forward curve, which is then converted into a discount curve. This curve is interpolated via zero coupon rates to calculate the prices of the five gilts. The Solver is used to estimate a forward curve with minimum "roughness" whilst ensuring that the curve prices match the market prices.

Worksheet 5.23 constructs the forward inflation curve, using the set of indexed gilts shown below. Note that they are ordered in terms of increasing maturity:

1 2 3 4 5 6 7 8

coupon 2.500% 2.500% 4.375% 2.000% 2.500% 2.500% 2.500% 2.500%

maturity 26-Aug-0l 20-May-03 21-Oct-04 19-Jul-06 20-May-09 23-Aug-11 16-Aug-13 26-Jul-16

Indexed gilt data set issue initial date index 26-Aug-82 78.3 27-Oct-82 78.8 22-Sep-92 135.6 69.5 08-Jul-81 19-Oct-82 78.8 28-Jan-82 74.6 21-Feb-85 89.2 81.6 19-Jan-83

market prices 202.81 201.77 127.86 231.71 210.90 221.33 185.88 205.42

last coupon 26-Aug-99 20-May-99 21-Apr-99 19-Jul-99 20-May-99 23-Aug-99 16-Aug-99 26-Jul-99

dirty prices 203.40 203.75 130.29 232.74 212.88 222.00 186.53 206.41

Working with the first, shortest, bond. If we guess a growth rate g1 of 2.165% from September 1999 until December 2000, then the cashflows are shown in the box below, and the value equals the dirty price: cashflow dates 26-Aug-82 06-Oct-99 26-Feb-OO 26-Aug-OO 26-Feb-0l 26-Aug-0l

indexing months Jun-99 Dec-99 Jun-00 Dec-00

inflation index 78.3 164.90 166.69 168.48 170.29

non-indexed cashflows

1.25 1.25 1.25 101.25

indexed cashflows

DF

203.401 2.63 2.66 2.69 220.21

1 0.978571 0.950454 0.919697 0.889258

We then move on to the second bond. This matures in May 2003, so that the last index required is September 2002. We know the index up till December 2000, so we now guess a growth rate g2 that will apply from January 2001 to September 2002, and which will correctly price this bond. This bootstrapping process is repeated for the entire index set.

163

More Complex Swaps Worksheet 5.23

Pricing an inflation swap: Step 2. Building an inflation curve

Today's date

06-Oct-99

Market data 3: Index Gilts maturity coupon 1 2.500% 26-Aug-0l 2 2.500% 20-May-03 3 4.375% 21-Oct-04 4 2.000% 19-Jul-06 5 2.500% 20-May-09 6 2.500% 23-Aug-ll 7 2.500% 16-Aug-13 8 2.500% 26-Jul-16

issue initial date index 26-Aug-82 78.3 27-Oct-82 78.8 22-Sep-92 135.6 08-Jul-81 69.5 19-Oct-82 78.8 28-Jan-82 74.6 21-Feb-85 89.2 19-Jan-83 81.6

market prices 202.81 201.77 127.86 231.71 210.9 221.33 185.88 205.42

last coupon 26-Aug-99 20-May-99 21-Apr-99 19-Jul-99 20-May-99 23-Aug-99 16-Aug-99 26-Jul-99

dirty prices 203.40 203 75 130.29 232.74 212 88 222 00 186 53 206.41

estimated prices difference 0.000 203.40 0.000 203.75 130.29 0.000 232.74 0.000 0.000 212.88 0.000 222.00 0.000 186.53 206.41 0.000

Forward inflation index 2.165% 4.101% 3.879% 6.087% 2.741% 1.607% 1.079% 0.897%

RPI re-based to 100 in February 1987

Index Bond #2 cashflow dates 27-Oct-82 06-Oct-99 20-Nov-99 20-May-OO 20-Nov-OO 20-May-0l 20-Nov-0l 20-May-02 20-Nov-02 20-May-03

Index Bond #3 cashflow dates 22-Sep-92 06-Oct-99 21-Oct-99 21-Apr-00 21-Oct-00 21-Apr-0l 21-Oct-0l 21-Apr-02 21-Oct-02 21-Apr-03 21-Oct-03 21-Apr-04 21-Oct-04

indexing months Mar-99 Sep-99 Mar-00 Sep-00 Mar-01 Sep-01 Mar-02 Sep-02

indexing months Feb-99 Aug-99 Feb-00 Aug-00 Feb-01 Aug-01 Feb-02 Aug-02 Feb-03 Aug-03 Feb-04

inflation non-indexed cashflows index 78.8 164.00 165.80 167.58 169.39 172.01 175.51 179.07 182.70

1.25 1.25 1.25 1.25 1.25 1.25 1.25 101.25

inflation non-indexed index cashflows 135.6 163.70 165.50 167.28 169.08 171.44 174.92 178.47 182.09 185.62 189.19 192.82

2.1875 2. 1875 2. 1875 2. 1875 2. 1875 2. 1875 2. 1875 2. 1875 2. 1875 2. 1875 102. 1875

indexed cashflows

Z-c rates

203.7514 2.60 2.63 2.66 2.69 2.73 2.78 2.84 234.75

5 .125% 5.647% 5.822% 6 .119% 6 .258% 6.314% 6 .332% 6 .335%

indexed cashflows

Z-c rates

130.29 2.64 2.67 2.70 2.73 2.77 2.82 2.88 2.94 2.99 3.05 145.31

4.904% 5 .629% 5.762% 6 .087% 6.245% 6 .309% 6.331% 6 .335% 6 331% 6 316% 6 274%

DF 0 12 0.62 1.13 1 62 2 13 2.62 3 13 3.62

1 0.993702 0.965490 0.936549 0.905515 0.875419 0.847437 0.820412 0.794985

DF

0.04 0.54 1 04 1.54 2.04 2.54 3.04 3.54 4 04 4 55 5 05

1 0.997987 0.969925 0.941628 0.910387 0.880181 0.851805 0.824728 0.798985 0.774115 0.750445 0.728603

164

Swaps and Other Derivatives

NS curve Piecewise constant

Jan-99

Sep-01

Jun-04

Mar-07

Dec-09

Sep-12

Jun-15

Mar-18

Figure 5.6 Inflation growth curve

From the curve, the market anticipates inflation growing over the next few years and then declining. Assuming a piecewise constant growth curve is probably the most widely used assumption. Other approaches include fitting a piecewise linear and even a parametric curve such as Nelson—Siegel (which only has four parameters and therefore will not correctly price all bonds but would be very smooth) — see worksheet on CD and Figure 5.6 which compares the output from the two approaches. We are now in a position to discuss inflation swaps. As mentioned earlier, there are two main types of inflation swap, fixed-floating inflation or inflation against Libor. The typical users of the former are pensions funds, who often offer inflation-linked annuities such as guaranteeing a growth in the pension of min{RPI growth, 5%}. These funds find hedging with indexed gilts is difficult due to the long reference lag and the large redemption payment on maturity of the bond. Other users in the UK include organizations who have access to inflation-linked governmental funding. Typical fixed-floating inflation swap Maturity: 17 years (usually between 10 and 50) Principal: £1 million (a medium size) To receive: fixed inflation growth rate of 3.22% pa quarterly cashflow = £1 million x (1+3.22%) T To pay: floating inflation quarterly cashflow = £1 million x { I T - i / I 0 - i } Need to the reference lag T. For a Sterling swap, this is usually set to 2 months as explained agree: above the daycount of T

Worksheet 5.24 demonstrates that this example swap is indeed a fair swap. The starting point is in column [3], which defines the cashflow dates. We can therefore work out the indexing months and forward inflation index given the 2 month reference lag, as shown in columns [1] and [2]. Next, the two sets of cashflows are constructed in columns [4] and [5]. Whilst these swaps are using a governmental curve as a reference, the cashflows carry the usual Ibor credit risk, and hence are discounted off the normal Ibor curve as given in column [7]. The overall swap has a net value of zero.

165

More Complex Swaps

Inflation—Libor swaps are less common, and are effectively a play on real interest rates, as we can see from the Fisher effect. This is approximately defined as (1 + i nominal ) = (1 + ireal) x (1 + inflation) or even more simply /real = / nominal - inflation: Typical inflation-Libor swap Maturity: 5 years (typically shorter than 30 years) Principal: £100 million To receive: 6mo. Libor sa, i.e. £100m x Libor x year fraction To pay: £100m x 6.12% x {/7-__T//0_T} sa x year fraction The price could be the fixed rate, a spread on Libor, or even an upfront fee is not uncommon. The only structure not seen is Libor x{/ r _ T // 0 _ T } as this would have significant correlation effects. In Worksheet 5.25, as before the first three columns are the same. The cashflows are then calculated using, on the Libor side the implied forward rates, and the indexation on the inflation side. Finally the net cashflows are discounted using the Libor curve to get a fair value of zero.

5.11 5.11.1

EQUITY AND COMMODITY SWAPS

Equity Swaps

Equity derivatives, traditionally written on the individual stock, have a long history. They have frequently been used in association with capital raising in the guise of: • convertible bonds, i.e. bonds that are convertible into equity, usually at the option of the bond-holder; • equity warrants, i.e. long-dated equity options; and so on. Modern equity derivatives are very often written on equity indices, for example a bond that pays coupon according to the rise in a particular index is targeted at a traditional investor who does not wish to risk capital in the equity market but is prepared to take a view on the index. For the more adventurous, there are structures where the principal redemption itself is linked to the level of the index. One typical issue is shown in the box below; the investor can buy either tranche, depending on their view of the likely direction of the index: Bull and bear issues An issuer, very often Scandinavian such as the Kingdom of Denmark, would issue two bonds simultaneously: Principal: Y10 billion Maturity: 5 years Coupon: 8% ann pa Y10bnx{l+[I 5 -I 0 (l+r>]// 0 } "Bull" bond redemption: "Bear" bond redemption: Y10bnx{l+[/ 5 -/ 0 (l-*)]//<,} where: I0 = Nikkei index on date of issue (22,720) 75 = Nikkei index at maturity X = anticipated growth in index (set to 14.7% in this issue) and subject to a minimum of Y6bn and a maximum of Y 11.054bn.

Pricing an inflation swap: Step 4. Pricing

Worksheet 5.24

Indexing Estimated dates index

[1]

Aug-99 Nov-99 Feb-00 May-00 Aug-00 Nov-00 Feb-01 May-01 Aug-01 Nov-01 Feb-02 May-02 Aug-02 Nov-02 Feb-03 May-03 Aug-03 Nov-03 Feb-04 May-04 Aug-04 Nov-04 Feb-05 May-05 Aug-05 Nov-05 Feb-06 May-06 Aug-06 Nov-06 Feb-07

165.50 166.39 167.28 168.18 169.08 169.99 171.44 173.17 174.92 176.69 178.47 180.27 182.09 183.86 185.62 187.40 189.19 191.00 192.82 195.69 198.61 201.56 204.56 207.60 210.69 213.83 215.28 216.74 218.21 219.69 221.18

YC

Index year fractions

Payment dates

0.2521 0.5041 0.7507 1.0027 1.2548 1.5068 1.7507 2.0027 2.2548 2.5068 2.7507 3.0027 3.2548 3.5068 3.7507 4.0027 4.2548 4.5068 4.7534 5.0055 5.2575 5.5096 5.7534 6.0055 6.2575 6.5096 6.7534 7.0055 7.2575 7.5096

06-Oct-99 06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 06-Jan-0l 06-Apr-0l 06-Jul-0l 06-Oct-0l 06-Jan-02 06-Apr-02 06-Jul-02 06-Oct-02 06-Jan-03 06-Apr-03 06-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05 06-Jul-05 06-Oct-05 06-Jan-06 06-Apr-06 05-Jul-06 06-Oct-06 06-Jan-07 06-Apr-07

curve length

[3]

[2]

quarterly Act/365 2 months 0.00

Frequency Daycount convention Indexing lag Swap value

06-Oct-99 1m GBP 17 years 3.22% pa

Today's date Principal Maturity Fixed rate

0.252 0.501 0.751 1.003 1.255 1.501 1.751 2.003 2.255 2.501 2.751 3.003 3.255 3.501 3.751 4.003 4.255 4.504 4.753 5.005 5.258 5.504 5.753 6.005 6.258 6.504 6.753 7.005 7.258 7.504

Variable cashflow [4]

Fixed cashflow [6]

Net

Libor

cashflow

DF

,005,369.33 ,010,767.49 ,016,194.63 ,021,650.92 ,027,136.50 ,035,886.58 ,046,346.35 ,056,911.73 ,067,583.80 ,078,363.63 ,089,252.30 ,100,250.92 ,110,966.46 ,121,587.42 ,132,309.92 ,143,134.92 ,154,063.42 ,165,096.39 ,182,434.60 ,200,030.84 ,217,888.93 ,236,012.77 ,254,406.32 ,273,073.59 ,292,018.64 ,300,782.46 ,309,605.71 ,318,488.81 ,327,432.16 ,336,436.18

1,008,031.22 ,016,126.93 ,024,109.57 ,032,334.42 ,040,625.32 ,048,982.81 ,057,131.63 ,065,621.69 ,074,179.93 ,082,806.90 ,091,218.49 ,099,982.30 ,108,816.49 ,117,721.64 ,126,404.46 ,135,450.85 ,144,569.91 ,153,762.20 ,162,826.08 ,172,164.99 ,181,578.91 ,191,068.42 ,200,321.02 ,209,961.06 ,219,678.52 ,229,474.02 ,239,024.96 ,248,975.84 ,259,006.64 ,269,117.99

2,661.89 5,359.45 7,914.94 10,683.50 13,488.82 13,096.22 10.785.28 8,709.95 6,596.13 4,443.27 1,966.19 -268.62 -2,149.96 -3,865.78 -5,905.46 -7,684.07 -9,493.51 -11,334.19 -19,608.52 -27,865.84 -36,310.02 -44,944.35 -54,085.30 -63,112.53 -72,340.13 -71,308.44 -70,580.75 -69,512.97 -68,425.53 -67,318.19

[7] 0.98494917 0.97020604 0.9538296 0.93881015 0.92293787 0.90724029 0.89120655 0.87488096 0.85925771 0.84401346 0.8286364 0.81316721 0.79890475 0.78515202 0.77143355 0.75777019 0.74505083 0.73272578 0.72064131 0.70867074 0.69725358 0.68634118 0.67555008 0.66487406 0.65431272 0.64420763 0.63420851 0.62430546 0.61475327 0.60562296

Discounted cashflows

[8] 2,621.82 5,199.77 7,549.50 10,029.78 12,449.34 11,881.42 9,611.92 7,620.17 5,667.77 3,750.18 1,629.25 -218.44 -1,717.62 -3,035.22 -4,555.67 -5.822.76 -7,073.15 -8,304.85 -14,130.71 -19,747.71 -25,317.29 -30,847.15 -36,537.33 -41,961.88 -47,333.07 -45,937.44 -44,762.91 -43,397.33 -42,064.82 -40,769.44

May-07 Aug-07 Nov-07 Feb-08 May-08 Aug-08 Nov-08 Feb-09 May-09 Aug-09 Nov-09 Feb-10 May- 10 Aug-10 Nov-10 Feb-11 May-11 Aug-11 Nov-1 1 Feb-12 May- 12 Aug-12 Nov-12 Feb-13 May-13 Aug-13 Nov-13 Feb-14 May-14 Aug-14 Nov-1 4 Feb-15 May-15 Aug-15 Nov-15 Feb-16 May-16 Aug-16

222.68 224.19 225.71 227.24 228.78 230.34 231.47 232.39 233.32 234.25 235.19 236.13 237.07 238.02 238.97 239.71 240.36 241.00 241.65 242.30 242.95 243.60 244.26 244.84 245.39 245.94 246.49 247.04 247.59 248.14 248.70 249.25 249.81 250.37 250.93 251.49 252.05 252.61

7.7534 8.0055 8.2575 8.5096 8.7562 9.0082 9.2603 9.5123 9.7562 10.0082 10.2603 10.5123 10.7562 11.0082 11.2603 11.5123 11.7562 12.0082 12.2603 12.5123 12.7589 13.0110 13.2630 13.5151 13.7589 14.0110 14.2630 14.5151 14.7589 15.0110 15.2630 15.5151 15,7589 16.0110 16,2630 16.5151 16.7616 17.0137

06-Jul-07 06-Oct-07 06-Jan-08 06-Apr-08 06-M-08 06-Oct-08 06-Jan-09 06-Apr-09 06-Jul-09 06-Oct-09 06-Jan-10 06- Apr-10 06-Jul-10 06-Oct-10 06- Jan- 11 06- Apr-11 06-Jul-ll 06-Oct-ll 06-Jan-12 06-Apr-12 06-Jul-12 06-Oct-12 06-Jan-13 06-Apr-13 06-Jul-13 06-Oct-13 06-Jan-14 06- Apr- 14 06-Jul-14 06-Oct-14 06-Jan-15 06- Apr- 15 06-Jul-15 06-Oct-15 06-Jan-16 06- Apr-16 06-Jul-16 06-Oct-16

7.753 8.005 8.258 8.507 8.756 9.008 9.260 9.507 9.756 10.008 10.260 10.507 10.756 11.008 11.260 11.507 11.756 12.008 12.260 12.510 12.759 13.011 13.263 13.510 13.759 14.011 14.263 14.510 14,759 15.011 15.263 15.510 15.759 16.011 16.263 16.512 16.762 17.014

-1,345,501.27 -1,354,627.85 -1,363,816.34 -1,373,067.15 -1,382,380.72 -1,391,757.45 -1,398,608.06 -1,404,192.94 -1,409,800.14 -1,415,429.72 -1,421,081.78 -1,426,756.41 -1,432,453.70 -1,438,173.74 -1,443,916.62 -1,448,423.89 -1,452,314.41 -1,456,215.38 -1,460,126.83 -1,464,048.79 -1,467,981.27 -1,471,924.33 -1,475,877.97 -1,479,397.82 -1,482,703.37 -1,486,016.31 -1,489,336.65 -1 ,492,664 .41 -1,495,999.61 -1,499,342.26 -1,502,692.38 -1,506,049.98 -1,509,415.09 -1,512,787.71 -1,516,167.87 -1,519,555.58 -1,522,950.87 -1,526.353.74

1,278,976.90 1,289,248.64 1,299,602.88 1,310,040.27 1,320,331.87 1,330,935.74 1,341,624.78 1,352,399.66 1,362,905.52 1,373,851.31 1,384,885.01 1,396,007.33 1,406,851.95 1,418,150.68 1,429,540.16 1,441,021.11 1,452,215.41 1,463,878.47 1,475,635.20 1,487,486.34 1,499,171.95 1,511,212.13 1,523,349.00 1,535,583.35 1,547,512.24 1,559,940.65 1,572,468.87 1,585,097.71 1,597,411.25 1,610,240.40 1,623,172.60 1,636,208.65 1,648,919.23 1,662,162.06 1,675,511.24 1,688,967.64 1.702,236.07 1,715,907.10

-66,524.37 -65,379.21 -64,213.46 -63,026.88 -62,048.84 -60,821.71 -56,983.28 -51,793.29 -46,894.61 -41,578.40 -36,196.76 -30,749.08 -25,601.75 -20,023.06 -14,376.46 -7,402.79 -99.00 7,633.09 15,508.37 23,437.56 31.190.68 39,287.80 47,471.03 56,185.53 64,808.87 73,924.34 83,132.22 92,433.30 101,411,64 110,898.15 120,480.22 130,158.67 139,504.15 149,374,35 159,343.37 169,412.06 179.285.20 189,553.36

0.59659726 0.58766137 0.57893006 0.57048188 0.56222342 0.55405563 0.54607026 0.53843877 0.5308971 0.52342527 0.51567771 0.50825103 0.50089527 0.4939311 0.48644091 0.47958547 0.47279571 0.46605377 0.459759 0.45366081 0.44769677 0.44179109 0.43601019 0.43048283 0.4250178 0.41959314 0.41428944 0.40921968 0.4042079 0.39923146 0.39411134 0.38920808 0.38435554 0.37953208 0.37481758 0.37024301 0.36576596 0.36132572

-39,688.26 -38,420.84 -37.175.10 -35,955.69 -34,885.31 -33,698.61 -31,116.87 -27,887.51 -24,896,21 -21,763.19 -18,665.86 -15,628.25 -12,823.79 -9,883.24 -6,993.30 -3,550.27 -46.81 3,571.41 7,130.11 10,632.70 13,963.96 17,357.00 20,697.85 24,186.91 27,544.92 31,018.15 34,440.80 37,825.52 40,991.38 44,274.03 47,482.62 50,658.80 53,619.19 56,692.36 59,724.70 62,723.63 65,576.42 68,490.51

Worksheet 5.25

Pricing an inflation-fixed vs. Libor swap

Today's date:

06-Oct-99

Principal Maturity

100,000,000 GBP 5 years

Fixed rate Frequency Daycount convention Indexing lag Swap value

6.12% pa index-linked Semi annual act/365 2 months 0.00

YC Indexing dates

[1] Aug-99 Feb-00 Aug-00 Feb-01 Aug-01 Feb-02 Aug-02 Feb-03 Aug-03 Feb-04 Aug-04

Estimated index

[2] 165.50 167.28 169.08 171.44 174.92 178.47 182.09 185.62 189.19 192.82 198.61

Payment dates

curve length

[3] 06-Oct-99 06-Apr-OO 06-Oct-OO 06-Apr-0l 06-Oct-0l 06-Apr-02 06-Oct-02 06-Apr-03 06-Oct-03 06-Apr-04 06-Oct-04

Forward rate

Libor

[4]

[5]

DF

Variable cashflow

Effective fixed rate

Fixed cashflow

[6]

[7]

[8]

Net cashflow

Discounted cashflows

[9]

1 0.501 0.501 0.499 0.501 0.499 0.501 0.499 0.501 0.501 0.501

6.033% 6.297% 6.484% 6.674% 6.780% 6.888% 6.908% 6.930% 6.904% 6.880%

0.97064119 0.94093493 0.91146642 0.88195397 0.85311368 0.82463678 0.79717734 0.77041089 0.74463404 0.71980579

3,024,682.19 3,157,099.93 3,233,088.28 3,346,257.67 3,380,591.86 3,453,265.48 3,444,583.80 3,474,308.45 3,461,680.45 3,449,297.17

6.1822% 6 .2488% 6.3359% 6.4645% 6 .5957% 6.7295% 6.8600% 6 .9918% 7 .1262% 7 .3398%

-3,099,583.73 -3,132,958.46 -3,159,254.51 -3,241,088.03 -3,288,801.30 -3,373,990.45 -3,420,625.55 -3,505,497.00 -3,572,843.25 -3,679,972.00 Net PV =

-74,901.53 -72,702.51 24,141.47 22,715.55 73,833.77 67,297.00 105,169.64 92,754.78 91,790.56 78,307.78 79,275.03 65,373.11 23,958.26 19,098.98 -31,188.55 -24,028.00 -111,162.80 -82,775.61 -230,674.83 -166,041.08

0.00

More Complex Swaps

The coupon is above the current market level, as compensation for the equity risk. The factor X can be used to ensure that there is sufficient demand for both tranches. From the issuer's point of view, the total redemption amount to be paid for both bonds is simply 2 x (1 - X). In this case, X = 14.7% of the redemption amount is the compensation for paying the higher coupon. There seems to be a general perception that equity derivatives are widespread, possibly because of the raised profile they receive through embedded structures. In reality, they typically constitute less than 2% of the derivative market 15 . Equity swaps are a relatively new invention, and currently represent only 20% of the total equity OTC derivative market. But it is the fastest growing part because they can be an efficient means of moving investments between the equity and interest markets. For example, suppose an investor is currently receiving USD Libor on some assets, but believes that the stock market is about to rise. He could liquidate his investment and reinvest in stocks, but that is likely to incur considerable transaction costs, or he could enter into an equity swap: to pay USD Libor on a notional principal, and to receive the return on an equity index applied to a notional principal. An equity swap is really a form of basis swap, as we shall see in the discussion of the following example: Trade date: Notional principal: Maturity: To receive: To pay:

4 January 2000 $100 million 2 years USD 3mo. Libor S&P 500 Index quarterly + 10 bp pa.

Some typical cashflows are shown in Worksheet 5.2616. Hypothetical future values of the index and Libor are shown in columns [1] and [2] respectively. Therefore the return on the index over each period can be calculated as r, — (I, — /,_])//,_]: for example, the return over the first period is: (1,276.44- 1,209.62)/1,209.62 = 5.52% Note that this return may be either positive or negative, depending on the movement in the index. The index-related cashflow to be paid at the end of the first quarter is: $ 100 million x {5.52% + 10 bp x 0.253/10,000} = $5,523,707 + 25,278 = $5,548,985 as shown in columns [4]-[6]. Note that the return does not use a daycount fraction, whereas the margin is quoted on a per annum basis as usual and therefore uses the fraction. The Libor cashflows are calculated in the usual fashion, and finally a net cashflow is settled, as shown in column [8]. Such a swap could be priced by estimating future values of the index using a cost-ofcarry argument: 15

See Global OTC derivative report, BIS, November 2000. The spreadsheet contains two versions of each worksheet. The first randomly generates the index, interest and FX rates to demonstrate that the hedges will work successfully under any movements. However, for discussion purposes, this randomness has been removed from the second "fixed" versions. 16

Worksheet 5.26

Today's date: Maturity

Example of fixed notional equity swap

4-Jan-OO

2 years

Principal amount (DEM):

100m margin

Receive side: Pay side:

USD Libor S&P 500 Index

Frequency of pay side Frequency of receive side

Daycount Act/360

+0 bp +10 bp Qu Qu

Observed S&P Index

Act/360 Act/360

Observed Libor

Principal amount

Return on S&P Index

Cashflow on index

Cashflow from margin on index

Cashflow on Libor including margin

Net cashflow

[5]

[6]

[7]

[8]

per qu

06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253

[1]

[2]

1209.62 1276.44 1296.19 1276.28 1263.66 1321.16 1301.36 1343.01 1364.72

6.03% 6.31% 6.63% 6.75% 6.84% 6.96% 7.09% 7.22%

[3] 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000

[4] 5.52% 1.55% -1.54% -0.99% 4.55% -1.50% 3.20% 1.62%

-5,523,707 -1,547,231 1,536,027 988,197 -4,549,872 1,498,867 -3,200,945 -1,616,404

-25,278 -25,278 -25,556 -26,111 -24,444 -25,278 -26, 111 -25,278

,524,566 ,595,040 ,694,758 ,762,113 ,672,715 ,760,186 ,851,886 ,825,510

-4,024,419 22,532 3,205,229 2,724,199 -2,901,601 3,233,776 -1,375,171 183,828

C/J

P 3 O.

3

More Complex Swaps

171

It — IQ + expected growth — expected dividends and after adjustments for share issues, splits, etc. However a rather simpler approach is to consider how such a swap might be hedged. At the beginning of the swap, we borrow $100 million at 3mo. Libor flat. These interest payments will exactly match the Libor receipts on the swap. We will then buy $100 million of the components of the index. At the end of the first quarter, the index has risen by 5.52%, i.e. the holding is now worth $105,523,707. The increased amount of $5,523,707 is sold off, thereby generating a cashflow to meet the payment on the swap and simultaneously rebalancing the principal investment in the index back to $100 million. This process is then repeated each quarter. At the end, the investment in the index is totally liquidated, and the Libor borrowing repaid. See Worksheet 5.27. An alternative would be to use Eurodollar and S&P 500 Index futures to hedge the swap, in a very similar fashion to the method employed in money market swaps. Unfortunately there is only good liquidity in (at most) the three nearest index futures contracts, so using these futures to price and hedge a longer swap would involve quite a lot of basis risk, potential roll-over costs, as well as the funding of the initial and daily variation margins which will create convexity issues. So, in theory as with basis swaps, a Libor-equity swap should be priced very close to flat. The existence of the margin will reflect supply and demand, the bank's efficiency replicating the index, various costs of running the portfolios and of course the required return on capital employed. The above swap is called a "fixed notional" swap; each cashflow is estimated using the constant notional principal of $100 million. In practice, such hedging is not very efficient because it involves selling or buying potentially small amounts of the index each quarter. A more common structure is a "variable notional" which avoids this. See Worksheet 5.28. At the end of the first quarter the index has risen by 5.52%. Instead of liquidating part of the index investment, we borrow the money required to pay under the swap, namely $5,523,707: columns [8] and [9] show the periodic borrowings or repayments and the total effect. To match the future Libor interest payments, the notional principal of the swap is also increased as shown in column [3]. The swap cashflows are therefore calculated based upon the notional principal at the beginning of each period. At the end of the swap, the index investment is liquidated and used fully to repay the accumulated Libor borrowing, as shown in columns [8] and [11]. In summary, equity swaps are used to simulate the exposure to an index but without having to actually make the physical investment. The variable notional versions in particular mimic the cashflows that would have occurred. 5.11.2

Commodity Swaps

These are similar in structure to equity swaps with two main types: • fixed-for-floating commodity price swap, and • floating price against floating interest swap with other types of swaps being generated from these basic ones. Fixed-for-floating are popular "natural" multiperiod structures used by commodity producers and users to lock in common prices. Price-interest swaps would on the face of it also appear to have a

172

Swaps and Other Derivatives

Worksheet 5.27

Hedging a fixed notional equity swap

06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02

Libor borrowing

S&P cashflow

[9] 100,000,000 -1,524,566 -1,595,040 -1,694,758 -1,762,113 -1,672,715 -1,760,186 -1,851,886 -101,825,510

[10] -100,000,000 5,523,707 1,547,231 -1,536,027 -988,197 4,549,872 -1,498,867 3,200,945 101,616,404

S&P hedge before rebalancing

S&P hedge after rebalancing

[11] 100,000,000 105,523,707 101,547,231 98,463,973 99,011,803 104,549,872 98,501,133 103,200,945 101,616,404

[12] 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 0

173

More Complex Swaps Worksheet 5.28 Example of variable notional equity swap

Today's date:

04-Jan-00

Maturity 2 years Principal amount (USD): 100m margin Receive side: Pay side:

USD Libor S&P 500 Index

+ 0 bp + 10 bp

Qu Qu

Frequency of pay side Frequency of receive side

Act/360 Act/360

Observed Observed Daycount DAX Libor Act/360 Index 06-Jan-OO 06-Apr-00 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253

[1] 1209.62 1276.44 1296.19 1276.28 1263.66 1321.16 1301.36 1343.01 1364.72

Total Libor borrowing 06-Jan-OO 06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-Ol 07-Jan-02

[8] 100,000,000 105,523,707 107,156,402 105,510,451 104,467,799 109,220,951 107,583,874 111,027,575 -112,822,229

[2] 6.03% 6.31% 6.63% 6.75% 6.84% 6.96% 7.09% 7.22%

Principal amount [3] 100,000,000 105,523,707 107,156,402 105,510,451 104,467,799 109,220,951 107,583,874 111,027,575 112,822,229

Net Libor borrowing [9] 100,000,000 5,523,707 1,632,695 -1,645,952 -1,042,651 4,753,151 -1,637,077 3,443,701 1,794,654

Return on S&P Index

Cashflow on index including margin

Cashflow on Libor including margin

Net cashflow

[4]

[5]

[6]

[7]

5.52% 1.55% -1.54% -0.99% 4.55% -1.50% 3.20% 1.62%

-5,548,985 -1,659,369 1,618,567 1,015,102 -4,778,688 1,609,468 -3,471,792 -1,822,720

Libor interest

[10] -1,524,566 -1,683,146 -1,816,041 -1,859,213 -1,747,449 -1,922,492 -1,992,330 -2,026,820

1,524,566 -4,024,419 1,683,146 23,776 1,816,041 3,434,608 1,859,213 2,874,315 1,747,449 -3,031,239 1,922,492 3,531,961 1,992,330 -1,479,462 2,026,820 204,100

S&P cashflow

[11] -100,000,000

112,822,229

S&P principal

[12] 100,000,000 105.523,707 107,156,402 105,510.451 104,467.799 109,220,951 107,583,874 111,027.575 112,822.229

174

Swaps and Other Derivatives

natural purpose. For example, commodity producers are naturally long the commodity and short interest rates, and should therefore be prepared to pay the price and to receive Libor as a hedge against rising borrowing costs. But in practice this link between interest payments and production costs is not so simple, and hence such swaps are less popular. Overall, OTC commodity derivatives constitute about 0.5% of the total market17 and are not growing particularly rapidly: Typical fixed-floating commodity swap Trade date: 4 January 2000 Notional quantity: 100,000 bbl Commodity: WTI Light Sweet Crude Maturity: 6 July 2000 Frequency: Monthly Period end date: 6th day of each month Settlement date: 5 days after period end To pay: $20 per bbl To receive: Arithmetic daily average of reference price over each period Settlement method: Net cash payment

Commodity swaps are usually priced off commodity futures or by estimating the forward price of the commodity. The former can involve considerable basis risk as the range of reference commodities under futures contracts is considerably restricted compared to the possible references for OTC contracts, although frequently the dates are matched to reduce the basis risk. Forward pricing is in theory relatively simple to calculate as a commodity may be purchased today (spot) and held until fixing date18: current price + cost of funding position + cost of physical storage This suggests that forward price > spot price, i.e. said to be in "contango". Unfortunately supply and demand, especially for seasonal commodities, can distort this relationship quite considerably, and it is feasible to observe "backwardation", i.e. when forward price < spot price. Some producers, wishing to entice consumers to commit to forward purchases and hence provide a guaranteed demand, will offer a "convenience yield" or discount on the theoretical forward price which also needs to be taken into account. Commodity prices do exhibit high volatility, and simplistic forward pricing carries a very real risk. There is a range of practical commodity swaps. Swaps involving oil prices are probably the most common, but it is feasible to get OTC swaps against most commodities. Currently an increasing market is weather derivatives, trading such structures as fixedfloating temperature or rainfall. Another one that fluctuates in popularity is property swaps; for example, trading the percentage change in a property index over a period against Libor. One might also include structures described later in this chapter, such as inflation and volatility swaps. Other forms of structured swaps include: • basis swaps with reference to the spread between two linked commodities such as Brent and WTI crude oils; 17 18

BIS (2000). Except for commodities which cannot be stored, such as electricity.

More Complex Swaps

• spread swaps with reference to either side of production such as WTI crude oil against refined products like jet fuel or kerosene; • curvelock swaps, which lock in the backwardation/contango spread (very similar to spreadlock swaps described above).

5.12

VOLATILITY SWAPS 19

A swap is effectively an exchange of cashflows between two counterparties. There are a wide range of different ways in which these cashflows may be calculated, as we have already seen. One class of swaps are the volatility and variance swaps, whereby the cashflows are calculated with reference to the volatility or variance of some market entity. For example: • if a is the annualized volatility (standard deviation of returns) of some dynamic measure such as share price, FX rate, etc. then a single period volatility swap, also known as a "realized volatility forward contract" has a payoff at expiry of: N x {aT-

Fvo[}

where: <jr is the realized volatility over the lifetime of the swap, N is the notional amount of the swap ($ per annualized volatility point), Fvol is the annualized fixed volatility (volatility delivery price). Consider a 1 year volatility receiver's forward on IBM's stock price for $250,000 per point and a fixed rate of 30%. If the observed volatility over the year was 25%: payout = $250,000 x (30% -25%) = $1,250,000 Option traders are often described as "trading volatility", but in practice this is not entirely true as the price of an option depends upon many other factors as well. On the other hand, a volatility swap is a pure play on volatility! But apart from the directional trading of volatility levels, these swaps may also be used to trade volatility spreads. They may also be used as a hedge against a volatility exposure, possibly arising from an option portfolio. Dynamic delta hedging of option portfolios20 is common but is subject to a tracking error as the hedge is usually changed after the market (and therefore the portfolio delta) has moved. These tracking strategies are more active in periods of high volatility, hence incurring greater costs with less accuracy. Other exposures to volatility are less direct. For example, market relationships such as spreads are notoriously less stable during periods of high volatility; therefore running a spread strategy has an indirect exposure to volatility. Equity prices are generally negatively correlated to volatility, i.e. as equity drops, volatility rises which can act as a diversification strategy. Obviously it is critically important to specify the calculation of the volatility quite precisely. Some of the relevant factors are listed below: 19 Much of this section is drawn from "A Guide to Volatility and Variance Swaps", Derneterfi et al.. Journal of Derivatives, Summer 1999, pp. 9-32. The current volume of volatility swaps is extremely small, but this section was included to indicate how the swap markets are continually evolving. 20 See Chapter 8,

176_

Swaps and Other Derivatives

• • • •

source and observation frequency, e.g. daily close of S&P 500; if OTC, how is it defined? derivation of return, e.g. simple or compounded? calculation of standard deviation, assumption about mean return? usual assumption: zero mean (less argument and also permits easier risk management); • conversion factor (or formula) from observed standard deviation to annualized, e.g. no. of days in year? Variance swaps are closely related to volatility swaps, but defined by: payoff = N x {(
or v0T = /T x v0r +(1 - /.T) x vtT

where /.T = t/T. We know V 0T , hence the mark-to-market value of the swap is: N x [ T x {V0t - Fvar} + (1 - . T ) x {vrT - Fvar}] x DF r T These swaps are similar to average rate swaps, so that delta ->0 as T-* T. The existing swap could be hedged by using (1 — /.T) of a new off-setting swap. Unfortunately, none of these simple results apply to volatility swaps. In this section (which is considerably more complex than most of the book), we will first discuss the pricing and hedging of variance swaps, and then briefly discuss how volatility swaps might be approached. As usual, a fair price for the swap when it first starts would have a risk-neutral expectation of zero, i.e.:

We therefore need to estimate oT. This could either be historically (unconditionally or conditionally) measured, or we could use an implied volatility from the options market. If we used the latter, we would also want to use the option for hedging. Consider a normal call option with maturity T. If we define variance v = o 2 T, we can easily calculate the variance sensitivity, i.e.: aC/av = 1/2T/(2IIv)1/2S exp{-/12,(d1)2} It would be feasible to delta hedge the variance sensitivity of the swap with a single option. Unfortunately the hedge would need frequent rebalancing as the variance sensitivity of the option is also a function of the underlying, as shown in Figure 5.7, which would not affect the swap. So a simple hedge such as this is unlikely to be very efficient as it brings along other exposures as well.

177

More Complex Swaps

100

200

150 Current spot level

Figure 5.7

250

Variance sensitivity

Suppose we create a weighted portfolio of options with strikes K 1 ,K 2 , ... as follows:

n-

x

x dK

where &K is the steps between the strikes and f(•) is the amount of the ith option. The portfolio variance sensitivity is:

Vn = Eif(Ki) x {aC i /av} x &K 2

If f(K i ) l/(K i ) then we find that Vn becomes increasingly independent of S as the number of options in the portfolio increases21. We could at least in theory create a robust delta hedge. If all the options were priced off the same volatility, then that would be our best estimate for the swap volatility. But unfortunately this above result ignores any smile effects, so we need to go back to basics. Assume a stock price evolves as: r, = dS,/St = [udt + odz, where the drift u and continuously sampled volatility a are functions of time (and other parameters). Using Ito's lemma: d ln(St,) = (u- 1/2o 2 )dt + o dz, Therefore, by subtraction, we get: Substituting for {aC i /av} we get: Vn =

Define xi, = K i /S, we can write: Vn =1/2T/(2IIv)1/2

where di is now a function of xi only. We wish: aV n /aS = 0 => aVn/aKi,. aK i /aS = av n /aK i , xi, = 0 Differentiating the term [f(K i ) x (K i /x i ) 2 ] with respect to Ki, we get: a f / a K i x (K i /X i ) 2 The expression [•] is zero if/(K i )

1/(K i ) 2 .

+

2fK i ,(1/xi) 2 = [af/aK i , x K i , + 2f]K i (l/x i )2

178

Swaps and Other Derivatives dS t ,/S t -d In(S t ) = 0.5x o 2 d t

If we integrate from 0 to T, we get: v0T = (1/T) [o2 dt = (2/T)] [dSf,/St, - ln(Sr/S0)| Hence the fair fixed rate on the swap is: Fvar - E{v 0 T ] = (2/T)

E{dS i /S t }

-

E{ln(Sr/S0)}|

(5.6)

where E{ • } is the risk-neutral expectation. But dSt,/St, is the return over a time interval dt, hence the expected risk-neutral return over T is simply {rT}. The other term is more difficult. There is an identity that enables us to break a log-contract up into a portfolio of a simple forward contract plus (an infinite number of) put and call options, i.e.: (ST — S*)/S*

forward contract

s

ln(ST/S*) =

* 1 —rmax{K—ST,0}dK J0 K2

put options

— 1/K22 max{ST — K,0}dK Js* K

call options

—

for any given value S*. If we set S* = E{ST} = S0 erTrl (i.e. ATM forward), then: E{ln(ST/S0)} = l n ( S / S 0 )

Js*

or:

Notice that each integral is effectively a portfolio of increasingly OTM puts and calls which will rapidly decline in value as they move further OTM. Unfortunately, this can't work in practice either as options are only traded at finite steps and it is highly unlikely that there is an option traded with a strike S* exactly equal to E{ST], so we need to approximate. We can rewrite Equation (5.6) as F var - (2/T)[rT- (S0erT- S*)/S* - ln(S*/S0)] +f(S T ) where: J(ST) = (2/T)E({ST

- S*)/S* - ln(ST/S*)}

The function f[S T ) may be replicated at time Tby a portfolio of calls and puts expiring at time T:

More Complex Swaps

where it is assumed that: put strikes: call strikes:

S* — S* — .

> K1p > K2P > < K ... 2C < < < A1C1C<
Therefore: rT

For example, we want to estimate the expected variance for use in a 1 year swap. We can observe a strip of 1 year options, trading on a spot of 100, as shown in Table 5.7 below. We will initially assume that all these options are priced using a constant option volatility of 20% pa, together with a risk free rate of 10% pa. Therefore E{ST} is calculated to be 110.52, and we select S* = 111, and will use strike steps of 5. The shape of the function f(ST) is given in Figure 5.8. The ws can be estimated sequentially due to the asymmetric nature of the option payoff function, i.e.:

w 1C

w0c- [f(K 1C ) - . f ( K 0 ) ] / [ K 1 C - K0] w lC + w 0C = [f(K2c) - f(K l C )]/[K 2 c - K lc ] +w 1C+ w oc = [ f ( K 3 C ) - f ( K 2 C ) ] / [ K 3 C - K2C], etc.

For example: • for 111 < ST < 116, only the first call option is in-the-money, with a value of C0 = 5 when ST = 116: • f(111) = 0 by construction, and f(116) = 0.00197 • therefore f(116) = w 0 C x C 0 • which gives w0C = 0.00197/5 = 0.00039 • for 116 < ST < 121, the first two call options are in-the-money, with values C0 = 10 and C1 = 5 respectively when ST= 121: • f(121) = 0.00766 = w0C x C0 + wlC x C, = 0.00039 x 10 + w1C x 5 • this can also be written as:

46

66

86

106

126

146

Figure 5.8 Shape of function f(ST)

166

186

180

Swaps and Other Derivatives

f(121) -f(116) = woc x C0(121) + u- lc x C, - M-^ x C 0 (l 16) = (WQC + u- IC ) x C, • which gives vv lc = 0.00074, as shown in Table 5.7. The approximate value of E(f[ST)} is estimated to be: erT x (0.01661 + 0.01994) = 0.0403915 This gives: Fvar = -0.0000190 + 0.0403915 = 0.0403725 = 4.04% and volatility is 20.09%. This is slightly higher than the input volatility due to the linear interpolation of the log function over the finite step size. See Worksheet 5.29. Now suppose however that the strip was subject to a smile as shown in Figure 5.9. In this case, Fvar increases as expected to 6.46%, and therefore the "average" volatility to 25.42% as shown in Worksheet 5.30. In summary, this section has shown how "correct" variance and volatility may be implied out of the options market, and how a static replicating hedge for the swap may be constructed so that: • changes in volatility do not force the hedge to change; • hedge is model-independent; • hedge does not protect against jumps in the stock price. We have concentrated on variance swaps. Volatility swaps are much more difficult because there is no simple, i.e. linear, relationship between an observation and volatility. This Table 5.7 Call strikes

111 116 121 126 131 136 141 146 151 156 161 166 171 176 181 186 191 196 Total weighted cost

Call value

Weights

Put strikes

Put value

Weights

7.77 5.92 4.44 3.28 2.39 1.71 1.22 0.85 0.59 0.41 0.28 0.19 0.13 0.09 0.06 0.04 0.02 0.02

0.0004 0.0007 0.0007 0.0006 0.0006 0.0005 0.0005 0.0005 0.0004 0.0004 0.0004 0.0004 0.0003 0.0003 0.0003 0.0003 0.0003

111 106 101 96 91 86 81 76 71 66 61 56 51 46 41 36 31 26

8.20 5.93 4.07 2.64 1.59 0.88 0.44 0.20 0.08 0.02 0.01 0.00 0.00 0.00 0.00 0.00 0.00 0.00

0.0004 0.0009 0.0010 0.0011 0.0012 0.0014 0.0015 0.0017 0.0020 0.0023 0.0027 0.0032 0.0039 0.0048 0.0060 0.0078 0.0105

0.01661

0.01994

Worksheet 5.29 Spreadsheet to demonstrate variance hedging with no smile effect Summary Value of call portfolio Value of put portfolio Total value

maturity = interest rate = current stock price = E{ST} = S* step length

Call option strikes vol = dl d2 Cost of call f(ST) = gradient of f(ST) = Weights = Weighted cost = Put option strikes dl d2 Cost of put

f(ST) = gradient of f(ST) = Weights = Weighted cost =

0.0166 0.0199 0.0365

Fvar Volatility

1 10% 100

4.04% 20.09%

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

116 20.000% -0.14 -0.34 5.92

121 20.000% -0.35 -0.55 4.44

126 20.000% -0.56 -0.76 3.28

131 20.000% -0.75 -0.95 2.39

136 20.000% -0.94 -1.14 1.71

141 20.000% -1.12 -1.32 1.22

146 20.000% -1.29 -1.49 0.85

110.5170918 111 5 111 20.000% 0.08 -0.12 7.77

0.00766 0.01677 0.00000 0.00197 0.00039 0.00114 0.00182 0.00245 0.00063 0.00039 0.00074 0.00068 0.003060131 0.004403467 0.003034671 0.002066634 96 111 101 106 20.000% 20.000% 20.000% 20.000% 0.80 0.08 0.31 0.55 0.60 -0.12 0.11 0.35 2.64 4.07 8.20 5.93 0.00000 0.00042 0.00042 0.00343

0.00209 0.00131 0.00089 0.00528

0.00864 0.00229 0.00098 0.00400

0.02009 0.00338 0.00109 0.00287

0.08248 0.06208 0.04420 0.02903 0.00455 0.00408 0.00358 0.00303 0.00047 0.00050 0.00054 0.00058 0.00139185 0.000927811 0.000612675 0.000401 1 1 1 76 81 91 86 20.000% 20.000% 20.000% 20.000% 1.65 1.97 1.35 1.07 1.77 1.45 0.87 1.15 0.44 0.20 1.59 0.88 0.12696 0.05992 0.08962 0.03698 0.00920 0.00747 0.00459 0.00594 0.00174 0.00135 0.00153 0.00121 0.00034 0.00067 0.00119 0.00192

182

Swaps and Other Derivatives 80 70 60. 50. 40. 30' 2010-

0100

50

200

150

250

Strike Figure 5.9

Volatility smile

means that there is no simple replication strategy, and the swap would have to be dynamically hedged. Define f(x) = Jt1/2. We can therefore write, using a Taylor's expansion: ./(v)

(v - v0)f' +1/2(v -

v0)2f

This gives: {(v + v0)/2v01/2} - {(v - v0)2/8v03/2 If we expand f(•) around v0 = £{v}, we get: 3/2

E{v 1/2 }=E{v} 1/2 -Var{v}/8vo 3/2 Thus the expected volatility is less than the square root of the expected variance as calculated above, i.e. there is a convexity adjustment. To estimate the size of the adjustment, we need a more complex model of the volatility process, and there can be no simple replicating hedge22. Exchange and spread options, i.e. options which involve two or more assets, possess correlation effects. Namely, the value of the option depends upon the behaviour of the net portfolio of assets, which obviously depends on the correlations between the assets. Similarly correlation effects can arise in FX option portfolios, where the options are being traded on, for example, $-Yen, $-€ and Yen-=€. Movements in two of the currencies will be reflected in the third, and hence its volatility must reflect both the volatilities of the other two plus their correlation. Such correlation risks would not be controlled by straightforward "greek" hedging. In a very similar fashion to above, it is feasible to design covariance and correlation swaps that would enable these risks to be managed. As before, covariance swaps are relatively straightforward, correlation swaps are less so23. 22

See Brockhaus et al., "Volatility swaps made simple". Risk, January 2000. pp. 92-95 for some results using some models. 23 See Brockhaus et al. (2000).

]•a* o

nb

Worksheet 5.30

3 *SL ><

Spreadsheet to demonstrate variance hedging with a smile effect

Summary Value of call portfolio Value of put portfolio Total value

maturity = interest rate = current stock price = E{ST} S* step length

Call option strikes vol = dl d2 Cost of call f(ST) = gradient of f(ST) = Weights = Weighted cost = Put option strikes vol = dl d2 Cost of put f(ST) = gradient of f(ST) = Weights = Weighted cost =

C/l

0.0283 0.0302 0.0585

Fvar Volatility

6.46% 25.42%

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

1 10% 100

116 20.161% -0.14 -0.34 5.98

121 20.560% -0.34 -0.54 4.65

126 21.210% -0.51 -0.72 3.70

131 22.109% -0.66 -0.88 3.04

136 23.258% -0.78 -1.01 2.62

141 24.657% -0.86 -1.11 2.36

146 26.306% -0.93 -1.19 2.24

0.04420 0.06208 0.01677 0.02903 0.00000 0.00197 0.00766 0.00358 0.00408 0.00114 0.00182 0.00245 0.00303 0.00039 0.00054 0.00050 0.00074 0.00068 0.00058 0.00039 0.00063 0.003062023 0.004450837 0.003178637 0.002330449 0.001774947 0.001416392 0.001189264 81 91 86 96 111 106 101 24.367% 23.016% 21.915% 20.464% 21.065% 20.012% 20.113% 1.20 1.40 0.54 0.08 0.77 1.00 0.31 1.15 0.97 0.34 0.56 0.78 -0.12 0.11 1.00 2.04 1.42 8.21 4.23 2.95 5.97 0.08962 0.05992 0.00864 0.02009 0.03698 0.00209 0.00000 0.00594 0.00747 0.00459 0.00338 0.00042 0.00131 0.00229 0.00153 0.00121 0.00135 0.00109 0.00042 0.00089 0.00098 0.00152 0.00192 0.00320 0.00247 0.00532 0.00343 0.00415

0.08248 0.00455 0.00047 0.00105072

1 10% 100

110.5170918 111 5 111 20.012% 0.08 -0.12 7.77

76 25.968% 1.57 1.31 0.72

0.12696 0.00920 0.00174 0.00124 00

184

Swaps and Other Derivatives

APPENDIX: MEASURING THE CONVEXITY EFFECT Complex non-generic swaps suffer from a convexity effect, namely the value does not move linearly with respect to some movement in interest rates but exhibits a second-order effect. Why does this matter? Suppose we write the value of a swap as V(F^r^,r^r^. . .) where F is the current fixed rate and rI, etc. are a mixture of spot and forward rates: obviously the latter are yet to be determined and are therefore stochastic. If this is a "fair" swap we would expect E{V(F,r1,r2,r3,. . .)} = 0 where the expectation is under some (riskneutral) measure and is taken only using information available today. If F is determined so that the expectation is not zero, then because swaps are a zero-sum game, this effect will benefit one counterparty and penalize the other. The one that is penalized will demand that F is adjusted until the expectation is indeed zero. Ultimately, the most obvious way to estimate the adjustment is to simulate the value of the swap under a range of conditions, and select F that will give a zero mean. However this is often impractical, and analytic approaches are commonly used. This Appendix describes a range of approaches that have been used in practice to estimate the theoretical size of the adjustment, depending upon the specific type of swap. The Appendix first describes two approaches applicable to a range of swaps, and shows some results. The convexity effect in yield curve swaps is then considered separately, as this produces some particular problems. The analytic results will then be compared to the simulated values. Finally the convexity bias in futures will be discussed. Two Approaches to Measuring the Convexity Effect The first approach is simple and crude, but effective in many circumstances and makes few underlying assumptions. The second approach is more generally applicable, but is based upon a specific stochastic generating process. However it is demonstrated that the two approaches are in fact consistent with each other, and produce the same results for a range of swaps. Approach 1 Consider a normal floating swap payment on nominal principal of $1:

r j-1

The fixing of the floating rate r(t j-I ,t j ) takes place at tj-1 , and the payment at tj. The value of the cashflow at time tj is: V(tj,tj) = r(tj) x dj

where

dj = tj - tj-1

Define p(t j-1 ,,t j ) to be the price of a discount bond at time tj-1 maturing to 1 at /,. We can write:

1.e.

More Complex Swaps

185

The value of the cashflow at time 0 is given by: V(0,t j; ) = V(0,t j ) x DF(tj)

(A5.2)

We can express DF(tj) as a sequence of compounded discount bond prices: DF(tj) = II p(t i ,t i+l ) where i = 0 to j - 1

(A5.3)

Combining Equations (A5.1)-(A5.3), we get: (A5.4) which is of course a well-known result. If we repeat the same analysis for an in-arrears fixing, using the same notation, we get: the fixing takes place at tj, and is given by r(tj+1); the payment is also at tj; i.e. V(tj,tj) = r(tj+l) x dj where dj = tj - (tj-1 or V(tj,tj) = [p(t j ,t j+l ) – r -1](d j /d j + 1 ) and V(0,tj) = [p(t j ,t j + 1 )- 1 – l](dj+dj+1) x DF(tj)

(A5.5)

which does not simplify as before. Consider a function y = f ( x 1 , . . ., xn) where the ,xSare random variables. We can use the following approximation: 1/2(a 2 f/ax i a X j )s i s j

E{y} =f(E{x1}, . . ., E{x n }) +

pij

where si is the st. dev. of xi and pij the correlation between xi, and Xj. If we assume all the xs are independent, then this reduces to: E{y] = f ( E { x 1 , } , . . . ,

E{xn}) +

1/2E i (a 2 f/ax i ax j )v i

(A5.6)

where vi, is the variance of xt. If we apply Equation (A5.6) to Equation (A5.4) above, treating the discount bond prices as the random variables, we get: aDF(tj)/ap(ti-j,ti) = DF(tj)/p(ti-j,tj)

and

a 2 DF(t j )/ap(t l _ l ,t i ) 2

i.e. no convexity effect. Applying Equation (A5.6) to the in-arrears expression (A5.5), we get:

and

Thus:

=0

186

Swaps and Other Derivatives

If we assume that each p is distributed lognormally, the relationship between variance and volatility is given by, where it is assumed also that the expected discount bond price is given by the implied price: Vj = p(tj,ti+j,)2{exp[(oj)2tj] - 1} Substituting and rearranging: E{ V(0,tj)} = [p(tj,ti+j)–1exp[(oj)2tj] - \](dj/dj+l) x DF(tj)

(A5.7)

where the volatility oj is on the forward bond price. This of course is very similar to Equation (A5.5) but with the convexity factor exp[(oj)2tj]. If the volatility is set to zero. then this factor is equal to 1, and the convexity adjustment disappears.

Approach 2 This approach is based upon Heath-Jarrow-Morton modelling, and can produce more general results24. As before let p(t,T) be the price of a discount bond at time t which matures at time T to a par value of 1 . We assume that the price follows the process: dp(t,T)/p(t,T)

= r(t)dt + ap(t,T)dW

For convenience, only one stochastic Weiner source has been assumed, although the results may easily be extended. Because the bond is a traded security in a risk-neutral world, its expected return is given by r(t). The bond price volatility is op(t,T), with of course op(t,t)=0. Using Ito's lemma, we get: d[ln p(t, T] = WO - &(t,T) 2 dt + o p (t,T)dW Define the continuously compounded forward rate from t to T observed at time t0 as f(t 0 ,t,T). This will satisfy the relationship: p(t0,T) = p(t 0 ,t)exp{-f(t 0 ,t,T)(T

- t)}

i.e.

f(t 0 ,t,T) = [In p(t0,t)- In p(t0,T)]/(T- t) i.e.

As t T, then f{t0,t,T) f (t0,T), i.e. a forward rate of infinitesimal tenor at time T. This is given by: (A5.8) where o(t 0 ,T) = ao p (t 0 ,T)/dT. Integrating this last expression, we get: 24

Following discussions with Stuart Turnbull, 1994–5.

More Complex Swaps

187

An obvious interpretation of o (t0,T) is an instantaneous volatility. There are two common assumptions: (a) constant instantaneous volatility a which gives: op = oT. In practice, we might observe the volatility of a 3 month bond price to be 0.25–0.5% pa which would imply a to be within the range 1-2%; (b) reverting instantaneous volatility oe which gives op = ( o / ) [ 1 — e–T] where is some reversion factor. Note that this definition of an instantaneous forward rate clearly shows a link between the drift and the variance of the instantaneous rate, as given in Equation (A5.8). We can now go back and price a forward discount bond in terms of the instantaneous forward rate:

r - f7' fMduli i Jt j and also a money account: r(u)du where r(t) =f(t,t] is the riskless rate of return. HJM show that Z(t,T) = p(t,T)/B(i) is a unique martingale under conditions of no arbitrage, which implies Z(0,T) = EQ{Z(t,T)} where the Q indicates expectation with respect to risk-neutral probabilities. This leads to: B(t) = -r^exp -if P(0,t)

I

Z>(v,02dv - f b(v,t)d WQ(v} ]

2JQ

JQ

where

b(v,i) = - I o(v,«)dw

]

J}.

and W® is a Weiner process, and to: #(0 T}7 '

-

f r f'' fyy fr 'f' a(v,s) cr(v,u)dudvds a(v, Jr Jo Jv Jt Jo

Now let's consider an in-arrears swap: as before the fixing takes place at th i.e. r(f /+1 ), and the payment at tj, i.e.: V(tj,tj) = r(tj+l)dj = \p(tj,tj+lr{

- \}(dj/dj+l)

where

d} = tj - *,._,

Discounting back using the money account, and also the fact that this is a martingale, we can write:

If we make some assumptions about the shape of a(v,u) as discussed above, then we can take expectations by:

188

Swaps and Other Derivatives

1. substituting for p(tj,tj+l) and B(tj) 2. evaluating the integrals; 3. taking expectations over the Weiner process dWQ using the result that:

to get:

.) This is the standard result but with an adjustment factor e'. If we assume o(v,u) = o, i.e. constant with no reversion, then p = o 2 tj(dj + l ) 2 = o p (t,T) 2 tj, i.e. the same result as under the first approach. See the next section for a fuller statement of . In the main text, we priced a 5 year swap to receive 3mo. Libor in arrears, and to pay 3mo. Libor —6.5 bp in advance. The margin was negative because of the rising forward curve. Applying the above formula, if a = 1%, then the margin is increased to just over 7bp. The spreadsheet (Worksheet 5.31) has been slightly rearranged from the earlier one, and uses the relationship: PVin-arrears + PVin-advance + m x PVO1 = 0 to estimate m dynamically. Figure 5.10 shows how the margin increases with volatility. Is the adjustment appropriate? The simulation worksheet (Worksheet 5.32) is built as follows: [1] contains the usual IBOR DFs; [2] quarterly discount bond prices; [3] simulated bond prices using the formula pt =p0exp{—1/2(o P } 2 t + o p t . e } where e is [4] recalculated DFs; [5] implied forward rates from the new DFs; [6] in-arrears cashflows; [7] in-advance cashflows including 6.5 bp margin; [8] PV of 1 bp using the original DFs.

and oP = 1%

Convexity-adjusted Non-adjusted

6.0

1.0

1.1

1.2

1.4

1.5

1.6

Instantaneous volatility (%)

Figure 5.10 5-Year in-arrears/in-advance basis swap

More Complex Swaps

189

The net value of this swap is calculated by present valuing columns [6] and [7] and summing. The net value was simulated 500 times, the average calculated and divided by PV01 to estimate the average convexity adjustment. The example shows that the in-arrears side is still valued above the in-advance side, with an average convexity adjustment of 0.67bp compared to a theoretical one of 0.65 bp using a bond price volatility of 1%. Figure 5.11 shows the results of the simulation.

A General Mismatch Swap

The two approaches above are in fact very similar: the first one uses a discrete framework whilst the latter uses a continuous one. This means that the latter may be used to model more general situations. For example, consider a floating interest payment25: let rj be an interest rate fixed on time tt, rj has a tenor of rj, interest is payable calculated over a period qj, interest is paid at time Tj > tj. This is a very general statement as to how the value of a cashflow resulting in a floating reference rate may be estimated. The value of the cashflow at 7} is given by, as before:

Thus, we can write the present value of this as:

Using the same approach as before, we find:

where p=1/2(o 2 /A 3 )(l - exp[-Ary])( 1 - exp[-2A(,.]){exp[-A(7} - /,.)] - exp[-AFy]}. If A = 0, then (p reduces to o^iy//,- + F, - Tj). Some special cases follow. (a) If a — 0, then no volatility and hence no convexity. (b) For a conventional fixing and payment, Tj = tj 4- rj, i.e. (p reduces to zero as expected. (c) For an in-arrears payment, tj = Tj, i.e. cp simplifies to 1/2(a*/l?)(\ - exp[— 2Ar;]) x(l — expf-AFy])2 and for A = 0, (p reduces to cr2rjtj as shown above. Consider an average rate swap, which may be described in the following terms: • partition a period of the time [t,t] into k slices t — tl < • • • < tk = T; As discussed in Section 5.6.

190

Swaps and Other Derivatives

Worksheet 5.31 Pricing an in-arrears swap

Today's date = 4-Jan-00 Swap structure Maturity 5 years Principal 100m USD To receive 3m L in-arrears To pay 3m L-margin in-advance

Swap dates

06-Jan-00 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 06-Apr-05

Time Daycount count (Act/360) (Act/360)

0.25 0.51 0.76 1.02 1.27 1.52 1.78 2.03 2.29 2.54 2.79 3.04 3.30 3.55 3.80 4.06 4.31 4.56 4.82 5.08 5.33

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250

IBOR DFs

No convexity adjustment

Implied forward rates

[1] 1 0.984983 6.03125% 6.310% 0.969519 6.557% 0.953540 0.936853 6.821% 6.804% 0.921526 6.953% 0.905611 7.106% 0.889113 7.256% 0.873099 7.024% 0.857868 7.089% 0.842766 7.154% 0.827796 0.812962 7.219% 7.054% 0.798719 7.084% 0.784668 7.114% 0.770808 0.756987 7.144% 7.137% 0.743572 0.730349 7.162% 0.717174 7.189% 0.704190 7.215% 0.691754 7.19133% Present value=>

Swap cashflows Swap receive cashflows side pay side [2]

PV01 6.50

[3]

[4]

.5246 .5950 .6758 .7811 .6632 .7575 .8555 .8341 .7755 .7919 .8084 .8248 .7832 .7907 .7982 .8258 .8041 .8105 .8371 .8437

-0.0025 -0.0025 -0.0026 -0.0026 -0.0024 -0.0025 -0.0026 -0.0025 -0.0025 -0.0025 -0.0025 -0.0025 -0.0025 -0.0025 -0.0025 -0.0026 -0.0025 -0.0025 -0.0026 -0.0026

29.8576 -29.5810

-0.0426

1.5950 1.6576 1.7432 .7766 .6995 .7963 .8946 .7755 .7919 .8084 .8248 1.7832 1.7907 1.7982 1.8059 1.8239 1.8105 1.8172 1.8437 1.8378

-

191

More Complex Swaps

Convexity adjustment with zero reversion

Convexity adjustment with reversion Instantaneous Volatility =

1%

Reversion factor =

2%

phi [5] 0.000000 0.000002 0.000003 0.000005 0.000006 0.000008 0.000010 0.000011 0.000012 0.000014 0.000015 0.000017 0.000018 0.000020 0.000021 0.000023 0.000024 0.000025 0.000027 0.000028 0.000029

exp(phi) [6] 1 1.0000016 1.00000325 1.00000508 1.00000596 1.00000785 1.00001 1.00001093 1.00001242 1.00001389 1.00001535 1.00001679 1.00001822 1.00001964 1.00002104 1.00002292 1.00002382 1.00002518 1.00002711 1.00002848 1.00002856

Swap Convexity adjusted cashflows receive forward side rates 7.05 [7]

[8]

6.311% 6.559% 6.823% 6.807% 6.956% 7.110% 7.260% 7.029% 7.095% 7.160% 7.226% 7.062% 7.092% 7.122% 7.153% 7.147% 7.173% 7.200% 7.226% 7.203%

1.5952 1.6577 1.7435 1.7772 1.7001 1.7971 1.8956 1.7766 1.7932 1.8098 1.8263 1.7849 1.7925 1.8002 1.8081 1.8263 1.8129 1.8197 1.8465 1.8407 29.8810

1%

Instantaneous Volatility =

phi [9] 0.000000 0.000002 0.000003 0.000005 0.000006 0.000008 0.000010 0.000011 0.000013 0.000015 0.000016 0.000018 0.000019 0.000021 0.000023 0.000025 0.000026 0.000028 0.000030 0.000031 0.000032

Swap Convexity adjusted cashflows forward receive side rates exp(phi) 7.15

[10] 1 1.000002 1.000003 1.000005 1.000006 1.000008 1.000010 1.000011 1.000013 1.000015 1.000016 1.000018 1.000019 1.000021 1.000023 1.000025 1.000026 1.000028 1.000030 1.000031 1.000032

[H]

[12]

6.311% 6.559% 6.823% 6.807% 6.956% 7.110% 7.260% 7.029% 7.095% 7.160% 7.226% 7.062% 7.092% 7.123% 7.154% 7.147% 7.174% 7.201% 7.227% 7.204%

1.5952 1.6579 1.7437 1.7773 1.7003 1.7973 1.8958 1.7769 1.7934 1.8100 1.8266 1.7851 1.7928 1 .8005 1.8084 1.8266 1.8133 1.8202 1.8469 1.8411 29.8852

Worksheet 5.32

Pricing an in-arrears swap via simulation

Today's date = 4-Jan-00 Swap structure 5 years Maturity 100m USD Principal 3m Libor To receive 3 m Libor margin To pay Swap dates

Time count (Act/360)

Daycount (Act/360)

in-arrears in advance Curve discount bond prices

Curve IBOR

DFs

Random discount bond prices

Implied forward rates

IBOR

DFs

Swap Swap cashflows cashflows receive side pay side 6.50

PV01

1% 06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-01 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05 ()6-Apr-05

[1] 0.25 0.51 0.76 1.02 1.27 1.52 1.78 2.03 2.29 2.54 2.79 3.04 3.30 3.55 3.80 4.06 4.31 4.56 4.82 5.08 5.33

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250

1

0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962 0.798719 0.784668 0.770808 0.756987 0.743572 0.730349 0.717174 0.704190 0.691754

[2] 0.984983 0.984300 0.983518 0.982501 0.983640 0.982729 0.981783 0.981989 0.982554 0.982396 0.982238 0.982079 0.982481 0.982408 0.982336 0.982070 0.982279 0.982217 0.981960 0.981897 0.982339

[3] 0.984983 0.984727 0.980615 0.985273 0.984587 0.984079 0.979797 0.975370 0.985462 0.981342 0.985722 0.991070 0.973920 0.981221 0.986527 0.986690 0.990040 0.975917 0.979584 0.980386 0.982904

[4]

1

0.984983 0.969940 0.951138 0.937130 0.922686 0.907996 0.889652 0.867740 0.855125 0.839170 0.827188 0.819802 0.798422 0.783428 0.772872 0.762585 0.754990 0.736808 0.721765 0.707609 0.695512

[5] 6.03125% 6.136% 7.735% 5.725% 6.404% 6.400% 7.897% 9.990% 5.836% 7.521% 5.730% 3.565% 10.593% 7.571% 5.403% 5.279% 3.980% 9.763% ' 8.155% 7.828% 6.957%

Present value=> Net value of swap

[6]

[7]

[8]

.5510 .9553 .4630 .6721 .5645 .9961 2.6084 .4752 .9013 0.4485 0.9011 2.6778 1.9139 1.3657 1.3343 1.0170 2.4677 2.0615 2.0006 1.7779

.5410 .5674 .9934 .5117 .5813 .6343 -:2.0789 -22.5416 .4917 .9177 .4649 -().9175 -;2.6942 .9303 .3821 .3656 .0224 -2.4842 -2.1007 -2.0172

0.0025 0.0025 0.0026 0.0026 0.0024 0.0025 0.0026 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0026 0.0025 0.0025 0.0026 0.0026

29.5103

-29.5163 -0.0059

0.0426

193

More Complex Swaps 12

10 8 6

Mean = 0.67 bp

i\

Illl 111111IT Illlllllll Hi

T- T - O O O O O O O

O O O O O OO

OO

O <- T-

Figure 5.11 In-arrears simulation showing distribution of convexity adjustments

define an average payment that has to be paid at time Tj, not necessarily the same as T, by:

where the vector w contains the known averaging weights that sum to 1. An average rate swap was priced in the main text. Its convexity adjustment is shown in Worksheet 5.33. The adjustment is extremely small, only a fraction of a basis point; this is hardly surprising as the timing difference (tj, + rj - Tj) is relatively small for each fixing. If we wish to value the average part way through a period, then we could partition the time slices into {1, . . ., j} and { j + 1 , . . . , k } where the fixings on the first partition have already been observed. Obviously the convexity effect will then disappear for 1, . . ., j. The convexity effects for in-arrears and average rate swaps are relatively small. Turbo or power swaps (see Worksheet 5.34), e.g. swaps that pay fixed rate and receive (Libor)n where n is usually set to 1.5 or 2, have a much greater effect. The same analysis as above produces the following result: exp[-2A/,.])(l - exp[-/4])2 or for A = 0: V

V

//

/

Notice that this expression is exactly the same as for the in-arrears, but the jth cashflow is given by: V ( T j , T j ) = [1 +1/2{n(n - 1)/(1 - p)2 (e - 1)] x L"/100 x dj where

p=p(tj,Tj)

Turning to the worksheet: [1] calculates the PV01 (i.e. present value of 1 bp) of the fixed side of the swap; [2] calculates the turbo Libor rate, notice that this is calculated by raising the rate expressed as a whole number, e.g. 6.031, to the power and then converting to a percentage;

Worksheet 5.33 Pricing a customized average rate swap with convexity adjustment

Today's date = Structure of Debt Next payment date 20-Jan-00 07-Feb-OO 21-Feb-00 03-Mar-OO 15-Mar-00

04-Jan-OO

Principal 100 50 75 100 25

Margin (bp) 25 35 30 25 50

Last fixing date 20-Oct-99 08-Nov-99 22-Nov-99 03-Dec-99 15-Dec-99

Last fixing 6.0875% 6.0938% 6.1000% 6.1033% 6.1050%

Weights 28.57% 14.29% 21.43% 28.57% 7.14%

Estimated Average fixed rate = 6.7926% Instantaneous Volatility = 1% Swap dates

[16] 06-Jan-OO 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-JuI-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-()3

Daycount (Act/360)

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

Time count (Act/360)

0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792 3.044

3 mo. formal rates

5.986% 6.123% 6.251% 6.381% 6.452% 6.525% 6.601% 6.674% 6.706% 6.738% 6.770% 6.802%

DFs

Weighted average forward rates

Average rate cashflows

[17]

[18]

[19]

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

PV01 6.7926% [20]

6.097% 6.192% 6.421% 6.671% 6.827% 6.868% 7.017% 7.168% 7.163% 7.048% 7.114% 7.178%

.5411 .5652 .6410 .7417 .6687 .7360 .8322 .8118 .8106 .7816 .7981 .8145

-25.2778 -25.2778 -25.5556 -26.1111 -24.4444 -25.2778 -26.1111 -25.2778 -25.2778 -25.2778 -25.2778 -25.2778

Present value =

18.5711

-273.4027

195

More Complex Swaps

[3] the cashflows are calculated in the usual way, and the fixed rate is then estimated using PVfloating + F x P V 0 1 = 0 ; [4] calculates p with reversion; [5] calculates the convexity adjustment as given above; [6] hence the PV of the adjusted floating side, and the fair fixed margin. This is repeated in columns [7] to [9] for zero reversion; the impact is relatively small. Table 5.8 shows the theoretical size of the fair fixed rate for a 4 year qu/qu swap. As we can see, the adjustments are large and highly sensitive to the volatility. Turbo swaps have a convexity comparable with interest rate options such as caps, and may be risk managed together. Why would anybody enter into these swaps, other than for speculation? Consider a typical company paying floating debt; as rates rise, obviously there is an adverse impact. But also, as rates rise, this has a dampening effect on the economy and therefore is likely to reduce demand. The likely impact on the company's bottom line (P&L or cashflow) is not linear with rate movement but greater, roughly L1.2. Yield Curve Swaps

As one may expect from the above discussion, yield curve swaps also require convexity adjustments. We could apply the same formula as above, but this would be incorrect as the reference rate is not a zero coupon rate as tacitly assumed. Traditionally the financial markets have approached these swaps in a very different manner to the analysis above, so we will first discuss the market method, and then try to understand whether it is realistic. Consider a typical fixed coupon bond; there is obviously a non-linear relationship between the bond price P and its yield-to-maturity y given by:

where CF, represents the bond cashflows. Assume we are buying the bond at time T in the future, so that the actual bond price will be PT and the forward bond yield yT. For a given bond curve today, we can imply P0(T) and y 0 (T). We can approximately write: y0(T)]P0(T)' + i[v -

PT = P0(T)

y0(TjfPQ(T)"

where P 0 (T)' and P 0 (T)" are the first and second derivatives with respect to yield at y0(T) — see later. Taking expectations: Table 5.8 Convexity adjustment volatility —> Power

1.0 1.2 1.4 1.6 1.8 2.0

No convexity adjustment

1.0%

1.2%

1.4%

1.6%

1.8%

2.0%

6.90% 10.15% 14.95% 22.01% 32.40% 47.71%

6.90% 10.20% 15.11% 22.42% 33.31% 49.59%

6.90% 10.22% 15.18% 22.60% 33.72% 50.41%

6.90% 10.25% 15.27% 22.81% 34.19% 51.38%

6.90% 10.27% 15.36% 23.06% 34.74% 52.51%

6.90% 10.31% 15.47% 23.34% 35.36% 53.78%

6.90% 10.34% 15.60% 23.65% 36.05% 55.20%

196

Swaps and Other Derivatives

Worksheet 5.34

Date:

Turbo swaps — pricing a 4 year qu/qu swap

4-Jan-00

Power of Turbo (n) = reversion

Swap dates

6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-00 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 8-Oct-03 6-Jan-04

Period count

1.2

No convexity

Daycount (d)

Time count

3 4 5 6

7 8 9 10

11 12 13 14

15 16

0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792 3.044 3.297 3.550 3.803 4.058

Implied forward rates

Discount bond price

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256

0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962 0.798719 0.784668 0.770808 0.756987

Turbo implied forward rates

[1]

1

0 1 2

Libor dfs interpolated (zero bonds)

6.031 6.310 6.557 6.821 6.804 6.953 7.106 7.256 7.024 7.089 7.154 7.219 7.054 7.084 7.114 7.144

0.98498 0.98430 0.98352 0.98250 0.98364 0.98273 0.98178 0.98199 0.98255 0.98240 0.98224 0.98208 0.98248 0.98241 0.98234 0.98207

8.640% 9.121% 9.552% 10.015% 9.984% 10.247% 10.519% 10.785% 10.373% 10.488% 10.604% 10.719% 10.427% 10.479% 10.532% 10.587%

'pv =

197

More Complex Swaps

Convexity adjustment with reversion adjustment 1.00% pa Instantaneous Volatility = 2.00%

Reversion Factor =

Turbo cashflows

PV01

phi

convexity adjustment

[3]

0.0218 0.0231 0.0244 0.0261 0.0244 0.0259 0.0275 0.0273 0.0262 0.0265 0.0268 0.0271 0.0264 0.0265 0.0266 0.0271

0.0025 0.0025 0.0026 0.0026 0.0024 0.0025 0.0026 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0025 0.0026 ' 0.0352

0.3577

turbo cashflows

phi

convexity adjustment

[4]

0.000000 0.000002 0.000003 0.000005 0.000006 0.000008 0.000010 0.000011 0.000012 0.000014 0.000015 0.000017 0.000018 0.000020 0.000021 0.000023 PV =

[5]

[6]

0 0.08% 0.14% 0.20% 0.27% 0.32% 0.36% 0.40% 0.49% 0.54% 0.58% 0.63% 0.71% 0.76% 0.81% 0.86%

0.0218 0.0231 0.0244 0.0262 0.0245 0.0260 0.0276 0.0274 0.0263 0.0267 0.0270 0.0273 0.0265 0.0267 0.0268 0.0273 0.3593

turbo cashflows 10.2015

10.1989

10.1545 [2]

Convexity adjustment with zero reversion

[7]

0.000000 0.000002 0.000003 0.000005 0.000006 0.000008 0.000010 0.000011 0.000013 0.000015 0.000016 0.000018 0.000019 0.000021 0.000023 0.000025 PV =

[8]

[9]

0 0.08% 0.15% 0.20% 0.27% 0.33% 0.37% 0.42% 0.51% 0.57% 0.62% 0.67% 0.76% 0.82% 0.87% 0.93%

0.0218 0.0231 0.0244 0.0262 0.0245 0.0260 0.0276 0.0274 0.0264 0.0267 0.0270 0.0273 0.0266 0.0267 0.0269 0.0273 0.3594

198

Swaps and Other Derivatives

E{P(y)} - P0(T) = (E{y} - y0(T)]P0(T)' + { E { [ y -

y 0 (T)] 2 }P 0 (T)"

If we assume that E{P(y)} = P 0 (T), i.e. a world that is forward risk-neutral with respect to prices, then: E{y} = y0(T) -1/2E{\y-

y0(T)f}P0(T)"/P0(T)'

As we can write E{[y - y0(T)f} % y0(T)2(av)2T, the expected yield is: E{y] = y0(T) -

{y0(T)2(avfTP0(T)"/P0(T)'

Therefore, for a yield curve swap, we can regard the forward swap as equivalent to a forward par bond, estimate the forward rate off the current curve, but then apply the adjustment26. Swaption volatility of the appropriate forward time and underlying length is typically used in the adjustment. As the convexity C of a bond is P 0 (T}"/P 0 (T), and the modified duration D is P0(T)'/Po(T), the ratio P0(T)"/P0(T)' is equivalent to C/D which increases with the maturity of the bond. Hence the size of the adjustment depends upon: the tenor of the underlying reference rate, the volatility of the reference rate, the time to the fixing. In the main text, we priced a 4 year CMS which was receiving the 5 year swap rate less 25.6 bp. If we assume the volatility curve is a flat 20% pa then the convexity adjustment can be seen in Worksheet 5.35, with an overall margin of 34.5 bp. The convexity details are: [5] unadjusted CMS yield; [6] calculates (1 +y 0 (T)/2) –1 ; remember that the reference swap is semiannual; [7]–[9] these columns calculate P1 and P" having differentiated the general bond formula; [10] finally the convexity adjustment in bps is calculated using the above formula. Despite being widely used, this approach however has some practical shortcomings in that it concentrates purely on adjusting the CMS fixing, and does not measure the effect on the value of the swap. Furthermore, the swap reference rate fixes in-advance at the beginning of each period, but the cashflow is discounted back from the end of the period. There is therefore an impact due to both the CMS fixing and the Libor fixing over the period, which is likely to reduce the overall adjustment27, followed by discounting. Pugachevsky28 has recently developed an alternative approach which incorporates both adjustments, and shows that the yield-based approach described above can overestimate the adjustment quite considerably. 26

A fuller description is given in many texts, such as J. C. Hull, Options, Futures and Other Derivatives. 4th edn.. Prentice Hall, 2000, pp. 554–555 and Appendix 20A. 27 Hull (2000) pp. 554–555 suggests a double adjustment which requires an estimate of the correlation between the two fixings. He suggests using a figure of 0.7, which seems a touch high. 28 D. Pugachevsky, "Forward CMS rate adjustment". Risk. March 2001. pp. 125-128.

More Complex Swaps

Furthermore, we would like to be consistent in our approach to convexity, and earlier we used independent discount bond price volatilities which are related to cap volatilities. There is obviously a theoretical relationship between cap and swaption volatility which could be used29. Alternatively we could simulate the swap as before. The end result appears very different when we assume: • that only the CMS fixing is stochastic, and • that the Libor fixing and discounting are also stochastic. Worksheet 5.36 (on the CD) shows that, for a parallel shift in the discount bond curve, the convexity effect if only the CMS fixing is adjusted is more than 16 times greater than the effect if all rates are adjusted. Worksheet 5.37 (on the CD) performs the same calculations using random simulations of the discount bond curve, and this time the convexity effect is much smaller and closer to Pugachevsky's results.

Convexity Bias in Futures

We saw in Chapter 2 that a tailed hedge for the money market swap possessed a very desirable property, namely the net value of the swap plus hedge was positive irrespective of rates going up or down. Another way of expressing this is to say that the combination possessed positivity convexity or, using option terminology, positive gamma. Where did this come from? Let us consider a very simple situation, namely selling an IMM FRA at .FFRA and selling a single deposit futures at Ffutures to hedge. Obviously the tenor of the rates and the dates all match. So would we expect FFRA and Ffulures to match as well? Consider what happens when rates move:

Rates

Lose on FRA

Receive margin on futures

Invest the margin at a high rate

Rates

Gain on FRA

Pay margin on futures

Borrow the margin at a low rate

See Chapter 6.

Swaps and Other Derivatives

200

Worksheet 5.35

Pricing a CMS with convexity adjustment

Date: Maturity: Principal: To pay: To receive:

Swap dates

06-Jan-OO 06-Jul-00 08-Jan-0l 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05 06-Jul-05 06-Jan-06 06-Jul-06 08-Jan-07 06-Jul-07 07-Jan-08 07-Jul-08 06-Jan-09 06-Jul-09 06- Jan- 10

Daycount (Act/360)

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511 0.503 0.517 0.497 0.514 0.506 0.508 0.503 0.511

04-Jan-OO 4 year swap 100 million USD 6mo. Libor 5 year CMS less

Curve IBOR DFs [1]

1 0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564114 0.543575 0.523588 0.504100 0.484913

34.52 bp margin

Q

Implied Libor

Libor cashflows discounted

CMS yield

[2]

[3]

[4]

[5]

0 0.4901 0.9742 1.4245 1.8732 2.2992 2.7102 3.1069 3.4917 3.8609 4.2209 4.5624 4.8972 5.2145 5.5285 5.8199 6.1098 6.3846 6.6507 6.9042 7.1520

6.219% 6.749% 6.938% 7.246% 7.119% 7.252% 7.132% 7.194%

PV =

-3.143.923.61 -3,486.733.92 -3.449,914.84 -3.723,663.72 -3.599.290.91 -3.666.118.81 -3.605.774.39 -3.656.803.79

24.301.323.10

7.008% 7.125% 7.187% 7.238% 7.260% 7.295% 7.320% 7.356%

201

More Complex Swaps

Unadjusted margin Adjusted margin Difference Volatility = n=

20% 10

u

sum

— P'

P"

Convexity adjustment (bp)

[6]

[7]

[8]

[9]

[10]

0.9661 0.9656 0.9653 0.9651 0.9650 0.9648 0.9647 0.9645

25.55 34.52 8.97

41.9055 41.7219 41.6241 41.5448 41.5111 41.4566 41.4175 41.3622

4.1574 4.1452 4.1387 4.1334 4.1312 4.1276 4.1250 4.1213

20.9527 20.8609 20.8120 20.7724 20.7556 20.7283 20.7088 20.6811

0.0000 2.5832 5.3110 8.0012 10.7684 13.5699 16.3792 19.2770

Adjusted rate

[11] 7.008% 7.151% 7.241% 7.318% 7.368% 7.431% 7.484% 7.548%

CMS cashflow

Less margin on CMS

[12]

34.52 bp [13]

3,543,072.72 3,694,615.12 3,600,182.01 3,760,773.33 3,724,694.73 3,756,555.71 3,783,472.88 3,837,101.01

25,506,668.69 Fair value =

–174,518.85 –178.354.42 -171,642.16 -177,395.53 -174,518.85 –174,518.85 -174,538.85 –175,477.74

–1,205,345.59 0.0000

Swaps and Other Derivatives

202

Remembering that the gain or loss on the FRA occurs effectively at the end of the FRA, the margin payment or receipt is invested or borrowed until then. But this is a secondorder win-win situation, investing at high rates, borrowing at low rates. If we think of a futures contract in the same way as an FRA: Libor Seller of futures

Buyer of futures

futures

If the seller is gaining from this effect, the buyer must be losing. As compensation therefore, the buyer will want to receive a higher rate, which suggests that in practice, F futures > FFRA. We can determine the theoretical adjustment that should be made using either of the methods discussed above. We get approximately: futures

where: r is the tenor of the rate (in years), t is the time to the start of the forward rate, v is the instantaneous volatility, estimated by
Flesaker. "Arbitrage Free Pricing of IR Futures and Forward Contracts". Journal of Futures Markets. 13(1). 1993. 77–91. 31 Dean Witter Institutional Futures. "The Convexity Bias in Eurodollar Futures". September 1994.

More Complex Swaps

203 Table 5.9 Futures

Futures price

Forward volatility

Adjustment (bp)

Mar-95 June-95 Sep-95 Dec-95 Mar-96 Jun-96 Sep-96 Dec-96 Mar-97 Jun-97 Sep-97 Dec-97 Mar-98 Jun-98 Sep-98 Dec-98 Mar-99 Jun-99 Sep-99 Dec-99 Mar-00 Jun-00 Sep-00 Dec–00 Mar-01 Jun-01 Sep-01 Dec-01 Mar-02

9254 9334 9315 9310 9302 9296 9287 9290 9286 9283 9276 9277 9272 9269 9261 9262 9257 9252 9244 9245 9238 9232 9224 9225 9218 9212 9204 9208 9292

9.71% 10.09% 10.97% 12.49% 13.70% 14.55% 15.04% 15.13% 15.36% 15.73% 16.26% 16.94% 17.29% 17.29% 16.95% 16.26% 15.73% 15.36% 15.15% 15.10% 15.05% 15.00% 14.95% 14.89% 14.83% 14.76% 14.70% 14.63% 14.56%

0.01 0.06 0.16 0.36 0.70 1.15 1.72 2.25 2.97 3.89 5.12 6.60 8.17 9.56 10.78 11.25 12.04 13.04 14.43 15.86 17.69 19.72 21.84 23.52 25.76 28.04 30.57 32.23 27.44

course have done exactly the reverse, namely taken the change in cashflow under the FRA and present valued it. In this case, we would have found that the FRA would have had negative convexity relative to the futures contract. In a PV world, constant margin futures contracts are strictly linear with the movement in rates (sometimes called "tangential" contracts), and all non-margined contracts possess convexity. However the original discussion was from the angle that the futures had convexity, to place them consistently into a non-margined world.

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Cross-currency Swaps

OBJECTIVE This chapter reintroduces cross-currency swaps. It starts by discussing that most fundamental building block, namely the cross-currency basis swap, and considers its pricing, hedging and its role in creating a proper foreign currency discount curve. Subsequently fixed-floating cross-currency swaps are described in the context of swapping a fixed coupon bond into a floating one in another currency. In particular, the impact of changing the terms of the swap on the floating margin is explored. Diff and quanto diff swaps are then introduced, as examples of cross-currency swaps without exchange of principal. This leads on to fixed-fixed swaps, of which the best known example are long-term FX forwards. Swap valuation is then discussed again, this time from the point of effectively settling the swap after a short period of time, and resetting it to the new exchange rate. The chapter concludes by a brief look at two rarer structures, namely dual currency and equity swaps.

6.1 FLOATING—FLOATING CROSS-CURRENCY SWAPS Cross-currency swaps (CCS) were briefly discussed in the introductory chapter. Generic CCS all have the same fundamental three-part structure betraying their back-to-back loan origins: • the initial exchange of principal amounts; • periodic exchanges of interest payments; • re-exchange of the principal amounts at maturity. A floating-floating or cross-currency basis swap (CCBS) possesses this structure where both of the reference interest rates are floating. Consider the following example, based on a swap executed in 1994: • from party A's point of view, pay $100 million and receive £30 million; • every 6 months, receive 6mo. $ Libor on $100 million, and pay 6mo. £ Libor on £30 million; • at maturity, receive back $100 million and pay £30 million. First we shall value the swap. If we consider the two sides of the swap separately, as shown below, then each one is effectively a rolled money account (or par FRN depending on your perspective) which has, as we argued in Chapter 3, zero economic value. Therefore in theory a CCBS with no spreads on either side should have a zero value. If we took two discount curves, one USD and the other GBP, created from the relevant IRS markets, calculated the implied Libor rates, created the cashflows, and discounted, we would indeed get zero values on both sides. In practice, as we shall see in the next two sections, CCBS are not quite as simple as this.

206

Swaps and Other Derivatives $100m

£30m 6mo. $ Libor on $100m

6mo. £ Libor on £30m

$100m

£30m

The swap was freely entered into in 1994. The prevailing spot exchange rate was $1.7/£, therefore we can see that the initial exchange is considerably away from the spot rate. Does this matter? From the point of view of market "fairness", i.e. does the swap have an initial value of zero?, the answer must be that the exchange rate is irrelevant. By breaking the swap into the two FRNs, each of which is fair in its own right, the combination must also be fair. Of course, as soon as the swap starts, the first Libors are fixed and then the exchange rate becomes important at least to net value the known cashflows as the markets move. Obviously also, if the swap included any margin on either side, then the exchange rate is relevant from the beginning. What about from a credit perspective? This is a very different story. Imagine a situation that, immediately after the initial exchange, counterparty B absconds! C/p A is effectively out [$100m —£30mx 1.7] = $49m. To make the swap credit fair as well, the principal amounts should be $51m and £30m. Suppose we rewrite this swap as shown below: $49m

>

$51m

B

£30m 6mo. $ Libor on $49m 6mo. $ Libor on $51m 6mo. £ Libor on £30m

$49m $51m £30m

207

Cross-currency Swaps

Then we can see that it is actually made up of a $51 million "at-market" swap plus a $49 million loan. The swap is said to be "off-market" with the relationship: off-market = at-market + loan or deposit but effectively off-balance sheet. Coming back to the original swap for a moment, and thinking of it as back-to-back rolled money accounts, there is no natural maturity to the contract. The arrangement could last in perpetuity provided that the two counterparties are happy with the ongoing credit exposures. In the late 1980s there were at least two banks in London that used to offer "perpetual swaps", i.e. CCBS with no contractual maturity. Either counterparty could declare at the time of a floating rate fixing that the swap would terminate at the next payment date with a re-exchange of the principal amounts. Whilst there is a reasonably active market in CCBS, it is estimated that some 75% of the CCS market has a fixed-floating (usually USD Libor) structure. But CCBS are extremely important as a fundamental building block, as we can construct a fixed-floating swap using a CCBS plus a simple IRS: Fixed Yen

Fixed Yen Floating Yen

Floating Yen Floating USD

Floating USD

As a theoretical CCBS swap is priced at Libor-Libor flat, this suggests that the market rates in the fixed-floating CCS market should be very close to those in the IRS market. For highly liquid pairs this is generally true, although distortions do occur. The suggested bidoffer spread should be wider than in the IRS market because CCBS carry quite a high potential credit exposure due to the terminal principal exchange, and hence a much higher capital charge than IRS.

6.2 PRICING AND HEDGING OF CCBS Table 6.1 shows quotes for freely available CCBS. The quotations are showing the basis point spread on the 3mo. non-USD Libor side. For example, the 3 year Yen swap is {— 6, — 9} meaning that: 3mo. JPY Libor-9 bp Bank

3mo. JPY Libor-6 bp Bank

3mo. $ Libor

3mo. $ Libor

In practice therefore it would be argued that the CCBS at the mid-rate of {—7.5 bp} would have zero value — not quite the theoretical argument used before. There are at least three possible reasons for the existence of this spread.

208

Swaps and Other Derivatives

Table 6.1 Example of quotes for CCBS (against USD - 3m Libor on both sides)

JPY

Euro Years

1 2 3 4 5 7 10 15 20 30

GBP

Rec

Pay

Rec

Pay

Rec

Pay

1.00 0.50 -0.50 –1.00 –1.00 –1.75 -2.00 –1.00 -0.25 0.25

–1.00 –1.50 -2.50 -3.00 -3.00 -3.75 -4.00 –3.00 -2.25 -2.00

0.50 -2.50 -6.00 -10.00 –13.00 –19.00 -23.00 -25.50 -27.50 -29.50

-2.50 -5.50 -9.00 –13.00 –16.00 -22.00 -26.00 -28.50 -30.50 -32.50

-0.50 –1.50 –1.50 –1.50 –1.50 -2.00 -2.50 -5.00 -5.50 -6.00

-3.50 -4.50 -4.50 -4.50 -4.50 -5.00 -5.50 -8.00 -9.50 –10.00

Source: ICAP 4 January 2000. Reproduced by permission of ICAP plc

1. Think of the CCBS as back-to-back money market accounts. If JPY Libor is the rate at which Japanese banks would lend to each other, then better credit non-Japanese banks that could raise money at $ Libor flat could borrow at rates below Libor. In other words, the spread is a reflection of the relative funding cost in USD and JPY over the maturity of the CCBS for a good credit bank. 2. An imbalance in the supply and demand for CCBS. If the market was perfectly balanced, the bid-offer prices would be equidistant from zero. 3. The spread is due to the skewness in the short-term FX forward market, which is used to hedge CCBS. The calculations below show that the skew in the short-term S/Y market, measured in terms of bp adjustment to the Yen rates, is not trivial relative to the CCBS spread. The first is likely to have the greatest effect. Today's date: 04-Jan-OO

3mo. 6mo. 12mo.

06-Apr-OO 06-Jul-OO 08-Jan-0l

Spot rate: Act/360 0.253 0.506 1.022

mid-JPY 0.109% 0.141% 0.219%

102.985 mid-USD 6.000% 6.172% 6.563%

theoretical forwards 101.4744 99.9399 96.7265

skew quoted forwards (bp) 101.485 + 7.20 99.957 + 7.99 96.750 + 8.25

The following approach may be used to hedge a CCBS, and hence produce an estimate of the cost of carry. In practice, as hedging would be done on a portfolio basis, this estimate is likely to be far too high unless the portfolio was completely one-sided. Consider the 3 year swap quoted above, but assume we don't know the spread s: Time

USD

JPY

0 3 6 9

+ 1m —Libor —Libor —Libor

-102.985m + (Libor + s) + (Libor + s) + (Libor + 5)

—Libor

+ (Libor + 5) + 102.985m

36

-1m

209

Cross-currency Swaps

Rewrite it as a series of 3-monthly forward FX swaps, i.e. with a principal exchange every 3 months. The first swap could be hedged using a reverse spot 3-monthly swap as shown: Cross-currency basis swap JPY Time USD

First hedge Time USD

JPY

0

+ lm

-102.985m

0

-1m

+ 102.985m

3

— Libor -1m

+ (Libor + 5) + 102.985m

3

+ Libor + 1m

—(Libor + sk) -102.985m

3

4 1m

-102.985m

6

—Libor -1m

+ ( Libor + s) + 102.985m

6

+ 1m

-102.985m

9

—Libor -1m

+ (Libor + s) + 102.985m

33

+ lm

-102.985m

36

—Libor -1m

+ (Libor + s) -102.985m

where sk is the implied skew. Hence the Yen cashflow at the end from the first contract plus hedge is + (s-sk)x 102.985m x $ l m x 0.25. Consider the second contract. We could think of hedging this using a forward-forward FX swap; this strategy might work in this case because it is only 3 months out, but forward-forward contracts are unlikely to be available as we consider contracts further out. Suppose we do nothing for 3 months, and then hedge with another spot 3 month forward swap, i.e.: Second CCBS contract Time USD JPY

Second hedge Time USD

0

+lm

-102.985m

0

3

—Libor -1m

+ (Libor + s) + 102.985m

3

-1m

+

Libor + 1m

JPY + S3,3 –(Libor 4+sk) – S3.3.

where S3,3 is the observed spot rate in 3 months' time, and it is assumed that the skew has remained constant. There is obviously a Yen principal mismatch at the beginning due to the movement in the spot rate. If: (S3,3 — S0,0) > 0 then we have excess Yen to be deposited at Libor + md (the margin is likely to be negative); (S3,3 — S0,0) < 0 then we have a Yen shortfall to be funded at Libor + mb.

210

Swaps and Other Derivatives

Therefore the cashflow at the end is (Note: the Libor cashflows cancel): {s x S0.0 — sk x S3.3 + (S3.3 — S0.0) x mx] x $lm x 0.25

where x depends on the sign of (S 3,3 — 50,0). If the future spot rates could be estimated, then all the cashflows at the end of each quarter could be calculated, and hence the cost of the hedge. It is of course a foolish person who would try to predict future spot rates. Assume that the change in the spot S follows a lognormal process: dS 1 /S =

where u is the drift, a the volatility, and e is N(0,1), or: S1.0 = S0.0 Market practice varies for the next step, because there are no good real predictors for the future spot rate: different approaches use • £{S1,0} = S0,0exp{(u — |cr2)dt} as shown, where u = (rY - rs) and r is a continuously compounded rate estimated from the market curves; • E{S 1,0 } =F1,0, the forward rate quoted at time 0; • E{S 1,0 } = S0,0 as various studies have shown that the current spot rate is as good a predictor of future spot rates as anything else. We can estimate a probability range for S1,0 .For example, there is a 90% probability that a normally distributed variable will lie within ± 1.65 standard deviations, i.e.: £{S1,0}exp{-Wdr 1.65} ^ S1.0 ^ £{S1.0}exp{+a\/d/ 1.65} This can be used to generate a probability envelope as shown in Figure 6. 11. Here the central line is the anticipated forward rates. Worksheet 6.1 calculates this graph as follows: • columns [1] and [2] are the current discount curves; • F1,0 is estimated by S0,0(DF?/DF7) in column [3]; • columns [4] and [5] calculate the upper and lower curves. Suppose the future spot rate followed the upper curve2. According to the construction above, there would be a continual excess of Yen that would have to be deposited for 3 months — see column [6]. The example worksheet has assumed a skew of 4.3 bp and a negative deposit margin of 5 bp pa, and calculated the total PV of the cost of hedging to be just over Y300,000 on a principal of Y 102 million based on an annualized volatility of l

This methodology is applied by the risk department of a major bank in London. They made an interesting practical observation. If E{S 1.0 ] is estimated by S0.0 exp{(u – ^er2)dt}, the upper curve will start to turn down again at t* = {\s.al(u — \cr2)}2. For example, if e = 1.65, u — 0 and a = 50% (say for a commodity), then t* is just under 11 years. This is one argument for using a different estimator for E{S 1.0 }; alternatively it is suggested that the upper curve is held constant for t > t*. 2 This is a very similar approach to that used to estimate the PFE of a swap, and hence its credit-adjusted price — see Section 5.5.

211

Cross-currency Swaps 250

0.5

1

1.5 Time (years)

2.5

Figure 6.1 90% FX envelope, assuming 30% pa volatility

30%. This can be converted into a spread of 9.7 bp pa by dividing the cost by the 3 year quarterly PV01, which is estimated in column [10]. The process is then repeated following the lower curve with a funding margin of 2 bp, giving a PV of Y92,000. In the real world, we don't know what path the spot rate will take in the future. However, we know that the cost of any path lying wholly within the envelope (and indeed any reasonable path extending below the lower curve) must be less than the worst case of the two curves. Therefore the cost of hedging is estimated to be 9.7 bp pa with a 5% chance that this might be exceeded. The approach described above is for micro-hedging, and hence pricing, a single CCBS very conservatively. In practice, a portfolio is likely to be reasonably balanced, so there would be a substantial amount of netting, and a lower probability such as 50% (corresponding to a multiplier of 0.67) would be used, leading to a substantially reduced required spread. This worst-case simulation approach can also be used for more complex swaps such as quanto diff swaps (see Section 6.5).

6.3 CCBS AND DISCOUNTING Consider the following situation: you are a US bank, expecting to receive Ylbn with certainty in 5 years' time. What is it worth today? You have two alternatives:

1. calculate the Yen PV using the Yen discount curve, and then convert at spot; 2. convert from Yen to USD using the quoted 5 year forward outright, and then present value using the dollar discount curve. Will the two produce the same valuation? It is highly unlikely unless the forward rate is calculated using purely the two discount curves and does not have any inherent skew. Consider a generic mid-rate CCBS and a generic mid-rate Yen IRS to which has been added the notional principals:

212

Swaps and Other Derivatives

Worksheet 6.1 Example pricing a cross-currency basis swap Today's date

4-Jan-00

Current spot rate =

102.985

FX volatility = Envelope =

30% pa 1.65

Assume a principal of

1.00m USD 102.985m Yen

Period

Daycount

6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-00 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03

USD DFs

0 0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792 3.044

Yen DFs

Estimated forward FX rates

–(LY P$

Net Value = 0

90% up

90% down

[1]

[2]

[3]

[4]

[5]

1 0.984983 0.969519 0.953465 0.936853 0.921280 0.905235 0.888806 0.873099 0.857653 0.842475 0.827576 0.812962

1 0.999605 0.999053 0.998251 0.997133 0.995757 0.994009 0.991881 0.989525 0.986887 0.983934 0.980635 0.976959

102.985 101.479 99.941 98.365 96.759 95.282 93.788 92.283 90.868 89.499 88.179 86.911 85.697

102.985 130.154 142.100 151.491 159.604 166.326 172.638 178.640 184.058 189.173 194.046 198.729 203.266

102.985 79.121 70.289 63.869 58.660 54.584 50.951 47.672 44,861 42.343 40.071 38.009 36.130

Yen IRS

C-C basis swap

–P$ + L$ + L$ + L$ + L$ + L$ + L$ +

Obp 4.3 bp 3bp –5bp

Spread on CCBS 3mo. Forward skew Borrowing margin Depositing margin

+PY — (LY + s) + s) —(LY + s) —(LY + s) —(LY + s) — (LY + s) — PY

–PY + LY +LY + LY + LY + LY + LY + PY

+ PY

—FY -FY -FY

-FY -FY -FY- PY

Net Value = 0

Each are freely traded in the financial markets, and will initially have zero value. Being a US bank, we assume that you can access your domestic money market efficiently, and are able to fund or deposit USD at Libor flat; this is the same assumption as in the previous chapter. Therefore the USD leg of the CCBS above has no value, and hence the Yen leg including the spread must have zero value as well. But if this is true, we can no longer argue

Cross-currency Swaps

cost of excess per period

excess (Yen)

[6]

27,169,060 39,115,236 48,506,199 56,618,979 63.340,681 69,653,127 75,655,344 81,073,433 86,188,006 91,061,226 95,744,275 Total = Equivalent margin =

[9]

[7]

(11,189) (17,564) (20,577) (23,275) (23,596) (25,928) (28,246) (28,676) (29,856) (30,950) (31,970) (32,925)

PV01

cost of shortfall per period

shortfall (Yen)

23,864,151 32,695,766 39,115,951 44,325,118 48,401,068 52,033,879 52,313,026 58,124,270 60,642,439 62,914,469 64,975,976

(301,577)

[10]

(11,189) (9,375) (8,795) (8,474) (7,542) (7,478) (7,426) (6,926) (6,697) (6,489) (6,299) (6,122)

2,603 2,603 2,632 2,689 2.517 2,603 2,689 2,603 2,603 2,603 2.603 2,603

(92,096)

31,076

2.96 bp

9.70

that the floating leg of the Yen IRS including the notional principals has zero value (except in the trivial case when s = 0). Quietly ignoring potential difficulties such as differences in frequencies and daycount conventions, the spread could be added to both sides of the IRS without affecting its net value. Both individual sides would now value to zero, and it would be feasible to repeat the bootstrapping process to derive the CCBS-adjusted discount curve, as shown in Box 2 of Worksheet 6.2 (not printed here). Yen IRS

+ Py —(Fy + s)

+ (Ly + s —(Fy —(Fy —(Fy —(Fy

Net Value = 0

+ s) + s) + s) + s) -Py

Worksheet 6.2

Bootstrapping a Yen swap curve

Today's date:

4-Jan-00

Current spot

102.985

market rates Yen IRS sa Act/365

[1] 6-Jan-00 13-Jan-OO 7-Feb-00 6-Apr-OO 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05 6-Jul-05 6-Jan-06 6-Jul-06 8-Jan-07 6-Jul-07 7-Jan-08 7-Jul-08 6-Jan-09 6-Jul-09 6-Jan-10 6-Jul-10 6-Jan-11

0.09375% 0.12500% 0.15625% 0.18750% 0.28125% 0.52500% 0.77500%

mid

days Act/365

-1 -4 -7.5

1.32500%

-14.5 -17.5

1.7450%

-20.5

1.8950%

-21.8

2.0150%

-23.2 -24.5

Yen IRS sa Act/365 unadjusted

[3]

[2]

-11.5

2.1150%

days Act/360

CCBS spread

1.05500%

1.5650%

1. Deriving DFs from a generic curve using optimization

6-Jan-OO 6-Apr-OO 6-Jul-OO 6-Oct-00 8-Jan-Ol 6-Apr-0l 6-Jul-Ol 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05 6-Apr-05 6-Jul-05 6-Oct-05 6-Jan-06 6-Apr-06

3mo. Yen forwards DFs unadjusted smoothed unadjusted

valuation smoothing of generic constraint swaps 38.36

Q [6]

[4]

[5]

0.15625% 0.21866% 0.31418% 0.42949% 0.56529% 0.69595% 0.82151% 0.94203% .05752% .18720% .33085% .48856% .66032% .82021% .96832% 2.10474% 2.22933% 2.35147% 2.47118% 2.58846% 2.70331% 2.78601% 2.83627% 2.85392% 2.83899%

0.999605 0.999053 0.998251 0.997133 0.995757 0.994009 0.991881 0.989525 0.986887 0.983934 0.980635 0.976959 0.972876 0.968420 0.963625 0.958470 0.953099 0.947467 0.941521 0.935334 0.929055 0.922558 0.915920 0.909288 0.902880

[7]

[8]

1 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256 0.250

0.249 0.249 0.252 0.258 0.241 0.249 0.258 0.249 0.249 0.249 0.249 0.249 0.249 0.249 0.249 0.252 0.249 0.249 0.252 0.252 0.247 0.249 0.252 0.252 0.247

0.28125%

0.5250%

0.7750%

1.0550%

1.3250%

1.5650%

0.498

1.006

0.000000

—

1.494

—

1.995

0.000000

—

2.486

._ —

2.973

0.000000

— 3.456 3.936

— 0.000000

4.409 4.880

0.000000

— 5.338 5.796

----0.000000

0.000001 0.000001 0.000002 0.000002 0.000002 0.000001 0.000001 0.000002 0.000002 0.000002 0.000003 0.000003 0.000002 0.000002 0.000002 0.000001 0.000001 0.000001 0.000001 0.000001 0.000000 0.000000 0.000000

. S?a. 2

I

Cross-currency Swaps

o o o o oo o ooooo

—< r-

oo

Q O Q Q O ~ O

.

o o o o o o , ON NO

n(N NO

o o o

CS

in in r-

co t-r-> ON

o o o o o o ON

oo m oo

So"

in ON roo

o o o o ^_

0

ON

Tf

ON

ON

^D OO

in

ON

ON

NO NO

ro O

S

Tj-

>—•—^i ^

CO

o o o o o o

^* r~~ o

—ifNjrn^t^ininininTj-Tf

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—1

"5f

NO

ro

oo •— 1

in (N

oc

o o o o o o

(N

NO

ON.

in oo >—• ON (N

(N

rj-Tj-rorncorornror^

O O O O O O O O O O O O O O O C 3 O O O O O O C 3 O O O O O O O O C 5 O O

OOOOONONONOOOO

c^r^r>irN{rNjc^rNir^rNjrv|r^r>jr^r^r^rNjr^r^r^rNjrsjc>{r>{rN|r^CNjr^ oooooooooociooooodooooodoodoooocsodod

rsjrs|r^r^r^r^r^o4r^r^r^r^r^r^rNiojr^rNjr^r^r^r^rsjr^r^r^(NrNjr^r^r^

Cr-t

>n O

o

ddddddddddddddddddddddddddddddddddd

C-i

215

216

Swaps and Other Derivatives 4.0 3.5 3.0 2.5

1.5 — Unadjusted curve (from IRS) — Adjusted curve (from CCBS)

1.0 0.5 0.0 6

9

12

15

Time (yrs)

Figure 6.2 Adjusted and unadjusted 3mo. forward curves

However, bootstrapping requires a CCBS-adjusted zero coupon rate to start the process off, and probably a better approach is to model the CCBS directly having already estimated the Libor rates off the unadjusted curve. Box 1 of Worksheet 6.2 first derives the implied discount factors from the generic Yen IRS curve using optimization (as described in Chapter 3); the forward rates are estimated in column [4], the discount curve calculated in [5], and each of the generic swaps valued in [7]. Changes in the forward curve are minimized, hence producing a smooth curve, whilst ensuring that all the generic values are zero. We now want to estimate a smooth discount (or forward curve) that will value all the CCBSs to zero. One way to do this3 is as follows — see Box 3: columns [11] and [12] show the unadjusted 3 month forwards and the CCBS spread respectively; we are going to derive a new forward curve, as shown in [14], so that the derived DFs will value the CCBSs to zero; however it would seem sensible for this new curve to be as close as possible to the old forward curve; assume the CCBS spreads have maturities t1, t2 , . . ., tn: define a vector of margins {mk, k=1, ...,n} so that Fnew = F°ld + mp where tp–1 <j^tp as shown in column [13]. (These may be thought of as forward spreads compared to the quoted s which are par spreads. This builds a piece-wise constant forward spread curve; it would be simple to build, say, a piece-wise linear or even a Nelson-Siegel one instead.); the objective is to calculate m so that the CCBSs are all zero, and that the new forwards are still smooth. Column [16] values the CCBSs, using the expression: V =-

x DFi + DFk

and column [17] contains the smoothing conditions. This procedure is easy to implement in practice, and yet provides good forward and discount curves. See Figure 6.2. 3

There are alternative and more complex approaches; see for example, E. Fruchard et al.. "Basis for Change". Magazine, October 1995, 8(10), pp. 70–75.

217

Cross-currency Swaps

To see the adjusted curve in use, we will asset package a USD bond into a 5 year par Yen package; the bond details are shown below: Today's date =

04-Jan-00

USD bond details: maturity = coupon = redemption — yield = fraction of year = dirty price = accrued interest = clean price = Current spot rate =

15-Sep-05 6.75% ANN 100 7.7935% 0.719 to next coupon 97.1702 1.8938 95.2764 102.985

First, let us assume we are going to swap the bond into a par 5 year USD 3mo. Libor package. The cashflows on the swap are shown below: Bond side

Package side

6 Jan 00

6 Jan 00

15 15 15 15

6 6 6 8

Mar 00 Sep 00 –6.75 Mar 01 Sep 0l –6.75

15 Mar 05 15 Sep 05 –106.75

Apr 00 Jul 00 Oct 00 Jan 01

6 Jan 05

–(100-DP) L +m L +m L +m L +m

L + w+100

Bond side

6 Jan 00 + DP 15 Mar 00 15 Sep 00 -6.75 15 Mar 01 15 Sep 01 -6.75

Package side

6 Jan 00 6 Apr 00 6 Jul 00 6 Oct 00 8 Jan 01

15 Mar 05 6 Jan 05 15 Sep 05 -106.75

m m m m m

Notice that the maturity dates are different, and therefore the principal cashflows at the end actually happen. The swap may be written into the more usual form on the right-hand side by setting the par FRN to zero. The bond has a current value of —$2.52 million on a principal amount of $100 million when priced off the swap curve; see Box 1 of Worksheet 6.3. If this were to be swapped into a par 5 year USD 3mo. Libor package, we find that the fair margin would be + 59.2 bp: see columns [2] and [3] in Box 2. We now want to swap this package into Yen. Box 3 swaps the bond into Yen in two stages. It first builds the floating Yen side of a cross-currency swap as {Libor + margin} off the IRS curve using the implied forward rates. The margin is calculated by ensuring that the value of this leg is equal to the value of the USD leg above: this gives a margin of 52.4 bp. This is effectively converting the USD margin of 59.2 bp into a Yen margin of 52.4 bp. Second, we now recognize the existence of the CCBS. The 5 year CCBS margin of —14.5 bp is added to the 52.4 bp margin, giving a net margin of 37.9bp. An alternative approach is to use the adjusted discount curve, as shown in Box 4. Notice that the discount factors in columns [4] and [8] are different; however the implied Yen Libor rates must be the same because these are only quoted once — see columns [5] and [9]. We now calculate the {Libor + margin} cashflows as before. The margin is set so that the value of this leg is again equal to the USD value, but of course this time using the adjusted discount curve. The resulting margin is 39.4 bp, slightly higher than before.

K) 00

3. Deriving DFs from a generic curve: using the CCBS curve directly forward dates

days Act/360

3mo. forwards unadjusted smoothed

[11] 0.25 0.50 0.75 1.00 1.25 1.50 1.75 2.00 2.25 2.50 2.75 3.00 3.25 3.50 3.75 4.00 4.25 4.50 4.75 5.00 5.25 5.50 5.75 6.00 6.25 6.50 6.75 7.00

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84

6-Jan-OO 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05 6-Apr-05 6-Jul-05 6-Oct-05 6-Jan-06 6-Apr-06 6-Jul-06 6-Oct-06 8-Jan-07

CCBS spreads

[12]

spread to 3mo. forward curve

[13]

Yen DFs

new 3mo. forward curve

adjusted

[14]

[15]

0.14625% 0.20866% 0.30421% 0.41972% 0.52964% 0.62557% 0.75039% 0.87234% 0.97319% .04281% .18370% .34537% .48375% .59038% .73962% .88512% .98946% 2.11640% 2.24026% 2.36002% 2.40501% 2.47844% 2.54088% 2.56352% 2.49402% 2.49653% 2.51723% 2.56023%

0.999630 0.999103 0.998327 0.997234 0.995945 0.994373 0.992428 0.990245 0.987814 0.985217 0.982278 0.978949 0.975291 0.971386 0.967133 0.962496 0.957680 0.952584 0.947162 0.941483 0.935856 0.930030 0.924030 0.918016 0.912327 0.906606 0.900811 0.894829

valuation of CCBS

smoothing constraint

41.34 [17]

[16]

1 0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256 0.250 0.253 0.256 0.261

0.15625% 0.21866% 0.31421% 0.42972% 0.60013% 0.69606% 0.82089% 0.94283% 1.11946% 1.18908% 1.32997% 1.49164% 1.72168% 1.82832% 1.97756% 2.12306% 2.25882% 2.38575% 2.50962% 2.62937% 2.73860% 2.81203% 2.87447% 2.89711% 2.89141% 2.89393% 2.91463% 2.95763%

-1.00

-1

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-7.05

-7.50

-14.63

-11.50

-23.79

-14.50

-26.94

-17.50

-33.36

-20.50

-39.74

— — 0.000000 — — — 0.000000 — — 0.000000 — 0.000000

0.000000

0.000000

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220

Swaps and Other Derivatives

Worksheet 6.3 Creating a synthetic foreign asset: swapping a USD bond into floating Yen Today's date = 04-Jan-OO

Current spot rate= 102.985

1. USD bond details maturity = coupon = redemption = yield = fraction of year = dirty price = accrued interest = clean price = Bond dates

15-Sep-05 6.75% ANN 100 7.7935% 0.719 to next coupon 97.1702 1.8938 95.2764 USD IBOR

DFs

Swap cashflows pay side

[1] 06-Jan-00 15-Mar-00 15-Sep-00 15-Mar-0l 15-Sep-0l 15-Mar-02 15-Sep-02 15-Mar-03 15-Sep-03 15-Mar-04 15-Sep-04 15-Mar-05 15-Sep-05

1 0.988726 0.957166 0.925188 0.892827 0.861718 0.831164 0.802243 0.773928 0.746780 0.720124 0.694781 0.669866

97.1702 -6.75 -6.75 -6.75 -6.75 -6.75 -106.75

PV (m $) on 100m nominal =

—2.5207

PV (mYen)=

–259.5950

221

Cross-currency Swaps

2. USD swap details

maturity principal 3mo. Libor+

Swap dates

06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05

5 years 100m USD 59.2 bp margin

Libor cash flows

Margin cash flows

[2]

[3]

-100,000,000 1,524,566 1,595,040 1,683,723 1,773,156 1,690,431 1,772,439 1,848,462 1,798,937 1,801,057 1,801,541 1,800,383 1,797,606 1,793,206 1,791,953 1,793,890 1,818,765 1,807,304 1,813,260 1,836,821 101,838,051

0.150 0.150 0.151 0.155 0.145 0.150 0.155 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.150 0.151 0.150 0.150 0.151 0.151

PV (m $) on 100m nominal =

0.00000

2.5207

Net PV =

0.0000

Daycount

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

USD IBOR DFs

1 0.984983 0.969519 0.953465 0.936853 0.921280 0.905235 0.888806 0.873099 0.857653 0.842475 0.827576 0.812962 0.798640 0.784581 0.770755 0.756987 0.743549 0.730306 0.717134 0.704190

Implied forwards

6.031% 6.310% 6.588% 6.791% 6.915% 7.012% 7.079% 7.117% 7.125% 7.127% 7.122% 7.111% 7.094% 7.089% 7.097% 7.117% 7.150% 7.173% 7.188% 7.192%

222

Swaps and Other Derivatives

Worksheet 6.3 continued 3. Yen swap details Uses unadjusted Yen IRS curve maturity principal 3mo. Libor +

5 years l0bn Yen 52.4 bp margin

less CCBS =

37.9 bp margin

Swap dates

IBOR DFs

Implied forwards

Libor cashflows

Margin cashflows

[4]

[5]

[6]

[7]

0.156% 0.219% 0.314% 0.429% 0.565% 0.696% 0.822% 0.942% .058% .187% .331% .489% .660% .820% .968% 2.105% 2.229% 2.351% 2.471% 2.588%

-10,000,000,000 3,949,653 5,527,331 8,029,068 11,214,511 13,818,176 17,592,162 21,450,552 23,812,340 26,731,861 30,009,882 33.640,819 37,627,450 41,969,132 46,010.897 49,754,865 53,787,788 56,352,478 59,439,975 63,152,434 10,066,149,485

13,238,353 13,238,353 13,383,829 13,674.782 12,801,924 13.238,353 13,674.782 13.238.353 13.238.353 13.238,353 13,238,353 13,238.353 13,238.353 13.238.353 13.238.353 13,383.829 13.238.353 13.238.353 13.383.829 13.383.829

0

259.595.013

Yen

6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03 07-Apr-03 07-Jul-03 06-Oct-03 06-Jan-04 06-Apr-04 06-Jul-04 06-Oct-04 06-Jan-05

1 0.999605 0.999053 0.998251 0.997133 0.995757 0.994009 0.991881 0.989525 0.986887 0.983934 0.980635 0.976959 0.972876 0.968420 0.963625 0.958470 0.953099 0.947467 0.941521 0.935334 PV(Yen) = Net PV =

0.0000

223

Cross-currency Swaps

4. Yen swap details Uses Yen IRS curve adjusted for CCBS 5 years l0bn Yen 39.4 bp margin

maturity principal 3mo. Libor +

Yen DFs

Implied forwards

Floating cashflows

[8]

[9]

[10]

IBOR

— 10,000,000,000

1 0.999630 0.999103 0.998327 0.997234 0.995945 0.994373 0.992428 0.990245 0.987814 0.985217 0.982278 0.978949 0.975291 0.971386 0.967133 0.962496 0.957680 0.952584 0.947162 0.941483

0.156% 0.219% 0.314% 0.429% 0.565% 0.696% 0.822% 0.942% 1.058% 1.187% 1.331% 1.489% 1.660% 1.820% 1.968% 2.105% 2.229% 2.351% 2.471% 2.588%

13,918,994 15,496,672 18,107,963 21,512,513 23,458,858 27,561,504 31,748,554 33,781,681 36,701,202 39,979,224 43,610,160 47,596.792 51,938,473 55,980,239 59,724,207 63,866,683 66,321,820 69,409,317 73,231,329 10,076,228,380

PV (Yen) =

259,595,013

Net PV =

0.0000

224

Swaps and Other Derivatives

Which approach is more "correct"? Probably (although it is not clear-cut to me) the first as it uses a tradable spot instrument, but the second is much easier to implement within a system, and to apply to a portfolio. The vast majority of CCBSs have a USD Libor leg. This is reducing in a similar way to the spot FX market, namely crosses are becoming more common as Europe and the Far East increasingly use Euro and Yen reference rates respectively instead of USD. The usual convention is to follow that of the USD market, namely a two-day settlement period. This can cause some small estimation difficulties if the other currency has a different settlement period. For example, if the currency was SA Rand with same day settlement, then the Rand leg of the CCS is strictly a forward-start contract. All the discount factors implied from the SA market would be same day, and therefore have to be adjusted for the two-day period.

6.4 FIXED-FLOATING CROSS-CURRENCY SWAPS Whilst CCBSs are fundamental financial instruments, some 75% of the CCS market is fixed-floating, in many cases as above originating from swapping a bond issue into a floating reference currency. When swapping a bond issue, the issuer requires the entire bond structure to be reflected in the swap, so that there is no residual exposure to the issuance currency. Some typical examples of the types of manipulations will be discussed through an example, which is based upon a real bond issue by a German bank in the late 1990s. The issuer had a target to: • raise USD 40m, • at Libor —25 bp or better. Notice that the bond was issued with a coupon below the current swap rate. This was possible because, being a German issuer, it was sold into the retail base in Germany, Switzerland and Benelux who cannot access the SA swap market very efficiently:

date: 7 June 1999 Details of bond issue Size: Term: Coupon: Issue price: Fees: Expenses: Payment:

ZAR250m 5 years 13.75% ann 30 360 101.25 2% 0.25% 4 weeks from issue

Details of swap Type: All-in swap:

ZAR fixed, US$ floating 13.935% qu Act/365

Current spot exchange rate:

1US$ = ZAR6.108

Cross-currency Swaps

225

There are a number of issues to be dealt with. 1. Bond proceeds are raised 4 weeks after issue • therefore the initial exchange has to be delayed although the swap starts today. 2. Net bond issue at 101.25–2.25 = 99 • i.e. raises ZAR247.5m • but of course has to repay ZAR250m on maturity • so there is a principal mismatch. 3. Issuer only wishes to raise USD40m • although at the current spot rate ZAR250m is worth USD40.93m • hence long USD0.93m which could subsidize the margin. 4. Bond proceeds are raised 4 weeks after issue • but the German bank wants USD40m immediately the bond issue is underwritten • effectively requiring a 4 week loan. The issuer wants all these taken into account in the swap structure, and to receive an estimate of the sub-Libor margin. As a starting point, assume that the bond is issued today (7 June 1999) at par. The bond has a value of ZAR8,060,748 or USD 1,319,703 as shown in Box 1 of Worksheet 6.4. This is equivalent to a USD stream based on a principal of USD250m/6.108 = USD40.93m. The fair margin is 75 bp pa below USD 6m Libor. Is this correct? The current 5 year ZAR swap rate is 13.935% quarterly. This converts into 14.680% using the quick and dirty formula (1 + ±rqu)4 = 1 +r ann or 14.700% if the discount curve is used; see Box 2. The bond has therefore been issued at 14.70% — 13.75% = 95bp below SA curve. 1 bp annual coupon on the SA side is worth (DF] H r DFs) = 3.39 bp upfront. The equivalent calculation of USD 1 bp sa is worth Q5=4.30 bp. Hence ZAR 95 bpx (3.39/4.30) = USD 75 bp sa so the margin appears correct. The ratio (3.39/4.30) = 0.789 is called the "conversion factor" and indicates how much 1 bp in ZAR is worth in USD. Conversion factors depend upon the two curves and the maturity, and as such change frequently. Their main use is to calculate the impact of changes in the issuance level of the bond on the funding margin. However, the principal exchange does not happen for 4 weeks: what is the impact? This may be estimated in two parts: • late receipt of ZAR250m costs the swap c/p ZAR250m x (1 — DF1 month ) =250m x (1-0.9894) = Z2,645,015 = $433,041 • late payment of $40.93m benefits the swap c/p $40.93m x (1 –0.9962) = $154,392 The net balance of $278,650 is a cost to the c/p. The value of 1 bp on 40.93m = 40.93m x Q5 x 1 bp = $ 17,607; therefore the balance corresponds to a margin of 278,650/17,607 or 15.8 bp. Box 3 confirms this result. The principal amounts are now exchanged on 5 July 1999. The value of the Rand cashflows has reduced quite significantly, and this is compensated by a smaller sub-Libor margin on the USD side. However the issuer only wants to raise $40m, not $40.93m, i.e. a reduction in the USD principal of 2.3%. As the SA leg hasn't changed, the USD margin should increase by the same amount, i.e. from 59 bp to 59 x (1 +2.3%) = 60.4bp. Note: this is only approximate

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as principal exchange is not at start of swap. Box 4 confirms this argument, increasing the margin to 60.3 bp. Next the issuer wants the $40m upfront, although he will not have ZAR bond proceeds for 4 weeks. The cost of this effective loan to the swap c/p is $40m x (1-DF 1 month) = $150,884. The value of 1 bp on 40m = 40m x Q5 x 1 bp = $ 17,207, therefore this cost is equivalent to a reduction in the margin of 150,884/17,207 —8.7 bp, as shown in Box 5. Finally the issuer does not receive par value of bond, but only 99%, i.e. Z247.5m. The cost to the swap c/p of being 1 % short in 4 weeks' time is Z250m x 1 % x (l-DF 1 month ) = Z2,473,550 or $404,969. This will reduce the margin by 404,969/17,207 = 23.54bp; see Box 6. The objective of this section was to demonstrate how a cross-currency swap may be manipulated, and in particular how the impact of actions may be quickly verified.

6.5 FLOATING-FLOATING SWAPS (CONTINUED) As we have already described, investors at the short end of the US curve were having a torrid time in 1993, especially as the year drew to a close, rates fell and no increase was anticipated. The same investors saw a very different picture in Germany. The need to fund reunification and to support the Deutschmark had inverted the curve, with short-term money rates at about 9%, some 600 bp above the US ones. But how could they take advantage of this situation? The obvious way is to liquidate the US investment, convert the money into DEM and reinvest. However, the same outcome may also be achieved by entering into an at-money CCBS to pay US Libor and receive DEM Libor (plus or minus a small margin which will be ignored in the following discussion). Because the initial exchange is done at the current spot rate, it has no economic value and can be omitted; the spot rate being used to determine the relative principals for the interest payments and the final exchange. The cashflows are as shown in the diagram below: USD Libor

DEM Libor US Investor

USD Principal at end

DEM Principal at end

USD Libor

USD Principal at end

230

Swaps and Other Derivatives

A similar situation today from a Japanese investor's point of view would be looking across at USD. The current forward curves are some 600 bp apart at the short end, closing to 500 bp at the longer end, as shown in Figure 6.3. Again the investor could pay away Yen Libor plus Yen principal at the end, and receive USD Libor plus USD principal. The Japanese investor is now receiving a higher rate of interest, but has entered into two sources of FX exposure. 1. Exposure on the principal exchange at the end. Because US interest rates are higher than Yen rates, theoretically the USD would be expected to weaken against the Yen during the swap, and therefore the USD principal received at maturity is likely to be worth less than the Yen principal paid away. 2. Exposure on the periodic payments of interest. It is likely, unless the swap has a very long maturity, that the exposure on the principal is greater than on the interest receipts. Two swap structures were devised to remove the FX exposures and make the transactions rather cleaner for the investor. The first were called "diff" swaps, or CCBS with no exchange of principals. This removed the first exposure to the investor, but of course the counterparty would demand compensation in the form of a margin, in this case deducted from the USD side. Worksheet 6.5 prices a 5 year Yen-USD diff swap. It is in three parts; columns [1] and [2] estimate the implied Yen Libor rates off the unadjusted curve; columns [3] and [4] value the Yen side without any principal cashflows off the adjusted discount curve; columns [5] to [7] construct the USD cashflows without principals, and discount them off the USD curve. The fair margin is 547 bp under the USD curve, representing the (theoretically) anticipated strengthening of the Yen. See Figure 6.3. Diff swaps still expose the investor to some FX risk. To eliminate all FX risk, "quanto diff" swaps were devised. From the investor's point of view, these are extremely simple, i.e.: to pay Yen Libor on PY, to receive ($ Libor —margin) on a Yen principal PY. Because the currency of the cashflow is determined by the principal, this is really a single currency swap, albeit using a foreign USD reference curve. Worksheet 6.6 has the same

USD Libor —— Yen Libor - - - • USD Libor-margin

0.5

1.5

2.5

3.5

Figure 6.3 USD-Yen diff swap curves

4.5

Cross-currency Swaps

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Worksheet 6.6 Pricing a 5 year Yen-USD quanto diff swap Today's date = 04-Jan-00 Current spot rate =

102.985

Principal amount Principal amount Maturity To pay To receive

l0bn Yen 97.10m USD 5 years 6mo. Yen Libor 6mo. USD Libor+

-572.11 bp 2. USD-related receive side

1. Yen pay side Daycount

Yen IBOR unadjusted dfs [1]

06-Jan-OO 06-Jul-00 08-Jan-0l 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511

PV(Yen) on l0bn nominal = Net PV =

1 0.999053 0.997133 0.994009 0.989525 0.983934 0.976959 0.968420 0.958470 0.947467 0.935334

Implied forwards [2]

0.187% 0.373% 0.632% 0.882% 1.124% 1.412% 1.744% 2.042% 2.297% 2.538%

Yen IBOR adjusted dfs [3]

1 0.999103 0.997234 0.994373 0.990245 0.985217 0.978949 0.971386 0.962496 0.952584 0.941483

Yen Libor cashflows [4]

-9,479,167 -19,252,583 -31,434,647 -45,313,971 -56,821,965 -71,394,851 -88,173,133 -103,810,274 -116,127,412 -129,719,669 -648,990,813

USD IBOR dfs [5] I 0.969519 0.936853 0.905235 0.873099 0.842475 0.812962 0.784581 0.756987 0.730306 0.704190

Implied forwards [6]

USD-related Libor cashflows [7]

6.219% 25,157,632 6.749% 53,081,848 7.025% 64,816,042 7.162% 74,062,766 7.190% 74,269,797 7.181% 73,800,626 7.155% 72,494,543 7.171% 73,704,172 7.226% 76,098,735 7.256% 78,450,174 648,990,813 0

233

Cross-currency Swaps

7 6 5

USD Libor Yen Libor

g 4 3 2 1

USD Libor-margin

0 0.5

1,5

1

2

2.5

3

3.5

4

4.5

Figure 6.4 USD-Yen quanto diff curves Table 6.2 Swap CCBS USD margin (bp) Risks IR FX on principal FX on interest

Diff

Quanto diff

-547bp

-572 bp

structure as the previous one, but applies the USD Libor rates to the Yen principal. The fair margin is 572 bp. This is higher than before as expected, but only 25 bp which confirms the earlier statement that the FX exposure on the principal is likely to be greater than on the interest cashflows. Table 6.2 summarizes the position. Notice the difference between the diff and the quanto diff curves in Figures 6.3 and 6.4 respectively. In the former, the investor is still paying away the stronger currency, and hence the USD curve less margin is mainly above the Yen curve. But with the quanto swap, the Yen and {USD —margin} curves are on average equal, and the investor is effectively taking a view on the relative movement of them. If they are roughly parallel, then there is no immediate return enhancement, and hence little demand. If the curves are substantially non-parallel, as was the case in 1993, then the US investor would receive a return enhancement albeit at the expense of potential losses later. More recently in 1995, the Japanese-Australian curve spread was wider at the short end than the long end, and there was some investor interest. Now consider the quanto diff swap from the point of view of the swap counterparty. How might he hedge it? The Yen Libor side and the margin are routine and will be ignored:

USD Libor on PY

Swap counterparty

->• sell USD futures

234

Swaps and Other Derivatives

Assume it is to be hedged by selling USD deposit futures. If USD rates rise, the USDrelated payment on the swap increases, but the futures margin is received. Conversely, if rates fall, the swap payment is less but margin has to be paid. But of course the swap payments are in Yen, whilst the margin is in USD. So the hedge will off-set fluctuations in the swap payments, but in the wrong currency, i.e. the counterparty has an FX exposure in addition to the interest rate exposure4. This is difficult to hedge in any static fashion because its size and direction depend on the movement in interest rates. Furthermore, FX rates and interest rates are generally correlated, so shifts in the correlation structure will impact the hedge efficiency. The normal practice is to dynamically hedge in a similar fashion to FX options, but recognizing that this carries a substantial amount of basis risk which should be reflected in the pricing. A revised formula and some sample results are outlined in the Appendix to this chapter; note that for the example above, the margin is further increased indicating the possible impact of these effects.

6.6 FIXED-FIXED CROSS-CURRENCY SWAPS Conceptually these are very straightforward, simply consisting of two known cashflows in different currencies which have a net value of zero. Each cashflow may be derived by reference to a fixed rate of interest, i.e. 6% Act/360 annual on $100 million with or without principal cashflows, or simply be a stream of cashflows possibly determined by some other activity. They are widely used, but very often under the guise of a long-term FX forward contract (LTFX). Consider a normal FX outright contract such as the one discussed above: 6mo. FX out of 4 Jan 2000: to sell $100 million on 6 Jul 2000, and to buy Y9.9957 billion. This was priced theoretically off the two money market curves, and the skew estimated to be 7.99 bp on Yen Libor. The structure of the outright, using swap terminology, is two zero coupon legs with bullet payments at the end of each. But when the maturity of this outright is increased beyond 12 months, zero coupon cash rates seldom exist and we must resort to swap techniques. For example, a 5 year LTFX has the following structure: USD

Yen

0 0 0 0

0 0 0 0

0 +1

0 -S5

Today

Year 5

4

Hardly surprising as the exposure was removed from the investor, and had to go somewhere.

Cross-currency Swaps

235

namely, to buy USD and to sell Yen in 5 years' time at the rate of Yen S$ per USD. If SQ is the current spot rate, then 1 x df* x SQ — SS xdf^ = 0, i.e. S5 = SQ x d f I f d f } where the Yen discount factor is off the CCBS-adjusted curve. Using the 5-yr DFs from Worksheet 6.6, this gives: S5 = 102.985 x 0.704190/0.941483 = 77.028 Pricing LTFX and similar structures is relatively simple. Because they are often very large one-off transactions, they may be priced and hedged as a single deal rather than merely managed within a portfolio. A technique that is widely used is as follows. First consider the USD side on its own: we will be receiving (say) $1 million in 5 years' time as shown in column [2] of Worksheet 6.7. What transactions can we do today that will create a matching liability in 5 years? (a) Suppose we borrow some money $/*s at, given we are still a US bank, $ Libor flat. (b) And simultaneously enter into a 5 year swap to receive floating, and to pay $jF5 = 7.135% annual Act/360 — this effectively converts the borrowing from floating to fixed. (c) The amount to be borrowed is: $P5 = $1 million/(l + 1.017 x 7.135%) = $932,367 where (6 Jan 05-6 Jan 04)/360= 1.017. At the end of the last period, the liability is $^5x0 + 1.017 x7.135%) = $lm. (d) Of course, interest has to be paid in each of the earlier periods; this gives rise to negative cashflows as shown in column [3]. (e) The net effect is in [4], where it can be seen that the 5 year cashflow of the LTFX has now been reduced to an upfront transaction plus only four future cashflows. (f) The steps are now repeated: as the fourth net cashflow = —$67,448 is negative, we deposit $67,448/(l + 1.014 x 7.085%) = $62,928 which generates principal plus interest receipts in the last period which exactly off-set the fourth cashflow: see column [5], and so on. At the end of the process, we have effectively entered into five money market transactions and five IR swaps with differing maturities. The net amount of money to be borrowed upfront is $704,190 (see column [12]) which is of course equal to $lm x dfi In practice, it would work somewhat differently, and Worksheet 6.8 demonstrates this. The swap principals may incidentally be estimated rather more easily than above: • create a matrix A such that {a;y = 0 if / < /, S1/ x dj if /' >j, and 1 + S1, x d,- if /=/'}, where S/ is the /th swap rate, dj the length of the /th period; • P — A-1CF where CF is the vector of original USD cashflows arising from the LTFX (column [2]). This result is shown in column [3] of Worksheet 6.8. We can replicate the transaction as follows, see Box 1: • Borrow $704,190 at 12mo. Libor for 5 years and simultaneously enter into the swaps. At the end of the first year:

236

Swaps and Other Derivatives

Worksheet 6.7 Pricing a USD-Yen LTFX: analysing the USD leg (I) Today's date: 4-Jan-OO Current spot

102.985

USD Daycount mid-market rates act/360 0 1.022 1.011 1.011 1.014 1.017

6.59375% 6.8950% 7.0250% 7.0850% 7.1350%

Upfront transaction 5-year NetCF [3]

[2]

[1]

6-Jan-OO 8-Jan-Ol 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05

USD cashflow

1,000,000

+ 932367 -68,003 -67,264 -67,264 -67,448 -1,000,000

Upfront transaction 4-year Net CF

[4]

932,367 -68,003 -67,264 -67,264 -67,448 0

[6] 869,439 -63,445 -62,756 -62.756 0

[5] -62,928 4,558 4,508 4,508 67.448

Worksheet 6.8 Pricing a USD-Yen LTFX: analysing the USD leg (II)

Daycount act/360

USD mid-market rates

USD DFs

volatility => 6-Jan-OO 8-Jan-Ol 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05

0 1.022 1.011 1.011 1.014 1.017

6.59375% 6.8950% 7.0250% 7.0850% 7.1350%

1 0.936853 0.873099 0.812962 0.756987 0.704190

USD forwards

1% [1]

6.59375% 7.26388% 7.30190% 7.37911% 7.19415%

Matrix A for calculating the hedging swaps

1 6.5938%

2 6.8950%

3 7.0250%

4 7.0850%

5 7.1350%

1.067 0.000 0.000 0.000 0.000

0.070 1.070 0.000 0.000 0.000

0.072 0.071 1.071 0.000 0.000

0.072 0.072 0.072 1.072 0.000

0.073 0.072 0.072 0.072 1.073

237

Cross-currency Swaps

Upfront transaction 3-year [7]

Upfront Upfront transaction transaction 1-year Net CF Net CF 2-year Net CF [8]

[10]

[9]

-58,594 810,845 4,208 -59,238 4,162 -58,594 62,756 0

-54,775 756,070 3,861 -55,377 58,594 0

[12]

[11]

-51,880 704,190 55,377 0

Swap transactions

[13] to receive fixed -54,775 to receive fixed -58,594 to receive fixed -62,928 to receive fixed to pay fixed 932,367 -51,880

Note: the forwards may be randomly simulated by pressing F9 1. Money account transactions USD cashflow [2]

0 0 0 0 1,000,000

Swap transactions [3]

-51,880 -54,775 -58,594 -62,928 932,367 704,190

to to to to

receive fixed receive fixed receive fixed receive fixed to pay fixed

Initial money account

Surplus from swaps

[4]

[5]

704,190 -47,464 -51,720 -51,991 -52,685 -755,695

Money account reinvested

[6]

-4,416

755 1,271 2,120

561

-51,880 -106,655 -165,249 -228,176 -1,000,000

238

Swaps and Other Derivatives

Worksheet 6.9 Pricing a USD-Yen LTFX

Daycount Act/360

Unadjusted market rates Daycount Yen IRS Act/365 sa Act/365 [4]

6-Jan-00 6-Jul-00 8-Jan-0l 6-Jul-0l 7-Jan-02 8-Jul-02 6-Jan-03 7-Jul-03 6-Jan-04 6-Jul-04 6-Jan-05

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511

0 0.499 0.510 0.490 0.507 0.499 0.499 0.499 0.501 0.499 0.504

Yen DFs volatility =» [5]

0.19010% 0.28488% 0.40109% 0.52500% 0.64629% 0.77500% 0.91379% 1.05500% 1.19151% 1.32500%

1 0.999053 0.997133 0.994009 0.989525 0.983934 0.976959 0.968420 0.958470 0.947467 0.935334

Yen forwards 1% [6]

Yen cashflow [7]

0.18750% 0 0.37663% 0 0.62973% 0 0.87176% 0 1.13043% 0 1.39623% 0 1.78725% 0 2.11259% 0 2.27253% 0 2.48333% -76,963364

Estimated forward

16.963

Actual forward off adjusted curve

77.028

Theoretical forward

77.535

239

Cross-currency Swaps

Note: margin may be adjusted Libor Margin

(bp) -14.5 2. Yen money account transactions

Initial money account

Swap transactions

[9]

[8]

490,263 501,513 483,339 500,523 493,717 495,308 497,222 502,232 502,129 -76,452,886 to

to pay to pay to pay to pay to pay to pay to pay to pay to pay receive

fixed fixed fixed fixed fixed fixed fixed fixed fixed fixed

-72,521,048 15,582 86,790 174,788 270,846 361,292 458,744 602,105 725,348 780,024 73,387,780

Surplus from Money account swaps reinvested

[10]

[11]

422,025 360,478 254,833 171,424 71,026 -30,257 -180,003 -308,008 -370,460 -459,720

437,607 885,399 1,317,155 1,764,344 2,205,452 2,647,891 3,091,976 3,540,242 3,987,884 76,963,605

240

Swaps and Other Derivatives

•

interest has to be paid based upon the current 12mo. rate of 6.59375%: this is $47,464 — see column [4]; • the swaps will generate a surplus or deficit — in this case a deficit of only $4,4165 in column [5]; • therefore there is a total cash shortfall of $51,880 as in [6], which will be funded by a new Libor borrowing. • The new Libor rate is also fixed at the end of the first year. This is of course currently unknown, Worksheet 6.8 calculates a forward rate from the implied curve using: F,.\/2 = Fo.i/2 cxp{aVte] whereFt, 1/2 is the 1/2 forward rate observed at time /, a the annualized volatility, and e a random sample from a unit normal. The new forward curve is in column [1]. The worksheet on the CD will permit the Libor curve to be randomly simulated, to demonstrate the hedge working under a range of situations. At the end of the second year: • as before, interest has to be paid based on F1, 1/2 • the new swap surplus or deficit generated; • the cash shortfall rolled over. Notice that the total cash shortfall at this point is constant; as F1, 1/2 changes, fluctuations in the interest payments are exactly off-set by the cash generated by the swaps. At the end of 5 years, the total shortfall is $1 million, i.e. precisely matching the inflow from the LTFX. The above discussion has assumed mid-swap rates, and all borrowing and lending takes place at Libor flat. Very often bid-offer spreads are included in the swap rates, especially if the transaction is being hedged at arm's length. It is quite simple to modify the calculations accordingly. We have agreed that we needed to borrow $704,190 upfront to create a liability which exactly off-sets the $1 million that will be received in 5 years' time. But what shall be done with the borrowed money? We can enter into a spot FX transaction to sell the USD and receive Yen S0X 704,190 = 72,521,048, and then deposit these proceeds using the Yen money market. Using the same technique, Yen IRSs may be used to guarantee the value of the asset in 5 years' time — see column [8] of Worksheet 6.9 and also Box 2. The quoted forward rate is therefore: S$ = Value of Yen asset in 5 years' time/Value of USD liability in 5 years' time If it is assumed that the Yen deposit will earn Libor flat, then S$ = 77.535 calculated either using the method above or directly from SoxDFs/DF^ using of course the unadjusted Yen DFs. The structure of the transaction plus hedge is shown diagrammatically below. The spot and forward transactions are shown on the outside, then the two sets of IRS hedges, and 5 For the kth period, given by d'k x Lk £ P, - dk ^2(PjS}) where d'k and dk are the lengths of the kih period on the floating and fixed side of the swaps which may differ due to daycount conventions, and the summation fory ^ k.

241

Cross-currency Swaps

finally the two money-market transactions are shown in the middle. But we know that back-to-back money market transactions can be replicated by a structured CCBS, which is effectively rolled over each period. Hence the total hedge for an off-BS forward would also off-BS. USD spot sale of 704, 190

$ IR swaps

USD forward receipt of 1,000,000

USD borrowing of 704, 190

Yen deposit of S0x704,190

periodic $ Libor interest payments

periodic Y Libor interest receipts

USD liability of 1,000,000

Yen spot buy ofS 0 x704,190

Yen IR swaps

Yen asset of S5 x 1 ,000,000

Yen forward payment of S x 1 ,000,000

However, we know from the 5-year CCBS market that a bank capable of raising money at USD Libor flat would pay (a mid-spread of) 14.5bp below Yen Libor. Incorporating this reduces S$ to 76.963 — the margin may be entered into the worksheet as indicated. In these circumstances the first CCBS transaction is: to receive a USD principal of 704,190 to pay a Yen principal of 72,521,048 and to pay $ Libor and to receive Yen Libor—14.5 bp respectively; the cashflows are shown in column [4] of Worksheet 6.8 and column [11] of Worksheet 6.9. The period cashflows plus the surpluses or shortfalls from the two IRS strips, see columns [6] and [11], are then also paid into CCBSs all of which mature on 6 Jan 2005. The overall outcome is a USD liability of 1 million and a Yen asset of 76,963,605. Notice that there is a very small difference between the Yen asset used to estimate the size of the IRS hedge and the resulting balance on the money market account of some Yen 200. This is because the argument here is circular, and only converges to within a small error. If the adjusted Yen discount factors are used in SQ x DFs/DF^, as discussed above, we get a very similar result, namely 55 = 77.028. The latter is, as before, a very quick method for pricing LTFXs whilst still reflecting the relative costs of funds.

6.7 CROSS-CURRENCY SWAP VALUATION This is very similar to interest rate swap valuation, namely each side of the swap is valued separately in its own currency in the usual fashion; these values are then netted by

Swaps and Other Derivatives

242

Worksheet 6.10

Cross-currency swap valuation: original data 4-Jan-00

Today's date:

102.985

Current spot rate

100m 10,298.50m - 14.50 bp

USD principal JPY principal CCBS margin

6-Jan-00 6-Apr-OO 6-Jul-00 6-Oct-OO 8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

USD 3mo. forwards

JPY 3mo. forwards unadjusted

JPY 3mo. forwards adjusted

Daycount

USD DFs

6.03125% 6.31005% 6.58848% 6.79081% 6.91540% 7.01185% 7.07921% 7.11667% 7.12506% 7.12698% 7.12240% 7.11141% 7.09400% 7.08904% 7.09671% 7.11691% 7.14977% 7.17334% 7.18756% 7.19237%

0.15625% 0.21866% 0.31421% 0.42972% 0.60013% 0.69606% 0.82089% 0.94283% 1.11946% 1.18908% 1.32997% 1.49164% 1.72168% 1.82832% 1.97756% 2.12306% 2.25882% 2.38575% 2.50962% 2.62937%

0.14625% 0.20866% 0.30421% 0.41972% 0.52964% 0.62557% 0.75039% 0.87234% 0.97319% 1.04281% 1.18370% 1.34537% 1.48375% 1.59038% 1.73962% 1.88512% 1.98946% 2.11640% 2.24026% 2.36002%

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

0.984983 0.969519 0.953465 0.936853 0.921280 0.905235 0.888806 0.873099 0.857653 0.842475 0.827576 0.812962 0.798640 0.784581 0.770755 0.756987 0.743549 0.730306 0.717134 0.704190

243

Cross-currency Swaps

Original Valuation JPY DFs adjusted

[2]

[1] 1 0.999630 0.999103 0.998327 0.997234 0.995945 0.994373 0.992428 0.990245 0.987814 0.985217 0.982278 0.978949 0.975291 0.971386 0.967133 0.962496 0.957680 0.952584 0.947162 0.941483

Value =

JPY cashflows

USD cashflows

100,000,000 -1,524,566 -1,595,040 -1,683,723 -1,773,156 -1,690,431 -1,772,439 -1,848,462 -1,798,937 -1,801,057 -1,801,541 -1,800,383 -1,797,606 -1,793,206 -1,791,953 -1,793,890 -1,818,765 -1,807,304 -1,813,260 -1,836,821 -101,838,051

-10,298,500,000 292,785 1,917,556 4,453,306 7,656,171 11,457,369 14,345,319 18,174,901 20,769,271 25,367,453 27,179,633 30,847,468 35,056,107 41,044,635 43,820,575 47,705,652 52,059,142 55,027,519 58,331,971 62,232,876 10,363,884,603

0.0000

0.0000

244

Swaps and Other Derivatives

Worksheet 6.11

Cross-currency swap valuation: new market data

Historic data for simulation FX vol =»

20%

See correlation sheet for further data USD forward volatilities

JPY forward volatilities

Uncorrelated normal rv

Correlated normal rv

Length of time =>• 9.715% 10.086% 10.975% 12.493% 13.695% 14.552% 15.038% 15.126% 15.359% 15.734% 16.260% 16.942% 17.288% 17.294% 16.955% 16.261% 15.729% 15.356% 15.145% 15.101%

14.572% 15.128% 16.462% 18.740% 20.543% 21.828% 22.556% 22.689% 23.038% 23.601% 24.390% 25.414% 25.932% 25.941% 25.432% 24.392% 23.593% 23.034% 22.718% 22.651%

8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02

6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

-1.0771 -0.7456 1.6893 0.1852 0.1080 -1.4439 0.6959 -0.4691 1.1274 0.1670 -1.0770 -0.4167 -0.5908 0.5553 0.4868 -0.9683 1.6720 -1.4595 -0.3065 0.0469

-1.0771 -1.1041 1.1308 0.6081 0.3348 -1.2021 0.1769 -0.3642 0.8988 0.5016 -0.7992 -0.6933 -0.8130 0.1977 0.5254 -0.6898 1.2752 -0.8528 -0.6124 -0.1936

new USD 3mo. forwards

new USD DFs

1.02

1

7.24352% 5.87524% 7.27217% 6.73109% 8.19226% 7.71903% 6.24546% 6.31505% 6.15426% 7.33839% 7.76548% 6.35392% 8.75707% 6.28376% 6.54416% 6.98293%

0.982602 0.968222 0.950180 0.934283 0.915329 0.897810 0.883857 0.869970 0.856643 0.841042 0.824851 0.811671 0.794093 0.781677 0.768819 0.755340

245

Cross-currency Swaps

97.43

New spot FX rate =

Uncorrelated normal rv

Correlated normal rv

new JPY 3mo. forwards unadjusted

new JPY 3mo. forwards adjusted

1.02

0.0445 0.6740 -1.0271 -0.7523 1.0162 0.0774 -1.6875 0.9845 1.3985 0.9371 -1.7851 -0.4906 -0.3837 0.0881

1.6914 -0.3475 0.0150 -0.9308 -1.2080 -0.7925

0.0445 0.6284 -0.6600 -0.9622 0.5074 0.2794 -1.4112 0.3006 1.3902 1.4349 -1.0114 -0.8656 -0.7148 -0.2232 1.4378 0.2885 0.1391 -0.7847 -1.4248 -1.3198

0.66683% 0.74034% 0.59500% 1.01014% 1.54755% 1.67461% 1.03639% 1.19418% 1.42746% 1.72438% 2.86216% 2.27958% 2.33503% 1.98730% 1.80919% 1.94349%

0.58850% 0.66536% 0.54390% 0.93462% 1.34535% 1.46862% 0.92241% 1.07708% 1.23019% 1.49997% 2.51779% 2.02411% 2.05659% 1.76293% 1.61501% 1.74440%

New Valuation

new JPY DFs adjusted

USD cashflows

JPY cashflows

[3]

[4]

1 -1,770,638 0.998564 -1.485,131 0.996887 -1,898,844 0.995473 0.993127 -1,701,470 -2,070,821 0.989761 -1,951,200 0.986100 -1,578,712 0.983806 -1,596,305 0.981135 -1,555,660 0.978093 -1,854,981 0.974399 -1,962,941 0.968237 -1,623,779 0.963254 -2,213,593 0.958272 -1,588,394 0.954021 -1,672,396 0.950100 0.945883 -101,784,527

16,786,866 19,272,659 15,999,820 26,296,326 40,286,412 43,594,098 26.979,590 31.087,382 37,160,128 44,889,638 74,508,574 59,994,943 60,786,263 51,733,985 47,615,067 10,349,649,626

Value (Yen) = Value (USD) = Net value (USD)

10,370,345,039 -100,000,000

106,435,029 6,435,029

246

Swaps and Other Derivatives

converting into a single currency using the current spot FX rates. In theory, either the notional principal or the implied forward method may be used to value the floating side if there is one. If an adjusted foreign curve is being used for discounting, then only the implied method is appropriate. For example, Worksheet 6.10 values a 5 year CCBS which was originally traded at: to pay 3mo. USD Libor to receive 3mo. Yen Libor— 14.5 bp Discounting off an adjusted curve, this swap initially has a zero value, as shown in columns [1] and [2] of the worksheet. Worksheet 6.11 then simulates what might happen after 1 year. It takes the two existing forward curves, and randomly simulates them using the formula: F(r,t,T) = F(0,/,r)exp{cr,y?£,} where T is the length of time moving forward, a is the forward rate volatility (actually taken off a cap curve—see Chapter 7), £ is a unit normal random variable6, as well as the spot rate. Finally, the new cashflows are calculated in columns [3] and [4], note that the valuation is being done on 8 January 2001, but immediately after the Libor cashflows on that date have been completed. The two sides of the swap are then discounted and the Yen side converted into USD at the current spot rate. The USD side, being both estimated and discounted off the same curve, is always valued at $100 million, but the value of the Yen side fluctuates. However CCS are often treated differently to IRS in one important aspect. The potential credit exposure of a CCS is much higher than an IRS due to the large reexchange of principals at the end, which of course an IRS does not possess. This is recognized in the old Basel Accord, which requires five times as much capital for a CCS with a maturity greater than 1 year as for an equivalent IRS. Therefore many CCS are traded on the basis that the principals will be adjusted to new current spot rate at regular intervals, such as annually7. Consider a simple generic CCBS with no margin, as shown below. It will initially be assumed that both estimating and discounting are off the same Yen curve: Time 0

USD + l00m

3 6 9 12 15 18

60

6

JPY -10,298.5m + LY + Z.Y

+ LY

-L$- 100m

+LY+ 10,298.5m

The worksheet actually uses correlated sampling. First a vector of independent unit normal random variables is generated, and then a correlated vector t = \- where A-A' = correlation matrix — see Chapter 9 for more details. 7 The Accord has a cut-off, whereby a 1% capital charge is imposed for up to and including 1 year, and 5% beyond. Hence a long swap with annual revisions will, in theory at least, only carry a 1 % charge. However the regulators are somewhat wary of this, and usually demand more than a paper revision.

Cross-currency Swaps

247

At the end of year 1, the current value of the Yen side is simply Yl 0,298. 5 million. But the spot rate has shifted from SQ = Y 102.985 to Si = Y96.95, therefore valuing the Yen side at ${10,298.5 million/96.95} = $106,226,370. The swap has a positive net value of $6,226,370, which is a credit exposure — see column [1] of Worksheet 6.12. More generally, the net value is given by: ${(S 0 /S,)-l}x/» $ The swap could therefore be settled by paying this amount, and restarted at the new exchange rate by (paying YSj x P$, receiving P$}. Equivalently, by simply receiving the payment of Y{So — S\} x P$, the swap principal is now rebalanced at the new exchange rate as shown in column [2]. This process could be repeated each year, receiving Y{S,_i — St} x P$, and rebalancing the swap to the new exchange rate. When a swap has a fixed rate or a margin, then the process is not quite so simple. If the margin of — 14.50bp was included, but the discounting and estimation were still done off the same curve, then the value of the swap is lower. To rebalance, the present value of the margin on the change in principal, i.e.:

Y{S0 - S1} x P$ x (- 14.50 bp) x d, (see column [3] of Worksheet 6.13) has to be included in the payment to be made. Finally, if the valuation uses different estimation and discounting curves, then a further allowance has to be made for this. Worksheet 6.14 shows the actual valuation of the Yen cashflows, including the margin, off the adjusted curve to be Y10,308,847,502 or at the new spot rate $106,333,102. This may be replicated by an upfront payment of: -S1]} x P$ = Y603,637,984— see column [2]. PVadjusted of Y{S0-S1} x / » $ x ( - 14.50 bp)xrf,= -Y3,467,056 as shown in column [3]. 3. The change in value due to the use of the adjusted rather than the unadjusted curve, this is calculated by: (a) PVadjusted — PVunadjusted of {Yen cashflows using new principal S1 x P$}; (b) PVadjusted-PVunadjusted of {Y{S0 - SJ x P$ x (-14.50 bp) x d,}(c) PVadjusted — PVunadjusted of {Yen cashflows using old principal S0 x P$). The total change in value = (a) + (b) - (c) = -Y4,073,566. This gives a net receipt of Y603,637,984- 3,467,056 + 4,073,566 = Y604,244,494. This receipt plus the rebalanced cashflows are shown in column [4].

6.8 DUAL CURRENCY SWAPS Investing requires a judicious balance between return and risk, whilst issuance is almost invariably about raising money as cheaply as possible. Securities are structured to meet the risk-return requirements of a group of investors, but almost inevitably swapped into simple debt for the issuer. Dual currency issues are a perfect example of this. Consider the dilemma of Japanese investors since the crash of the Nikkei in 1989. Equity has given very poor returns, and the 10 year benchmark bond yield has been

248

Swaps and Other Derivatives

Worksheet 6.12

Cross-currency swap valuation: new market data

Historic data for simulation FX vol =»

20%

USD principal JPY principal

100m 10,298.50m

Daycount

new USD 3mo. forwards

Length of time =>

1.02

0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

6.34400% 6.09060% 4.92515% 6.53014% 5.73057% 5.81287% 7.39951% 6.76986% 9.68864% 6.05924% 6.76319% 7.00372% 8.07383% 7.12637% 7.36942% 7.92493%

See correlation sheet for further data USD forward volatilities

JPY forward volatilities

new USD DFs

6-Jan-OO 9.715% 10.086% 10.975% 12.493% 13.695% 14.552% 15.038% 15.126% 15.359% 15.734% 16.260% 16.942% 17.288% 17.294% 16.955% 16.261% 15.729% 15.356% 15.145% 15.101%

14.572% 15.128% 16.462% 18.740% 20.543% 21.828% 22.556% 22.689% 23.038% 23.601% 24.390% 25.414% 25.932% 25.941% 25.432% 24.392% 23.593% 23.034% 22.718% 22.651%

8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

1

0.984729 0.969799 0.957485 0.941937 0.928487 0.915042 0.898241 0.883128 0.862017 0.849013 0.834742 0.820064 0.803663 0.789442 0.774849 0.759468

249

Cross-currency Swaps

Old spot FX rate = New spot FX rate =

102.985 96.95

New Valuation

2. Rebalanced

1. The original principals new JPY 3mo. forwards unadjusted

new JPY DFs unadjusted

USD cashflows

1.02

0.58424% 0.52981% 0.75728% 1.09119% 1.12600% 1.71106% 1.41844% 1.49203% 1.43568% 1.61709% 1.37820% 1.77082% 1.64636% 2.22331% 2.79598% 2.84180%

[1]

1 0.998574 0.997238 0.995270 0.992533 0.989716 0.985453 0.981933 0.978243 0.974706 0.970738 0.967368 0.963010 0.959019 0.953659 0.946893 0.940066

-1,550,755 -1,539,568 -1,286,010 -1,650,675 -1,448,560 -1,469,365 -1,870,432 -1,711,270 -2,449,074 -1,531,641 -1,709,583 -1,789,839 -2,040,884 -1,801,389 -1,883,296 -102,025,261

Value (Yen) = Value (USD) = Net value (USD) =

JPY cashflows

-100,000,000

JPY cashflows [2]

14,707,776 13,792,308 20,363,554 28,406,329 29,312,464 44,542,867 36,925,354 38,840,956 37,374,061 42,096,611 35,877,692 46,605,176 42,858,614 57,877,947 73,585,706 10,373,291,517

603,637,984 13,845,692 12,983,883 19,169,962 26,741,316 27,594,338 41,932,024 34,761,005 36,564,326 35,183,411 39,629,154 33,774,751 43,873,453 40,346,493 54,485,480 69,272,541 9,765,269,691

10,298,500,000

10,298,500,00

106,226,370

106,226,370

6,226,370

6,226,370

250

Swaps and Other Derivatives

Worksheet 6.13

Cross-currency swap valuation: new market data

Historic data for simulation FX vol =>

20%

USD principal JPY principal CCBS margin

See correlation sheet for further data USD forward volatilities

JPY forward volatilities

Daycount

14.572% 15.128% 16.462% 18.740% 20.543% 21.828% 22.556% 22.689% 23.038% 23.601% 24.390% 25.414% 25.932% 25.941% 25.432% 24.392% 23.593% 23.034% 22.718% 22.651%

8-Jan-0l 6-Apr-0l 6-Jul-0l 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

new USD 3mo. forwards

new USD DFs

1.02

Length of time => 6-Jan-OO 9.715% 10.086% 10.975% 12.493% 13.695% 14.552% 15.038% 15.126% 15.359% 15.734% 16.260% 16.942% 17.288% 17.294% 16.955% 16.261% 15.729% 15.356% 15.145% 15.101%

100m 10,298.50m - 14.50 bp

0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

6.34400% 6.09060% 4.92515% 6.53014% 5.73057% 5.81287% 7.39951% 6.76986% 9.68864% 6.05924% 6.76319% 7.00372% 8.07383% 7.12637% 7.36942% 7.92493%

1 0.984729 0.969799 0.957485 0.941937 0.928487 0.915042 0.898241 0.883128 0.862017 0.849013 0.834742 0.820064 0.803663 0.789442 0.774849 0.759468

251

Cross-currency Swaps

Old spot FX rate = 102.985 New spot FX rate = 96.95

New Valuation 1 . The original principals

new JPY 3mo. forwards unadjusted 1.02

0.58424% 0.52981% 0.75728% 1.09119% 1.12600% 1.71106% 1.41844% 1.49203% 1.43568% 1.61709% 1.37820% 1.77082% 1.64636% 2.22331% 2.79598% 2.84180%

new JPY DFs unadjusted

USD cashflows

1 0.998574 -1,550,755 0.997238 -1,539,568 0.995270 -1,286,010 0.992533 -1,650,675 0.989716 -1,448,560 0.985453 -1,469,365 0.981933 -1,870,432 0.978243 -1,711,270 -2,449,074 0.974706 -1,531,641 0.970738 0.967368 -1,709,583 -1,789,839 0.963010 0.959019 -2,040,884 0.953659 -1,801,389 0.946893 -1,883,296 0.940066 --102,025,261

Value (Yen) off unadjusted curve = Value (USD) = Net value (USD) =

-100,000,000

2. Rebalanced

JPY cashflows [1]

JPY cashflows CCBS [2]

margin CFs on delta-FX [3]

11,057,530 10,017,621 16,464,428 24,631,643 25,537,777 40,768,180 33,150,667 35,066,270 33,599,374 38,321,925 32,103,005 42,789,010 39,083,928 54,103,261 69,769,540 10,369,475,350

600,180,812 10,409,402 9,430,447 15,499,379 23,187,880 24,040,902 38,378,587 31,207,569 33,010,890 31,629,975 36,075,717 30,221,315 40,280,968 36,793,056 50,932,043 65,680,056 9,761,677,206

-213,956 -221.250 -228,544 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -223,681 -221,250 -221,250 -223,681 -223,681

10,239,518,156

10,239,518,156

-3,457,172

105,617,988

105,617,988

5,617,988

5,617,988 i

252

Swaps and Other Derivatives

Worksheet 6.14 Cross-currency swap valuation: new market data Historic data for simulation FX vol =»

20%

See correlation sheet for further data USD forward volatilities

USD principal JYP principal CCBS margin

100m 10,298.50m - 14.50 bp

Daycount

new USD 3mo. forwards

JPY forward volatilities

Length of time =>• 6-Jan-OO 9.715% 10.086% 10.975% 12.493% 13.695% 14.552% 15.038% 15.126% 15.359% 15.734% 16.260% 16.942% 17.288% 17.294% 16.955% 16.261% 15.729% 15.356% 15.145% 15.101%

14.572% 15.128% 16.462% 18.740% 20.543% 21.828% 22.556% 22.689% 23.038% 23.601% 24.390% 25.414% 25.932% 25.941% 25.432% 24.392% 23.593% 23.034% 22.718% 22.651%

8-Jan-Ol 6-Apr-Ol 6-Jul-Ol 8-Oct-0l 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05

0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256

Old spot FXrate = New spot FXrate =

102.985 96.95

new JPY 3mo. forwards unadjusted

new JPY 3mo. forwards adjusted

new USD DFs

1.02

1.02

6.34400% 6.09060% 4.92515% 6.53014% 5.73057% 5.81287% 7.39951% 6.76986% 9.68864% 6.05924% 6.76319% 7.00372% 8.07383% 7.12637% 7.36942% 7.92493%

1 0.984729 0.969799 0.957485 0.941937 0.928487 0.915042 0.898241 0.883128 0.862017 0.849013 0.834742 0.820064 0.803663 0.789442 0.774849 0.759468

0.58424% 0.52981% 0.75728% .09119% .12600% .71106% .41844% .49203% .43568% .61709% .37820% 1.77082% 1.64636% 2.22331% 2.79598% 2.84180%

0.51562% 0.47616% 0.69225% 1.00961% 0.97888% 1.50058% .26244% .34572% .23727% .40664% .21238% .57236% .45004% .97229% 2.49589% 2.55068%

253

Cross-currency Swaps

New Valuation 1 . The original principals new JPY DFs unadjusted

1 0.998574 0.997238 0.995270 0.992533 0.989716 0.985453 0.981933 0.978243 0.974706 0.970738 0.967368 0.963010 0.959019 0.953659 0.946893 0.940066

new JPY DFs adjusted

1 0.998741 0.997541 0.995741 0.993206 0.990754 0.987011 0.983871 0.980535 0.977478 0.974015 0.971039 0.967153 0.963621 0.958841 0.952763 0.946593

USD cashflows

JPY cashflows

JPY cashflows CCBS

[1]

[2]

-1,550,755 11,057,530 10,017,621 -1,539,568 -1,286,010 16,464,428 -1,650,675 24,631,643 25,537,777 -1,448,560 -1,469,365 40,768,180 -1,870,432 33,150,667 -1,711,270 35,066,270 -2,449,074 33,599,374 -1,531,641 38,321,925 -1,709,583 32,103,005 -1,789,839 42,789,010 -2,040,884 39,083,928 -1,801,389 54,103,261 -1,883,296 69,769,540 -102,025,261 10,369,475,350

Value (Yen) off adjusted curve =

2. Rebalanced

603,637,984 10,409,402 9,430,447 15,499,379 23,187,880 24,040,902 38,378,587 31,207,569 33,010,890 31,629,975 36,075,717 30,221,315 40,280,968 36,793,056 50,932,043 65,680,056 9,761,677,206

margin CFs on delta-FX [3]

-213,956 -221,250 -228,544 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -221,250 -223,681 -221,250 -221,250 -223,681 -223,681

Net JPY cashflows [4]

604,244,494 10,409,402 9,430,447 15,499,379 23,187,880 24,040,902 38,378,587 31,207,569 33,010,890 31,629,975 36,075,717 30,221,315 40,280,968 36,793,056 50,932,043 65,680,056 9,761,677,206

10,308,847,502

10,308,240,991

-3,467,056 10,308,847,502

106,333,102

106,291,084

106,333,102

6,333,102

6,291,084

6,333,102

10,239,518,156

10,242,975,328

-3,457,172

Change in valuation off the curves

(a) 69,329,346

(b) 65,265,664

(c) -9,885

Total change in valuation off the curves

-4,073,566

Value (USD) =

-100,000,000

Net value (USD) = Value (Yen) off unadjusted curve =

Swaps and Other Derivatives

254

Worksheet 6.15 Swapping a dual currency A$-Yen bond into floating USD Today's date =

26-Jun-96

Current Yen/$ spot rate = Current $/A$ spot rate =

109.085 1.429 76.360 58.990

Current Yen/A$ spot rate = Theoretical Yen/A$ forward rate =

26-Jun-96 26-Dec-96 26-Jun-97 26-Jun-98 26-Jun-99 26-Jun-00 26-Jun-0l

3-year zeros 2.09% 10.58%

USD data

USD DFs

5.750% 6.156% 6.550% 6.760% 6.900% 7.010%

1 0.971601 0.941250 0.879110 0.819095 0.761901 0.707799

Yen data

Yen DFs

AUD data

1

1 1.156% 1.419% 1.670% 1.895% 2.120%

0.988413 0.967182 0.938283 0.905832 0.870541

AUD DFs

10.41% 10.87% 11.25% 11.51% 11.63%

0.905501 0.812930 0.724857 0.644230 0.573733

255

Cross-currency Swaps

Yen principal = USD principal =

l0bn 100m

Maturity = coupon (Yen)redemption (AUD) =

3.00 years 5.65% ANN 100

Bond dates

Bond cashflows Yen AUD [1]

26-Jun-96 26-Dec-96 26-Jun-97 26-Jun-98 26-Jun-99

USD IBOR DFs

Swap dates

USD Libor

USD cashflows

margin =>•

-375.89 [3]

[2]

10,000,000,000

j | I !

26-Jun-96 26-Dec-96 26-Jun-97 26-Dec-97 26-Jun-98 26-Dec-98 26-Jun-99

0.508 0.506 0.508 0.506 0.508 0.506

1 0.971601 0.941250 0.910446 0.879110 0.848935 0.819095

5.75% 6.38% 6.66% 7.05% 6.99% 7.21%

100,000,000 -1,012,143 -1,324,252 -1,472,563 -1,664,142 -1,643,763 -101,742,712

-565,000,000 -565,000,000 -565,000,000

-130,959,475

PV of bond cashflows

8,364,959,233

-94,926,879

PV(USD) =

76,682,947

-66,448,815

10,234,131

10,234,131

0

NetPV(USD)=

Swaps and Other Derivatives

256

Worksheet 6.16

Swapping a dual currency A$-Yen bond into floating USD

Today's date =

26-Jun-96

Current Yen/$ spot rate = Current A$/$ spot rate =

109.085 1.429

26-Jun-96 26-Dec-96 26-Jun-97 26-Jun-98 26-Jun-99 26-Jun-0 26-Jun-0l

USD data

USD DFs

5.750% 6.156% 6.550% 6.760% 6.900% 7.010%

0.971601 0.941250 0.879110 0.819095 0.761901 0.707799

Yen data

Yen DFs

AUD data

AUD DFs

1

1 1.156% 1.419% 1.670% 1.895% 2.120%

0.988413 0.967182 0.938283 0.905832 0.870541

10.41% 10.87% 11.25% 11.51% 11.63%

0.905501 0.812930 0.724857 0.644230 0.573733

257

Cross-currency Swaps

Yen principal = USD principal =

lObn 100m

Maturity = coupon (AUD)= redemption (Yen) =

3.00 years 2.53% ANN 100

USD Bond dates

Bond cashflows Yen AUD

Swap dates

IBOR DFs

USD Libor

USD cashflows

0.00

margin =» [1]

26-Jun-96 26-Dec-96 26-Jun-97 26-Jun-98 26-Jun-99

[3]

[2]

10,000,000,000

26-Jun-96 26-Dec-96 26-Jun-97 26-Dec-97 26-Jun-98 26-Dec-98 26-Jun-99

0.508 0.506 0.508 0.506 0.508 0.506

1 0.971601 0.941250 0.910446 0.879110 0.848935 0.819095

5.75% 6.38% 6.66% 7.05% 6.99% 7.21%

100,000,000 -2.922,917 -3,224,585 -3,383,337 -3,564,474 -3,554,538 -103,643,045

-10,000,000,000

-3,308,011 -3,308,011 -3,308,011

PV of bond cashflows

617.169,961

-8,082,425

PV(USD)-

5,657,698

-5,657,698

0

0

0

NetPV(USD)-

258

Swaps and Other Derivatives

considerably below 2%. During the next 10 years, a number of dual currency bonds have been issued, for example: 25 September 1995 Issuer: SNCF (French Railways) Maturity: 3 years Principal: l0bn Yen Coupon: Y5.65% ann Principal repayment in A$ Issuer: Maturity: Principal: Coupon:

Asfinag (German autobahn financing company) 20 years 20bn Yen either A$5.70% or DM5.31% ann, the issuer had option to select currency

In each case, the investor is taking a currency risk in return for a greater return. In both cases, the issuer then swapped the bond into plain USD Libor less a margin. There are two main types of dual currency bond: • coupon is paid in a foreign currency, but the principal is in the domestic (i.e. currency of issue); • coupon is paid in the domestic currency, but the principal is repaid in a foreign currency. Generally, unless the bond is extremely long, the latter are considerably riskier than the former as the principal itself is at risk. Consider the French issue: this was paying about 400 bp over the curve to the investor for taking on the currency risk that the A$ will weaken against Yen. The Yen/A$ exchange rate in 1995 was about 76.4, and based on the interest rate differentials was expected to weaken to 59 over the 3 years. In other words, losing about 22% of the principal amount whilst only gaining a total of 12% in coupon. Hardly surprising that this swaps into a substantial margin below Libor. Looking at Worksheet 6.15, the bond cashflows are shown from the point of view of the bond issuer, i.e. receiving the principal, and then paying away the coupons and redemption as shown in columns [1] and [2]. This has a total positive value to the issuer of just over $10 million. If this is given away on the swap, the issuer would expect a substantial margin of 376 bp below Libor on a USD principal of 100 million. The "fair" breakeven coupon, i.e. the coupon that gives the bond a zero value, is 9.51% suggesting that the issued coupon is some 400 bp too low, as we have already surmised. What happened over the 3 years? The AUD actually strengthened for most of the time, and the lucky investors received both the high coupon plus a valuable principal. If we structure the bond the other way, namely with AUD coupon and a Yen principal repayment, then the fair coupon is only 2.53%. A much smaller increase for much less risk. See Worksheet 6.16. As we can see from the Asfinag issue, these complex bonds frequently contain embedded options. That issue only contained a single 1 year option, whether to select to pay the PV{AUD stream} or PV{DEM stream}. Other bonds of this type have FX-related options on each cashflow, very often both protecting the issuer from paying large amounts, and

Cross-currency Swaps

259 8

protecting the investor from ever receiving effectively a negative payment . These complex structures often cause a moral debate, as it is very hard for typical investors to assess their "fairness", and there is no doubt that some issues are overly complex to obscure their true value. On the other hand, they could be viewed as the high risk-high return component of a diversified portfolio which is relatively easy to buy. If they did not exist, the range of investing opportunities would be significantly reduced.

6.9 CROSS-CURRENCY EQUITY SWAPS We saw in the previous chapter how equity swaps may be constructed. It is perfectly feasible to extend the construction into cross-currency swaps: Trade date: Notional principal: Maturity: Current FX rate: To receive: To pay:

4 January 2000 $100 million 2 years 102.985 Yen 3mo. Libor S&P 500 Index quarterly

At the current spot rate, the Yen principal is Yl 0.2985 billion, i.e. we are assuming an atmarket swap. The cashflows are calculated in the same way as before, but notice that there is a final exchange of principals at the original spot rate — see columns [7] and [8], The hedging is also very similar: the Yen principal is borrowed at Yen Libor flat, converted into USD and invested in the index. At the end of each period, the hedge is rebalanced to $100 million and the surplus or shortfall paid to the counterparty. See Worksheet 6.17. This hedge suffers from the same problem, namely it requires odd index-based transactions at the end of each quarter which may be inefficient, and so a variable notional structure may be more appropriate. However because of the need to exchange principals and the movement in the FX rate, CCVN structures are more complex. Consider how it might work. Over the first quarter, the index rises by 5.52% implying a payment of $5,523,707. In the single currency version, this payment is funded by increasing the Libor principal by the same amount. In the cross-currency version, we do the same but in Yen which has to be converted into USD at the prevailing FX rate, i.e.: $5,523,707 x S{ = Y560,537,884 using

5, = 101.48

Therefore the Yen principal is increased by this amount. Turning to Worksheet 6.18, columns [4] and [7] show the USD principal and cashflow for a hypothetical series of index movements. Column [5] shows the new Yen principals calculated by:

and column [8] the resulting cashflows. Note that there is an exchange of principals at the swap maturity. 8

A good source describing many of the different structures is Structured Products and Hybrid Securities by Satyajit Das. published by Wiley, 2001.

Worksheet 6.17 Example of cross-currency fixed notional equity swap Today's date Maturity Principal Amount (USD) Current Yen/USD spot rate

Daycount Act/360

04-Jan-00 2 years Receive side: 100m Pay side: 102.985 Frequency of pay side Frequency of receive side

Observed S&P Index

Observed Yen Libor

Spot FX Rate (Yen/USD)

Yen Libor S&P 500 Index

Principal Amount

Principal Amount

USD

Yen

margin Obp Obp Act/360 Act/360

+/+/Qu Qu

Return on S&P Index

Cashflow on index including margin

Cashflow on Libor including margin

USD [1]

06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253

[2]

,209.62 ,276.44 ,296.19 ,276.28 ,263.66 ,321.16 ,301.36 ,343.01 ,364.72

0.156% 0.219% 0.314% 0.429% 0.565% 0.696% 0.822% 0.942% 1.058%

[3]

102.985 101.4786 99.9406 98.3539 96.7592 95.2989 93.8152 92.3070 90.8680

[4]

[5]

100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000

10,298,500,000 10,298,500,000 10,298,500,000 10.298,500,000 10,298,500,000 10,298,500,000 10,298,500,000 10,298,500,000

[6]

[7]

5.52% 1.55% -1.54% -0.99% 4.55% -1.50% 3.20% 1.62%

-5,523,707 -1,547,231 1,536,027 988,197 -4,549,872 1,498,867 -3,200,945 -101,616,404

Yen [8] 4,067,550 5,692,322 8,268,736 11,549,264 14,230,648 18,117,288 22,090,851 10,323,023,138

Hedging a cross-currency fixed notional equity swap

S&P

Libor borrow

[9] 06-Jan-OO 06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-Ol 06-Apr-Ol 06-Jul-Ol 08-Oct-Ol 07-Jan-02

10,298,500,000 -4,067,550 -5,692,322 -8,268,736 -11,549,264 -14,230,648 -18,117,288 -22,090,851 -10,323,023,138

cashflow

[10] -100,000,000 5,523,707 1,547,231 -1,536,027 -988,197 4,549,872 -1,498,867 3,200,945 101,616,404

S&P hedge before rebalancing [11] 100,000,000 105,523,707 101,547,231 98,463,973 99,011,803 104,549,872 98,501,133 103,200,945 101,616,404

S&P hedge after rebalancing [12] 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000 100,000,000

1'

I

Worksheet 6.18 Example of cross-currency variable notional equity swap Today's date Maturity Principal Amount (USD) Current Yen/USD spot rate

Daycount Act/360

06-Jan-OO 06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253

04-Jan-00 2 years Receive side: 100m Pay side: 102.985 Frequency of pay side Frequency of receive side

Observed Observed S&P Yen Index Libor

Spot FX Rate (Yen/USD)

P] 0.156% 0.219% 0.314% 0.429% 0.565% 0.696% 0.822% 0.942% 1.058%

[3] 102.99 101.48 99.94 98.35 96.76 95.30 93.82 92.31 90.87

[1] 1,209.62 1.276.44 1,296.19 1,276.28 1,263.66 1,321.16 1,301.36 1,343.01 1,364.72

Yen Libor S&P 500 Index

Principal Amount USD

Principal Amount Yen

[4] 100,000,000 105,523,707 107,156,402 105,510,451 104,467,799 109,220,951 107,583,874 111,027,575 112,822,229

[5] 10,298.500,000 10,859,037,884 11,022,210,387 10,860,324,554 10,759,438,390 11,212,408,310 11,058,825,613 11,376,703,353 —

+/— +/Qu Qu Return on S&P Index [6]

5.52% 1.55% -1.54% -0.99% 4.55% -1.50% 3.20% 1.62%

margin 0 bp 0 bp Act/360 Act/360 Cashflow on index including margin USD [7]

Cashflow on Libor including margin Yen [8]

-5,523,707 -1,632,695 1,645,952 1,042,651 -4,753,151 1,637,077 -3,443,701 -112,822,229

4,067,550 6,002,149 8,849,808 12,179,323 14,867,581 19,725,050 23,721,792 11,403,793,945

Hedging a cross-currency variable notional equity swap periodic Libor borrow/depo [9]

06-Jan-OO 06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-Ol 06-Apr-0l 06-Jul-0l 08-Oct-Ol 07-Jan-02

560,537,884 163,172,503 -161,885,834 -100,886,164 452,969,921 -153,582,697 317,877,740

total Libor borrow/depo

[10] 10,298,500,000 10,859,037,884 11,022,210,387 10,860,324,554 10,759,438,390 11,212,408,310 11,058,825,613 11,376,703,353

total Libor interest

[11] -4,067,550 -6,002,149 -8,849,808 -12,179,323 -14,867,581 -19,725,050 -23,721,792 -11,403,793,945

S&P cashflow

S&P principal

[12] -100,000,000

[13] 100,000,000 105,523,707 107,156,402 105,510,451 104,467,799 109,220,951 107,583,874 111,027,575 112,822,229

112,822,229

262

Swaps and Other Derivatives

The swap can be hedged by borrowing $100 million x S0 in Yen at Libor flat, converting the proceeds to USD and investing in the index. Each period, the Yen borrowing is increased or decreased by PY./ — ^Y./-I = ^s./-i x (1 + r(-) x S, as shown in column [9]. The interest being paid in column [11] exactly matches the interest received in column [8]. At the end, the index investment is liquidated, converted to Yen at the prevailing spot rate and used to repay the total borrowing. However, the CCVN do expose the investor to movements in the FX rate as well as movements in the index. It is feasible to get currency-protected ("quanto") swaps where both sides would be denominated in, say, USD. These are either dynamically delta-hedged or hedged using simple quanto instruments such as FX forwards — see the Appendix.

6.10

CONCLUSION

This chapter has discussed the construction and pricing of cross-currency swaps. Whilst the market for them is considerably smaller than for IRS, it is still an extremely important market. CCS are extensively used by organizations who borrow in a "cheap" currency, and then swap the proceeds into their desired currency. Exchange rates have become more volatile over the last 50 years due to the abolition of many fixed-rate regimes, and demand for currency exposure management has increased accordingly. The use of CCS, and particularly long-term FX forward contracts which are merely a special type of CCS, to provide medium to long-term risk management is increasing each year on the back of increasing currency deregulation.

APPENDIX: ADJUSTMENTS TO THE PRICING OF A QUANTO DIFF SWAP The HJM approach, as outlined in the Appendix to Chapter 5, may also be used to adjust the pricing of a quanto diff swap. The following results are based upon the assumptions9: 1. the domestic term structure has one source of uncertainty, call it W\; 2. the foreign term structure has two sources of uncertainty, W\ and W2. i.e. one source in common plus one additional one; 3. the spot FX rate S(t) has three sources of uncertainty, W1, W2 and W3, i.e. the FX rate is related to the two term structures plus one additional source. This permits three correlations between the two term structures and spot rate. Using the same notation as in the Appendix to Chapter 5, the expected present value of a domestic payment at time tj+ 1 based on a foreign reference rate may be written as: J 1_ \rtj+\}BA(tj}

Bd(t

i.e. constructing the foreign discount bond pf(t j ,t j+1 ), but applying a domestic money account B d (t j ). By evaluating the integrals, we get:

9

Details are given in S. M. Turnbull. Pricing and hedging diff swaps. Journal of Financial Engineering. 2(4). 1993.

297–333.

163

Cross-currency Swaps

This is the usual expression with an adjustment term e^, where q> consists of the following expressions: ,-//])(! — exp[—A f/ fl?y +1 ]) 2

= I K-)2(Af,r3( 1

for i= 1,2

exp[-Af(7,-])(l - exp[-;.f,4+,])

$ «d,4/+1) J} f ^-exp(-/ exp[-Af,4+1]) If As = 0, i.e. no reversion:

)2

for / = 1 ,2

If the input data are the following: 1. volatility of domestic term structure, in terms of discount bond prices, reversion factor, Adi, 2. volatility of foreign term structure, oy, plus reversion factors, An and AO, 3. spot FX rate volatility, crp\, 4. correlations between the three components, i.e. pdf, pdFX and

i, plus

then the parameters for the above formulae are calculated by: a,2 = crf[l - (pdf)2]1/2 f5

i.e. from (af)2 = (a n ) 2 + (an)2

l = PdFX^FX

<52 = (pfpxaFX°'f

~ ^i^nV^o i- e - from covol(f,FX) = 5\an +

In the main text, we priced a Yen USD QDS to have a margin on — 572 bp on the USD side. Worksheet 6.19 contains the following data: volatility

lambda 1

lambda!

domestic IR: foreign IR: FX rate:

1% 1% 12%

10% 10%,

10%

correlation:

d-f 0.5

d-FX 0.3

f-FX -0.3

The margin increased by 9bp to — 581bp. The worksheet also contains sensitivity graphs with respect to the main parameters. The biggest impact, not surprisingly, following the discussion about the difficulty of hedging, is due to the FX/foreign IR correlation.

264

Swaps and Other Derivatives

Worksheet 6.19

Pricing a 5 year Yen-USD quanto diff swap

Today's date =

04-Jan-00

Current spot rate =

102.985

Principal amount Principal amount Maturity To pay To receive

l0bn Yen 97. 10m USD 5 years 6mo. Yen Libor 6mo. USD Libor +

-580.87 bp

1. Yen pay side

Dates

Daycount (domestic)

06-Jan-00 06-Jul-00 08-Jan-0l 06-Jul-Ol 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511

2. USD-related receive side Accumulative domestic time

0.506 1.022 1.519 2.033 2.539 3.044 3.550 4.058 4.564 5.075

Yen IBOR unadjusted dfs

Implied forwards

[1]

[2]

1 0.999053 0.997133 0.994009 0.989525 0.983934 0.976959 0.968420 0.958470 0.947467 0.935334

0.187% 0.373% 0.632% 0.882% 1.124% 1.412% 1.744% 2.042% 2.297% 2.538%

Yen IBOR adjusted dfs

Yen Libor cashflows

USD IBOR dfs

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511

0.506 1.022 1.519 2.033 2.539 3.044 3.550 4.058 4.564 5.075

1 0.969519 0.936853 0.905235 0.873099 0.842475 0.812962 0.784581 0.756987 0.730306 0.704190

[4]

[3]

1 -9,479.167 0.999103 0.997234 -19,252.583 0.994373 -31.434.647 0.990245 -45.313.971 0.985217 -56.821.965 0.978949 -71.394.851 0.971386 -88.173.133 0.962496 -103.810.274 0.952584 -116.127.412 0.941483 -129.719.669

PV (Yen) on l0bn nominal —

Daycount (foreign)

Accumulative foreign time

-648.990.813

Net PV =

Sensitivity of a quanto diff swap to changes in the parameters of the convexity adjustment QDS: Impact of changing IR volatility •550

*-— »— —»--

-^

^— - ^ -700 -

0

1

3

2

4

5

(%)

QDS: Impact of changing IR correlation -560 -565 -570 -575 -580 ^^^^^^^^^^^^^^ ' — -585 ~ ^^"•^ -590 1 0 -0.6

) / -*-.- ••

-0.2

+- — *

0.2

« -

^—

0.6

1.0

265

Cross-currency Swaps Convexity data

vol. 1% 1% 12%

lambda 1 10% 10%

lambda 2

Domestic: Foreign: FX rate:

d-f Correlations:

0.5

d-FX 0.3

f-FX -0.3

Implied forwards set if lambdas >0 a(f.l)

6.219% 6.749% 7.025% 7.162% 7.190% 7.181% 7.155% 7.171% 7.226% 7.256%

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0001 0.0001 0.0001

a(f,2)

Transformations sigma(f,l) 0.0050 delta(l) sigma(f,2) 0.0087 delta(2) delta(3)

10%

set if lambdas = 0

a(f,d) a(f,FX)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0001 0.0000 0.0001-0.0001 0.0001-0.0001 0.0002-0.0001 0.0002-0.0001 0.0003-0.0002 0.0003-0.0002

0.0000 0.0001 0.0002 0.0003 0.0003 0.0004 0.0005 0.0005 0.0006 0.0007

a(f,l)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000

a(f.2)

a(f,d) a(f,FX)

0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.0000-0.0001 0.0000-0.0001 0.0000-0.0002 0.0000-0.0003 0.0000-0.0004 0.0000-0.0005 0.0000-0.0006

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0006 0.0007 0.0008

0.0360 -0.0624 0.0960

adjustment USD-related Libor factor cashflows phi exp(phi)

USD-related margin cashflows

314,392,361 349,693,016 351,270,513 371,129,501 367,532,259 368,053,272 367,724,527 371,527,051 373,244,615 379,803,822

-293,660,728 -300,114,810 -288,820,167 -298,501,290 -293,660,728 -293,660,728 -293,660.728 -295,274,249 -293,660,728 -296,887,769

0.0000 0.0001 0.0002 0.0003 0.0004 0.0005 0.0006 0.0007 0.0008 0.0009

1 1.0001 1.0002 1.0003 1.0004 1.0005 1.0006 1.0007 1.0008 1.0009

3,529,965,652 -2,880,974,839 0

QDS: Impact of changing FX volatility o/O 1

0/0

-•

_ con _

^-— ^_ — —_

0

- 540 _| - 550 - 500 . - 570 - "iRn -

finn ' -1 0

4

8

1

2

1

6

"••« .

20

QDS: Impact of changing FX/foreign IR correlation ^^ '

^^-— •-•*""

*" ^- —*" -0.6

-0.2

0.2

0.6

1.0

This page intentionally left blank

7 Interest Rate OTC Options

OBJECTIVE Many banks trade, i.e. price and hedge, a range of interest options such as caps and floors alongside swaps. One can also get options on interest rate swaps, although options on cross-currency swaps are very rare and not discussed in this chapter. It was therefore thought appropriate to discuss the more common forms of OTC options, and especially in the context of the swap market. The chapter starts by very briefly discussing the Black option pricing model, the one that is universally used for the pricing of European interest rate options. The practical estimation of volatility from both historic data and implied from an existing options market is described. The concepts of par and forward volatilities are introduced, and a bootstrapping approach for transforming between them demonstrated. Different forms of caps, floors and collars are described, including examples of digital and embedded swap structures. More complex non-European structures are briefly discussed but pricing is not covered. The terminology and pricing of swaptions are then looked at, especially when embedded in swaps. The relationship between cap and swaption volatility is explored with some examples. Finally, there is a brief section on FX options, looking at their pricing and replication.

7.1

INTRODUCTION

Interest rate options have been widely traded over-the-counter since the mid 1980s, following the growth in the swap market. The "first" generation of options, as it is often described, constituted European options such as caps/floors and swaptions on the level of forward interest rates and swap rates. They are invariably priced using Black's 1976 formula, which provides a closed-form pricing model requiring a small number of market inputs. The formula is "single" factor, implying that the level is the only source of uncertainty. Second and third generation options, possessing path-dependency and barrier characteristics respectively, have now been developed to take more complex views on the movement of interest rate curves, such as rotation (steepening) or twisting. The variety of such options is extremely wide, and being extended daily. The pricing and hedging of such instruments requires multifactor models, which are capable of incorporating the correlation effects along a curve. The wide practice however of hedging these options with a portfolio of single generation options has resulted in many unexpected losses. Let us establish some boundaries. First, this is not a book primarily about options; therefore it will not attempt to discuss the wide variety of options that are now at least theoretically available. Second, most of this chapter will concentrate on the practical

268

Swaps and Other Derivatives

implementation of first generation options, and especially in conjunction with swaps and other securities. Third, it is assumed that all readers are familiar with simple Black & Scholes option pricing models, so it will be rapidly introduced. Finally, the discussion of numerical models for the pricing of more complex options will be short, with references to more appropriate texts.

7.2 THE BLACK OPTION PRICING MODEL A "caplet" is defined as a single call option on a forward interest rate F(r, T), which starts at time r and finishes at time T. If we assume that the option has a strike of K% and is written on a principal P, then: • at time r, the fixing of F is observed to be L%; • if L> K, payout = [L — K]x(T— r)xP conventionally paid at time T; • if L^ K, pay out = zero. More generally the payout can be expressed as: max[0, L — K\x(T — T)xP. If the payout were to be made at time i instead, adopting the FRA convention, then it would have to be discounted back in the usual fashion, i.e: max[0, L-K\x(T-i) x P [l+Lx(r-t)] Consider now an option on a discount bond. Let p(t, T, T) be the estimate at time t of the price at time T of a discount bond that matures to pay 1 at time T. The payoff of S put options with strike pK at maturity T is defined as max[0, pK — p?(T, T, T)] x S. Using the definitions: p(r, T, T) = [l+Lx(T-r)]-1

and pK = [1 + K x (T- t)]"1

and substituting into the payout, we get: {max[0, L - K\ x (T - T) x /»} x 5 x pK [l+Lx(r-T)] Setting S= \/pK, we get an identical payoff to the caplet. We can therefore either represent a caplet as a call option on a forward interest rate or a put option on a discount bond. This latter result is especially useful as there are many closed-form solutions available for forward discount bond prices and for options on discount bonds using "normal" models. The Black model for the caplet on F(r, T) may be written as: C=P x DFT x {F(r, T) x #(
N(d2)} x (T- T)

where a is the volatility of the forward rate, dl = [\n(F/K) + 0.5 x cr2T}/o\/T, d2 = d{ — a*fi and N(x) is the cumulative unit normal distribution. Notice that the payout is discounted back from time r, following the convention that the payout occurs at the end of the period, although the option matures at time T. Therefore the volatility, which would be quoted on the basis of some standard time

269

Interest Rate OTC Options

period such as a year, is scaled by >/? and not by \/T—see later for a more detailed discussion. As an example, to price a caplet on a 3mo. forward rate:

Today's date: Principal amount: Forward rate: Strike: Volatility:

4 January 2000 $100 million start date 6 July 2001 end date 8 Oct 2001 7.25% 17% pa

The dates for the forward rate use the normal daycount convention; as 6 Oct 2001 is a Saturday and therefore a non-business day, the end moves to the next business day. The discount factors are (see Chapter 3 for their derivation): 6 July 2001 8 Oct 2001

0.905611 0.889113

which implies that F= (0.905611/0.889113-1)/0.261=7.106%. Substituting into the formula, we get: i— 1.519 (this uses a basis of Act/360, which was used to annualize the volatility) d1 - {m(7.106%/7.25%) + 0.5 x 17% x 17% x 1.519}/(17% x 1.232) = 0.0092 d2 = 0.0092 - 17% x 1.232 = -0.2003 #(
'Remember Chapter 3, which discussed the reference rate approach. This has not been applied in the following examples, although in theory should be.

270

Swaps and Other Derivatives

Worksheet 7.1 Cap pricing model Settlement date:

06-Jan-00

Start date [1]

End date [2]

Strike (%) [3]

06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250%

Principal (m) [4] 100 100 100 100 100 100 100 100 100 100 100

Ann Vol (%) [5]

17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

Caplet (bp) [6]

0.31 2.31 5.77 6.55 9.28 12.30 14.52 12.61 13.% 15.25 16.47

Total cost (bp)= 109.335 Total cost ($) = 1,093,350

The Black formula is frequently criticized, but nevertheless is the de facto standard for first generation options. Its use implies that the percentage returns on the forward rate are distributed normally with zero mean (or drift). One criticism is the apparent inconsistency that on the one hand the model treats the forward rate as stochastic, and yet uses an expected rate for discounting. It would seem more intuitive that high payouts, corresponding to high forward rates at maturity, should be discounted at a correspondingly high rate, whereas low payouts should be discounted at a lower rate. This is very much what happens in the tree approaches such as Black-Derman-Toy (BDT). However. as Rebonato shows2, BDT implies a drift to the evolution of the forward rates which exactly compensates for the different discounting processes, so that the results from Black and BDT are consistent with each other. In theory, the payout of a caplet should be discounted back from time r at a risk-free rate. In practice the market seldom (never?) does this, but uses Ibor-based discount factors that reduce the price of the option. One could argue that this is simple convenience, as such factors are readily available. It is probably also a pragmatic recognition that the other underlying model assumptions are not satisfied in practice and therefore option pricing models are providing at best price "indications". In the long run, interest rates exhibit mean reversion, ie. if high, then they are more likely to fall and conversely, if low, then more likely to rise. This suggests that the scaling

2

Rebonato, Interest Rate Options Models. Wiley. 1996. p. 122.

271

Interest Rate OTC Options Worksheet 7.1

(Continued)

Discount factors near

[7] 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796

[9]

implied forward [10]

time to exercise [11]

0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219%

0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792

far [8]

daycount

0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

dl [12]

N(dl) [13]

N(d2) [14]

-1.582 -0.770 -0.337 -0.283 -0.123 0.009 0.117 -0.009 0.041 0.086 0.127

0.057 0.221 0.368 0.388 0.451 0.504 0.547 0.496 0.516 0.534 0.550

0.048 0.186 0.314 0.324 0.377 0.421 0.456 0.401 0.415 0.427 0.438

of volatility by */i is likely to be an over-estimate, especially for a long-dated option, and that the Black model therefore over-prices options. Practitioners sometime suggest, with tongue firmly in cheek, that the impact of Ibor discounting is to adjust the price for meanreversion, but that's just wishful thinking.

7.3 INTEREST RATE VOLATILITY The estimation of volatility is of course central to the pricing of options. The Black model is based upon the following evolution; that the return on a factor x, which is defined as the percentage change in that factor, over a period of time dt, is given by: rv = dx/x — n dt + a \/d7 e, where K ~ N(0, 1) Thus cr\/cU is the standard deviation of rx over the time period dt. In theory the returns should be defined using r(t) = ln(;tf/.*,_!), i.e. continuously compounded returns. In practice they are often defined using simple returns, i.e. r(f) = (x, — x t - 1 ) / x t - 1 , Numerically, for short periods of time, the results are virtually identical. For example, a sample of 1 day returns on USD 12mo. cash rate gave a c-c volatility of 1.288% and a simple volatility of 1.291%. It is assumed that the returns are independently distributed, thus giving rise to the "square-root" rule. This is equivalent to assuming that trends in x do not exist. Whilst this is generally true for liquid markets over short time horizons, it is frequently untrue for longer time horizons. A trend has effectively constant returns, i.e. its volatility is close to

272

Swaps and Other Derivatives

zero. Thus the square-root rule will overestimate volatility in the presence of trends, and the Black model is likely to overprice long-term options. Obviously the factor xt has to be observed periodically. Most practitioners would use closing prices, so that a daily return would be based on the close-to-close pricing. This is because closing prices are often recorded for risk management and P&L calculations independently of actual trading activities. However, closing prices are frequently subject to distortion as there may be very little market activity at that time of day, and therefore the prices are likely to be "indicative" and not represent actual transactions. Sometimes closing prices are recorded over a period such as an hour before the normal market closing time, and some average calculated. If there is an official fixing for the factor, such as for Libor rates, then this may be used instead. However fixings are usually relatively early in the day, for example Libor fixes at 11am, and therefore may not be representative of transactions throughout the day. A single observation per day is a crude indication of what may have happened during the day, and increasing use is being made of intra-day price data. High frequency data, driven by either time or event (i.e. tick movement or transaction) sampling, are becoming widely available. As a result, as expected, measured volatility appears to have increased. Some people use open-to-close prices as a surrogate but a better measure is:

where Hi and Li are the high and low for day i. This is supposedly five to six times more accurate than using closing prices alone3. The highs and lows may be adjusted numerically to compensate for the fact that reported high < continuous high, etc. The definition of the "period of time" requires careful consideration. Naively one might argue that a 10 day period would run, for example, from Wednesday 21 June to Saturday 1 July 2000. However this period includes two weekends. Are the returns from, say, Saturday to Sunday statistically indistinguishable from the returns from Monday to Tuesday or Thursday to Friday? Early academic studies4 found that the average return over a 3 day weekend, i.e. from Friday to Monday, is only about 10% higher than the average return between two consecutive business days. This suggests that market rates only move when the markets are active. Most practitioners therefore use business days to define the time period, so that 10 days from 21 June would be Wednesday 5 July. Now that some markets are truly global, trading in all time zones, such as USD/Euro spot rate or Eurodollar futures contracts, the above discussion suggests that they should exhibit greater volatility than a domestic market open only 8 hours per day.

3 Derived by Parkinson, "The extreme value method for estimating the variance of the rate of return". Journal of Business, 53(1), 1980, pp. 61–65. For a more detailed description, see Tompkins, Options Explained, Macmillan. 1994, p. 133. If using high/low is more accurate than using merely closing prices, how about combining the two? See Garman et al., "On the estimation of Security Price Volatilities from Historical Data", Journal of Business. 53(1), 1980, pp. 67-78 and is based on Parkinson's working paper. 4 E. F. Fama, The behaviour of stock prices, Journal of Business, 38, 1965, 34–105: K. R. French. Stock returns and the weekend effect. Journal of Financial Economics, 8, 1980, 55–69.

Interest Rate OTC Options_

_

__

__

273

''Economic" days, i.e. days during which important economic figures are released, are likely to exhibit higher volatility than "normal" business days. Some practitioners modify their volatility formula to include the number of economic days in the sample period. The estimation of unconditional volatility from historic data is more of an art than a science, with individuals favouring many approaches: see for example Tompkins (Chapters 4 & 5) for a more in-depth discussion. To price an option, the volatility of returns from today until the maturity of the option is required. The Black model assumes constant volatility over this period, unlike the stochastic volatility and some of the numeric models. There are three ways in which this future volatility may be estimated. (a) To assume volatility is stationary, and to calculate historic volatility. This is probably the most common approach. After calculating the returns, almost invariably over a 1 day time period, the standard deviation is estimated using the sampling expression:

where u is the average return over these observations. The volatility for the option is then estimated by crVT where T is the number of business days from today until maturity. The choice of n is arbitrary but crucial. On the one hand, the standard error of a is proportional to *Jn; therefore the larger the number of observations selected, the smaller the error. But of course the assumption of stationarity is likely to be less true with increasing observations. A common rule-of-thumb is to match n with the maturity of the option, i.e. set n roughly equal to T. Use of the square-root rule cr\/T, embedded in the Black model, assumes that the returns are independent, or alternatively that market returns do not exhibit trends. In practice this is not true. One could argue that trends are eventually arbitraged away, and therefore the more liquid the market the shorter the trend; 30 minutes is a long time for a trend to exist in spot-FX. The existence of trends reduces volatility, and therefore it is likely that this rule will overestimate the actual volatility. One approach that is increasingly used, especially for short-dated options, say under 6 months maturity, is to weight the returns on the basis that the more recent returns are likely to be more relevant than returns that occurred further ago: 0* = IX x (r,. - A0 2 /(« - 0 i=\

where IX = 1

If we assume that the returns are ordered such that rt was the 1 day return observed / days ago, then the weights constructed are vv, > w2 > vv3 > • • - . An exponentially weighted scheme would set w, = A'/Z^ where A < 1; in the range of 0.90–0.95 is common. If we assume an infinite series of returns, then the following recursive relationship may be derived:

where the subscript refers to the time of the last available data, indicating that we are now treating volatility as conditional or time-variant. The estimate is adjusted as a new return is observed, and hence may be used for short-term forecasting.

Swaps and Other Derivatives

274

Unweighted Weighted, lambda = 0.94

0.0

601

1001

801

1201

Figure 7.1 180-Day volatility of 12mo. USD cash rate

(b) To model volatility based upon historic information to provide a forecast. We have already seen how the traditional method of calculating volatility may be modified to incorporate a weighting scheme, and this may be interpreted as a forecasting model. A simple weighted scheme is modelling the "responsiveness" of volatility to changes in the market returns. New returns are given greater weight, which will cause the volatility estimates themselves to be more volatile as shown in Figure 7.1. ARCH (AutoRegressive Conditional Heteroskedastic) modelling takes this approach one stage further. A generalized ARCH(1,1) model is:

where v is the unconditional volatility estimated above, and where a > 0, 1 > ß > 0. a + ß ^ 1. The parameters may be interpreted as: • v is the long-run volatility; • a indicates the persistence of shocks, i.e. the larger, the longer a shock lasts; • ß measures the reactivity of the market to shocks, i.e. the larger, the faster. Together, these two parameters model the rise and fall of volatility, unlike the single weighted scheme which treats the two the same. A typical result is: USD/GBP spot JPY/USD spot

a 0.931 0.839

ß

0.052 0.094

i.e. cable reacts slowly but persistently, $-Yen faster but drops off.

275

Interest Rate OTC Options

However, these methods are seldom applied to option pricing. Because of the need to estimate three parameters, a lot of historic data is required — say 500 days minimum. The resulting forecasts only appear to be better than either an unweighted or single weighted estimates for about 20 days ahead, which is hardly significant in the lifetime of most options. ARCH models are of course useful in option trading, trying to position an option portfolio to anticipate the movement in volatility, and in risk management5. (c) To imply volatility from other options already trading in the marketplace. This suggests a circular argument, namely deriving the volatility from existing options. It can however act as an extremely useful check on where other participants see volatility, but must always be interpreted carefully. Many option markets that are highly liquid, for example at-themoney USD or GBP cap markets, will quote volatilities rather than option prices. This is because all the other pricing parameters required for the Black model are available elsewhere such as in the swap market. Therefore volatility is the only unknown parameter, and there is a precise relationship between it and the option price. However the following sequence of events may be theoretically true:

True volatility

Model option price

i Implied volatility

Market option price

such that the true volatility matches the implied volatility. But of course this is only true if the price of the option in the market matches the model price, which in practice is highly unlikely. Whilst the Black model is universally used, its underlying assumptions and limitations are also well understood. Model prices are invariably adjusted to reflect a wide range of factors, which in turn means that the volatility implied from the market price also incorporates these factors. There would appear to be two different arguments describing the source of factors at work. The first argument runs as follows: the Black model assumes that returns follow a normal distribution; in reality, large returns appear more frequently than theoretically justified; hence the "correct" distribution has ''fatter tails" than the normal distribution; therefore the chance of an option having to make a large payout is greater than suggested by theory; 3

There is an enormous volume of literature on GARCH modelling, and the above discussion does not do it justice. See for example T. Bollerslev et al., ARCH models, in Handbook of Econometrics, Vol. 4, R. F Engle et al. (Eds.), North Holland, 1994.

276

Swaps and Other Derivatives Table 7.1 Annualized par volatilities for GBP caps against 3mo. Libor (March 1999)

%ATM 50% 60% 70% 80% 90% 100% 110% 120% 130% 140% 150%

1

yr

214.73% 152.08% 105.13% 68.07% 40.37% 23.60% 17.49% 17.90% 18.18% 18.47% 18.80%

5

yr

47.73% 39.33% 33.72% 29.21% 26.43% 24.72% 24.12% 23.82% 23.75% 23.73% 23.75%

10

yr

34.02% 30.12% 27.10% 24.69% 23.06% 21.99% 21.41% 21.04% 20.80% 20.68% 20.60%

%ATM is defined as (strike/ATM forward swap rate); 50% means a strike equal to half the forward swap rate. i.e. a cap which is heavily in the money, and so on.

• hence the market option price should be increased over the model price; • which in turn would lead to a higher implied volatility. The fat-tailed effect will be greater for options that are away from the money than ATM, and therefore the price adjustment is likely to be greater. This phenomenon gives rise to the well-known "smile" effect, when implied volatility appears to be a function of strike and not solely a property of the underlying returns. Because a cap is a strip of independent options, what denotes ATM for a cap is not obvious. It is usually interpreted for the strike of each caplet to be constant and equal to the fixed swap rate of the same maturity as the cap, remembering of course that if the cap does not include the first fixing then this should be a forward swap. The swap rate may easily be calculated (see Chapter 4) by: Fw(s,e) = Edi x Li x DF i /Ed i x DFi = (DFS — DF e )/(Q e — Qs) Table 7.1 is often referred to as a "volatility surface", with the two dimensions of cap maturity and either absolute strike or relative at-the-moneyedness. Notice that the surface is not symmetric around ATM, but is rather lop-sided (sometimes called the "sneer" effect!). The second argument says that there are a number of serious practical omissions from the model, such as: • • • • •

fixed costs of undertaking a transaction such as salaries, systems, rates and rents; variable costs of undertaking a transaction such as all the back office processing; costs of risk management, especially the potential cost of imperfect hedging; capital charge on a transaction, and the required return on capital; the real cost of funding all these activities.

The market price has to reflect such factors. In addition, market prices are adjusted by perceived supply and demand for the options, forecasted movements in the market rates especially volatility, and not least a general desire whether or not to do a particular transaction. By the time all the adjustments have been completed, it is hardly surprising that the market price may bear little resemblance to the model price. Both arguments have validity, and reality is probably a mixture of them both. This has not stopped some people, mainly academics, from assuming that the first argument is the

Interest Rate OTC Options Table 7.2 ATM caps based on 3mo. USD Libor Cap maturity 1 2 3 4 5 7 10

yr yr yr yr yr yr yr

Volatility (pa) 10.37% 12.87 14.12 15.12 15.25 15.13 14.88

Source: Bloomberg, L. P.

sole source of price adjustment and backing out the implied distribution with its fat tails from smile data6. In summary so far, of the three alternative ways of estimating volatility for option pricing, forecasting is seldom used as the time period is too great. Ultimately historic data are used, but market-traded option prices frequently provide an additional check. Traders often calibrate their volatilities to the generic prices in the market; in this case, their models are effectively interpolation devices.

7.4 PAR AND FORWARD VOLATILITIES Most volatilities quoted in the marketplace are "par" volatilities applied to generic instruments: see Table 7.2 for caps based on 3mo. USD Libor. The par or average volatility of 10.37% would apply for all three caplets in the 1 year cap (not forgetting that there is no option on the first Libor fixing), the par volatility of 12.87% would apply to all seven caplets in the 2 year cap, and so on. This makes the act of quotation in terms of volatility very much easier. However we know that the first three caplets in the 2 year cap are identical to the caplets in the 1 year cap. Therefore for consistency, it would seem sensible that they should always be priced using the same volatilities. Let us define "forward" volatility as the volatility that would apply to a single caplet. The forward volatility for the very first caplet would be the volatility of a 3mo. rate which will be fixed in 3 months' time. The forward volatility for the second caplet would be the volatility again of a 3mo. rate, but this time fixed in 6 months' time, and so on. Par volatility is therefore some average of the relevant forward volatilities, but based upon an underlying generic cap structure, such as: • • • •

notional principal is constant, the tenor of all the rates are constant, the caplets are contiguous, the strike is ATM.

Forward volatilities are required for pricing non-generic or structured caps, when each caplet may have different parameters. 6

M. Rubenstein, Implied binomial trees, Journal of Finance, 49(3). 1994, 771–818; B. Dupire, Pricing with a smile. Risk, 7, 1994, 18-20.

278

Swaps and Other Derivatives

Chapter 4 described how non-generic swaps are priced using discount factors, which in turn were derived from generic swaps. A similar operation is performed in the cap market, whereby forward volatilities are derived from quoted par volatilities. Let T denote the maturity of the Tth generic cap for which we have par volatilities. Define the price of a cap of maturity T using par volatility VT as:

where c,(VT) is the price of a single caplet of maturity t. For arbitrage reasons, the same cap using the forward volatility curve should have the same price:

where v, is the single period forward volatility. Hence we can define a recursive relationship: CT=CT_1+

E

c t (v t )

(7.1)

A crude but common assumption is to set v, equal to a constant for T — 1< t < T. Then we can solve sequentially for the forward volatilities. Worksheet 7.2 first builds the current discount curve; see columns [1] and [2]. Column [4] contains the start date for each caplet, 3 months apart starting out off 23 June 2000 but obviously adjusted to be on business days. Column [6] contains the end date of the 3mo. Libor rate, again adjusted for business days. Notice carefully however that the adjusted end dates are not necessarily the same as the next start date. The appropriate discount factors are shown in columns [5] and [7], and the 3-monthly forward rates in column [9]. Because the par volatilities are most appropriate for ATM options, the forward swap rates are then calculated in column [10] using the above expression. Step 2 is to price the caps based upon the par volatilities, using the forward rates and forward swap rates. Columns [12, 14, ..., 24] calculates d1, and the caplets are priced in columns [13, 15, ..., 25]. Step 3 estimates the forward volatility curve. A simple way is as follows: • Guess a forward piece-wise constant volatility curve; see column [26]. • Price each of the caps using this curve; see columns [27, ..., 40]. • Adjust each segment of the volatility curve, starting at the short end in a bootstrapping fashion, so that the price of each cap based off the forward volatility curve matches the original price. The result is shown in Figure 7.2. Whilst such a curve is arbitrage free, a smoother curve would be better. The approach may be modified to use the optimization technique discussed earlier, as shown in Box 4. The main difference is that each point on the forward volatility curve is guessed in column [41]. Column [42] defines a smoothness criterion E(ff, - cr,_,) 2 which has to be minimized whilst still retaining the arbitrage-freeness. The final result is shown in Figure 7.3. Notice the very typical "humped" structure over the 2-5 year region; this is likely because of the traditional high demand by end-users for interest rate protection over those maturities. These curves are often combined with statistical confidence bands. In practice it is found that volatilities do revert to a long-run level (as suggested by the ARCH model), which

279

Interest Rate OTC Options 18 16

/

14 -

/ 10

10

•

Par

— Forward 8 C)

1

2

Figure 7.2

3

4

5

6

7

8

9

1

Volatility curves: USD 21 June 2000

Figure 7.3 Volatility curves: USD 21 June 2000

means that the confidence bands are wider at the short end than at the longer end. The bands are often called "volatility cones" due to their shape, and are used by traders to imply the likely movement of volatility through time7. We have just derived forward volatilities from a single ATM par volatility curve. It is however common practice to use volatility surfaces, i.e. a matrix of {strike vs. forward start date}, when pricing and valuing caps and floors. This allows the smile effect to be incorporated. IR options on 3 month Libor are the most common, probably reflecting the fact that one can get exchange-traded options on 3 month deposit futures for hedging (see Section 10, Chapter 8). Therefore the most liquid volatility surface would also be on 3 month Libor, and volatility surfaces for other tenors represented by an off-set surface from the 3 month one. A more complete approach therefore would be to model the entire twodimensional surface. This surface is likely to contain gaps due to missing maturities and also missing volatilities for particular strikes. If the underlying forward interest rate curve is rising, a strike that is below but close to the money for a short maturity will be a long way from the money at a long maturity. As caps are usually only quoted relatively close to the money, there would be no long volatility quoted. If we represent the surface as a bi-cubic 7 See Tompkins (1994), Chapter 5 or Burghardt et al., "How to tell if options are cheap". Journal of Portfolio Management, Winter 1990, pp. 72–78.

280

Swaps and Other Derivatives

Worksheet 7.2 Constructing a forward volatility curve Today's date:

21-Jun-00 Market data

23-Jun-00 24-Jul-00 23-Aug-00 25-Sep-00 23-Oct-OO 23-Nov-OO 27-Dec-00 25-Jun-01 24-Jun-02 23-Jun-03 23-Jun-04 23-Jun-05 23-Jun-06 25-Jun-07 23-Jun-08 23-Jun-09 23-Jun-10

[1]

0.086 0.169 0.261 0.339 0.425 0.519 1.019 1.011 1.011 1.017 1.014 1.014 1.019 1.011 1.014 1.014

6.650% 6.693% 6.765% 6.824% 6.874% 6.920% 7.185% 7.285% 7.285% 7.285% 7.289% 7.299% 7.305% 7.305% 7.306% 7.306%

DFs

Cap maturities

[3]

[2]

1 0.994306 0.988787 0.982642 0.977398 0.971616 0.965302 0.931752 0.866943 0.807466 0.751786 0.699942 0.651333 0.605901 0.564226 0.525258 0.489066

Par volatilities

1 2 3 4 5

10.37% 12.87% 14.12% 15.12% 15.25%

7

15.13%

10

14.88%

281

Interest Rate OTC Options Step 1: Calculate the forward rates and strike

Start

3 6 9 12 15 18 21 24 27 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 75 78 81 84 87 90 93 96 99 102 105 108 111 114 117 120

[4] 23-Jun-00 25-Sep-OO 27-Dec-OO 23-Mar-Ol 25-Jun-01 24-Sep-01 24-Dec-01 25-Mar-02 24-Jun-02 23-Sep-02 23-Dec-02 24-Mar-03 23-Jun-03 23-Sep-03 23-Dec-03 23-Mar-04 23-Jun-04 23-Sep-04 23-Dec-04 23-Mar-05 23-Jun-05 23-Sep-05 23-Dec-05 23-Mar-06 23-Jun-06 25-Sep-06 25-Dec-06 23-Mar-07 25-Jun-07 24-Sep-07 24-Dec-07 24-Mar-08 23-Jun-08 23-Sep-08 23-Dec-08 23-Mar-09 23-Jun-09 23-Sep-09 23-Dec-09 23-Mar-10 23-Jun-10

DFs on start [5]

1 0.982642 0.956302 0.949285 0.931752 0.915279 0.898985 0.882874 0.866943 0.851674 0.836675 0.821940 0.807466 0.793092 0.799127 0.765409 0.751786 0.738372 0.725336 0.712667 0.699942 0.687373 0.675153 0.663271 0.651333 0.639390 0.628032 0.617235 0.605901 0.595202 0.584692 0.574368 0.564226 0.554141 0.544341 0.534820 0.525258 0.515890 0.506788 0.497945 0.489066

End

DFs on end

[6]

[7]

27-Dec-OO 27-Mar-01 25-Jun-Ol 25-Sep-01 24-Dec-Ol 25-Mar-02 25-Jun-02 24-Sep-02 23-Dec-02 24-Mar-03 24-Jun-03 23-Sep-03 23-Dec-03 23-Mar-04 23-Jun-04 23-Sep-04 23-Dec-04 23-Mar-05 23-Jun-05 23-Sep-05 23-Dec-05 23-Mar-06 23-Jun-06 25-Sep-06 25-Dec-06 26-Mar-07 25-Jun-07 25-Sep-07 24-Dec-07 24-Mar-08 24-Jun-08 23-Sep-08 23-Dec-08 23-Mar-09 23-Jun-09 23-Sep-09 23-Dec-09 23-Mar-10 23-Jun-10 23-Sep-10

1 0.965302 0.948539 0.931752 0.915098 0.898985 0.882874 0.866774 0.851508 0.836675 0.821940 0.807308 0.793092 0.779127 0.765409 0.751786 0.738372 0.725336 0.712667 0.699942 0.687373 0.675153 0.663271 0.651333 0.639390 0.628032 0.616870 0.605901 0.595085 0.584692 0.574368 0.564116 0.554141 0.544341 0.534820 0.525258 0.515890 0.506788 0.497945 0.489066 0.471541

Daycount Act/360 [8]

0.258 0.250 0.261 0.256 0.253 0.253 0.256 0.256 0.253 0.253 0.256 0.256 0.253 0.253 0.256 0.256 0.253 0.250 0.256 0.256 0.253 0.250 0.256 0.261 0.253 0.253 0.261 0.256 0.253 0.253 0.256 0.256 0.253 0.250 0.256 0.256 0.253 0.250 0.256 0.256

Forward rates [9] 6.954% 7.069% 7.207% 7.121% 7.170% 7.220% 7.268% 7.093% 7.092% 7.092% 7.092% 7.092% 7.091% 7.090% 7.091% 7.109% 7.110% 7.111% 7.114% 7.156% 7.160% 7.165% 7.172% 7.153% 7.155% 7.158% 7.164% 7.112% 7.111% 7.111% 7.111% 7.122% 7.122% 7.121% 7.123% 7.106% 7.105% 7.104% 7.104% 14.543%

Forward swap rates [10]

7.075%

7.141%

7.124%

7.116%

7.115%

7.127%

7.123%

282

Swaps and Other Derivatives

Step 2: Pricing the generic caps 1 yr Cap

Strike Par Volatilities Cap prices (bp) Option maturity

2 yr Cap 7.141% 12.87%

7.075% 10.37% 15.18 d1

[11]

[12]

caplet pricing [13]

0.258 0.512 0.748 1.005 1.255 1.504 1.753 2.003 2.252 2.501 2.751 3.000 3.252 3.501 3.751 4.003 4.255 4.504 4.751 5.003 5.255 5.504 5.751 6.003 6.260 6.510 6.751 7.008 7.258 7.507 7.756 8.005 8.258 8.507 8.753 9.005 9.258 9.507 9.753 10.005

-0.303 0.024 0.250

2.35 4.89 7.94

3 yr Cap 7.124% 14.12%

57.88 dl

[14] -0.374 -0.064 0.138 0.043 0.100 0.148 0.189

caplet pricing [15] 2.62 5.37 8.57 8.35 9.68 10.98 12.30

115.10 d1

[16] -0.302 -0.027 0.155 0.068 0.119 0.163 0.200 0.078 0.085 0.091 0.098

caplet pricing [17] 3.17 6.14 9.52 9.36 10.76 12.14 13.53 11.98 12.35 12.79 13.34

283

Interest Rate OTC Options

7.115% 15.25%

[18] -0.263 -0.008 0.162 0.080 0.129 0.170 0.205 0.092 0.098 0.105 0.112 0.118 0.123 0.129 0.134

caplet pricing [19]

3.58 6.71 10.22 10.12 11.57 13.00 14.45 12.92 13.32 13.79 14.38 14.75 14.92 15.21 15.64

[20]

caplet pricing [21]

-0.258 -0.005 0.163 0.082 0.130 0.171 0.206 0.094 0.100 0.107 0.114 0.120 0.125 0.130 0.136 0.149 0.155 0.160 0.166

3.64 6.78 10.31 10.21 11.68 13.12 14.57 13.05 13.44 13.92 14.51 14.89 15.06 15.35 15.79 16.21 16.25 16.25 16.79

dl

10 yr Cap 7.123% 14.88%

7.127% 15.13% 251.82

184.61 d1

7 yr Cap

5 yr Cap

4 yr Cap 7.116% 15.12%

579.21

386.76

dl [22]

caplet pricing [23]

-0.281 -0.021 0.151 0.071 0.121 0.163 0.198 0.085 0.092 0.099 0.106 0.112 0.118 0.124 0.129 0.143 0.148 0.154 0.160 0.181 0.187 0.193 0.199 0.196 0.200 0.205 0.210

3.50 6.60 10.10 10.01 11.47 12.90 14.35 12.83 13.23 13.71 14.29 14.67 14.84 15.14 15.57 15.99 16.03 16.04 16.57 17.11 17.08 17.01 17.51 17.75 17.24 17.29 17.91

dl

[24] -0.281 -0.019 0.155 0.073 0.122 0.165 0.201 0.085 0.092 0.099 0.106 0.112 0.117 0.123 0.128 0.142 0.147 0.152 0.158 0.180 0.186 0.191 0.198 0.194 0.198 0.203 0.208 0.193 0.196 0.200 0.203 0.210 0.213 0.216 0.220 0.218 0.221 0.223 0.227

caplet pricing [25]

3.44 6.52 9.99 9.88 11.32 12.74 14.18 12.65 13.04 13.51 14.08 14.45 14.62 14.91 15.33 15.75 15.79 15.80 16.32 16.86 16.83 16.77 17.26 17.49 16.99 17.04 17.65 16.82 16.62 16.60 16.76 16.81 16.58 16.35 16.66 16.46 16.20 15.94 16.22

284

Swaps and Other Derivatives

Step 3: Calculating a stepped volatility curve

Par vol curve

Forward

Strike

1 yr Cap 7.075%

2 yr Cap 7.141%

3 yr Cap 7.124%

vol curve Pricing Difference d1

0% 0% 10.37%

0% 0% 0% 12.87%

0% 0% 0% 14.12%

0% 0% 0% 15.12%

0% 0% 0% 15.25%

0% 0% 0% 0% 0% 0% 0% 15.13%

0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 0% 14.88%

[26]

[27]

10.37% 10.37% 10.37% 14.03% 14.03% 14.03% 14.03% 15.63% 15.63% 15.63% 15.63% 17.12% 17.12% 17.12% 17.12% 15.62% 15.62% 15.62% 15.62% 14.91% 14.91% 14.91% 14.91% 14.91% 14.91% 14.91% 14.91% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40% 14.40%

-0.303 0.024 0.250

15.18 0.00 caplet pricing [28]

0.00 d1

[29]

2.35 -0.478 4.89 -0.099 7.94 0.147 0.051 0.104 0.150 0.188

57.88 0.00 caplet pricing [30] 1.82 4.18 7.07 9.12 10.52 11.89 13.27

0.00 dl

[31] -0.434 -0.068 0.173 0.067 0.119 0.163 0.200 0.091 0.098 0.105 0.112

285

Interest Rate OTC Options

4 yr Cap 7.116%

115.10 0.00 caplet pricing [32]

1.94 4.35 7.29 9.31 10.70 12.07 13.46 13.29 13.70 14.20 14.80

0.00

dl [33] -0.413 -0.053 0.185 0.075 0.126 0.170 0.206 0.096 0.103 0.110 0.116 0.137 0.143 0.149 0.155

184.61 0.00 caplet pricing [34]

2.01 4.44 7.39 9.39 10.79 12.16 13.55 13.37 13.78 14.27 14.87 16.72 16.91 17.24 17.72

5 yr Cap

0.00

dl [35] -0.410 -0.051 0.187 0.076 0.127 0.171 0.207 0.097 0.103 0.110 0.117 0.137 0.143 0.149 0.155 0.153 0.159 0.164 0.170

Cap 7.127%

7.115% 251.82 0.00 caplet pricing [36]

2.01 4.45 7.40 9.41 10.80 12.17 13.56 13.38 13.79 14.28 14.88 16.73 16.92 17.25 17.73 16.60 16.64 16.64 17.19

0.00

dl [37] -0.440 -0.072 0.169 0.065 0.117 0.161 0.199 0.090 0.097 0.104 0.111 0.132 0.138 0.144 0.151 0.148 0.154 0.159 0.165 0.179 0.185 0.190 0.197 0.193 0.197 0.202 0.207

386.76 0.00 caplet pricing [38]

1.93 4.33 7.26 9.28 10.68 12.05 13.43 13.27 13.68 14.18 14.78 16.63 16.82 17.15 17.64 16.51 16.55 16.56 17.10 16.87 16.83 16.77 17.27 17.50 17.00 17.05 17.66

10 yr Cap 7.123%

0.00

dl [39] —0.431 -0.066 0.174 0.068 0.120 0.164 0.201 0.092 0.099 0.106 0.113 0.133 0.140 0.146 0.152 0.150 0.155 0.160 0.166 0.180 0.186 0.192 0.198 0.194 0.198 0.203 0.208 0.186 0.190 0.193 0.196 0.203 0.206 0.209 0.213 0.210 0.213 0.216 0.219

579.21 0.00 caplet pricing [40]

1.95 4.36 7.30 9.32 10.71 12.08 13.47 13.30 13.71 14.21 14.81 16.66 16.85 17.18 17.67 16.53 16.57 16.58 17.13 16.89 16.86 16.80 17.29 17.52 17.02 17.07 17.68 16.28 16.09 16.07 16.22 16.27 16.05 15.83 16.13 15.93 15.68 15.43 15.70

286

Swaps and Other Derivatives

Step 4: Calculating a smooth volatility curve

Par vol curve

Forward

vol

Smoothness condition

Strike

1 yr Cap 7.075%

2 yr Cap 7.141%

3 yr Cap 7.124%

curve 7.82

0 0 10.37%

0 0 0 12.87%

0 0 0 14.12%

0 0 0 15.12%

0 0 0 15.25%

0 0 0 0 0 0 0 15.13%

0 0 0 0 0 0 0 0 0 0 0 14.88%

[41]

[42]

9.71% 10.09% 10.97% 12.49% 13.70% 14.55% 15.04% 15.13% 15.36% 15.73% 16.26% 16.94% 17.29% 17.29% 16.95% 16.26% 15.73% 15.36% 15.15% 15.10% 15.05% 15.00% 14.95% 14.89% 14.83% 14.76% 14.70% 14.63% 14.56% 14.51% 14.46% 14.41% 14.38% 14.35% 14.32% 14.31% 14.29% 14.29% 14.28%

0.0014% 0.0079% 0.0231% 0.0144% 0.0073% 0.0024% 0.0001% 0.0005% 0.0014% 0.0028% 0.0047% 0.0012% 0.0000% 0.0011% 0.0048% 0.0028% 0.0014% 0.0004% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000% 0.0000%

Pricing Difference d1

[43] -0.327 0.023 0.241

15.18 0.00 caplet pricing [44] 2.13 4.75 8.29

0.00 d1

[45]

57.88 0.00 caplet pricing [46]

-0.514 1.61 -0.104 4.04 0.144 7.44 0.041 8.10 0.103 10.27 0.151 12.29 0.188 14.11

0.00

dl [47] -0.467 -0.072 0.168 0.059 0.118 0.164 0.200 0.087 0.096 0.106 0.118

287

Interest Rate OTC Options

115.10 0.00 caplet pricing [48]

1.73 4.22 7.65 8.28 10.46 12.47 14.30 12.85 13.46 14.29 15.40

0.00

d1 [49] -0.444 -0.057 0.180 0.068 0.126 0.170 0.206 0.092 0.101 0.111 0.122 0.135 0.144 0.151 0.153

184.61 0.00 caplet pricing [50]

1.80 4.30 7.75 8.37 10.54 12.56 14.39 12.93 13.53 14.36 15.47 16.55 17.08 17.42 17.55

0.00

dl [51] -0.441 -0.055 0.182 0.069 0.126 0.171 0.206 0.093 0.101 0.111 0.123 0.136 0.145 0.151 0.154 0.160 0.160 0.161 0.164

7.123%

7.127%

7.115% 251.82 0.00 caplet pricing [52]

1.80 4.31 7.76 8.38 10.56 12.57 14.40 12.94 13.54 14.37 15.48 16.56 17.09 17.42 17.56 17.28 16.75 16.36 16.67

10 yr Cap

7 yr Cap

5 yr Cap

4 yr Cap

7.116%

0.00

dl [53] -0.473 -0.077 0.165 0.057 0.116 0.162 0.198 0.085 0.094 0.105 0.117 0.130 0.140 0.146 0.149 0.155 0.155 0.156 0.160 0.181 0.186 0.191 0.197 0.193 0.196 0.200 0.205

386.76 0.00 caplet pricing [54]

1.72 4.19 7.62 8.26 10.43 12.45 14.27 12.83 13.44 14.27 15.38 16.46 16.99 17.33 17.47 17.19 16.66 16.28 16.58 17.08 16.99 16.87 17.31 17.48 16.91 16.89 17.41

0.00

d1 [55] —0.464 -0.070 0.170 0.060 0.119 0.164 0.201 0.087 0.096 0.107 0.119 0.132 0.141 0.148 0.150 0.156 0.156 0.158 0.161 0.182 0.188 0.193 0.198 0.194 0.197 0.201 0.206 0.190 0.192 0.194 0.197 0.203 0.206 0.209 0.212 0.209 0.211 0.214 0.217

579.21 0.00 caplet pricing [56]

1.74 4.23 7.66 8.29 10.47 12.49 14.31 12.86 13.47 14.30 15.41 16.49 17.02 17.36 17.50 17.21 16.69 16.30 16.61 17.10 17.02 16.90 17.33 17.50 16.93 16.91 17.44 16.53 16.27 16.19 16.28 16.29 1 6.03 15.77 1 6.04 15.83 15.57 15.31 15.57

288

Swaps and Other Derivatives

function, i.e. it is a cubic polynomial in both the maturity and strike directions, then the smoothness approach could be generalized8. This issue may seem academic, but actually has major practical ramifications. Suppose a trader sells a 10 year ATM cap. After 5 years, interest rates could have moved significantly so that the remaining caplets are nowhere close to ATM — yet they still need to be valued daily! Where do the risk controllers obtain the appropriate volatilities, as they are unlikely to be readily available in the current quotes9. In the discussion below (and in Section 8.10 where the risk management of IR options is discussed), a single volatility curve has been used throughout for ease of exposition.

7.5 CAPS, FLOORS AND COLLARS We have already discussed the pricing of caps using Black's model. Obviously the same model can be used to price a floor, which is a strip of put options on forward interest rates. Using the same notation as before: • • • •

define a forward interest rate F(r, T), which starts at time T and finishes at time T: assume that the option has a strike of K% and is written on a principal P; at time T, the fixing of F is observed to be L%; payout of the floorlet is max[0, K — L] x (T— T) x P.

Black's model for the floorlet on F(t, T) may be written as: Fl = P x DFT x {K x N(–d 2 ) - F(T, T) x N(—d1)} x (T— T) As an example, to price a floorlet on a 3mo. forward rate (i.e. the same period as above): Today's date: Principal amount: Forward rate: Strike: Volatility:

4 January 2000 $100 million start date 6 July 2001 end date 8 Oct 2001 6% 17% pa

d1 = {ln(7.106%/6%) + 0.5 x 17% x 17% x 1.519}/(17% x 1.232) = 0.9123 d2 = 0.0092 - 17% x 1.232 = 0.7027 N(—d1) = 0.1808 N(—d2) = 0.2411 Fl = $100 million x 0.889113 x {6% x 0.2411 – 7.106% x 0.1808} x 0.261 = $3,756 or 3.756 bp An alternative approach would be to create a portfolio consisting of: • short a caplet, and • long a floorlet 8

See Coleman et al. for details of a specific application {T. F. Coleman, Y. Li and A. Verma. "Reconstructing the unknown local volatility function". Journal of Computational Finance. 2(3), 1999, pp. 77–102}. 9 This problem has provided the basis for a number of well-publicized losses by banks, as they have invariably been forced to rely on traders for the information. There are various "closed clubs" whereby banks anonymously share information.

Interest Rate OTC Options

on the same forward rate F(i, T), with the same strike K and principal amount, and using the same volatility. The payout is proportional to max[0, K — L] — max[0, L — K] = [K — L] where L is the Libor fixing. Now consider a one period forward swaplet starting at T and finishing at T: • to receive fixed K, • to pay floating. The net settlement at the end of the period is again proportional to [K— L], As this is identical to the portfolio payout, today's value of the swaplet and portfolio should also be equal. This gives: price of floorlet = value of swaplet + price of caplet As the best estimate for the Libor fixing is the implied forward rate, the value of the swaplet is [K — F] x ( T — t ) x DFT. Using the same example: value of swaplet = 10,000 x [6% – 7.106%] x 0.261 x 0.889113 = –25.683 bp value of caplet = 29.439 bp (struck at 6%) value of floorlet = –25.683 + 29.430 = 3.756 bp (as before) Details of the calculations are shown in Worksheet 7.3. It is divided into two parts: the first calculates caplet prices and subsequently floorlet prices using the above put—call parity expression, whilst the second calculates floorlet prices followed by caplet prices. The columns are the same as before, with the introduction of column [15] which contains the new pricing. A full relationship between caps, floors and forward swaps may be derived in an exactly analogous fashion. The value of a forward swap to receive a fixed rate K is given by: value = K x (Qe — Qs) — (DFS — DFe) for a notional principal of 1. Thus, for the above example: Start date: End date:

6 April 2000 6 January 2003

DFS =0.984983 DFe =0.812962 Qe — Qs = 2.485045 Value of swap =10,000 x {6% x 2.485045 - (0.984983 — 0.812962)} = –229.189 bp Value of cap = 273.214 bp Value of floor = 273.214–229.189 = 44.030 bp The floor market is often less liquid than the cap and swap markets, and these relationships are widely used to act as an arbitrage check between the markets. Many borrowers buy caps as protection against rising interest rates. The example cap above was struck at 7.25% at a cost of 109 bp, and Figure 7.4 shows the constant strike compared with the forward rate curve. Is the cap providing good protection? At the short end, interest rates could rise by 50 to 100 bp before the appropriate caplets came into the money. So the protection is not very tight. An alternative structure is a "curve cap", in which the strike is set to be a constant spread above the forward curve. Setting the strike spread to 26.4 bp, the cost of the curve cap is exactly the same as the original cap (see Figure 7.5 and worksheet on CD for precise

290

Swaps and Other Derivatives

Worksheet 7.3 Cap and floor pricing with USD data Cap Pricing Model Settlement date:

06-Jan-00

Start date

End date

Strike (%)

Principal (m)

[1] 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

[2] 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

[3] 6.000% 6.000% 6.000% 6.000% 6.000% 6.000% 6.000% 6.000% 6.000% 6.000% 6.000%

[4] 100 100 100 100 100 100 100 100 100 100 100

Ann Vol (%)

Caplet (bp)

[5] 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

[6] 9.81 16.07 22.57 21.45 25.38 29.44 31.41 27.49 28.73 29.89 30.97

Total cost (bp) = Total cost ($) =

Floorlet (bp) [15]

2.21 2.48 2.48 3.34 3.57 3.76 3.70 5.28 5.53 5.75 5.92

273.219 2,732,192

44.030 440,301

Floorlet (bp) 2.21 2.48 2.48 3.34 3.57 3.76 3.70 5.28 5.53 5.75 5.92

Caplet (bp) 9.81 16.07 22.57 21.45 25.38 29.44 31.41 27.49 28.73 29.89 30.97

44.03 440,301

273.22 2,732,192

Floor Pricing Model Settlement date: Start date 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-Ol 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

06-Jan-00 End date

06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

Strike (%)

Principal (m)

6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00% 6.00%

100 100 100 100 100 100 100 100 100 100 100

Total cost (bp) = Total cost ($) =

Ann Vol (%) 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

291

Interest Rate OTC Options

Discount factors

far

near

[7] 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796

[8] 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

daycount

implied forward

[10]

time to exercise

d1

N(d1)

N(d2)

[12]

[13]

[14]

0.632 0.795 0.939 0.818 0.866 0.912 0.951 0.771 0.777 0.785 0.793

0.736 0.787 0.826 0.793 0.807 0.819 0.829 0.780 0.782 0.784 0.786

0.708 0.750 0.785 0.741 0.750 0.759 0.766 0.702 0.699 0.696 0.695

[9] 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219%

[11] 0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792

daycount

forward

time to exercise

dl

N(–d1)

N(–d2)

0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792

0.632 0.795 0.939 0.818 0.866 0.9123 0.951 0.771 0.777 0.785 0.793

0.264 0.213 0.174 0.207 0.193 0.1808 0.171 0.220 0.218 0.216 0.214

0.292 0.250 0.215 0.259 0.250 0.2411 0.234 0.298 0.301 0.304 0.305

Discount factors near 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796

far 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219%

292

Swaps and Other Derivatives

Worksheet 7.4 Pricing a mid-curve cap, fixings set at the end of year 1 and year 2 Settlement date:

06-Jan-00 Principal

Start date

End date

Strike (%)

[1] 6-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

[2] 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

[3] 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250%

(m)

Ann Vol (%)

[4] 100 100 100 100 100 100 100 100 100 100 100

[5] 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

Total cost (bp) = Total cost ($) = Price saving =

Caplet (bp) [6] 0.31 2.31 5.77 6.55 9.28 12.30 14.52 12.61 13.96 15.25 16.47

109.34 1,093,350

Mid-curve caplet (bp) [7] 0.31 2.31 5.77 6.55 8.06 9.82 11.03 12.61 13.08 13.56 14.02

97.12 971,175 11.2%

293

Interest Rate OTC Options

Mid-curve cap

Conventional cap Discount factors far near [8] 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796

[9] 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

day- implied time to count forward exercise [10] 0.253 0.256 0.256 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

[11] 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219%

[12] 0.253 0.506 0.761 1.022 1.267 1.519 1.781 2.033 2.286 2.539 2.792

d1

[13] –1.582 –0.770 -0.337 -0.283 -0.123 0.009 0.117 -0.009 0.041 0.086 0.127

N(d1)

N(d2)

time to exercise

[14] 0.057 0.221 0.368 0.388 0.451 0.504 0.547 0.496 0.516 0.534 0.550

[15] 0.048 0.186 0.314 0.324 0.377 0.421 0.456 0.401 0.415 0.427 0.438

[16] 0.253 0.506 0.761 1.022 1.022 1.022 1.022 2.033 2.033 2.033 2.033

dl

N(d1)

N(d2)

[17] -1.582 -0.770 -0.337 -0.283 -0.158 -0.031 0.091 -0.009 0.029 0.066 0.103

[18] 0.057 0.221 0.368 0.388 0.437 0.488 0.536 0.496 0.511 0.526 0.541

[19] 0.048 | 0.186 | 0.314 0.324 0.371 0.420 0.468 0.401 0.415 0.430 0.445

294

Swaps and Other Derivatives 8.0

7.5

7.0

6.5

6.0

0.00

0.50

1.00

1.50

2.00

2.50

3.00

Years

Figure 7.4 Illustration of standard cap using implied 3mo. forwards

details). Each caplet is out-of-the-money but to the same absolute amount. One might argue that a traditional cap is for a borrower that cannot afford to pay above a certain level; a curve cap for somebody that wants an insurance policy against rates rising but is content to pay if the actual fixings follow the forward curve. Another interesting extension to the vanilla cap is the "mid-curve" cap. Using the same notation as before, the payout of a cap was defined as proportional to max[0, L — K] where L is the fixing of the forward rate F(r, T). Modifying the notation for the forward rate to F(t, T, T) where t < t is the observation time. Obviously F(r, T, T) equals the actual fixing L, but when t < T, then F(t, T, T) is an observation of a forward rate. Consider now a call option of maturity t on F(t, T, T), i.e. where the payout is proportional to max[0, F(t, T, T) — K]. Thus, comparing this option with a conventional

8.0-

7.0-

Strike spread = 26.4 bp 6.0

0.00

0.50

1.00

1.50

2.00

2.50

Years Figure 7.5 Illustration of curve cap using implied 3mo. forwards

3.00

295

Interest Rate OTC Options

Cap-floor strategy "Do nothing" strategy Actual interest rate

Floor strike Figure 7.6

caplet, the payout is based on a forward rate instead of a spot rate. The maturity of the mid-curve option is less, and therefore likely to be cheaper. However it is also riskier because the forward rate at maturity may not be an accurate estimate of the spot rate at the fixing. Like many things, it's a trade-off. Worksheet 7.4 shows the details. It prices both the conventional cap struck at 7.25% for 109 bp and a mid-curve cap. This latter has been defined as follows: • • • • •

total maturity of 3 years; conventional caplets in year 1; caplets in year 2 are based upon F(l, T, T) where i = 1, 1.25, 1.5 and 1.75; caplets in year 3 are based upon F(2, T, T) where r = 2, 2.25, 2.5 and 2.75; the payout for each option matches the conventional cap. i.e. at time T as before.

The time of payout needs to be defined; the usual choices are either as above or all made at the maturity of the options, as soon as the forward rates are fixed. From a bank's perspective, the latter is probably more acceptable as it would obviously reduce the credit exposure; it is the same argument as for discounting an FRA. From an end-user's point of view, the former would match the timings of physical interest payments. Column [16] shows the date of the fixings, and then the option parameters calculated in columns [17]–[19]. The mid-curve caplets are priced in column [7], discounting back from the actual payment date. The price saving is just over 11%. When obtaining protection through the purchase of a cap, a common strategy to reduce the overall cost is to sell a floor at a lower strike. The impact of this is shown in Figure 7.6. The strategy is often called a "collar" or "cylinder" because the effective interest rate to be paid by a borrower is constrained to lie between the upper cap strike and the lower floor strike. The overall cost of the strategy depends upon the relative cost of the cap and floor, which in turn depends upon the positioning of the two strikes. Remember the basic sensitivity results:

296

Swaps and Other Derivatives

Original cap-floor strategy Participation

6.807%

7.25%

7%

Actual interest rate

Figure 7.7

cost of cap decreases as strike increases; cost of floor increases as strike increases. The usual approach to locating the strikes is first to set the cap strike, say at 7.25%. This gives a 3 year cap cost, as we have seen, of 109 bp. Secondly to decide upon the maximum overall cost that you wish to pay, say 50 bp. Therefore the floor must have a price of 59 bp which implies a strike of 6.205%. See Worksheet 7.5. The borrower has bought the cap and sold the floor back to the bank. It would be unusual for the bank to buy the floor back on the same volatility; normally the bid (or ask) volatility would be used. If we assume a 2% spread, then the example would use 17% for the cap but only 15% for the floor. As expected, the floor would be struck at a higher strike of 6.357% implying that the borrower would be paying a higher effective interest rate if the floating rate dropped. Zero cost collars are particularly popular as these involve no upfront payments at all. In this case, and using the {17%, 15%} volatility spread, the floor would be struck at 6.807%. An alternative structure to collars is "participations", which are constructed by adjusting the amounts of the cap and floor. For example, assume the zero cost collar above was based on a constant principal amount of $100 million. Instead of buying the floor struck at 6.807%, suppose it was struck at 7%, hence increasing its value from 109 bp to 136bp. The collar is no longer zero cost. However, by only selling $80 million of the floor, the overall structure is again zero cost, i.e. $100 million x (109 bp/136 bp) = $80 million. The net effect is that there is full protection still against interest rates rising, the borrower would gain if interest rates fell between 7.25% and 7%, and would also gain on 20% of the borrowing if rates fall below 7% (Figure 7.7). Introducing the idea of participations increases the flexibility of these structures considerably, and there are many varieties. See Worksheet 7.6.

297

Interest Rate OTC Options Worksheet 7.5

Collar pricing

Strike Volatility Principal Cost (bp) Cost ($)

Cap 7.250% 17.0% 100 109.34 1,093,350

Floor Upfront Payment Overall Net 6.357% 15.0% 100 59.34 50 0 500,000 0 593,350

06-Jan-OO

Settlement date: Cap Pricing Model Start date

End date

Strike (%)

Principal (m)

Ann Vol (%)

[1] 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-01 06-Apr-01 06-Jul-Ol 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

[2] 06-Jul-OO 06-Oct-00 08-Jan-Ol 06-Apr-01 06-Jul-Ol 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

[3] 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250%

[4] 100 100 100 100 100 100 100 100 100 100 100

[5]

[6]

17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

0.31 2.31 5.77 6.55 9.28 12.30 14.52 12.61 13.96 15.25 16.47

Total cost (bp) = Total cost ($) =

Caplet (bp)

109.335 1,093,350

Floor Pricing Model Start date

End date

Strike (%)

Principal (m)

Ann Vol (%)

Floorlet (bp)

06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-Ol 06-Apr-Ol 06-Jul-Ol 08-0ct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

06-Jul-01 06-Oct-OO 08-Jan-Ol 06-Apr-Ol 06-Jul-OO 08-Oct-OO 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

6.357% 6.357% 6.357% 6.357% 6.357% 6.357% 6.357% 6.357% 6.357% 6.357% 6.357%

100 100 100 100 100 100 100 100 100 100 100

15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0%

5.27 4.54 3.92 4.81 4.84 4.84 4.57 6.44 6.59 6.71 6.80

Total cost (bp) = Total cost ($) -

59.335 593,350

298

Swaps and Other Derivatives

Worksheet 7.6

Structuring a participation

Cap Strike Volatility Principal Cost (bp) Cost ($) Settlement date:

7.250% 17.0%

100 109.34 1,093,350

Floor 7.000% 15.0% 80.17 136.38 1,093,350

Upfront Payment Overall Net

0

0

Principal (m)

Ann Vol (%)

Caplet (bp)

[4] 100 100 100 100 100 100 100 100 100 100 100

[5]

[6]

17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

0.31 2.31 5.77 6.55 9.28 12.30 14.52 12.61 13.96 15.25 16.47

06-Jan-00

Cap Pricing Model Start date

End date

[2]

[1] 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

Strike (%)

[3] 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250% 7.250%

109.335 1,093,350

Total cost (bp) = Total cost ($) = Floor Pricing Model Start date

End date

Strike (%)

06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02

06-Jul-00 06-Oct-00 08-Jan-01 06-Apr-01 06-Jul-01 08-Oct-01 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

7.000% 7.000% 7.000% 7.000% 7.000% 7.000% 7.000% 7.000% 7.000% 7.000% 7.000%

Principal (m)

Ann Vol (%)

80.17 80.17 80.17 80.17 80.17 80.17 80.17 80.17 80.17 80.17 80.17 Total cost (bp) = Total cost ($) =

15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0% 15.0%

Floorlet (bp) 17.38 13.69 11.18 11.77 11.29 10.87 9.92 12.69 12.62 12.53 12.43 136.379 1,093,350

299

Interest Rate OTC Options

7.6 DIGITAL OPTIONS Digital or binary caps are a very fundamental structure, and are often embedded in more complex products. The payout from a digicaplet is simply D{× (T —T) × P} if L > K and zero otherwise, where D is some predetermined constant usually quoted as a percentage or in basis points, as shown below:

Payout from ordinary caplet Payout from digicaplet

K

L

Using a Black model to price such an option is straightforward, i.e.: DC = DFT × D{×(T—T ) × P} × N(d2) This formula may be interpreted as "discounted constant payout" x "probability of being in the money at maturity". Digicaplets are usually cheaper than the equivalent ordinary caplet because the payout is limited; the exception arises when the option is close to ATM and has relatively low volatility. Digital options may be regarded as fundamental building blocks, and in theory can be used to replicate ordinary options, or of course vice versa. They can also be used to price a European option with a complex payout strategy. For example, consider an ordinary call option with strike K. Its payout can be replicated by a series of digital call options struck at K+ ih for i = 1, 3, 5 , . . . each with payout P = 2h. This ensures that the digital portfolio will approximate the call option with a maximum error of h. Figure 7.8 shows the asymptotic cost accumulation of a series of digital options to a call option struck at 100.

K

K+h

Hedging a single digital option can be difficult only due to the discontinuous nature of its payout, which is likely to be far less evident in a portfolio. Probably the most popular

300

Swaps and Other Derivatives 16 T

100.3

115.3

130.3

145.3

160.3

175.3

190.3

Strike of the digitals Figure 7.8 Approximating a standard call option by a strip of digitals: h = 0.5

method for hedging (say) a sold digicap struck at K with payout P would be to buy N ordinary caps with strike Kh and to sell N caps struck at K + h. The parameters would be determined by P = 2Nh: for example, if K = 100 and P = 10, then five caps would have to be bought and sold if h = 1.

7.7 EMBEDDED STRUCTURES Many structures are provided with embedded options. For example, many investors during the 1980s entered into pension arrangements with Equitable Life, a large UK insurance company. At maturity these pension schemes delivered annuities with guaranteed minimum levels; in essence the company had provided the investors with a floor. When these pensions were being sold, the floor was considerably OTM and was effectively ignored by the company. Unfortunately interest rates declined significantly during the 1990s and the floor became a long way ITM. The company had apparently paid no attention to the impact of declining rates until 1999; after a subsequent investigation, it was declared effectively insolvent! It is obviously important to be able to price and replicate such structures correctly. In this section we will discuss a number of structures, in particular bringing together swaps and options. As the background, we will assume a company is raising $100 million for 3 years at 3mo. Libor flat. The company then wishes to enter into various structures to manage the interest rate risk. For example, the company could buy a cap struck at 7% for 132.48 bp. As Worksheet 7.7 shows, spreading this cost over 3 years results in a margin of 48.5 bp, i.e.: Libor + 48.5 bp 7.485%

for Libor < 7% for Libor > 7%

This strategy is a common option on many retail mortgages, namely to buy a separate option over and above the mortgage.

301

Interest Rate OTC Options

Worksheet 7.7 Capped loan (1)

Settlement date:

06-Jan-OO

Company enters into a $100 million 3 year loan @ 3mo. Libor Cap the Libor @

7%

Cost of cap:

1 32.48 bp (using a vol. of 17%)

Hence borrowing cost: Libor + 48.5 bp Date

Daycount

Discount factors

Margin over LIBOR 48.5bppa

06-Jan-OO 06-Apr-OO 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 Present value — Difference =

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

12.249 12.249 12.383 12.653 11.845 12.249 12.653 12.249 12.249 12.249 12.249 12.249 132.48 0.000

302

Swaps and Other Derivatives

Table 7.3 Demonstration of iterative method for calculating the cap strike and margin, target 7% Iteration:

1

2

3 4

5 6

7 8

9 1 0

1 1

1 2

Final

Strike 7.000% 6.515% 6.304% 6.189% 6.122% 6.080% 6.053% 6.035% 6.024% 6.016% 6.010% 6.005% 5.991% Cap cost 132.5 190.3 221.6 240.2 251.7 259.0 263.7 266.8 269.0 270.7 272.0 272.9 275.9 Margin (bp) 48.5 69.6 81.1 87.8 92.0 94.7 96.5 97.6 98.4 99.0 99.5 99.8 100.9 Ceiling 7.48% 7.21% 7.11% 7.07% 7.04% 7.03% 7.02% 7.01% 7.01% 7.01% 7.00% 7.00% 7.00%

Suppose however the company wishes to place a cap on the total interest rate payable of 7%, including the cost of the cap. In other words, cap strike + margin = 7%. The calculation is not so simple as the margin is obviously a function of the strike, and so an iterative method has to be used. Table 7.3 uses a very simple rule, namely Ki = 7% —margin,.], which converges quite slowly. The final cap is struck at 5.991% which costs 100.9 bp pa, and therefore the sum is equal to 7%, as shown in Worksheet 7.8. One practical issue that must be taken into consideration: a conventional cap does not include the first fixing as discussed above, whereas almost invariably embedded constraints will apply across all fixings. In this case, the first fixing was 6.03125%, i.e. above the cap strike, so an additional cost of DF1 x{F(t1, T1) — K x (T1, — r 1 ) = 1 bp has to be included in the cost of the cap. In summary, this structure would be: Libor+ 100.9bp 7%

for Libor< 5.991% for Libor> 5.991%

This second structure is costing the company an additional 50 bp but limits the maximum rate to 7%. Which structure is preferred depends upon its view of interest rates over the next 3 years and the likely impact on the performance of the company. Swaps are frequently provided with embedded options on the floating side. In this case, the cost of the option is invariably integrated into the effective fixed rate. For example, consider a generic 3 year swap to receive the fixed rate and pay 3mo. Libor; the current rate is 7.025% ann. We wish to cap the Libor at 7%; the option price should be reflected in the fixed rate which of course should be lowered. Looking at Worksheet 7.9, the cost of the cap is 132 bp. Spreading this out over 3 years: 132 bp/Q3 = 132/2.66 = 49.76 bp or this margin can be estimated by constructing the cashflows. Therefore the new rate is 7.025%–49.76 bp = 6.527%. Let's now turn the swap around the other way, and receive Libor subject to a floor at 6.5%. The same approach can be employed, but there is one additional complication. The first fixing of Libor was 6.03125%, i.e. below the floor. Conventionally however, these embedded option structures include the first fix. Therefore the total cost of the floor is: cost of conventional floor (2 6.5% = 87.54bp cost of first fix, (6.5% - 6.03125%) x 6 Apr–6 Jan)/360 x DF3 month = 11.67 bp The adjustment to the fixed side is 99.21 bp/2.66 = 37.26 bp which of course in this case has to be added to the fixed rate as the floating rate receiver/fixed rate payer is benefiting from the option.

303

Interest Rate OTC Options

Worksheet 7.8 Capped loan (2)

06-Jan-OO

Settlement date:

Company enters into a $100 million 3 year loan @ 3mo. Libor Cap the Libor Cost of cap:

@

5.991% 275.87 bp (using a vol. of 17%)

Hence borrowin g cost: Libor + 100.901 bp Provides an effective cap (@ 7.000% Date

Daycount

Discount factors

Margin over LIBOR 100.901 bp pa

06-Jan-00 06-Apr-OO 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 Present value =

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

25.505 25.505 25.786 26.346 24.665 25.505 26.346 25.505 25.505 25.505 25.505 25.505 275.87

Worksheet 7.9 Spreadsheet to price a capped/floored swap Capped Settlement date:

06-Jan-00

Floored Settlement date:

06-Jan-OO

Structure Maturity To pay To receive First Libor fixing

3 years 3mo. L subject to 6.527% Act/360 annually 6.03125%

Structure Maturity To receive To pay First Libor fixing

3 years 3mo. L subject to 7.398% Act/360 annually 6.03125%

7.00% cap

Generic 3 year swap rate 7.0250% Value of cap Value of first fix Total value

132.482 0.000 132.482

Generic 3 year swap rate 7.0250% Q3= margin

2.66 49.759

06-Jan-OO 06-Apr-00 06-Jul-00 06-Oct-OO 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

Daycount

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

Spreading the cost of the cap 49.759

1.022

50.865

1.011

50.312

1.011

50.312

PV = Difference

Value of cap Value of first fix Total value

87.540 11.672 99.211

Q3

=

margin =

2.66 37.263

Note: in this case. the first fixing should be included

Note: in this case, the first fixing should be included DFs

6.50% floor

132.482 0.000

DFs

06-Jan-OO 06-Apr-OO 06-Jul-OO 06-Oct-OO 08-Jan-Ol 06-Apr-Ol 06-Jul-Ol 08-Oct-Ol 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

Daycount

1 .022

Spreading the cost of the floor 37.263

38.091

1.011

37.677

1.0 II

37.677

PV =

Difference

99.211 0.000

Interest Rate OTC Options

305

At the time of writing, interest rates are relatively low, and their future direction uncertain. For a more complex example, assume that the company issues a reverse floating rate note to raise money; this strategy is attractive to investors who anticipate rates declining further. The coupon is set at 13.5%—3mo. Libor and is subject to a cap to prevent it going negative. The company has a funding target of 25 bp below Libor, and to achieve this it is prepared to consider inserting a floor onto the note: L–25bp

13.5% - L subject to: F< L < 13.5%

13.5% - L subject to: F< L < 13.5%

The objective therefore is to calculate the level of floor such that the swap has an overall value of zero, taking into account the embedded options. Looking at Worksheet 7.10, we can see that the swap, with zero sub-Libor margin and ignoring the options, has a current value of —$498,302 from the point-of-view of the issuer. The fair breakeven margin is only 18.2bp below Libor, so to achieve 25 bp below Libor is worth $185,205 to the issuer. The existence of the cap, by preventing the receive cashflow from going negative, adds further value of $6,771 to the issuer. Therefore the counterparty would demand compensation worth $185,205 + 6,771 = 191,976, which is equivalent to a floor struck at 5.49%. The key to pricing these structures is always to ask "who benefits?". Starting with the basic fair swap to receive {13.5% — L} and to pay {L— 18.2bp} first add the cap. This benefits the issuer, and therefore the counterparty wishes to receive a higher cashflow. Then add the floor, this benefits the counterparty and hence the issuer will pay a lower cashflow. There are a wide variety of embedded option structures. For example, range structures are popular when interest rates are deemed to be relatively stable, without any obvious direction of movement. In the second half of 1993, USD cash rates were very low, and the market anticipated them to stay low for 1994 as well. General Electric Capital Corp (GECC) issued a range FRN with the following details and then swapped it into vanilla sub-Libor: issue date: 22 February 1994, led by Kidder Peabody maturity: 2 years size: $100m issue price: 99.80, net of fees and expenses coupon: 3mo. Libor + 75 bp interest accrued daily only when: first 6 months: 3.00% < 3mo. Libor <4.00% second 6 months: 3.00%<3mo. Libor <4.75% third 6 months: 3.00% < 3mo. Libor< 5.50% fourth 6 months: 3.00% < 3mo. Libor<6.00% and did not accrue otherwise

Worksheet 7.10

Spreadsheet to price a reverse floating swap with a collar

Settlement date: 06-Jan-OO Structure (from maturity issue size to pay to receive subject to

the point of view of the issuer) 3 years 100 million USD 3mo. L less 25 bp margin 13.5% less 3mo. L cap at 13.5% and a floor at 5.490%

First Libor fixing Generic 3 year swap rate

6.03125% 7.0250% With embedded options

With no embedded options

Value of Pay side Value of Receive side Value of cap Value of floor Total Value

Net PV with zero margin = -498,302 Breakeven margin -18 23 bp's (with no embedded options)

-18,020,327 18,205,532 6,771 -191,976

0

Note: in this case, the first fixing has been included in the pricing of the options DFs 06-Jan-OO 06-Apr-00 06-Jul-00 06-Oct-00 08-Jan-0l 06-Apr-0l 06-Jul-0l 08-Oct-0l 07-Jan-02 08-Apr-02 08-Jul-02 07-Oct-02 06-Jan-03

Day count

Implied Libor

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253

6.03125% 6.310% 6.557% 6.821% 6.804% 6.953% 7.106% 7.256% 7.024% 7.089% 7.154% 7.219%

Pay side

Receive side

,478,495 ,548,969 ,629,218 ,733,501 ,618,668 ,711,393 ,807,935 ,788,035 ,729,462 ,745,871 ,762,282 ,778,696

,887,934 ,817,460 ,774,204 ,743,909 ,636,779 ,655,036 ,669,475 ,578,394 ,636,967 ,620,558 ,604,147 1,587,733

-18,205,532

18,205,532

Pay side

Receive side

,461,372 ,531,846 ,611,907 ,715,813 ,602,109 ,694.269 ,790.247 ,770,912 ,712,338 ,728.747 ,745,159 ,761.573

,887,934 ,817,460 ,774,204 ,743,909 ,636,779 ,655,036 ,669,475 ,578,394 ,636,967 ,620,558 ,604,147 ,587,733

-18,020,327

18,205,532

1 0.984983 0.969519 0.953540 0.936853 0.921526 0.905611 0.889113 0.873099 0.857868 0.842766 0.827796 0.812962

PV = Net PV =

-

0

-

185.205

307

Interest Rate OTC Options

The payoff structure for a single fixing is shown below:

3%

4%

3mo. Libor

The payoff during the first 6 months may be replicated by GECC: • • • • •

paying Libor + 75 bp; buying a normal cap struck at 4%; selling a normal floor struck at 3%; buying a digital cap struck at 4% with a payout of 4.75%; buying a digital floor struck at 3% with a payout of 3.75%;

Each cap and floor should consist of a series of caplets and floorlets with, at least in theory, one maturing each business day. In practice, the structure may be hedged and hence priced off cap/floorlets maturing (say) each week. If we assume there are 60 working days in the first quarter, then each fixing in the range will contribute {(Libor + 75 bp) x $100m x d} x (1/60), i.e. the notional principal for each cap/floorlet is {$100m/60}. This was one of the last range structures to be done in that period, and was made more attractive by the rising cap strikes over its maturity. Interest rates rose quite rapidly shortly after the issue, and most investors buying these structures lost substantial amounts of money. The notorious swap executed between Procter & Gamble and Bankers Trust was an extreme example, as discussed in Box 7.1.

7.8 MORE COMPLEX STRUCTURES There is a wide range of more complex caps and floors, and three of the most popular types are described below. Unlike an ordinary equity or FX option, caps offer more possibilities because they are multi-options, and it is this property that is highlighted below. The common factor for all of these structures is that they are cheaper than vanilla instruments, and yet offer some form of interest rate protection.

308

Swaps and Other Derivatives

Box 7.1

Procter & Gamble Swap

The date is early November 1993. P&G wished to replace a maturing swap that achieved CP—40 bp. Obviously to repeat this rate would require assuming some risk, and they were prepared to gamble on USD interest rates remaining relatively constant over the next 6 months. They entered into a 5 year swap with Bankers Trust: • • • • • • •

notional principal: $200m to receive: 5.30% to pay for first 6 months: daily average 30-day CP—75 bp to pay for remaining 4.5 years: daily average 30-day CP—75 bp + .P where P was defined as: max{0, S} and where S =[17.04x5yr bond yield — 30yr bond price/100] with an option to buy back P within the first 6 months at market value.

P&G has sold an embedded option P to BT. The latter demonstrated that if both interest rates and volatility remained stable for 6 months, then the buy-back option would cost about 37 bp pa for the remainder of the 5 years, thus nearly achieving P&G's target. Notice of course, S is not a "spread"; if rates go up, S would increase as the bond yield increases and as the bond price decreases. The "leverage" factor of S is not 17, as superficially suggested, but closer to 25. At the time the deal was signed, S was equal to just over —1,700 bp, with a 6 month forward value of —1,271 bp. Unfortunately, rates started to rise in early February 1994 to the effect that P&G became obligated to pay some l,400bp over the CP rate.

1. Barrier caps. Both knock-ins and knock-outs are common. Examples of the latter are, based on a cap struck at K: • the ith caplet is knocked out when Li > Kt > K; • the (i+ l)th caplet is knocked out when Li,: > K > K, i.e. the original protection is kept, but subsequent protection is weakened; • or even entirely lost if the remainder of the cap is knocked out. 2. Periodic caps (closely related to multi-forward and ratchet options). Under this structure, the caplets' strikes are determined dynamically. Assume that when the option is entered into, the strike of the first caplet (K1) is set to the current forward rate FI +m. If m is positive, which is most common, then the caplet is OTM and hence relatively cheap. The fixing of the floating rate is latterly observed (L 1 ) and the payout is max[0, L1, — K1]. The strike of the second caplet (K2) is simultaneously set to L1, + m, again ensuring that it is OTM, and so on. The structure protects against spikes in the floating rate, but not against small movements. 3. Chooser caps. Consider a 3 year cap on 3mo. Libor; this will consist of 11 caplets. Examples of a "5 chooser cap" would be: • you can choose which 5 out of the 11 original caplets to exercise, but only before or on the fixing date, i.e. no lookback;

309

Interest Rate OTC Options

• you can choose to exercise a contiguous strip of 5 caplets; • the first 5 caplets that are ITM are automatically exercised. Numerical model are really required to price these types of structures, although in practice variants of the Black model are frequently used.

7.9

SWAPTIONS

A swaption is a single option on a forward swap. Some terminology: Receiver's: Payer's:

forward swap to received fixed, pay floating forward swap to pay fixed, receive floating

FRA notation is generally used to characterize swaptions, i.e. {length of option/length of swap}. For example, 5/2 payer's is a 5 year option on a 2 year swap to pay fixed. How does a swaption compare in terms of risk management control with a cap or even a mid-curve cap? Consider our company that is paying 3mo. Libor on its debt: it buys a payer's. If it exercises the single option, it will be receiving Libor quarterly and paying a fixed rate which is effectively an average of the implied forward rates. Obviously if Libor decreases after exercise then the company cannot benefit. As we may see below, it is typically cheaper but provides less protection: Comparison between:

Cost

1 yr vanilla cap 1 yr mid-curve cap: exercise 8 Jan 01 1/1 payer's swaption

42.6 bp 35.5 bp 35.04

• • •

out of 6 Jan 00 on 3mo. Libor from 8 Jan 01 to 07 Jan 02 strike = 7.25%, vol = 17% pa

• •

the vanilla cap is most expensive, but has four exercise dates the mid-curve cap only has one exercise date, but four separate options the swaption is a single option on the same exercise date

•

European swaptions are usually priced using a Black model, just like caps and floors, which introduces some interesting issues. Consider a payer's swaption with strike K: let the value of the fixed and floating sides at maturity of the option be VK and VL respectively. The payoff is max[0, VL — VK]. As interest rates move during the lifetime of the option, both VL and VK change. Therefore the payoff is not in the usual form for a Black model, namely a stochastic underlying compared to a fixed strike, and spread option models such as Margrabe have been advocated by some practitioners10: 10

Tompkins (1994) pp. 455-456.

310

Swaps and Other Derivatives

Margrabe spread option model: Payer's = V0L x N(d 1 ) - VOK x N(d2) where: VOL is the PV of the floating side V0K is the PV of the strike side Vo, etc. oL and oK are the vols of the value of the floating and fixed sides respectively pLK is the correlation between the two sides

However, we can manipulate the Black formula if we rewrite the future values as: VL = (DFS - DFe)/DFs for a principal P equal to 1 , and ensuring of course that the Q factors are calculated using the correct frequency of cashflows. We know that the generic forward swap rate Fse can be estimated using market rates: F** = (DFs - DFe)/(ge - &) Substituting into the payoff, we get: max[0, Fse - K\ x (Qe — (2S)/DFS which gives a Black swaption pricing model for a payer's: payer's =

- K x N(d2)] x (Qe - QJ

where d\ and d2 are defined in the usual way, and the volatility refers to the forward swap rate. The apparent assumption, as in the cap model, is that the estimation F is stochastic whereas the discounting process is not, but this is resolved as discussed above. Indeed, as a one period swaption is a caplet, it would be surprising if the assumptions were not consistent. Having said that, there is one inconsistency between the cap and swaption markets, namely: • cap model assumes forward interest rates are lognormally distributed; • swaption model assumes forward swap rates are lognormally distributed; • a forward swap is a (approximately linear) function of forward interest rates. The statements are together logically inconsistent; nevertheless the financial markets invariably price both caps and swaptions using Black models! Worksheet 7. 1 1 demonstrates the pricing of a range of payer's swaptions struck at 7.5%. The sheet is constructed to calculate 5 year sa forward swap rates every 3 months. Using the discount factors (column [1]) constructed earlier, first the appropriate (^-factors are calculated ([2]) and then the forward swap rates ([3]). For example: start date: end date:

7 Jan 02 8 Jan 07

DF 0.873099 0.607724

Q

1.383007 5.038391

Forward 7.25984%

For a given volatility,
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Interest Rate OTC Options

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312

Swaps and Other Derivatives

Worksheet 7.12 Spreadsheet to transform forward interest rate volatility to forward swap volatility Model set-up to calculate 5 year sa swap volatility Settlement date:

06-Jan-00

Start date

Daycount

Original DF

06-Jan-00 06-Jul-00 08-Jan-0l 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05 06-Jul-05 06-Jan-06 06-Jul-06 08-Jan-07 06-Jul-07 07-Jan-08 07-Jul-08 06-Jan-09 06-Jul-09 06-Jan-10

0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511 0.503 0.511 0.503 0.517 0.497 0.514 0.506 0.508 0.503 0.511

1 0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190 0.679367 0.654891 0.631276 0.607724 0.585939 0.564114 0.543575 0.523588 0.504100 0.484913

Original sa Q 0 0.490146 0.974187 1.424476 1.873153 2.299217 2.710215 3.106908 3.491710 3.860942 4.220861 4.562432 4.897154 5.214546 5.528537 5.819878 6.109770 6.384578 6.650735 6.904186 7.152030

Original forward swap rate

Original forward interest rate

7.125% 7.187% 7.238% 7.260% 7.295% 7.320% 7.356% 7.388% 7.434% 7.481%

6.21875% 6.74852% 6.93838% 7.24605% 7.11948% 7.25166% 7.13230% 7.19371% 7.21435% 7.26792% 7.26737% 7.31240% 7.44033% 7.50079% 7.47760% 7.52854% 7.47396% 7.50954% 7.68890% 7.74159%

313

Interest Rate OTC Options

Forward swap volatilities

Calculated for 5/10 forward swap assumed forward rate vol

forward rate st dev

st dev * sensitivity

[3]

Correlation st dev vol

perfect 0.0127 16.970%

Correlation st dev vol

zero 0.0040 5.389%

Correlation lambda st dev vol

ridge 30% 0.0086 11.554%

[2]

[1]

17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0% 17.0%

forward rate sensitivity

1.2% 1.2% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3% 1.3%

0.1165 0.1141 0.1081 0.1069 0.0992 0.0987 0.0935 0.0906 0.0863 0.0844

0.0014 0.0014 0.0014 0.0014 0.0013 0.0013 0.0012 0.0012 0.0011 0.0011

314

Swaps and Other Derivatives 3

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1

25

30

Figure 7.9 ATM USD swaption volatility curves

column [4]. Notice that the price curve is rising with time; this is due both to the increasing time value and also the rising forward swap curve which would of course be received. The required volatility is that of the forward swap rate. Given that far forward rates are generally less volatile than near rates, and that long rates are also less volatile than short rates, we would expect swaption volatility to decline with both increasing option maturity and increasing swap maturity, as demonstrated in Figure 7.9. As swaps are a function of forward interest rates, and the cap market is usually considerably more liquid than the swaption market, it may make practical sense to imply swaption volatilities (albeit approximately) from the caplet vols. If y = f ( x 1 , -x2, . . . ) where f(•) is a known function, and x is a stochastic vector, then:

where is the variance of y, s the standard deviations, and pi.j the correlation between Xi and Xj. Using the approximation for a log-normally distributed variable z: v. = E{z} 2 [exp(o 2 Z t) — 1] % E{z} 2 cr 2 z t, where a. is its volatility, the swaption volatility can be estimated from the forward interest rate volatility. For a forward swap rate Fs,e, the sensitivity with respect to a forward rate fj for s < j < e is given by: dFs.e /afj = [d j /(1 + d j f j ) ] X (DFe

+ Fs.e(Qe - Q j–1 )

The details of the calculations are shown in Worksheet 7.12. The sheet calculates discount curve, sa Q-factors and a set of 5 year sa forward swap rates. For a single swap, in this case a 5/10 swap, the sensitivities to the 10 forward interest rates are calculated in column [2]. Assuming a volatility curve (column [1]), the standard deviation and hence volatility of the swap are given in column [3]. The volatility has been calculated with different correlation matrices. As expected, with perfect correlation, swaption vol is to all intents the same as the flat forward rate volatility curve. As the correlation reduces, the swaption vol reduces as well. Instead of continually recalculating a full correlation matrix, a common assumption is to assume a "ridge" structure such as pi.j = exp[— /.\i — j\] where /. indicates the drop off in correlation. The impact is shown in Figure 7.10: a typical value for /. would be about 0.3. In practice, this approach is adequate to provide an indication where the swaption volatility should be, but the swaption market has its own characteristics as distinct from

315

Interest Rate OTC Options

Lambda Figure 7.10 Impact of correlation on swaption volatility estimate

the cap market. Relying on this relationship for pricing, and more for risk management, would introduce considerable basis risk11. Volatility smiles are more complex than in the cap market, mainly because there are now three dimensions {strike, option maturity and swap length} compared to only two before. Local surface-fitting approaches are widely used. A receiver's swaption, namely to receive fixed and to pay floating, is equivalent to a floor and may be priced in a similar fashion either using a Black model directly: receiver's = [K x N(-d2) - Fs.e x N(-d 1 )] x (Qe - Qs) or by using a call-put parity argument. Consider a portfolio: -f-receiver's - payer's = max[0, VK — VL] - max[0, VL — VK] = VK - VL i.e. equivalent to a forward swap to receive fixed, pay floating. Therefore:

For example, the 2/5 receiver's swaption would be priced at: 218.68 bp + 10,000 x (7.5000% - 7.2598%) x (5.0384 - 1.3830) = 306.47 bp This result is of course replicated using the Black model, as shown in Worksheet 7.11. The above discussion has implied that when a swaption is exercised, a swap is delivered on the exercise date. It is feasible to get cash-settled swaptions, under which the cash value of the underlying swap is paid to the option purchaser. However the valuation convention is slightly different. Normal swap valuation is done off a discount curve, which itself is constructed from the swap curve in some fashion. However, if the 2/5 swaption above was cash-settled, then at maturity the current 5 year rate S5 is noted: • the cashflows on the fixed side of the swap, including the notional principal amounts at the start and end, are constructed; As various banks have found to their cost.

316

Swaps and Other Derivatives

• these cashflows are discounted using S5 only, treating it as if it were a bond yield; • and the swap value calculated. The reason for the convention is that there may well be disagreement about the construction of a discount curve and hence the implied cash amount, whereas the option counterparties are likely to agree on S5 as a visible traded rate.

7.10

STRUCTURES WITH EMBEDDED SWAPTIONS

Embedded swaptions are less common than embedded caps and floors. Nevertheless, some structures such as: extendibles — the ability to extend a swap at the same fixed rate retractibles — the ability to cancel the swap without penalty without any lump-sum payments are readily available. For example, we wish to price an extendible swap (see Worksheet 7.13, columns [1] and [3]): • 3 year sa swap • to pay fixed • extendible to 5 years at the option of the payer:

Receiver

Payer

The payer of the fixed rate has effectively bought a 3/2 payer's swaption. The premium is to be included in the fixed rate FE = 6.90% + m, where m is a margin such that PV(/w) over the first 3 years is equal to the premium. Note that the premium is spread over only 3 years, as those payments are certain to be made, and not over 5 years. The rate has to be calculated iteratively, and a good starting point is the current 3 year rate: Iteration 1 Strike = 6.90% Swaption = 149bp Margin = 55.1 bp

Iteration 2 Strike = 7.45% Swaption = 112bp Margin = 41. 4 bp

Iteration 3 Strike = 7.31% Swaption = 121 bp Margin = 44.6 bp

Iteration 4 Strike = 7.35% Swaption = 119bp Margin = 43. 8 bp

Iteration 5 Strike = 7.34% Swaption = 119bp Margin = 44.0 bp

The fair rate for the payer's extendible is 7.34%, 44 bp above the generic swap rate. A retractible would be priced in a similar fashion. For example, consider a 5 year swap, retractible to 3 years at the option of the payer: current 5 year rate: 7.01%

Worksheet 7.13

Pricing extendible and retractible swaps

Settlement date: 06-Jan-OO Daycount

Swap pricing ==»

DFs

Q

3 year payer's swap extendible to 5 years

[1] 06-Jan-OO 06-Jul-00 08- Jan-01 06-Jul-0l 07-Jan-02 08-Jul-02 06-Jan-03 07-Jul-03 06-Jan-04 06-Jul-04 06-Jan-05

1 0.506 0.517 0.497 0.514 0.506 0.506 0.506 0.508 0.506 0.511

0.969519 0.936853 0.905611 0.873099 0.842766 0.812962 0.784668 0.756987 0.730349 0.704190

Swaption pricing ==> 5 year payer's swap retractible to 3 years

Forward swap rate Strike (%) Ann Vol (%)

[2]

7.20% 7.3409% 17.0%

[3]

7.20% 7.3409% 7.3409% 17.0% [4]

0 0.490146 Generic fixed rate 0.974187 New fixed rate 1.424476 Target 1 .873 1 53 Swaption premium (bp) 2.299217 Annualized margin (bp) 2.710215 3.106908 3.491710 3.860942 4.220861

6.90% 7.3409% (0.0000) 119.15 43.96

7.008% 7.3409% (0.0000) 140.39 33.26

exercise dl N(dl) N(d2)

3.044 0.083 0.533 0.415

3.044 0.083 0.533 0.415

Payer's (bp) Receiver's (bp)

119.15 140.39

119.15 140.39

318

Swaps and Other Derivatives

Again the quoted price FR must equal 7.01 % + m1: this swap is also priced in the worksheet. The two rates, FE and FR, are of course identical because the two swaps have the same economic effect. There is therefore a potential arbitrage to be monitored. One of the earliest uses for swaptions was to assist in the swapping of callable bonds. Consider a 5 year bond that has a single call date in 3 years' time. Remember that the issuer has the right to call, and is likely to exercise this right if rates decrease over the 3 years, and therefore has to pay a higher coupon to the investors. The issuer wishes to swap the bond into floating, as shown: where the maturity of the swap must match the effective maturity of the bond

C, callable after 3 years

The naive approach is for the issuer to purchase a 3/2 swaption. This may be done in two ways: (a) enter into a 3 year receiver's swap plus a 3/2 receiver's swaption; (b) enter into a 5 year receiver's swap plus a 3/2 payer's swaption. One problem with either structure is that the issuer has effectively to pay two option premia, one to the investor for the call, and one to the swaption counterparty for the option. Another problem is that, whilst either structure will achieve the issuer's objective, neither makes economic sense. Consider the two scenarios: Action on bond Interest rates goup

Interest rates go down

no call -5 year bond

call -3 year bond

Action on swaption (a)

Action on swaption (b)

no exercise -3 year swap

exercise -3 year swap

exercise -5 year swap

no exercise -5 year swap

Exercising the swaptions rationally results in a maturity mismatch under either scenario. It is more appropriate for the issuer to sell a swaption. For example: (c) enter into a 3 year receiver's swap, and sell a 3/2 payer's swaption; (d) enter into a 5 year receiver's swap, and sell a 3/2 receiver's swaption.

319

Interest Rate OTC Options Action on bond Interest rates go up

Interest rates go down

no call -5 year bond

call -3 year bond

Action on swaption (a)

Action on swaption (b)

exercise -5 year swap

no exercise -5 year swap

no exercise -3 year swap

exercise -3 year swap

The issuer is paying a higher coupon to the investor, but also receiving the swaption premium. The swaption will be exercised so that the maturities also match. Because the swaption market is a wholesale market, the swaption is likely to be priced fairer than the call option in the bond market12, and therefore the issuer may be able to achieve sub-Libor funding. Callable bonds frequently have multiple call dates, very often coinciding with selected coupon dates. For example, a 7 year bond is issued paying annual coupon and with possible calls after the coupon is paid in 3, 4 and 5 years' time. Bermudan-style swaptions are required to swap such a bond. Indeed, unlike the cap market, Bermudan and to a lesser extent American swaptions are relatively common. There are two fundamental types: • fixed end, i.e. the underlying swap starts on the selected exercise date but has a constant maturity date; • fixed length, i.e. the underlying swap starts on the selected exercise date and has a constant length of maturity. For example, consider a 5/2 swaption with three exercise dates at the end of 3, 4 and 5 years: Exercise date 3 years 4 years 5 years

Fixed end 4 year swap 3 year swap 2 year swap

Fixed length 2 year swap 2 year swap 2 year swap

Fixed end swaptions are needed to swap multiple call bonds, and to risk manage a particular segment of the curve as it moves closer. Fixed lengths are used mainly when the time for an exposure to arise is unsure: for example, a company may want to enter into floating debt and swap it into fixed, but is unsure precisely when the debt will be called down. The reason for American swaptions being less common is that early exercise on any date may lead to the underlying swap having undesirable broken dates, which is not popular amongst the counterparties. '-Indeed for a long time, investors received very little compensation for embedded call options.

320

Swaps and Other Derivatives

400-350-300-

200-150-100-50-

15

10

20

No. of exercise dates Figure 7.11

Pricing Bermudan fixed end swaptions

A numerical model is really required to price these types of swaptions, although there is the old "binomial" trick that used to be widely used in the markets: max{price to first exercise date, price to last exercise date} It actually works surprisingly well, as usually a swaption is exercised at the beginning or end of the exercise period depending on the shape of the curve. As expected, American and Bermudan swaptions are more expensive than European. The price of a Bermudan approaches an American fixed end swaption as the number of exercise dates reaches about 10—see Figure 7.11. The pricing of fixed-length ones typically converges faster, at about 4 to 5 exercise dates.

7.11 FX OPTIONS Whilst this book is primarily about swaps and interest rate options, this is a brief section on FX options for completeness. Exotic options will not be described in any great detail as there are a large number of books available that describe both the theory and application of these options (see, for example, Haug in Footnote 18). The objective of this section is to cover briefly generic option pricing, so that it may be used in the risk management chapters later. Consider a simple call option on $-Yen with a strike AT of 105 Yen per USD. As with a normal call, this gives: • the right to buy or to receive the underlying numeraire currency, i.e. USD; • and the right to sell or to pay Yen at the strike rate. If the spot rate at expiry ST = Y107, then you can sell $1 and buy Y107 in the spot market and make a riskless profit of Y2 per $1. Generic FX options are typically priced using the "Garman-Kohlhagen" variant of the usual Black & Scholes model: C = S0 x DFfT x N(di) -Kx

DP? x N(d2)

Interest Rate OTC Options

where: DF^ is the discount factor on the foreign side, usually defined as exp{— rfT] where rf is continuously compounded risk free zero coupon rate DFdT is the same on the domestic side d1 = {ln(S0/K) + (rd - rf + 1/2o 2 )T} Jf d2 — d\ — avT

T is the time to expiry (in years) a is the annualized volatility of the spot rate N(x) is the cumulative unit normal This definition uses the usual (personally speaking, unhelpful) language of "foreign" and "domestic"; to translate, as the spot rate is quoted in terms of Yen per USD, Yen is the domestic currency, USD the foreign one13. The price of the option C will also be in the same units as the spot rate. Whilst the formula calls for risk free rates for discounting, in practice, Libor discount factors are used. This section will use USD/Yen examples, based out of 19 October 2001. As with most currencies, FX options are usually quoted with a two-day settlement period, so the options should start on 21 October. However that was a Saturday, therefore they start on 23 October. As described above, same day discount factors need to be adjusted for the settlement period. We wish to price a 6 month call option with the following data (the remainder are given on Worksheet 7.14): • • • •

strike: size annualized volatility: current spot rate:

105 SlOOm 10.35% 108.89

The steps are: • • • •

estimate the Act/360 Libor discount factors: columns [1] and [3]; calculate the Act/365 continuously compounded zero coupon rates: columns [2] and [4]; estimate maturity (on a consistent Act/365 basis); calculate d1 and d2, and hence price the call: 3.385 Yen per $1 or Y338.5m for SlOOm — see Box 1 of Worksheet 7.15 for details.

The volatility a is usually calculated using business days only, whereas interest is calculated using calendar days T, so the daycounts may be slightly different, i.e.: (r y -r $ )T +1/2o 2 T,etc. which was why the c-c rates were calculated on an Act/365 basis. Following the convention that in many countries exchange rates are quoted in terms of units of the domestic currency per unit of foreign.

Worksheet 7.14

Spreadsheet to calculate USD-Yen FX forward contracts (basic market data)

Today's date Settlement date Spot rate Interest rate0

19-Oct-00 23-Oct-00 108.89 Act/360

Act/365 USD

Libor Act/360

1 2 3 6 12

23-Oct-OO 23-Nov-OO 27-Dec-OO 23-Jan-0l 23-Apr-0l 23-Oct-0l

0.086 0.181 0.256 0.506 1.014

0.085 0.178 0.252 0.499 1.000

6.62500% 6.65625% 6.75000% 6.68750% 6.68750%

DFs

Yen

c-c rates

[1]

[2]

0.994327 0.988124 0.983043 0.967297 0.936502

6.698% 6.708% 6.785% 6.668% 6.560%

Libor Act/360

0.375% 0.469% 0.563% 0.563% 0.563%

DFs

c-c rates

[3]

[4]

0.999677 0.999154 0.998565 0.997164 0.994329

0.380% 0.475% 0.570% 0.570% 0.569%

Implied Actual Annual FX rates FX rates volatility

108.31 107.69 107.20 105.63 102.56

108.30 107.71 107.19 105.60 102.55

8.350% 8.900% 9.250% 10.350% 11.300%

C/5

o S"

3

Interest Rate OTC Options

323 Table 7.4 USD-Yen forward rates Maturity (months) 1 2 3 6 12

Implied FX rates

Quoted FX rates

108.31 107.69 107.20 105.63 102.56

108.30 107.71 107.19 105.60 102.55

See Worksheet 7.14 for details.

In the early days of the market, FX options were traded on futures exchanges; the Philadelphia exchange was the first in 1982. But as the FX spot markets moved to electronic trading around the world, the options market became predominantly OTC14. The price in the OTC market is usually quoted as a percentage of spot, i.e. C/S0: 3.385 Yen/108.89 = 3.108% of principal of $100m This method of quotation makes it independent of the size of transaction and of the currency of the premium. The option to put (or sell) the USD and receive Yen may be similarly calculated using: P = K x DFY7 x N(-d2) -S0x

DF$r x N(-d})

It has a price of 2.533% of principal. Whilst the G-K model is most popular, there is a "Black" equivalent using forward rates. The forward FX rate ST may be estimated using S0 and two sets of interest rates, as discussed above: ST = S0 exp{r$r}/exp{rY7] = S0 Substituting for S0 in the above formula gives: C = DFYr x (ST x Nfa) -Kx

N(d})}

2

where dx = {\n(ST/K) + \a T\fa^T. See Box 2 in Worksheet 7.15 for details. Because FX forward rates are traded, therefore market quotes for ST are available and it is feasible to substitute these directly into the formula. Unfortunately, the quoted rates may not be the same as the implied rates — see Table 7.4 — so the option prices will be different! So which should be used? The answer depends on the hedge: • if spot FX trades are used to hedge, use S0 model; • if FX forward trades are used to hedge, use ST model. Most people use the former because of the higher liquidity. The options model may also be described as a "carry" model, i.e. writing rb = rd — rf as the cost of carrying the hedge. This interest rate differential may be thought of as the expected rate of USD depreciation. Suppose a 1 year ATM spot (i.e. with strike equal to 14 In June 2001, the total open FX options on the Philadelphia exchange were 25,762. This is less than 5% of the total open contracts 10 years earlier.

Worksheet 7.15

Spreadsheet to calculate some vanilla FX options

Today's date 19-Oct-00 Settlement date 23-Oct-00 Spot rate 108.89 Yen/USD Pricing of a 6 month vanilla option Option to buy 100m USD to sell 10.5bn Yen

End date Strike Maturity Volatility

23-Apr-0l 105 Yen/$ 0.499 Act/365 10.35% pa (Act/365)

USD DF Yen DF USD c-c rate Yen c-c rate

0.967297 0.997164 6.6683% 0.5695% 2. Forward rate model

1. Spot model dl= d2 = C P

0.118 0.045 3.385 Yen/USD 2.758 Yen/USD

Percentage of strike 3.224% 2.627%

Percentage of spot 3.108% 2.533%

3. Proof of call-put parity

4. Using American quotes

PV of (K-S6) C-P

Strike* Spot rate*

0.626678 0.626678

F dl= d2= C P

105.628 0.118 0.045 3.385 Yen/USD 2.758 Yen/USD

Percentage of strike 2.533% 3.108%

Percentage of spot 2.627% 3.224%

0.009524 $/Yen 0.009184 $/Yen

Spot model dl= d2= C* P*

-0.045 -0.118 0.0002412 $/Yen 0.0002960 $/Yen

"8 I O

I I

I

Interest Rate QTC Options

the current spot rate) call option is purchased; this option would decline in value as the USD theoretically depreciates over the year. Call- put parity theorems obviously exist in FX options. For example: 1. Buy a call, sell a put on $1 at a strike of 105: • cost of strategy = 3.385–2.758 = Y0.626678. 2. Enter into a forward contract to buy $1, and sell 105 Yen in 6 months' time: • currently S6 = 105.628, and therefore the contract is off-market with an anticipated future value of 0.628 Yen; • or present value 0.628 x 0.9971 64 = Y0.626678. The call-put parity relationship is C - P = (ST - K) x DPYT. When K=ST, i.e. the options are ATM forward, then C = P. See Box 3 of Worksheet 7.15. The spot rate has been quoted as {Yen per $}: these are so-called "European" terms. But suppose we wished to quote it in American terms, i.e. as {$ per Yen} — do the same prices and relationships hold true? Define: • spotA = 1/108.89 = 0.009184 $/Yen, • strikeA = 1/105 = 0.009524 $/Yen. We can calculate dl and d2 in the usual way, but must reverse DF dT and DFfT because the numeraire domestic has been switched from USD to Yen. We can then price the call and put as shown in Box 4: CA = $0.0002412

and PA = $0.0002960 per Yl

Expressing these results as percentages, we get:

Call Put

% of strike 2.533% 3.108%

% of spot 2.627% 3.224%

which implies C = PA x K x S0, etc. So we can replicate the $ numeraire results, and all the earlier relationships remain true. The Garman-Kohlhagen model is the most widely used one for European options, despite various attempts to introduce "better" theoretical models. Probably the most popular extensions are as follows. • Assuming the interest rates are also stochastic; for example Hilliard et al.15 produced an identical expression to the G-K model but with a volatility term:

v2 = K) 2 r+1 r3{((7d)2 + (af)2 - 2
326

Swaps and Other Derivatives

Table 7.5

Days

Daycount

Daycount

Spot rate (Yen/$)

Call price (Yen/$)

Mid Delta

0 1 2

0.499 0.496 0.493

0.003 0.003

108.89 108.32 108.20

3.385 3.092 3.034

0.52915 0.50162 0.49593

Call dealing (Yen) 338.476.740

• Assuming the volatility is stochastic; for example Chesney et al.16 produced a model with a mean-reverting stochastic process for the volatility. After fitting the various parameters of the stochastic process, this model did produce significantly different results to the G-K model. Unfortunately they also found that market prices were consistent with G–K prices assuming constant volatility, so may have created an arbitraging opportunity. • Assuming that spot foreign exchange rates do not follow a Gaussian process but over the long-term are pulled by purchasing power parity, or one of the other broad macro assumptions. For example Cheung et al. 17 produced a modified G–K model but obviously with a number of additional parameters that require estimation. American options can (in theory) be exercised at any time up to maturity, although in practice this is seldom the case simply because of finite business hours. They are usually priced with a numeric model or by approximations such as Barone-Adesi and Whaley or Bjerksund and Stensland18. For example, the 6 month option above:

European Barone-Adesi and Whaley Bjerksund and Stensland

call

put

3.385 4.226 4.232

2.758 2.760 2.760

As expected the price is higher, especially for the call as it is ITM forward.

7.12 HEDGING FX OPTIONS Consider the 6 month USD Yen call option described above. We have just sold the option and will as usual perform a delta-neutralizing spot transaction. The delta for the option is: <5 = df x N(d1) = 0.52915 implying that if the spot rate shifted by 1 unit, the call price would increase by Y0.53 per USD or by Y52,915,285 in total. As we have sold the option, this will be a loss for us. We wish to enter into a reverse spot transaction that will off-set this change in value: 16 Chesney and Scott, "Pricing European currency options: a comparison of the modified B&S model and a random variance model", JFQA, 24(3), 1989. pp. 267-284. 17 Cheung and Yeung. Pricing FX options incorporating purchasing power parity, HK University Press. 1992. 18 For details, see one of the myriad books on options such as Haug. The Complete Guide to Option Pricing Formulas, McGraw-Hill. 1997.

327

Interest Rate OTC Options Table 7.5

(continued) Spot trades

Yen

$

–5,761,945,421 299,246,536 62,558,780

52,915,285 -2,762,572 -578,151

Net balance (Yen) -5,423,468,681 -5,124,306.767 -5,061,827,941

Net balance ($) 52,915,285 50,162,381 49,593,396

Close-out option

Current P&L (Yen)

-338,476,740 -309,187,706 -303,364,178

0 1 80,463 1,058,984

• buy $ 100m x6 = $52,915,285, • sell Y52,915,285 x S0 = Y5,761,945,421. If the spot rate shifts by 1 unit, the profit on this trade would be Y52,915,285. We now need to see what happens as the spot rate changes through time. Worksheet 7.16 is designed to perform random simulations of the spot, and Table 7.5 is a short extract. It assumes only one change per day, therefore at the end of the first day the current P&L, defined by the net balance in Yen plus the net balance in $ converted at the prevailing spot rate, is zero. The next day the spot rate shifts from Y108.89 to Y108.32 per USD. A number of things happen. • The option may be bought back at the new price C1 of Y3.092, thus representing a profit of Y100mx(C 0 - C1) =Y29,289,033. • However we estimated the new price should be C0 + 6 x (S1 – S0) = 3.084, so that we had expected a bigger profit. The difference of — Y782,182 is the negative gamma effect from which we of course lose. • We assume that we borrowed the balance of the Yen required to undertake the spot trade, and deposited the USD proceeds: • Yen interest payable = –5,423,468,681 exp{rY x 1/365} = Y84,622; • USD interest receivable = 52,915,285 exp{r$ x 1/365} = $9,668; • converting the USD to Yen at Sl gives a net balance of Y962,644; • this is (close to — strictly, should convert at S0) the positive theta effect. • On balance therefore, the trade shows a profit of Y180,463 as shown in Table 7.5. The hedge has to be rebalanced as the delta has shifted to 0.50162. The required delta hedge is to: • buy $100mx5, = $50,162,381, • sell ¥50,162,381xS1, = ¥5,433,674,936. However we already have balances in our two money accounts from the previous day's hedge plus interest: therefore we need to reduce the USD balance by selling $(52,924,953 —50,162,381) = $2,762,572 and buying the equivalent amount of Yen. Notice the resulting balances in the money accounts do not represent a perfect delta hedge now; this is mainly because the spot rate has not moved perfectly in line with the interest rate differential, but there are also some small theta effects. Worksheet 7.16 then repeats this process each day until the option matures. On the last day. if the option expires OTM then delta should be very close to zero and the money

Swaps and Other Derivatives

328

Worksheet 7.16 Spreadsheet to demonstrate option replication — based on option to receive foreign, put domestic Current FX rate (Yen/$) Strike = Size of transaction (in $) Vol= r(Yen)= r($)= T= Forward rate (Yen/$) Switch on simulation (Y/N?) Switch on discrete gapping (Y/N?) Unit size of discrete gaps

108.89 105 100,000,000 10.35% pa 0.5695% pa 6.6683% pa 182 days 0.499 years 105.63 Y N 0. 1

Residual error term (D) Residual error term (F)

37,589,820 399,535

Days 0 1 2 3 4 5 6 7 8 9 10 11 12 175 176 177 178 179 180 181 182

Yen USD

Daycount Daycount 0.499 0.496 0.493 0.490 0.488 0.485 0.482 0.479 0.477 0.474 0.471 0.468 0.466 0.019 0.016 0.014 0.011 0.008 0.005 0.003 0.000

0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003 0.003

Note: rows 13 to 174 are hidden

Base case

-0.7398 0.7458 -0.6868 -1.8499 1.1411 1.3628 -0.1977 0.7239 -0.2088 -0.8910 0.2939 0.0698 -0.5501 0.1878 -1.3653 -0.1580 0.3609 1.1428 0.8048 0.2603

Normal

rand( )

[A]

[B]

0.4168 -0.4097 -0.0693 -0.0340 -2.2407 0.0247 0.6850 -0.1825 0.3807 -0.2387 0.6971 -0.4336 0.9036 -0.8243 0.7962 -0.6172 1.1717 0.4123 -0.3677 0.3302

0.6943 0.1165 0.6470 0.3565 0.2157 0.5764 0.8538 0.8626 0.3434 0.0381 0.2689 0.4739 0.4268 0.6608 0.4729 0.3799 0.4173 0.9608 0.3810 0.9068

Spot rate (Yen/$) [1] 108.89 109.12 108.85 108.79 108.75 107.42 107.42 107.80 107.67 107.87 107.71 108.10 107.83 93.74 93.31 93.70 93.37 93.94 94.14 93.93 94.08

Mid call price (Yen/$) dl

0.118 0.149 0.119 0.113 0.111 -0.058 -0.057 -0.005 -0.020 0.009 -0.010 0.043 0.010 -7.985 -8.963 -9.465 -10.894 -11.906 -14.294 -20.588

[2] 3.385 3.507 3.369 3.338 3.318 2.660 2.658 2.837 2.777 2.875 2.798 2.989 2.854 0.000 0.000 0.000 0.000 0.000 0.000 0.000 NO

Delta [3] 0.52915 0.54107 0.52949 0.52757 0.52665 0.46176 0.46234 0.48220 0.47678 0.48787 0.48078 0.50130 0.48842 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000 0.00000

329

Interest Rate OTC Options

Spot trades — Call dealing (Yen) [4] 338,476,740

Yen

$

[5] (5,761,945,421) (128,983,589) 127,142,150 21,935,539 11,019,418 698,124,677 (5,330,463) (213,188,969) 59,333,313 (118,731,812) 77,418,909 (220,934,978) 139,832,485 0 0 — — — — —

[6] 52,915,285 1,182,074 (1,168,000) (201,625) (101,324) (6,498,874) 49,624 1,977,689 (551,062) 1,100,657 (718,741) 2,043,754 (1,296,795) (0) (0) — — — — —

Net balance (Yen)

Net balance ($)

Close-out option (Yen)

[7] (5,423,468,681) (5,552,536,892) (5,425,481,378) (5,403,630,493) (5,392,695,387) (4,694,654,852) (4,700,058,565) (4,913,320,869) (4,854,064,218) (4,972,871,768) (4,895,530,450) (5,116,541,813) (4,976,789,161) 37,585,715 37,586,301 37,586,888 37,587,474 37,588,061 37,588,647 37,589,234 37,589,820

[8] 52,915,285 54,107,028 52,948,914 52,756,963 52,665,278 46,176,027 46,234,087 48,220,224 47,677,972 48,787,340 48,077,513 50,130,051 48,842,416 0 _ — — — — —

[9] (338,476,740) (350,727,908) (336,857,997) (333,774,437) (331,791,684) (266,012,656) (265,762,514) (283,703,525) (277,671,209) (287,464,481) (279,779,974) (298,856,073) (285,365,671) 0 — .__ — —

Current P&L

[10] 0 694,050 1,392,421 2,231,835 3,076,983 (327,638) 516,672 972,269 1,796,466 2,530,970 3,340,253 3,787,613 4,488,476 37,585,715 37,586,301 37,586,888 37,587,474 37,588,061 37,588,647 37,589,234 37,589,820

Swaps and Other Derivatives

330

-59.9

-47.9

-35.9

Figure 7.12

-23.9

-11.8

0.2

12.2

24.2

36.2

48.3

Simulation of delta hedge of FX option

accounts run down to zero as well. Conversely, if the option finishes ITM then the money accounts should be close to — Y10.5bn and +$100m respectively. The simulation is performed using the expression:

where At = 1/365 years and e ~ N(0,1)19. The worksheet also permits the inclusion of a jump process K-(r3 —0.5) — see column [B] — to demonstrate how the delta hedging can break down in these situations. Figure 7.12 shows the final balance of the net money accounts over 100 simulations. The average of the distribution is only Y2.5 million, very close to zero as expected. However the distribution is not symmetric, having a long downside tail due to the unlimited negative gamma and a short upside due to the limited theta. Whilst the option had a lot of time value when it first started: ATM forward intrinsic value = YO. 626678 (see above) time value = Y2. 758089 most of it is off-set by the gamma losses when relatively large moves occur. We have seen how a vanilla FX option may be delta hedged with a succession of spot transactions. However there are a number of practical problems: • delta hedging is supposedly continuously rebalanced, but in practice always lagging one period behind. This doesn't matter if there is negligible gamma, i.e. the delta isn't changing very much, but for ATM short-dated options it becomes an increasing problem; 19 Generated using the expression {2 ln(r1)}1/ 2 sin(27r 2 ) where r1, and r2 are uniformly distributed variables between 0 and 1 — see column [A]. An alternative would be to use the Random Number Generator function in Tools Data Analysis: this is very useful as it gives a lot of control over the stream.

331

Interest Rate OTC Options

Figure 7.13 Delta of European and American call options

• transaction costs have been ignored in this model, although they would be very simple to add; • the volatility of the spot rate is assumed to be constant over the lifetime of the option; • the two interest risks are also assumed to be constant; • the hedge has a cost of carry which may not be the same as the c-c "risk free" rates as these are seldom (never?) available in practice. Delta hedging can also be used for American options, but there is one problem, as highlighted by Figure 7.13. Early exercise when heavily in-the-money is an optimal strategy for the owner of an American option, hence the American delta converges to 1 much faster than the European option. This causes a discontinuity in the delta at the early exercise boundary, which results in a "gamma spike".

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8 Traditional Market Risk Management

OBJECTIVE Traditional risk management, often called "desk-top" risk management, for derivatives has changed a lot over the past 20 years. This chapter discusses two main types of interest rate management, namely gridpoint which assumes that all rates behave independently, and yield curve which assumes they move according to some pattern. It also describes different interest representations, i.e. market, forward and zero coupon rates, and shows how different risk reports, sensitivity and equivalence, may be constructed for each of these. We then turn to yield curve risk management; this is discussed only in the context of market rates after an analysis of curve movements. Simple delta and delta-gamma hedging against parallel shifts are introduced, followed by the concepts of other shifts such as rotations. Swap futures are introduced as a potential hedging instrument instead of bonds. Theta risk, i.e. the risk of losses simply through the passage of time, is then briefly discussed. The risk management of CCS portfolios is not discussed as the sensitivities can be broken up into two IR risks and a spot FX risk. Typically these risks are then transferred into the relevant IR and FX portfolios for the reporting and management of risk. Finally the risk management of IR option portfolios and inflation swaps is considered. For IR options, their greeks are explored, concentrating particularly on long-term options where the greeks are less intuitive, and also how they change through time. The section also shows how to construct robust hedges using different mathematical programming formulations. For inflation swaps, a delta inflation hedge equivalent is constructed based upon the market data and swaps discussed in Chapter 5, and its effectiveness examined by simulation.

8.1 INTRODUCTION Risk management techniques have developed a lot in the last 20 years. When transactions were on-balance sheet and actually involved principal flows, the main concern was credit risk, i.e. will I get my money back? Market risk was deemed to be far less important and was often confined in the guise of interest rate risk to the ALM area. Here techniques such as gap analysis, namely identifying future periods of time during which the bank would be net lender or net borrower, were popular. With the explosive growth in off-balance sheet activities during the 1980s, which only involved notional principals, market risk increased in significance. Gap techniques became discredited at the same time because of their inability to handle complex structures and particularly options. So a whole new edifice of risk techniques has been developed during the 1980s and 1990s to manage the market risk of structured portfolios. This chapter will concentrate on the methods developed in the 1980s, which are very widely used. They are however being superseded by the newer techniques which are the topics of the next chapter.

334

Swaps and Other Derivatives

At the inception of any transaction, both sides are deemed to be valued equally. For a swap, the fundamental concept in pricing is that the value of the cashflows to be received equals the value of the cashflows to be paid. For an option, its (theoretical) price is the expected discounted value of the future payoffs. But as time passes, market rates move and one party gains in value to the detriment of the other party. The objectives of this chapter are to: • analyse how changing interest and exchange rates affect the valuation of swap portfolios; • construct hedging or equivalent portfolios. Before we start, it is probably worth reviewing just what we mean by risk in this context. For a company, risk is often synonymous with uncertainty. Hence a floating cashflow is a risky cashflow, which is why companies frequently want to swap from floating to fixed. It is not taking a view on the future direction of rates, but removing the uncertainty. Banks on the other hand very often fund themselves at floating rates, so from their perspective a floating cashflow that is received and passed on constitutes no market risk. This was the argument used to value the floating side of a generic swap, including the notional principal amounts, to zero. Banks however invariably mark their trading books to market, so for them, risk is any event that will cause a change in the present value. If we crudely characterize the valuation process as: PV = Z—// V CFt x DFt

then there are two sources of risk, namely changes in the future cashflows and changes in the discounting. These are frequently referred to as delta-0 and delta-1 risk respectively. In this chapter, we will investigate how derivatives respond to various types of changes in rates, and how we might manage those exposures. We will initially concentrate on interest rate swaps, and then briefly cross-currency swaps. Finally we will consider various strategies to hedge option portfolios. First, consider the single IR swap shown in the box below: Today's date:

4 January 2000

Swap details: start date: maturity: size: fixed rate: floating rate: last fixing:

15 November 1998 4 years $10m 7.956% sa receive 3mo. Libor pay 7.0125% on 15 Nov 99

This swap may be represented as a stream of cashflows, as shown in Table 8.1. The first cashflow on the floating side is of course known, but the remaining Libor values are currently unfixed. Replacing these Libors with the notional principals allows us to represent the swap as a single cashflow ladder. We could of course have represented the swap using implied forwards instead, and the overall risk management results would be identical. But there is a good practical reason for using the notional principal representation. As interest rates move around on 4 January, the cashflows do not change. Indeed the only time they change is when one rolls off and

Traditional Market Risk Management

335

Table 8.1 Cashflows of the swap ($ millions) Date

Pay

6 Jan 00 15 Feb 00 15 May 00 15 Aug 00 15 Nov 00 15 Feb 01 15 May 01 15 Aug 01 15 Nov 01 15 Feb 02 15 May 02 15 Aug 02 15 Nov 02

Receive

Net cash ladder

0.4022

-10.1792 +0.4022

0.4066

+0.4066

0.4000

+0.4000

0.4066

+0.4066

0.4000

+0.4000

0.4066

+ 10.4066

-0.1792 -L -L -L -L

-L -L —L -L -L

-L -L

there is a new fixing, i.e. every 3 months. In contrast, the implied method would have the cashflows changing whenever the rates change, which simply means a lot more work for no particular reward. However this principal representation will only provide total risk, and will not break the risk into delta-0 and delta- 1. We can obviously represent a range of other linear instruments such as FRAs, fixed or floating loans and deposits, bonds and FRNs, in a similar fashion. Futures present a small problem as they margin daily. We know that the tick value of a single Eurodollar contract is $25 per bp, paid each day. If this were to be paid at the end of the forward rate T, then it would be worth $25/DFr. Therefore a $1 million futures contract may be regarded as equivalent to $(l/DF T ) million of an FRA. For short-dated futures, the adjustment is relatively small and frequently ignored, but it may be very significant for longer dated transactions, as we saw in Chapter 2 hedging a money market swap with a futures strip. Similar adjustments should be made to incorporate bond futures. It would be extremely convenient if options, especially interest rate options such as caps, floors and swaptions, could be represented in a similar fashion. The Black price for a cap is: DF T x [ T - t ] x

- Kx N(d2)}

using the same notation as in Chapter 7. If we substitute for the forward rate F(t,T) using the usual formula and rearrange: C = {P x

x DFt -{Px

[N(d 1 ) + N(d2) x (T - t) x K]} x DFT

It is now in the desired form of two (variable) cashflows at time t and T, and may be entered into a cash ladder. If we also assume that N(d 1 = N(d2), which is patently not correct but approximately true, then the expression rearranges to:

C=

x {+P x D F t - P x [ 1 + ( T - t ) x K] x DFT}

The cashflows themselves are exactly the same as from a simple FRA, as shown in the diagram below:

336

Swaps and Other Derivatives +P _|

-Px[1+K x (T-t)] I

Therefore C = N(d) x {PV of the FRA} where N(d) is (approximately) the probability of exercise. If for example the option is heavily OTM, then N(d) is close to zero, the option carries little interest rate risk, and would not appear in the cash ladder. Conversely, if the option is heavily ITM, then it may be represented by the underlying contract. Thus, for risk management purposes, we can represent virtually all instruments in the same framework, namely as a stream of fixed or varying cashflows discounted back. For illustrative purposes, we will use a small USD swap portfolio represented in this fashion throughout the chapter. The next step is to value the portfolio: as shown in Worksheet 8.1, it is currently worth -$283,133.

8.2 INTEREST RATE RISK MANAGEMENT Before we can proceed with looking at some interest rate risk management techniques with this portfolio, we have two decisions to make. First, what form of interest rates shall we use? There are three common choices: market rates, forward rates and zero coupon rates. Each has advantages and disadvantages: Market rates: Forward rates: Zero coupon rates:

observable but not comparable because of differing tenors not observable except at the short end from the futures market, but may be constructed to be of the same tenor neither observable nor comparable, but most finance theory uses them

The latter were most popular when the theory of these newer risk techniques was being developed, and many risk systems still use them 1 . However, it is my personal observation that market rate methods are gaining in popularity, and I think that is due to the rise of the independent risk manager who wishes to ensure that all inputs are capable of being audited. We shall in fact analyse the portfolio using all three forms. Second, interest rates possess a term structure. What do we assume about this structure, and how it behaves? We can imagine a continuum: completely independent, zero correlation

imperfect correlation

completely dependent, perfect correlation

Gridpoint

Value-at-Risk

Curve

ranging from zero correlation with all rates moving independently along the curve through to perfect correlation when the curve moves in some form of pattern. This latter does not necessarily impose perfect positive correlation, i.e. all rates must move in the same direction, but would include movements such as rotations, where rates on one side of the pivot would be perfectly negatively correlated with rates on the other side. We will first of Usually the older systems.

Traditional Market Risk Management

337

all look at gridpoint hedging, then at curve hedging, and the next chapter considers imperfect hedging under the general value-at-risk topic.

8.3

GRIDPOINT RISK MANAGEMENT — MARKET RATES

The current market value of the portfolio was —$283,133. This was calculated by taking a set of market rates, bootstrapping the curve (interpolating where necessary) to derive the DFs, and finally discounting the cashflows (again interpolating where necessary). We have used a combination of nine cash and swap rates, but could easily have included futures or even FRAs as well. If a market rate shifts, then the DFs will change, and the value is likely to change as well. We can easily produce a market rate sensitivity report by taking each market rate in turn, shifting it by a predefined amount, and recording the resulting change in value (Table 8.2). Table 8.2 Market rate sensitivity report (based upon a 1 bp increase in each rate) Market rates 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap Syr swap 7yr swap 10yr swap

Change in value ($) 17.39 31.11 -22.44 -270.89 -2,236.27 -2,779.17 1,213.52 8,596.64 0.00

Such a report is often referred to as a PVBP (present value of a basis point) or a PV01 report. These sensitivities may either be calculated by perturbation — what is often called "blipping a curve" — or analytically. Obviously analytic methods are much faster, but they are also considerably more difficult to set up and maintain if there are any changes in the curve construction methodology. This is particularly true because of the inevitable presence of interpolation. Most modern systems use perturbation, and simply throw more computing power into the fray 2 . We can see that if the long end of the curve moved up, the portfolio gained in value; a move up in the medium term however would result in a loss. Summing the sensitivities gives $4,549.89. Strictly speaking, this is invalid but frequently done, and provides a good estimate of the change in value if the entire curve underwent a parallel movement of 1 bp — we will check this value later.

8.4 EQUIVALENT PORTFOLIOS The sensitivity report is expressed in terms of money. However, an alternative way of expressing these exposures is by creating an equivalence report. This takes a set of generic 2

Very often the older systems employ "mapping" techniques which allocate the actual cashflows onto the maturity gridpoints whilst maintaining various desired properties — see for example Chapter 8 in Miron & Swannell, Pricing and hedging swaps, Euromoney, 1991, also Section 8.6. Mapping is discussed further in Section 9.5.

Worksheet 8.1 Valuation of a USD portfolio Today's date: 4-Jan-00 Original market rates Original market rates Shift (ann) 0 3m cash 6m cash 12m cash 2yr swap 3yrswap 4yr swap 5yr swap 7yr swap 10 yr swap

6-Jan-00 6-Apr-00 6-Jul-00 8-Jan-01 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05 8-Jan-07 6-Jan-10

6.03125% 6.21875% 6.59375% 6.895% 7.025% 7.085% 7.135% 7.225% 7.335%

-

Interpolated market rates Daycount Cash & swap rates (act/360) (ann) 6-Jan-OO 6-Apr-OO 0.25 6.031% 6-Jul-OO 0.51 6.219% 8-Jan-Ol .02 6.594% 7-Jan-02 .01 6.895% 6-Jan-03 .01 7.025% 6-Jan-04 .01 7.085% 6-Jan-05 .02 7.135% 6-Jan-06 .01 7.180% 8-Jan-07 .02 7.225% 7-Jan-08 .01 7.262% 6-Jan-09 .01 7.298% 6-Jan-10 .01 7.335%

Swap portfolio Shifted market rates 6.03125% 6.21875% 6.59375% 6.895% 7.025% 7.085% 7.135% 7.225% 7.335%

USD DFs 1 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654672 0.607739 0.564318 0.523387 0.484927

USD cashflow (USD m) 6-Jan-OO 15-Feb-00 21-Feb-00 9-Mar-OO 15-May-00 21-Aug-00 11-Sep-00 15-Nov-00 9-Mar-01 15-May-01 21-Aug-01 in C~.» ni lu-aep-ui 15-Nov-01 ll-Mar-02 15-May-02 21-Aug-02 9-Sep-02 15-Nov-02 10-Mar-03 21-Aug-03 9-Sep-03 9-Mar-04 23-Aug-04 22-Aug-05 21-Aug-06

-10.1792 20.5929 -9.9296 0.4022 -1.5708 0.3707 0.4066 0.3568 0.4000 -1.5665 nu.joo/ 1Afi7 0.4066 0.3627 0.4000 -1.5665 0.3627 10.4066 0.3627 -1.5665 0.3647 10.3627 -1.5793 -1.5622 -21.5622

DFs 1 0.993399 0.992409 0.989604 0.978356 0.961440 0.957752 0.946337 0.926345 0.914610 0.897445 f\u.syjy^z 9Q1QA") 0.882382 0.862691 0.851952 0.835761 0.832622 0.821553 0.803300 0.778150 0.775236 0.747899 0.723809 0.673259 0.625643

PV=

-0.2831

Original

-0.2831

I

339

Traditional Market Risk Management

instruments, one per gridpoint, and constructs a portfolio that has exactly the same sensitivity as the original portfolio. This has two advantages, first it enables the reader to identify what transactions need to be done to reduce the exposures, and second equivalence reports are additive across systems. We will come back to this second point later. Consider a 5 year generic swap: the current rate is 7.135% and it obviously has a zero value as shown below: Example of a 5 year generic swap Generic rate 7.135% Rates

DFs

6-Jan-00 8-Jan-01 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05

1 0.936853 0.873099 0.812962 0.756987 0.704190

1.022 1.011 1.011 1.014 1.017

6.59375% 6.895% 7.025% 7.085% 7.135%

Cashflows

-100 7.2936 7.2143 7.2143 7.2341 107.2539 PV = 0.0000

Suppose the 3 year swap rate moves by 1 bp; we re-bootstrap the curve and revalue. The DFs have changed but the value of the 5 year swap remains zero: Example of a 5 year generic swap — changing the 3 year swap rate Generic rate 7.135% New Shift Cashflows in rates New rates DFs Rates DFs 6-Jan-OO 8-Jan-01 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05

1.022 6.59375% 1.011 6.895% 1.011 7.025% 1.014 7.085% 1.017 7.135%

-100 1 7.2936 0.936853 0.873099 7.2143 0.812962 7.2143 7.2341 0.756987 107.2539 0.704190 PV = 0.0000

0 0 1 0 0

6.59375% 6.895% 7.035% 7.085% 7.135%

Change in DFs

1 0 0 0.936853 0.873099 0 0.812713 -0.0002486 0.0000166 0.757003 0.0000156 0.704206

0.0000

This often seems counterintuitive at first glance but makes perfect sense: as the 5 year rate has not changed, a 5 year generic swap must retain its zero value irrespective of what else the swap curve is doing: Example of a 5 year generic swap — changing the 5 year swap rate Generic rate 7.135% Rates 6-Jan-OO 8-Jan-Ol 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05

1.022 1.011 1.011 1.014 1.017

6.59375% 6.895% 7.025% 7.085% 7.135%

DFs 1 0.936853 0.873099 0.812962 0.756987 0.704190 PV =

Cashflows -100 7.2936 7.2143 7.2143 7.2341 107.2539 0.0000

Shift in rates

New rates

0 0 0 0 1

6.59375% 6.895% 7.025% 7.085% 7.145%

New DFs 1 0.936853 0.873099 0.812713 0.757003 0.703804

-41,455.02

340

Swaps and Other Derivatives Table 8.3 Constructing an equivalent portfolio Portfolio 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap 10yr swap

17.39 31.11 -22.44 -270.89 -2,236.27 -2,779.17 1,213.52 8,596.64 0.00

Generic instruments -2,489.76 -4,901.22 -9,575.81 -18,402.99 -26,622.16 -34,296.44 -41,455.02 -54,284.90 -70,209.16

Equivalence (m USD)

0.70 0.63 0.23 1.47 8.40 8.10 2.93 15.84 -

to to to to to to to to

pay pay receive receive receive receive pay pay

The generic swap is in fact only sensitive to changes in its own rate — they are said to be "orthogonal". For a 1 bp increase in the 5 year rate, a $100 million receiver's swap loses $41,455 in value. From above, the portfolio gained $1,213 when the 5 year rate increased by 1 bp. Therefore this portfolio sensitivity must be equivalent to 1,213/(41,455/100) = 2.93 million of the 5 year swap to pay fixed. It is therefore straightforward to construct a delta equivalence portfolio of the nine market instruments that would match the sensitivity of our swap portfolio3, see Table 8.3.

8.5 GRIDPOINT RISK MANAGEMENT —FORWARD RATES We will now repeat the analysis, but this time using 3-monthly forward rates as the underlying factors. The first step is to build the forward curve: see column [1] of Box 1 of Worksheet 8.2. If a shift on a forward rate is imposed in column [2], then the new forward curve and new DFs are calculated in columns [3] and [4]. We can now calculate the sensitivity of the portfolio and each of the generic instruments to shifts in each of the forward rates. Unlike the market rates, the generic instruments are sensitive to all forward rates within their maturity range. This is because a blip in a forward rate F(t,T) is (approximately) equivalent to a parallel shift in all market rates with maturity equal to or greater than t. The results are shown in Box 2. We then bucket the sensitivities. The buckets are chosen to be the differences in the tenors of each neighbouring pair of market rates. The estimation of the total sensitivity in each bucket was done quite crudely by simply summing the sensitivities with respect to the forward rates in each bucket. This is not theoretically correct as one should generate the actual forward rates of the differing tenors, but the error will be very small. The effect of defining the buckets in this fashion means that the matrix of generic sensitivities as shown in Table 8.4 is square and upper-triangular. The sensitivity report for the portfolio, given on the left-hand-side of Table 8.4, is very different to the market rate sensitivity report. This report suggests that any upward movement in the forward curve would increase the portfolio value. But this is only to be expected as the imposed shifts in the rates are completely different. Sensitivity reports using different representations of interest rates cannot really be compared, even intuitively. 3 The worksheet "Generic Swaps" has not been printed here but is available on the CD. it clearly demonstrates the calculation of the sensitivity of each market instrument.

H

Worksheet 8.2 Forward rate sensitivity 1. Forward rates implied from market data Today's date: 4-Jan-00 Interpolated Forward DFs rate dates 6-Jan-00 6-Apr-00 6-Jul-00 6-Oct-OO 8-Jan-01 6-Apr-01 6-Jul-01 8-Oct-01 7-Jan-02 8-Apr-02 8-Jul-02 7-Oct-02 6-Jan-03 7-Apr-03 7-Jul-03 6-Oct-03 6-Jan-04 6-Apr-04 6-Jul-04 6-Oct-04 6-Jan-05 6-Apr-05 6-Jul-05 6-Oct-05 6-Jan-06 6-Apr-06 6-Jul-06 6-Oct-06 8-Jan-07

0.253 0.253 0.256 0.261 0.244 0.253 0.261 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.253 0.256 0.253 0.253 0.256 0.256 0.250 0.253 0.256 0.256 0.250 0.253 0.256 0.261

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190 0.691980 0.679635 0.667154 0.654672 0.643163 0.631525 0.619760 0.607739

Original forward rates [1]

Shift 0 [2]

Shifted forward rates [3]

DFs [4]

6.03125% 6.31005% 6.63166% 6.74852% 6.84293% 6.96337% 7.09233% 7.22180% 6.93151% 7.05513% 7.18323% 7.31607% 6.90960% 7.03242% 7.15970% 7.29314% 6.98127% 7.10668% 7.23814% 7.37455% 7.05796% 7.18617% 7.32061% 7.46018% 7.15807% 7.28997% 7.42836% 7.57529%

0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6.03125% 6.31005% 6.63166% 6.74852% 6.84293% 6.96337% 7.09233% 7.22180% 6.93151% 7.05513% 7.18323% 7.31607% 6.90960% 7.03242% 7.15970% 7.29314% 6.98127% 7.10668% 7.23814% 7.37455% 7.05796% 7.18617% 7.32061% 7.46018% 7.15807% 7.28997% 7.42836% 7.57529%

1 0.984983 0.969519 0.953362 0.936853 0.921440 0.905502 0.889038 0.873099 0.858065 0.843031 0.827996 0.812962 0.799006 0.785051 0.771096 0.756987 0.743860 0.730733 0.717462 0.704190 0.691980 0.679635 0.667154 0.654672 0.643163 0.631525 0.619760 0.607739 (continued)

2. Forward rate sensitivity of portfolio and each generic instrument to changes in the 3-monthly forward rates Forward rate 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28

Swap portfolio (USD m) 0 42.50 25.07 12.84 6.22 12.28 23.49 12.20 5.84 11.95 22.35 11.15 122.94 215.40 220.32 206.92 199.06 257.04 389.48 379.70 364.74 357.24 361.02 351.77 338.29 331.32 334.82 167.57 0

3m cash 0 -2,489.76 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

6m cash 0 -2,489.76 -2,488.03 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

12m cash 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

2yr swap 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 -2,245.42 -2,319.97 -2,394.29 -2,318.48 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

3yr swap 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 -2,242.43 -2,316.88 -2,391.10 -2,315.39 -2,163.00 -2,162.34 -2,161.65 -2,160.94 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0

4yr swap 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 -2,241.05 -2,315.45 -2,389.63 -2,313.97 -2,160.26 -2,159.59 -2,158.91 -2,158.19 -2,015.69 -2,015.08 -2,014.44 -2,035.50 0 0 0 0 0 0 0 0 0 0 0 0

Syr swap 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 -2,239.89 -2,314.26 -2,388.40 -2,312.78 -2,157.97 -2,157.31 -2,156.62 -2,155.91 -2,012.38 -2,011.77 -2,011.13 -2,032.16 -1,876.01 -1,875.42 -1,895.04 -1,894.39 0 0 0 0 0 0 0 0

7yr swap 0 -2,489.76 -2,488.03 -2,512.90 -2,565.83 -2,237.82 -2,312.12 -2,386.19 -2,310.64 -2,153.86 -2,153.19 -2,152.51 -2,151.80 -2,006.43 -2,005.82 -2,005.19 -2,026.15 - ,868.34 - ,867.76 - ,887.29 - ,886.65 - ,720.74 - ,738.97 - ,757.14 - ,756.53 - ,602.54 - ,619.50 - ,636.40 - ,670.67

Traditional Market Risk Management

343

Table 8.4 Bucketed forward rate sensitivities Buckets Portfolio $ 3 mo 3 mo 6 mo lyr lyr lyr 1 yr 2yr

42 25 19 54 168 842 1.391 2,242

4yr 2yr 3yr swap swap swap $ per $100 million notional

3m cash

6m cash

12m cash

-2,490

-2,490 -2,488

-2,490 -2,488 -5,079

-2,490 -2,488 -5,079 -9,278

-2,490 -2,488 -5,079 -9,266 -8,648

-2,490 -2,488 -5,079 -9,260 -8,637 -8,081

5yr swap

7yr swap

-2,490 -2,488 -5,079 -9,255 -8,628 -8,067 -7,541

-2,490 -2,488 -5,079 -9,247 -8,611 -8,044 -7,510 -13,502

We can calculate another equivalent portfolio. However its calculation is not as straightforward as before because the generic instruments are not orthogonal, and we have to use a technique known as "reverse bootstrapping" or "pyramiding". Starting with the generic instrument of longest maturity, we would require: 2,242/(-13,502/100) = –16.60 million of 7 year swap to pay fixed (note that the negative sign on the principal has been retained) as the equivalence over the final bucket. Working back, both the 5 and 7 year swaps have a sensitivity over the 4 to 5 year bucket, hence we have to adjust the sensitivity of the portfolio to account for the amount of the 7 year swap already done: net sensitivity = 1,391 - [(-16.60) x (–7,510/100)] = 144.34 and then calculate the residual equivalence: 144.34/(–7,541/100) = -1.91 million of 5 year swap to pay fixed The process is then repeated, working backwards. Alternatively, if A is the (square) matrix of generic sensitivities, and SP the vector of portfolio sensitivities, then the equivalence portfolio is given simply by 100A-1SP; see Table 8.5). Whilst the sensitivity reports were very different, the equivalence reports as one would expect are very similar and could within reason be combined. The reports are not identical because they are calculated under structures of different shifts. Table 8.5 Using forward rate sensitivity to construct an equivalent portfolio Equivalence (m USD) 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap

0.70 to pay 0.63 to pay 0.24 to receive 1.40 to receive 8.48 to receive 8.02 to receive 1.91 to pay 16.60 to pay

344

Swaps and Other Derivatives

8.6 GRIDPOINT RISK MANAGEMENT—ZERO COUPON RATES Finally, and briefly, we will analyse the portfolio using zero coupon rates. We will use continuously compounded rates for no particularly good reason other than that this form is extremely popular; i.e. DFt = exp{—z,t}. In exactly the same way as before, we calculate the zero coupon rates, shift them, recalculate the discount curve and revalue both the portfolio and the generic instruments. See Boxes 1 and 2 of Worksheet 8.3. One point to be careful about: we are using a 6 year zero rate because if we did not, we could not exactly replicate the original discount curve and hence we would find that the portfolio would have a slightly different value before any rate shift. As one would expect, the zero-coupon sensitivities are much closer to the market rate sensitivities than the forward rate ones. This is because the first two are comparable spot rates, and very different to forward rates. As before, the rates are not orthogonal, so we have to use reverse bootstrapping. However, we have to decide what to do about the sensitivity from the 6 year zero rate because the matrix of sensitivities in Box 2 is not square. We could simply add together the 6 and 7 year sensitivities as we did before. In the case of forward rates, this made practical sense as the result would have been the sensitivity due to a single 2 year forward rate from 5 to 7 years. But in this case, it is probably better to use a mapping technique. There are a wide variety of such methods, but the one described below is probably the most popular. Generalizing the problem, we have a sensitivity ST due to a rate of maturity T which we wish to allocate onto the gridpoints t and t' where t< T < t' in such a way that we wish to preserve some desirable properties. From the definition of DFT above, we can write PVT = CFT exp{— z T T} for some cashflow CFT The sensitivity of this value with respect to ZT is simply PV T T. Let us assume we wish to estimate St and St so that both the total value and total sensitivity are preserved, i.e.: St,+ S t , = S T and PV t + PVt' = PV T Substituting PV T = ST/T, etc. we can solve for St and S t >, i.e.:

St = STx((t'-T)/T\/[(t' -t)/t] and S, = ST x [ ( T - t)/T]/[(t' - t)/>'] Thus, we can allocate the 6-year sensitivity of both the portfolio and the generic 7 year swap to the 5 and 7 year gridpoints, as shown below: Mapping the 6 year sensitivity

5yr point 6yr point 7yr point

Time 5.075 6.089 7.108 Sum

Weights

Portfolio sensitivity

0.418

Allocated portfolio sensitivity 1,532.24

3,666.68 0.582 1

7yr swap sensitivity

Allocated 7yr swap sensitivity -1,219.86

-2,919.16 2,134.44

-1,699.30

We now have sensitivities for our nine market instruments as shown in Box 3. The equivalent portfolio can now be calculated using reverse bootstrapping again; see Table 8.6. The worksheet also calculates the resulting equivalent portfolio created by simply adding together the 6 and 7 year sensitivities. The portfolio at the long end is of course quite different, with increased weighting on the 7 year generic swap and actually receiving on the 5 year swap.

p: CX

Worksheet 8.3 Zero coupon rate sensitivity

1. Zero coupon rates implied from market data Today's date 4-Jan-00 Zero-C rate dates 6-Jan-00 6-Apr-00 6-Jul-00 8-Jan-01l 7-Jan-02 6-Jan-03 6-Jan-04 6-Jan-05 6-Jan-06 8-Jan-07

Interpolated DFs 0.253 0.506 1.022 2.033 3.044 4.058 5.075 6.089 7.108

1 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654672 0.607739

Original zero-C rates [1]

Shift 0 [2]

Shifted zero-C rates [3]

DFs [4]

5.98574% 6.123000% 6.38104% 6.67406% 6.80161% 6.86019% 6.91047% 6.95727% 7.00600%

0 0 0 0 0 0 0 0 0

5.98574% 6.123000% 6.38104% 6.67406% 6.80161% 6.86019% 6.91047% 6.95727% 7.00600%

1 0.984983 0.969519 0.936853 0.873099 0.812962 0.756987 0.704190 0.654672 0.607739

Worksheet 8.3 Zero coupon rate sensitivity (continued)

2. Zero-C rate sensitivity Swap portfolio (USD m) 3m rate 6m rate 12m rate 2yr rate 3yr rate 4yr rate 5yr rate 6yr rate 7yr rate

0 18 32 -17 -248 -2,035 -2,250

-73 3,667 5,759

3m cash

0 -2,528

0 0 0 0 0 0 0 0

6m cash

0 0 -5,055

0 0 0 0 0 0 0

12m cash

2yr swap

3yr swap

4yr swap

5yr swap

7yr swap

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

0 0 0

-10,222

-675 -18,989

-688 -1,261 -26,504

-694 -1,272 -1,773 -32,921

-698 -1,281 -1,785 -2,222 -38,320

-707 -1,297 -1,808 -2,250 -2,624 -2,919 -46,366

0 0 0 0 0 0

0 0 0 0 0

0 0 0 0

0 0 0

0 0

3. Bucketed Z-C rate sensitivities by mapping 6 year sensitivity onto 5 and 7 year points Buckets 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap

Portfolio

18 32 -17 -248 -2,035 -2,250 1,459 7,894

3m cash -2,528

0 0 0 0 0 0 0

6m cash

0 -5,055

0 0 0 0 0 0

12m cash

2yr swap

3yr swap

4yr swap

5yr swap

7yr swap

0 0

0 0

0 0

0 0

0 0

0 0

-10,222

-675 -18,989

-688 -1,261 -26,504

-694 -1,272 -1,773 -32,921

-698 -1,281 –1,785 -2,222 -38,320

-707 -1,297 -1,808 -2,250 -3,844 -48,065

0 0 0 0 0

0 0 0 0

0 0 0

0 0

0

3 pa

347

Traditional Market Risk Management Table 8.6 Using zero coupon rate sensitivity to construct an equivalent portfolio Equivalence (m USD) 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap

0.70 0.63 0.23 1.47 8.40 8.10 2.16 16.42

to pay to pay to receive to receive to receive to receive to pay to pay

In summary, the equivalence portfolios are all very similar, despite the fact that the sensitivity reports are very different, as highlighted in Worksheet 8.4. They will not be identical because the movements imposed on each of the three curves are not the same when translated into a comon framework. But the similarity is reassuring; a portfolio should be effectively hedged in the same fashion irrespective of how the sensitivity is calculated.

8.7 YIELD CURVE RISK MANAGEMENT Gridpoint risk management assumed that all interest rates were uncorrelated. The other extreme is to assume that the curve moves according to some structure. However, before we make this assumption, it is probably worthwhile to look at some historical evidence. Figure 8.1 shows the correlation between changes in the 5 year rate with the other points along a USD swap curve, represented in three different ways: • the correlation between the changes in the 5 year market rate and changes in other market rates is consistently high, except for the cash market; Daily data: 1988 to 1993

1.00

Market rates --*--

Forward rates

- - • * - - • Zero-C rates

0.00

-0.20 10

Figure 8.1 Correlation with 5 year point along a USD curve

Worksheet 8.4 Gridpoint sensitivity summary

Market instruments 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap

Market rate 17 31 -22 —271 —2,236 -2,779 1,214 8,597

Sensitivity Forward rate 42 25 19 54 168 842 1,391 2,242

Zero-coupon rate 18 32 -17 -248 -2,035 -2,250 -73 9,426

Market rate 0.70 0.63 0.23 1.47 8.40 8.10 2.93 15.84

Equivalence Forward rate Zero-coupon rate 0.70 0.70 to pay 0.63 0.63 to pay 0.24 0.23 to receive 1.40 1.47 to receive 8.48 8.40 to receive 8.02 8.10 to receive 1.91 2.16 to pay 16.60 16.42 to pay

0

•o

O.

I

Traditional Market Risk Management

349

• the correlation between the 5 year zero coupon rate and the other zero rates is not quite as high; • the correlation between 12 month forward rates is extremely low, except between the 5/6 and 6/7 rates. The results might be intuitively expected as both market and zero rates are spot rates, and hence possess considerable overlap, unlike forward rates. Indeed forward rate correlation matrices typically possess a sharp "ridge" structure, i.e. high correlation along the principal diagonal but dropping off rapidly — almost negatively exponentially — on either side. Similar results have been observed over other periods of time and with a range of different currencies. The Appendix describes some further analysis, including the use of principal components. The implication of this analysis is that yield curve risk management is most sensibly restricted to market rates and possibly zero rates, but should not be applied to forward rates. We will only consider market rates in the discussion below. Let us first assume that the market curve undergoes a parallel shift; i.e. if the curve is represented by the rate r, then it changes by Ar. If the current value of the portfolio is V, then its new value after the rate shift is given approximately by Taylor's theorem: V(r + Ar) % V(r) + Ar x (dV/dr) + ±Ar 2 x (d 2 V/dr 2 ) + ••• where (dV/dr) is the delta of the portfolio and (d 2 V/dr 2 ) is the gamma of the portfolio. If Ar is measured in basis points, then (dV/dr) is the PVBP of the portfolio estimated by shifting the entire curve up by a basis point: from Worksheet 8.5, PVBP = $4,5484. Strictly, this measures the delta at (r +1/2bp), so to calculate it more precisely, we should estimate the average by: delta-up: delta-down: average delta:

4,548.27 (as before) 4,552.83 (calculated by 1 bp shift downwards) 4,550.55

We can use the same results to estimate the gamma. This is defined as the change in the delta for a shift in the rate: i.e. (delta-up—delta-down) = –4.55 per bp as the distance between delta-up and delta-down is only 1 bp. As an example, suppose the entire curve shifts by —100 bp; the actual change in value is –$478,539. Substituting: (-100) x 4,550.55 + x (-100)2 x -4.55 = -$477,811, an error of 0.15%. Taylor's theorem lies at the heart of all risk management as it permits an estimation of the change in the value of an instrument/portfolio for given changes in its underlying parameters. Suppose, for example, we wish to hedge this portfolio with two bonds B1 and B2. We create a super portfolio SP= {V+n1 x B1 + n2 x B2} where n1 and n2 are the amounts of the bond we wish to buy or sell. We can write: ASP % Ar x {(dV/dr) + n1 x ( d B 1 d r ) + n2 x (dB 2 /dr)} + |(Ar)2 x {(d2 V/dr2} + n1 x (d 2 B 1 /dr 2 ) + n2 x (82B2/dr2)} 4

Compare this number with $4,549 calculated earlier by shifting each of the market rates in turn, and then summing the sensitivities. The error involved in the latter is negligible.

Worksheet 8.5

Valuation and market rate curve sensitivity of a USD portfolio Swap portfolio

Today's date: 4-Jan-00

3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap 10yr swap

Original market rates Cash& swap rates (ann) 6-Jan-OO 6.03125% 6-Apr-OO 6-Jul-00 6.21875% 6.59375% 8-Jan-01 7-Jan-02 6.895% 6-Jan-03 7.025% 6-Jan-04 7.085% 6-Jan-05 7.135% 8-Jan-07 7.225% 6-Jan-10 7.335%

Interpolated market rates Daycount (act/360) 6-Jan-OO 6-Apr-OO 0.25 6-Jul-OO 0.51 .02 8-Jan-Ol 7-Jan-02 .01 .01 6-Jan-03 6-Jan-04 .01 .02 6-Jan-05 .01 6-Jan-06 .02 8-Jan-07 7-Jan-08 .01 6-Jan-09 .01 6-Jan-10 .01

USD cashflow (USD m)

New

Shift (100.00)

rates

(100) (100) (100) (100) (100) (100) (100) (100) (100)

5.031% 5.219% 5.594% 5.895% 6.025% 6.085% 6.135% 6.225% 6.335%

Cash & swap rates (ann)

USD DFs

5.031% 5.219% 5.594% 5.895% 6.025% 6.085% 6.135% 6.180% 6.225% 6.262% 6.298% 6.335%

0.987442 0.974295 0.945912 0.889954 0.836564 0.786418 0.738595 0.693246 0.649772 0.609146 0.570429 0.533650

1

DFs 1

6-Jan-00 15-Feb-00 21-Feb-00 9-Mar-00 15-May-00 21-Aug-00 11-Sep-00 15-Nov-00 9-Mar-01 15-May-01 21-Aug-01 10-Sep-01 15-Nov-01 ll-Mar-02 15-May-02 21-Aug-02 9-Sep-02 15-Nov-02 10-Mar-03 21-Aug-03 9-Sep-03 9-Mar-04 23-Aug-04 22-Aug-05 21-Aug-06

-10.1792 20.5929 -9.9296 0.4022 -1.5708 0.3707 0.4066 0.3568 0.4000 -1.5665 0.3687 0.4066 0.3627 0.4000 -1.5665 0.3627 10.4066 0.3627 -1.5665 0.3647 10.3627 -1.5793 -1.5622 -21.5622

0.994480 0.993652 0.991306 0.981807 0.967275 0.964071 0.954152 0.936688 0.926388 0.911322 0.908248 0.898101 0.880713 0.871179 0.856805 0.854018 0.844191 0.827909 0.805377 0.802767 0.778186 0.756365 0.710268 0.666356

PV =

-0.7617

Original

-0.2831

Actual change in value Delta-up Delta-down Average delta Gamma Estimated change in value Error

-478,539.26 4,548.27 4,552.83 4,550.55 -4.55 -477,810.60 0.15%

3 Q.

I

351

Traditional Market Risk Management

An effective hedge in this case would mean that ASP — 0 for any movement in r. Ignoring for the moment the gamma terms, the only way we can set ASP = 0 for any Ar is by setting: {(dV/dr} + w, x (dBjdr) + n2 x (dB2 /dr)} = 0 If we arbitrarily set n2 — 0, then:

n, = -(dV / dr)/(dB 1 / dr) gives the amount of B1 required to delta hedge the portfolio. We have of course used exactly this expression earlier to estimate the equivalences. Turning to the actual portfolio, we are going to hedge this with a T-bond as follows: Maturity Coupon YTM Clean price

15-Nov-04 5.875% sa 6.40% 97.8438 Act/Act

In practice, such a bond would be valued off the bond curve. We want it to hedge a swap portfolio, and hence we have introduced some basis risk, namely that the swap spread may change. This is quite common; hedging seldom eliminates risk, but as in this case merely substitutes basis risk for absolute level risk. We then need to make a judgement which form of risk is smaller. So, for risk management purposes, we are going to assume that the spread remains constant. In that case it matters very little whether the bond is valued off the bond curve or the swap (equals bond plus spread) curve as we are only interested in the change in value: for convenience therefore, we will value everything off the swap curve. We calculate the average deltas for the portfolio and the bond, and hence the hedge ratio as above (see Worksheet 8.6):

Portfolio: Bond: Hedge ratio:

Delta ($) 4,550.55 —38,513.15 per 100m nominal 11.82 million to buy

The effectiveness of this delta hedge may be measured by creating: 11.82x5,} and measuring its change in value as r changes. For example, the worksheet shows that if the curve shifts up by 100 bp, SP loses $10,132 in value. For large changes up or down, the negative gamma from the portfolio appears to dominate, and losses would be made (see Figure 8.2). To remove this effect as well, returning back to the SP, we need to set both delta and gamma terms to zero, i.e.: (dV / dr] + H, x (dB 1 / dr) + n2 x (dB2 / dr) = 0 2

(d V / dr) + n1 x (d2Bl / dr2) + n2 x (d2B2 / dr2) = 0

352

Swaps and Other Derivatives

Worksheet 8.6 Creation of a delta curve hedge for a USD portfolio Swap portfolio USD cashflow (USD m) 6-Jan-00 15-Feb-OO -10.1792 21-Feb-00 20.5929 9-Mar-00 -9.9296 15-May-00 0.4022 21-Aug-00 -1.5708 11-Sep-00 0.3707 15-Nov-00 0.4066 9-Mar-0l 0.3568 15-May-0l 0.4000 21-Aug-0l -1.5665 10-Sep-0l 0.3687 15-Nov-0l 0.4066 ll-Mar-02 0.3627 15-May-02 0.4000 21-Aug-02 -1.5665 9-Sep-02 0.3627 15-Nov-02 10.4066 10-Mar-03 0.3627 21-Aug-03 -1.5665 9-Sep-03 0.3647 9-Mar-04 10.3627 23-Aug-04 -1.5793 22-Aug-05 -1.5622 21-Aug-06 -21.5622 PV= Original PV

Hedging bond DFs

1 0.992324 0.991173 0.987910 0.974931 0.955683 0.951526 0.938657 0.916223 0.903108 0.883926 0.880012 0.867093 0.845213 0.833342 0.815443 0.811973 0.799736 0.779655 0.752103 0.748911 0.719068 0.692955 0.638513 0.587777 0.1499 -0.2831

Change in value

4,329.93

Delta-up

4,548.27

Delta-down

4,552.83

Average delta

4,550.55

Maturity coupon

15-Nov-04 5.875% sa USD cashflow (USD m)

6-Jan-00 15-May-OO 15-Nov-OO 15-May-Ol 15-Nov-Ol 15-May-02 15-Nov-02 15-May-03 15-Nov-03 15-May-04 15-Nov-04 PV=

2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 102.9375

DFs

0.974931 0.938657 0.903108 0.867093 0.833342 0.799736 0.768567 0.737655 0.708592 0.679821 92.1034

Original

95.8537

Change in value Delta-up

-37,503.52 -38,502.84

Delta-down

-38,523.46

Average delta

-38,5 13.15

Hedge ratio = Value of SP ($ m)

11.82m bond 11.0324

Original value of SP ($m) Change in value of SP ($)for 100 bp shift in curve

11 .0425 - 10. 1 32.46

Traditional Market Risk Management

353

0 -50,000 - 100,000 -150,000 -200,000 -250,000 -300,000 ),000

-500 -400

-300

-200 -100

Figure 8.2

100

0

200

300

400

500

Effectiveness of a delta hedge

X 1 n nnn _ 5 000 —

\^

\^

"

- R nnn —

— -~— ^ —"— ~—*_

^•\^

"

_ -i K nnn- on nnn - ot; nnn-500

-400

-300

-200

-100

0

100

200

300

400

500

Figure 8.3 Effectiveness of a delta-gamma hedge To do this, we need a second T-bond: Bond 2 matures: coupon: yield:

15 August 2009 6.000% sa 6.50%

and to calculate the various sensitivity parameters (see Worksheet 8.7):

Average delta Gamma

Portfolio 4,550.55 -4.55

Bond 1 -38,513.15 20.63

Bond 2 -63,254.40 58.00

Notice that the SP gamma from the delta hedge alone is: -4.55+ 11.82 x (20.63/100) = -2.11 per bp i.e. as we saw, the delta hedge above halves the negative gamma of the portfolio, but does not fully off-set it. Solving the delta-gamma equations gives n{ — —2.58 million and

Worksheet 8.7 Creation of a delta-gamma curve hedge for a USD portfolio Swap portfolio USD cashflow (USD m) 6-Jan-OO 15-Feb-OO -10.1792 21-Feb-00 20.5929 9-Mar-OO -9.9296 15-May-00 0.4022 21-Aug-00 -1.5708 11-Sep-00 0.3707 15-Nov-00 0.4066 9-Mar-0l 0.3568 15-May-0l 0.4000 21-Aug-0l -1.5665 10-Sep-0l 0.3687 15-Nov-0l 0.4066 ll-Mar-02 0.3627 15-May-02 0.4000 21-Aug-02 -1.5665 9-Sep-02 0.3627 15-Nov-02 10.4066 10-Mar-03 0.3627 21-Aug-03 -1.5665 9-Sep-03 0.3647 9-Mar-04 10.3627 23-Aug-04 -1.5793 22-Aug-05 -1.5622 21-Aug-06 -21.5622 PV=

DFs

1 0.992324 0.991173 0.987910 0.974931 0.955683 0.951526 0.938657 0.916223 0.903108 0.883926 0.880012 0.867093 0.845213 0.833342 0.815443 0.811973 0.799736 0.779655 0.752103 0.748911 0.719068 0.692955 0.638513 0.587777 0.1499

Hedging bond 1

Hedging bond 2

Maturity

Maturity

coupon

15-Nov-04

coupon

5.875% sa USD cashflow (USD m)

6-Jan-OO 15-May-OO 15-Nov-OO 15-May-Ol 15-Nov-Ol 15-May-02 15-Nov-02 15-May-03 15-Nov-03 15-May-04 15-Nov-04

PV=

2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 102.9375

15-Aug-09 6.00% sa

DFs

0.974931 0.938657 0.903108 0.867093 0.833342 0.799736 0.768567 0.737655 0.708592 0.679821

92.1034

6-Jan-OO 15-Feb-00 15-Aug-00 15-Feb-0l 15-Aug-0l 15-Feb-02 15-Aug-02 15-Feb-03 15-Aug-03 15-Feb-04 15-Aug-04 15-Feb-05 15-Aug-05 15-Feb-06 15-Aug-06 15-Feb-07 15-Aug-07 15-Feb-08 15-Aug-08 15-Feb-09 15-Aug-09 PV =

USD cashflow (USD m)

DFs

3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 103.0000

0.992324 0.956871 0.920529 0.885101 0.849597 0.816539 0.783519 0.753111 0.722664 0.694206 0.665869 0.639531 0.613150 0.588591 0.564019 0.541334 0.518614 0.497395 0.476270 0.456649 87.4725 93.5178

Change in value (/bp)

4,329.93

95.8537 -37.503.52

Delta-up

4,548.27

-38,502.84

-63,225.40

Delta-down

4,552.83

-38,523.46

-63,283.60

Average delta

4,550.55

-38,513.15

-4.55

20.63

-63,254.40 58.00

Original

Gamma

-0.2831

-60,453.11

Traditional Market Risk Management

355

77, = 8.76 million. We are long the longest bond, as this will provide most gamma to offset the portfolio, and short the short bond as this creates the delta hedge. The effectiveness against the l00bp shift is: ASP =100 x (4,550.55 - 38,513.55 x (-2.58)/100 - 63,254.40 x (8.76)/10()} +1(100)2 x {-4.55 + 20.63 x (-2.58)/100 + 58.00 x (8.76)/100J = $116 The overall improvement in effectiveness is shown in Figure 8.3. The residual is no longer always negative, but changes sign, implying that the third-order term5 in the Taylor's expansion is negative. There is no reason why this type of analysis has to be restricted to parallel shifts. As we see in the Appendix, about 10% of the market curve movement can be attributed to rotational shifts. Hence, define a rotational delta as the change in the value of the portfolio as the market curve rotates around the 5 year point according to the formula: Ar, — .V x (/ — 5) bp

where we arbitrarily set s = 1

The parallel deltas and gammas, plus the rotational deltas have been calculated for the portfolio plus three bonds, and then hedge ratios estimated (see Worksheet 8.8):

Average delta Gamma Average rotation Hedge (millions)

Portfolio 4.550.55 -4.55 25.127.55

Bond 1 -38,513.25 20.63 3,196.57 -22.58

Bond 2 -63,254.40 58.00 -302,528.57 -5.60

Bond 3 -52,805.97 39.18 -130,048.46 31.79

All simultaneous parallel and rotational shifts can be described in terms of the pivot point moving plus a rotation around the standardized pivot. For example, suppose the curve shifts up by 100 bp and simultaneously rotates by —3bp around the pivot. The portfolio on its own gains $360,510.57 in value, but the SP only changes by —$183.45. To test the overall hedge effectiveness, movement in the curve was simulated by: Ar, = s x (t-P)

bp

where P is drawn from a uniform distribution between [0,10] and s from 7V(0,1.4%/V250)6. The resulting hedge effectiveness is shown by the distribution in Figure 8.4; the error is relatively small although demonstrates a negative rotational gamma.

8.8 SWAP FUTURES Hedging a swap portfolio with bonds appears to work extremely well. However the bonds are usually government bonds because of the need for liquidity, and therefore this style of hedging necessitates the assumption that the bond-swap spread remains constant. 3

Sometimes called the "omega" or last word in risk management!! This standard deviation was selected to be roughly equivalent to a volatility of 20% pa— see Worksheet 8.8 for details. 6

356

Swaps and Other Derivatives

Worksheet 8.8 Creation of a delta-gamma-rotational curve hedge for a USD portfolio Swap portfolio USD cashflow (USD m) DFs 6-Jan-00 1 15-Feb-00 -10.1792 0.992171 21-Feb-00 20.5929 0,909997 9-Mar-OO -9.9296 0.987670 15-May-00 0.4022 0.974460 21-Aug-00 -1.5708 0.954944 11-Sep-00 0.3707 0.950740 15-Nov-00 0.4066 0.937724 9-Mar-0l 0.3568 0.915105 15-May-0l 0.4000 0.901922 21-Aug-0l -1.5665 0.882639 10-Sep-0l 0.3687 0.878703 15-Nov-0l 0.4066 0.865717 ll-Mar-02 0.3627 0.843811 15-May-02 0.4000 0.831970 21-Aug-02 -1.5665 0.814117 9-Sep-02 0.3627 0.810656 15-Nov-02 10.4066 0.798451 10-Mar-03 0.3627 0.778500 21-Aug-03 -1.5665 0.751225 9-Sep-03 0.3647 0.748066 9-Mar-04 10.3627 0.718592 23-Aug-04 -1.5793 0.692926 22-Aug-05 -1.5622 0.693637 21-Aug-06 -21.5622 0.590299 PV =

0.0774

Original

-0.2831

Hedging bond 1 Maturity coupon

15-Nov-04 5.875% sa USD cashflow (USD m)

6-Jan-OO 15-May-OO 15-Nov-OO 15-May-Ol 15-Nov-Ol 15-May-02 15-Nov-02 15-May-03 15-Nov-03 15-May-04 15-Nov-04

PV=

2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 2.9375 102.9375

DFs

0.974460 0.937724 0.901922 0.865717 0.83'970 0.798451 0.767524 0.736923 0.708295 0.68016

92.0980 95.8537

3,605.11 4,548.27

-37,557.63 -38,502.84

Delta-down

4,552.83

-38,523.46

Average delta

4,550.55

-38,513.15 20.63 1,251,921.08

Change in value (/bp) Delta-up

Gamma Change in value

—4.55 130,170.19

Rotation-left

25,131.57

Rotation-right Average rotation

25,123.53 25,127.55

3.197.87 3.195.27 3,196.57

357

Traditional Market Risk Management

Hedging bond 2 Maturity coupon

Hedging bond 3

15-Aug-09

Maturity

6.00% sa

coupon

USD cashflow (USD m) 6-Jan-OO 15-Feb-00 15-Aug-00 15-Feb-0l 15-Aug-0l 15-Feb-02 15-Aug-02 15-Feb-03 15-Aug-03 15-Feb-04 15-Aug-04 15-Feb-05 15-Aug-05 15-Feb-06 15-Aug-06 15-Feb-07 15-Aug-07 15-Feb-08 15-Aug-08 15-Feb-09 15-Aug-09 PV=

3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 3.0000 103.0000

15-May-07 5.9375% sa

DFs

0.992171 0.956146 0.919434 0.883819 0.848183 0.815210 0.782325 0.752223 0.722127 0.694156 0.666342 0.640631 0.614911 0.591089 0.567283 0.545420 0.523549 0.503216 0.482995 0.464293 88.3259

6-Jan-OO 15-May-00 15-Nov-00 15-May-0l 15-Nov-0l 15-May-02 15-Nov-02 15-May-03 15-Nov-03 15-May-04 15-Nov-04 15-May-05 15-Nov-05 15-May-06 15-Nov-06 15-May-07

PV=

USD cashflow (USD m)

DFs

2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 2.9688 102.9688

0.974460 0.937724 0.901922 0.865717 0.831970 0.798451 0.767524 0.736923 0.708295 0.68016 0.653700 0.627562 0.603197 0.578980 0.556533

88.9715

93.5178

93.6878

-51,919.00

-47,162.89

-63,225.40

-52,786.38

-63,283.40

-52,825.56

-63,254.40

-52,805.97

58.00

39.18

1,730,633.30

1,572,096.30

-302,254.07

-130,027.73

-302,83.06

-130,069.20

-302,528.57

-130,048.46

Swaps and Other Derivatives

358 70 60 50 40 30 20 10 i

-5,657

i i i i i i i i i I i r r -5,062 -4,466 -3,871 -3,275 -2,680 -2,084 -1,489

i

i -893

i

i -298

i

Figure 8.4 Hedging parallel and rotational shifts

Unfortunately in practice, this spread can be quite volatile and, as we shall see in Chapter 9, the optimal hedge effectiveness can be quite low. An alternative would be to hedge with bonds that were much more in line with the swap curve, but again unfortunately such bonds are seldom available with adequate liquidity. Perhaps the only category that has been a serious contender is the Pfandbriefe7 bonds issued in DEM, which have traditionally had a very close correlation to the swap curve. It has long been an ambition by various futures exchanges to offer swap hedging products; with the massive growth in the swap market, this would be a substantial revenue generator. Various products have been tried in the past by both the Chicago and London exchanges, but have foundered due to a reluctance by the major swap-trading banks who wished to maintain their domination. In March 2001, LIFFE launched Swapnote futures, i.e. futures contracts on standardized forward swaps. These contracts, currently onlydesignated in Euros, have had an impressive start, with some 2.7 million contracts trading in its first 6 months. Putting this in context; this represents less than 10% of the trading in the Euro bond futures contracts, but it is an extremely good start compared to the early days of most derivative contracts. Buoyed by this success, LIFFE introduced options on the Swapnote futures in July 2001. There are also plans to expand the range of currencies. There are a number of factors that may have aided this growth. Despite the introduction of the Euro, European government bond markets are heterogeneous with supply /demand imbalances resulting from declining bond issuance in some countries. As a result, the Euro swap curve has become the de facto homogeneous benchmark. Furthermore, because of the relatively large size of this swap market compared to the stock of securities capable of being delivered under bond futures contracts ($16.6 trillion compared to S3.3 trillion at the end of 2000 respectively), and because the swapnotes are cash-settled, they are less susceptible to squeezes and other market distortions. Finally, the contracts may receive a boost as an increasing number of European national debt management offices are using 7

Highly liquid mortgage-backed bonds, typically issued in large sizes and with a good range of maturities.

359

Traditional Market Risk Management

swaps to manage the public debt. Even within the first 6 months, a number of European regional banks had been active traders, and the number of second tier or lower-graded institutions is likely to expand. Chicago followed suit with the launch of 10 year dollar swap futures contracts on 26 October 2001, with the intention of expanding the range to other maturities in the near future. Because this may become a significant hedging strategy in the future, a brief introduction is thought appropriate. There are three underlying swap contracts, namely with 2, 5 and 10 years maturity (matching exactly the underlying Schatz, Bobl and Bund bond futures). There are four delivery months, i.e. March, June, September and December, with the third Wednesday being the delivery date, and the last trading date two business days earlier. Currently all the open interest is in the near delivery month, with roll-overs to the next one occurring only as maturity of near futures approaches. The underlying is a swap with fixed rate of 6% ann 30/360 on a notional principal of €100,000 starting out of the delivery date. The example below shows the 10 year contract with a fair price of 107.29 per nominal €100 on 10 October 2001, using discount factors from the Euro swap curve: Today's date: 10-Oct-0l Delivery month: Dec-01 Delivery date: 19-Dec-01 Fixed rate 6% Notional principal €100,000 Swap Days dates 30/360 Cashflows 12-Oct-0l 19-Dec-0l -107,293.50 19-Dec-02 1.000 6,000 1.000 19-Dec-03 6,000 20-Dec-04 6,017 1.003 5,983 19-Dec-05 0.997 19-Dec-06 1.000 6,000 1.000 19-Dec-07 6,000 19-Dec-08 6,000 1.000 21-Dec-09 1.006 6,033 20-Dec-10 0.997 5,983 0.997 105,983 19-Dec-ll

DFs

0.993079 0.960177 0.924091 0.883912 0.842042 0.799285 0.756393 0.714378 0.673362 0.634878 0.598275

Discounted cashflows -106,551 5,761 5,545 5,318 5,038 4,796 4,538 4,286 4,063 3,799 63,407 PV = 0

The futures are cash-settled, paying €10 per 1 bp shift in the nominal price. For example, suppose we have a Euro swap portfolio that has a PV delta of € 10,000 per 1 bp parallel increase in the market curve. For the same movement, the price of the Swapnote futures decreases from 107.2935 to 107.2115, i.e. 8.2 price-bps or €82 per contract: note that this has not been rounded to the nearest contract yet. Therefore, the hedge would be to buy {10,000/82} = 122 contracts (this is rounded). The curve now shifts by +8 bp parallel shift: the futures price drops by a rounded 65 bp as may be seen in Worksheet 8.9: portfolio gains €80,000, futures has to pay 122 x 65 x €10= €79,300.

360

Swaps and Other Derivatives

Worksheet 8.9 Swapnote futures

Today's date: 10-Oct-0l Delivery month: Dec-01 Delivery date: 19-Dec-0l Fixed rate 6% Notional principal Swap dates 12-Oct-0l 19-Dec-Ol 19-Dec-02 19-Dec-03 20-Dec-04 19-Dec-05 19-Dec-06 19-Dec-07 19-Dec-08 21-Dec-09 20-Dec-10 19-Dec-ll Shifted price =

100,000

Days 30/360 1.000 1.000 1.003 0.997 1.000 1.000 1.000 1.006 0.997 0.997

Cashflows

z-c rates

DFs

106,639.85 6,000 6,000 6,017 5,983 6,000 6,000 6,000 6,033 5,983 105,983 106.6399

3.756% 3.456% 3.638% 3.889% 4.124% 4.335% 4.526% 4.689% 4.835% 4.950% 5.048%

0.992930 0.959287 0.922533 0.881738 0.839323 0.796084 0.752774 0.710400 0.669080 0.630341 0.593527 PV =

Original price =

107.2935

Delta =

-65.365

Discounted cashflows -105,886 5.756 5.535 5.305 5,022 4,777 4.517 4.262 4,037 3.772 62.904 0

The evidence over the past 6 months is that these contracts are showing much less basis risk with the generic swap curve than the equivalent Euro bond curve. But they currently lack good liquidity and it will be interesting to see if they gain it. LIFFE's transaction cost is 15p per side per lot at the moment, which is certainly quite cheap.

8.9 THETA RISK So far we have considered the possible losses that might be made if the market moves against the portfolio, and how these losses may be reduced by hedging. But just suppose that, as time passes, the market moves solely in accordance with the implied forward rates. What would be the impact on the value of the portfolio, and of course the hedge? There are a number of ways that are used to assess this impact. Probably the theoretically correct approach is as follows. Assume that we have a current forward rate curve FQ = {F0/r, FI/2, F 2/3 ,...}. As time passes from t = 0 to / = T the first forward rate F0/T falls away, and the new curve is FT = {Fr/2, F 2/3 ,...}. Notice that it is still the same forward rates as before, since they are still our best estimate of the curve, but all starting one period earlier:

361

Traditional Market Risk Management CR,

If we have a cashflow at time T > T, then its initial value is F0 = CFT x DF0 r, and after the passage of time FT = CFr x DFT T. The new discount factor DFT T is simply given by DF0 r /DF OT , hence Vr = F 0 /DF 0r . Therefore the theta of the cashflow is Vt — V0 = VQ x (l/DF 0 r — 1) over that period of time. Extending this to an entire portfolio is trivial. A 7 day period is frequently used for estimating theta, but it should vary depending upon the liquidity of the market. If the cashflow time T lies between 0 and T, then the cashflow would have to be either deposited (if positive) or borrowed (if negative) from time T until t = T. The discounting process implies that all interest accrues at Libor flat, therefore if the depositing or borrowing rate is away from Libor, this would incur an additional reward or penalty. The above approach effectively holds the forward curve constant, and moves the observer one period up the curve. An alternative popular, but in my view incorrect, approach is to hold the observer constant and move the curve as shown below: pc w 14.

pc wI_

T-T

In this case, the new value VT = CF7 x DF 0>r _ T = CFr x DF0.r/DFr_tT = jy DF r _ T , r . Extending this to an entire portfolio is less straightforward, as the adjustment term is specific to the timing of each cashflow, but not difficult. Table 8.7 shows the calculation of 7 day theta for the two approaches. Because of the curvature of the discount function, DF0 T > DF r _ T 7 and therefore theta from the second approach will always be absolutely greater than theta from the first approach. A third approach is to use an accrual concept as in the bond market. Suppose that this cashflow CFr has been calculated by P x d x r where d is the appropriate length of the interest period. If d 01 is the length of time from t = 0 to t = 1, then V1 — V0 + CFr x d01 /d. But this method: • ignores discounting; • tacitly assumes that the period d started some time before t = 0; • is difficult to apply to a portfolio of cashflows as it really concentrates only on the most immediate cashflow. It is therefore not really appropriate for derivative portfolios. Related to theta is of course the "cost-of-carry", which is usually discussed in the context of hedging a portfolio with market-based securities, as discussed in Chapter 2. In

362

Swaps and Other Derivatives

Table 8.7 Alternative methods for calculating theta Swap portfolio USD cashflow (USD m)

1 6-Jan-00 15-Feb-00 -10.1792 20.5929 21-Feb-00 -9.9296 9-Mar-00 0.4022 15-May-00 -1.5708 21-Aug-00 0.3707 11-Sep-00 0.4066 15-Nov-00 0.3568 9-Mar-0l 0.4000 15-May-0l -1.5665 21-Aug-0l 0.3687 10-Sep-0l 0.4066 15-Nov-0l ll-Mar-02 0.3627 0.4000 15-May-02 -1.5665 21-Aug-02 0.3627 9-Sep-02 10.4066 15-Nov-02 0.3627 10-Mar-03 -1.5665 21-Aug-03 0.3647 9-Sep-03 10.3627 9-Mar-04 23-Aug-04 -1.5793 -1.5622 22-Aug-05 21-Aug-06 -21.5622

DFs

Calculation of theta (1)

7 days 13-Jan-00

DF 0.998845

0.993399 0.992409 0.989604 0.978356 0.961440 0.957752 0.946337 0.926345 0.914610 0.897445 0.893942 0.882382 0.862691 0.851952 0.835761 0.832622 0.821553 0.803300 0.778150 0.775236 0.747899 0.723809 0.673259 0.625643

Calculation of theta (2)

7 days

USD cashflow (USD m)

6-Jan-00 -10.1792 8-Feb-00 20.5929 14-Feb-00 -9.9296 2-Mar-OO 0.4022 8-May-00 -1.5708 14-Aug-00 0.3707 4-Sep-00 0.4066 8-Nov-OO 0.3568 2-Mar-0l 0.4000 8-May-0l -1.5665 14-Aug-0l 0.3687 3-Sep-0l 0.4066 8-Nov-0l 0.3627 4-Mar-02 0.4000 8-May-02 -1.5665 14-Aug-02 0.3627 2-Sep-02 10.4066 8-Nov-02 0.3627 3-Mar-03 -1.5665 14-Aug-03 0.3647 2-Sep-03 10.3627 2-Mar-04 16-Aug-04 -1.5793 -1.5622 15-Aug-05 14-Aug-06 -21.5622

DFs 0.994554 0.993564 0.990759 0.979545 0.962670 0.958982 0.947566 0.927571 0.915836 0.898671 0.895168 0.883608 0.863847 0.853109 0.836918 0.833779 0.822709 0.804374 0.779223 0.776310 0.748909 0.724818 0.674208 0.626538

PV = -0.2844

PV = -0.2831

Theta = -0.0003

Theta = -0.00127

that case, a payer's swap was being hedged by purchasing a bond; the money used to buy the bond was borrowed under a repo agreement. The carry cost over a period of time is: -thetaswan + where thetaswap has been arbitrarily defined for a receiver's. For the usual positive curve, thetaswap is therefore positive.

8.10 RISK MANAGEMENT OF IR OPTION PORTFOLIOS The risk management of swap portfolios has already been discussed in some detail in the previous sections. Many of the techniques such as delta and delta-gamma hedging are equally applicable to options as well, but there are also some additional complications such as volatility risk. In this section, we will describe some of the practical problems involved in the hedging of IR option portfolios, and introduce some additional techniques.

363

Traditional Market Risk Management 0.25

5 mo. option 51/2 year option 0.00 -50

-40

-30

20

-10

0

10

20

30

40

50

Percentage away from ATM (%)

Figure 8.5

Caplet delta 0

Consider a single caplet, its price is given by: C = P x dfT x {F(r,T) x

-Kx

N(d2)} x (T - r)

The classic formulae for the "greeks" may easily be calculated. For example, we can estimate the option delta by differentiating with respect to F(t,T) which gives: <50 ^Px

df7 x N(ct}) x

(T-i)

Below the delta has been estimated for two options, one short and one long, over a range of strikes from —50% to +50% of the prevailing forward rate using the current market data out of 4 January 2000, as shown in Figure 8.5. Note that the delta does not range from 0 (for OTM options) through to 1 (for ITM options) as in the classical B&S formula because of the discounting and the tenor terms. The delta graph for the short-dated option is that typically found in most books on options. However many IR options are long-dated, and the behaviour of their delta is perhaps less intuitive. For example the impact of the discount term is much greater in the long option, so that delta reaches a lower maximum. Actually, one should be more careful because, under the usual caplet convention of fixing at the start of the forward period and paying at the end, the discount factor dfr is also a function of F(r,T). Writing dfr — df T /[l + F(r,7) x (T - T)] and differentiating we get: ddfr/<9F(T,T) = QF x dfr

where QF = -(T - r)/[l + F(t,7) x (T - T)]

Therefore d1 — Qf x C, and total delta is given by:

3 = P x dfr x N(di) x(T-

T) + Qf x C

This additional term has a very small effect, reducing the delta typically by less than 1 % except for heavily ITM very long-dated options, as shown in Table 8.8, and is generally ignorable (and will be in the ensuing discussion). In practice, it is often useful with long-dated options to understand how delta and other parameters will change through time. This enables the trader to anticipate how the hedge will have to be rebalanced if rates do not shift, but simply time passes. Figure 8.6 shows the impact of time on three options; each started out with a maturity of just over 6.5 years,

364

Swaps and Other Derivatives

Table 8.8 Calculating deltas for S-T (5 mo.) and L-T (5/$ yr) options Percentage ATM -50.00% -45.00% -40.00% -35.00% -30.00% -25.00% -20.00% -15.00% -10.00% -5.00% 0.00% 5.00% 10.00% 15.00% 20.00% 25.00% 30.00% 35.00% 40.00% 45.00% 50.00%

Delta 0

Delta 0 + 1

Percentage difference

S-T option

L-T option

S-T option

L-T option

S-T option

L-T option

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.125 0.225 0.246 0.247 0.247 0.247 0.247 0.247 0.247 0.247 0.247

0.006 0.011 0.017 0.025 0.035 0.045 0.056 0.068 0.079 0.090 0.100 0.109 0.118 0.126 0.133 0.139 0.144 0.149 0.153 0.157 0.160

0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.000 0.020 0.125 0.225 0.245 0.246 0.246 0.246 0.246 0.245 0.245 0.245 0.245

0.006 0.011 0.017 0.025 0.035 0.045 0.056 0.067 0.079 0.089 0.099 0.109 0.117 0.125 0.132 0.138 0.143 0.148 0.152 0.155 0.158

_ -0.03% -0.01% -0.01% -0.01% -0.02% -0.04% -0.09% -0.15% -0.23% -0.31% -0.38% -0.46% -0.54% -0.61% -0.69% -0.76%

-0.11% -0.14% -0.16% -0.19% -0.22% -0.25% -0.28% -0.32% -0.36% -0.40% -0.45% -0.50% -0.55% -0.61% -0.66% -0.72% -0.79% -0.85% -0.92% -0.99% -1.06%

0.25 ATM 75% ITM 125%OTM

0.20 0.15 0.10 0.5

Maturity (years) 0.0

Figure 8.6 Temporal delta and the strikes were set to be ATM, 75% of ATM and 125% of ATM respectively. As time passes, we assume that the implied forward rate does not shift but its volatility changes according to the forward volatility curve and the discount factor rolls off. As we can see, at short maturities the deltas are as expected heading to 50%, 100% and 0% respectively. However, when the options have long maturities, the deltas are much more similar. The long-dated heavily OTM option does not have a zero delta, because there is ample time for the option to move into the money, and therefore it requires almost as much delta hedging as the other options.

365

Traditional Market Risk Management 50 45 40 35 30 25 20 15 10

3.0

--2.5

+ 2.0 1.5

±

5 0

-50

1.0

—- 5 mo. option — 51/2 year option

0.5 0.0

-40

-20

-30

-10

0

10

20

30

40

50

Percentage away from ATM (%) Figure 8.7

Caplet gamma (scaled by 10,000)

The caplet gamma is given by: 70 = dfr x N'(d{) x(T-

r)/F(r,T)/(a VT) where N'(x) = exp{-ijr}/>/27r

This ignores the impact on the discounting as being negligible. The result as the options move from OTM to ITM is shown in Figure 8.7; the LH axis refers to the short option, the RH one to the long option. Gamma increases quite significantly over a short range as the maturity shortens, reflecting of course the increasing steepness in the delta curve. It is also virtually symmetric around ATM, whereas for long-dated options gamma is higher for OTM options. This suggests that dynamic delta hedging of long options is likely to be relatively successful. Vega, or the sensitivity with respect to the forward volatility, is given by: v = dfr x N ' ( d } ) x(T-r)x F(i,T) x VT This is graphed in Figure 8.8, again with the LH axis referring to the short option, and the RH one the long option. Changes in volatility, as one might expect, are far more significant for long-dated options, and slightly higher for OTM options.

d-60

40

5 mo. option 51/2 year option -50

-40

-30

-20

-10

0

10

20

Percentage away from ATM (%) Figure 8.8

Caplet vega (scaled by 10,000)

30

40

20 50

366

Swaps and Other Derivatives

The above discussion has concentrated on the behaviour of a single caplet. This of course is not terribly useful in practice, and we really should concentrate on the behaviour of multi-option caps and option portfolios. We will therefore use the following small option portfolio: Settlement date:

6-Jan-00

Buy/sell Start Principal ($ million) Reference rate Strike Maturity Last fixing Last fixing

Capl Sold 27-Aug-99

Cap 2 Sold 07-Jun-99

Floor 1 Bought 06-Dec-99

100

75

100

3mo. Libor 7.00% 7 years 29-Nov-99 6.15%

6mo. Libor 7.25% 8 years 07-Dec-99 6.35%

3mo. Libor 7.00% 10 years no fixing yet

The mark-to-market value of the portfolio is —$3,631,237 based upon the current forward interest rate and volatilities curves — see Worksheet 8.10 (Option Portfolio) on the CD for details. The next step is to estimate the sensitivity of the portfolio to changes in the two curves. Unlike the above, we are no longer going to consider the individual forward rates underlying each option, but will work with the two market curves themselves. Each forward interest rate and volatility was treated as independent; hence sensitivity vectors are created as shown in Table 8.9. The gamma vector is simply defined as (PVBP+ - PVBP~), and the vega estimated by perturbing each forward volatility by 1 bp upwards8. It would be feasible to dynamically delta hedge against shifts in the forward and volatility curves by creating equivalences and rebalancing, very much as described in Section 7.11. However we will use the opportunity to hedge the portfolio more robustly, and introduce some further techniques as follows. We are going to use a mixture of generic swaps and caps9 as shown below, the equivalent sensitivity vectors are calculated in the same worksheet: Generic hedging instruments (nominal principal $1 million) Swap 2 Swap 3 Swap 1 7 10 Maturity (yrs) 3 7.225% 7.335% ANN Act/360 Current rate 7.025% Cap 2 Cap 3 Cap 1 5 10 Maturity 3 7.003% 7. 177% against 3mo. Libor ATM strike 6.922% The strikes were calculated to be ATM forward quarterly swaps

There are not enough instruments for gridpoint hedging, even if one wanted to, and full yield curve hedging seems to be too gross. Therefore we will maturity band the sensitivity vectors into 0–3 years, 3–7 years and 7-10 years as shown in Table 8.10. 8

A detailed explanation of how the production of these sensitivities may be automated is provided in the Sensitivities worksheet on the CD. 9 Exchange-traded instruments such as deposit futures and short options could also be used in these methods using stack and strip concepts.

Traditional Market Risk Management

367

Table 8.9 Sensitivities of option portfolio to gridpoint changes in 3mo. forward rates and 3mo. forward volatilities No.

PVBP4

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40

-819 -1,968 -2,890 -2.716 -2,781 -2,916 -3,123 -2,970 -2,924 -2,889 -2,865 -2,814 -2,760 -2,722 -2,686 -2,675 -2,605 -2,568 -2,556 -2,518 -2,423 -2,415 -2,405 -2,376 -2,297 -2,292 -1,853 -1,287 -1,183 -1,204 -1,215 -972 -625 -621 -612 -619 -608 -422 0 0

PVBP"

-819 -1,967 -2,886 -2,715 -2,778 -2,914 -3,121 -2,969 -2,922 -2,888 -2,863 -2,814 -2,759 -2,721 -2,685 -2,675 -2,604 -2,568 -2,555 -2,517 -2,422 -2,415 -2,404 -2,375 -2,297 -2,292 -1,853 -1,287 -1,183 -1,204 -1,216 -972 -625 -621 611 -619 -607 -422 0 0

Gamma

Vega

0.06 -1.61 -3.88 -1.58 -3.58 -1.68 -2.57 -1.05 -1.94 -0.85 -1.45 -0.65 -1.24 -0.58 -0.98 -0.40 -0.92 -0.38 -0.87 -0.38 -0.85 -0.36 -0.76 -0.33 -0.68 -0.30 -0.26 0.77 0.34 0.68 0.37 0.40 -0.64 -0.35 -0.60 -0.33 -0.53 -0.25 0.00 0.00

0.05 -9.17 -18.87 -29.41 -56.15 -36.68 -78.36 -43.88 -81.76 -49.26 -93.00 -45.05 -98.49 -47.45 -106.25 -49.32 -106.65 -52.71 -111.36 -51.97 -109.35 -55.88 -112.58 -55.29 -112.78 -54.14 -68.91 56.29 -0.34 55.87 -2.48 15.17 -112.24 -51.20 -115.19 -52.19 -113.38 0.00 0.00 0.00

Suppose we create a super portfolio SP = {OP + Y^insi x $i + Y^ina x Q} from the option portfolio plus amounts of the hedging instruments. The resulting sensitivities of the SP in each band k are given by: delta: gamma: vega:

<5SP<jt = <$OP,A +

-ws/

x

<W +

/z°

x

^Ci-k

yspjc = yopk + ^.nsi x ysijc + J^.HO x 7a,k ySPi* = VOPJ( + T.%/ x vsijf + TV«o- x

y

a*

for k

~ *' 2l

3

368

Swaps and Other Derivatives

Table 8.10 Banded sensitivities of portfolio and hedges Option portfolio

Swap 1

Swap 2

Swap 3

Capl

Cap 2

Cap 3

-31,676 -38,438 -8,081

-280 0 0

-279 -291 0

-279 -289 -167

116 0 0

105 88 0

81 148 98

gamma Band 1: 0–3 years Band 2: 3-7 years Band 3: 7-10 years

-21 -9 -1

0 0 0

0 0 0

0 0 0

1 0 0

1 0 0

1 0 0

vega Band 1: 0–3 years Band 2: 3-7 years Band 3: 7-10 years

-542 -1,137 -376

0 0 0

0 0 0

0 0 0

7 0 0

7 8 0

7 17

IRrisk delta Band 1: 0–3 years Band 2: 3-7 years Band 3: 7–10 years

13

For hedging purposes we would like to select the ns such that the net sensitivities xSP,k = 0 for all k and where x = [6, y, v}. However we have nine equations but only six hedging instruments, so it is unlikely we can achieve this, and will adopt a robust approach using linear programming10. Let us create the following expressions: Sx x xspk + uxk — vxjc = 0

for all k and .Y

where uxk and vxk are non-negative variables. Hence the sum [uxjc + vxk] measures the absolute distance of xspk from zero. Sx is simply a scaling constant to ensure that the magnitudes of the sensitivities are similar — see below. The objective therefore is to minimize the penalty function ^2xk{wu,x,k x uxjc + H'V xlc x vxjc] where wuxk and \\\ .xk are optional (usually) positive weights that may be used to emphasize the importance of making sure that certain of the sensitivities are reduced to zero; for example, we might argue that it is essential that SP is delta-neutral, hence wu^js and u\. ^ could be an order of magnitude greater than the other weights. Positive gamma may well be acceptable; in this case, H'M.. *. would be positive, but \\\. ...k zero or even negative. Worksheet 8.11 is built in effectively the following steps: 1 . guess the hedging amounts n$j and ncf, 2. calculate the net sensitivities .XSP,A: of the SP; 3. calculate the u and v-variables using the goal expressions Sxxxspj + uXfk— v.v,*= 0; (note: as both u and v have to be ^0, one will be equal to zero and the other 4. calculate the value of the penalty function using the u—vweights. Then, using the Solver, change the hedging amounts to minimize the penalty function. For the n-s to play their role of controlling the relative importance of the various goals, the MS and vs must be of similar magnitudes — hence the scaling 5s. The worksheet has 10

Strictly speaking, "goal" programming.

P 0-

Worksheet 8.11 Plain goal programming 1. Estimat e the Hedging amounts Swaps Caps 1 2 3

16.61 -76.09 _44.48

-7.39 28.62 6.51

2. Set up t he equations SP sensithdties

Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10years

delta

gamma

vega

0.00 0.00 0.00

0.00 0.00 0.00

-335.91 -785.66 -288.38

0.00 0.00 0.00

0.00 0.00 0.00

0.49 1.15 0.42

V-variables Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10 years

0.00 0.00 0.00

vega

1.00 1.00 1.00

2.00 2.00 2.00

1.00 1.00 1.00

2.00 2.00 2.00

1.00 1.00 1.00

Weights

0.00 0.00 0.00

0.00 0.00 0.00

4. Weighted net sensitivities

0.00 0.00 0.00

gamma

3b. Define the GP weights Weights

3a. Define the GP variables U-variables Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10 years

delta

0.00 0.00 0.00

0.00 0.00 0.00

Penalty function =

2.06

1.00 1.00 1.00

Swaps and Other Derivatives

370

used Sx = (1/Vop.v) where ^Op.v is the absolute average sensitivity of the option portfolio with respect to x over all the buckets. The minimum value of the penalty function is 2.06; because it is positive, not all the net sensitivities of the SP are zero, as may be seen in the box below: Net sensitivities Band 1: 0–3 years Band 2: 3-7 years Band 3: 7-10 years

delta 0.00 0.00 0.00

gamma 0.00 0.00 0.00

vega -335.91 -785.66 -288.38

Whilst the delta and gamma net sensitivities are zero, vega is not because: (a) the penalty weights on gamma were two times greater than on vega. if the weights were switched, then the net vega sensitivities would become zero; (b) the hedging swaps will ensure a feasible delta hedge because they only possess delta. The resulting hedge is shown in the Hedging Portfolios box below. An alternative formulation is to adopt a minimax approach11, i.e. minimize /. defined as max{n- Mxk x uxk + wvxk x v v/t } 12 . Unlike the first formulation, which will try to achieve some of the goals whilst leaving others unsatisfied, this one attempts to satisfy each goal equally. Worksheet 8.12 has been formulated in a slightly more complex fashion, combining the two approaches. The penalty function is:

where: /. — {wuxk x uxk + H' V x k x v x k } ^ 0

for all .Y = {y,u} and k

This ensures that the net delta sensitivity is zero if possible, and that any remaining infeasibility is spread over both the gamma and vega sensitivities, as shown in the box below: Band 1: 0–3 years Band 2: 3-7 years Band 3: 7-10 years

delta 0.00 0.00 0.00

gamma 0.00 4.32 4.32

vega -323.80 -587.08 0.57

The resulting hedges are shown in the box below: Hedging Portfolios (expressed in $m of NPA) Swap 1 Swap 2 Swap 3 Cap 1 Cap 2 Cap 3 11 12

Hedge

Hedge

GP 16.61 -76.09 -44.48 -7.39 28.62 6.51

11.04 -84.16 -31.85 -5.62 7.32 28.00

RP

Sometimes known as "robust" programming. Which may be represented within a linear framework as /. — [wu fj. x wxt + w, xjl x r v t }

0 for all x and k.

1. Estimate the Hedging amounts Swaps Caps 1 2 3

11.04 -84.16 -31.85

-5.62 7.32 28.00

2. Set up ttle equations SP sensitiv ties

Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10 years

delta

gamma

vega

0.00 0.00 0.00

0.00 4.32 4.32

-323.80 -587.08 0.57

0.00 0.00 0.00

0.00 0.00 0.00

0.47 0.86 0.00

V-variables Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10 years

0.00 0.00 0.00

vega

1.00 1.00 1.00

2.00 2.00 2.00

1.00 1.00 1.00

2.00 2.00 2.00

1.00 1.00 1.00

Lambda = Lambda constraints

0.86

0.86 0.00 0.00

0.38 0.00 0.86

Penalty function =

0.86

Weights

0.00 0.43 0.43

0.00 0.00 0.00

4. Weighted net sensitivities

0.00 0.00 0.00

gamma

3b. Define the RP weights Weights

3a. Define the RP variables U-variables Band 1: 0–3 years Band 2: 3–7 years Band 3: 7–10 years

delta

Traditional Market Risk Management

Worksheet 8.12 Robust programming

0.00 0.00 0.00

0.00 0.00 0.00

1.00 1.00 1.00

371

372

Swaps and Other Derivatives

and make intuitive sense. The portfolio was a net seller of caps, hence the hedge should consist of net long caps plus swap payer's (as indicated by the negative NPA). The effectiveness of these hedges was then tested by imposing parallel shifts on both the forward rate and forward volatility curves simultaneously in steps of 50 bp up to ±250 bp. The full results of the 121 different scenarios are shown in the worksheet. If we define AFy.r.CT as the change in value of portfolio Y = {OP, GP hedge, RP hedge} under scenario {r,cr}, then a measure of the total change in value over all scenarios is AFr = {^ r(7 (AF rrcr ) 2 } 1/2 . The effectiveness of the hedge may be measured by [l-AF y /A> O P ]: " for GP: for RP:

effectiveness = 92.42% effectiveness = 93.74%

Both formulations would appear to provide good hedging over a wide range of scenarios. Instead of using the greeks to build the hedge, where because these are local measures the hedge must get worse for larger movements, an alternative is to use scenario analysis. Suppose a number of scenarios p = 1,2, . . ., TV are created; each scenario consisting of a defined (not necessarily parallel) shift in the forward rate and/or the forward volatility curves. Calculate the change in the value of the option portfolio AOP, and the changes in value of the hedging instruments {AS,, and AC,,} under each scenario, and hence the change in the value of SP: ASP, = I AOP, + ^./i5l- x AS,, + ^.w0

x AC,, 1 for each p =

1

N

As before, the ideal hedge would set ASP, = 0 under each p but this is unlikely to be feasible due to the limitations of the hedging instruments. But, using the same ideas as above, we could create the following equations:

ASP, + up - vp = 0 and then either minimize Y^P(w^,p x UP + H'>./> x VP\ or cre^te another lambda-style robust expression13. The advantage of this type of approach is that it can be manipulated to deal with a wide range of gapping or jump scenarios, which the greek approach is less suited to handle. More exotic options, especially those that involve a discontinuity in the payoff such as digitals and barriers, are considerably more difficult to risk manage. For example, the "greek" characteristics of a digital caplet are quite different to those of an ordinary caplet. As the price of a digital cannot exceed the payout, delta increases as the option moves from OTM to ATM but then drops back towards zero for ITM. This means that gamma is initially positive but then switches to be negative. From above, vega is always positive for a normal caplet because increased volatility increased the chances of a larger payout. If a digital option is OTM, then vega is also positive as expected because this has increased the probability of the option moving into the money. However for an ATM or ITM option, vega is typically negative because low volatility will keep the option ITM whereas high 13 The author first used techniques such as these to hedge IR option portfolios in the late 1980s. At that time Discount Corporation of New York Futures (later subsumed into Dean Witter) used to provide its clients with a service calculating robust exchange-traded hedges for their IR option books based upon similar techniques.

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373

volatility increases the probability that the option will go OTM. As an ATM option approaches maturity, these switches in sign become increasingly extreme which can result in costly and yet ineffective hedging. Thus the simple sensitivities for exotics are themselves relatively unstable, and delta hedging alone is seldom adequate, especially near the discontinuities. If a whole portfolio is to be risk managed, then the discontinuities arising from a single option may have little overall effect, and delta hedging may be appropriate. For the risk management of a small number of options with discontinuities, more robust hedging should be used, possibly using the methods described above, and possibly in combination with the concept of "static" hedging, which has been developed for this situation14.

8.11

HEDGING OF INFLATION SWAPS

This section is included to complement the section on inflation swap pricing, and to demonstrate a further use of the equivalence concepts. There are three sources of market risk in an inflation swap: 1. movement in the inflation rates, which may be hedged with index-linked bonds; 2. movement in the gilt discount rates, which may be hedged with conventional gilts; 3. movement in the Libor rates, used for discounting and possibly estimation, which may be hedged in the usual fashion in the swap book. Only the inflation risk is unique. We can use very similar techniques to those described above to create, for example, a portfolio of index-linked bonds that will delta hedge the inflation risk in a swap portfolio. For example, suppose we have a portfolio consisting of the two swaps described in Chapter 4, and we plan to delta hedge them using the eight index bonds we used to build the forward inflation curve. The first step is to calculate the delta sensitivities due to a small parallel shift in the inflation growth curve — see Box 1 of Worksheet 8.13. The next step is to calculate the delta hedge as: —{matrix of hedge sensitivities}"1 x {vector of swap sensitivities} as shown in Box 2. The effectiveness of the hedge can be demonstrated against ±1% shift in the growth curve as shown in Figure 8.9. The LH axis measures the change in value of the portfolio and the hedge separately, the RH axis the net effect. From a bank's perspective, the cost of carrying such a hedge can be quite expensive. Invariably the bank has to short the indexed bonds, which it will typically obtain on a repo contract. But because the market is relatively illiquid, and there is usually a considerable repo demand for these bonds, the repo rate the bank would earn on the purchase price of the bonds is normally considerably (i.e. 50 bp or so) below Libor.

14 See, for example, Derman et al., "Forever Hedged" and Carr et al., "Static Simplicity", both in Risk, 7, 1994, pp. 39–45 and 45–49 respectively.

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Change in inflation growth rate Figure 8.9

Effectiveness of hedging an inflation swap portfolio

APPENDIX: ANALYSIS OF SWAP CURVES The movement of interest rate curves through time has been intensively studied by many market practitioners; if one can develop a "good" model, then it is likely that safe profitable opportunities can be identified. There is also an extensive academic literature15 and it is not the purpose of this Appendix to replicate it. The objective is to briefly discuss an approach that has been gaining acceptance within the risk management community over the past decade. Assume we have available a set of interest rate curve data X = {xit, / = ! , . . . , « maturities, t = 1, . . ., T time} where xit represents the arithmetic change in rate i from time / — 1 to time t. We want to understand how the curve evolves through time. One way of doing this is to track each one of the ith gridpoints individually, as we did in gridpoint risk management. But this ignores any correlation structure along the curve, and is therefore likely to be inefficient. An alternative is to use the techniques of principal component analysis. Think of a curve at time t as a vector of rate changes x,, i.e. as a single point in an n-dimensional space. PCA transforms these data into Y — AX where Y is also dimension {n x T} and hence A is a square {n x n} matrix. Initially, think of this as simply a rotation of the original X axes into a set of orthogonal (i.e. at right angles) Y axes. We can therefore write yj = a'jX where y, and aj are the jth rows of Y and A respectively. The variance of yj- is V(yj) = ajV(X)a/ where V(X) is X's full covariance matrix. The total variance in the original data is given by ]T\ F(x,), i.e. the sum of the variances corresponding to each gridpoint. The above transformation cannot change the total variance, i.e. we know What we want to do is select the vectors a.(, a'2, . . ., a'n so that V(yl) > V(y7) > . . . > V(yn). Ideally, we want to identify a dimension m m are negligible. This can be done sequentially:

15 See, for example, Anderson et al., "New estimates of the UK real and nominal curves", Bank of England Quarterly Bulletin, Nov 1999, pp. 384–392 or Anderson et al., Estimating and Interpreting the Yield Curve, Wiley, 1997 as relatively modern introductions to both the academic and practitioner literature.

376

Swaps and Other Derivatives Table 8.11 Explanation Currency

USD GBP DEM CHF ITL JPY NLG BEF FRF ESP

Factor 1

Factor 2

Factor 3

Total explanation

3.6 6.9 7.2 4.5 4.6 6.1 6.7 8.5 6.8 6.4

0.7 1.0 1.1 1.0 0.8 1.2 1.2 1.5 0.9 1.4

99.3 98.9 99.1 98.6 99.4 99.0 98.8 98.6 99.3 98.6

95.0% 91.0 90.8 93.1 94.0 91.7 90.9 88.6 91.6 90.1

1. find a1 such that V(y 1 ) is maximized and that a1'a, = 1 (this is just a scaling constraint); 2. find a2 such that V(y2) is maximized and that a2a2 = 1 and a'1a2 = 0 (this says that the Y axes must be orthogonal); 3. find a3 such that V(y3) is maximized and that a3a3 = 1 and a'1a3 = 0 and a2a3 = 0. This may be thought of as projecting high dimensionality data onto a much smaller set of dimensions. The "explanation" of each factor is {^(y/)/^/^(x/)l and the total explanation cannot of course exceed 1. Niffikeer16 has reported the results of a PCA on a range of zero coupon swap rates over 10 different currencies: see Table 8.11. In summary, the total explanation using only three factors is uniformly high, over 98%. But there is a problem with such results, namely how can they be used in practice? The difficulty is that the a vectors will have no precise interpretation. The vectors for the USD data are shown below: Maturity

2yr 3yr 4yr 5yr 6yr 7yr 8yr 9yr l0yr

Vector 1 0.94210 0.96519 0.97917 0.98689 0.99519 0.98536 0.98606 0.97682 0.95590

Vector 3 Vector 2 0.30335 0.11583 0.24357 0.04931 0.14803 -0.05694 0.06814 -0.10037 -0.01824 -0.08994 –0.11355 -0.07781 -0.15837 0.01315 0.05592 -0.20651 -0.25644 0.12827

These vectors are usually "interpreted" as representing a parallel shift, a rotation and a change in curvature respectively. However they are not exactly these curve movements.

16 Niffikeer et al., "A synthetic factor approach to the estimation of VaR of a portfolio of IRS", Journal of Banking & Finance, 24, 2000, pp. 1903–1932.

Traditional Market Risk Management

377

Niffikeer proceeds to fit synthetic orthogonal factors, i.e. ones that are precisely defined in advance. A simple example would be to break the curve into three buckets and represent the movements as: Parallel Rotational Curvature

Bucket 1 +1 +1 +1

Bucket 2 +1 0 _2

Bucket 3 +1 _1 +1

The synthetic parallel factor is fitted to the data first, then the rotational factor to the residuals, and finally the curvature factor. Obviously the level of explanation with these synthetic factors cannot be as high as from the PCA, but the reduction is only about 1 % on average across all currencies. This analysis provides a strong rationale for curve risk management, although the explanation for the forward rate curve is very much lower. Hedging against parallel shifts alone is likely to remove some 90% of the fluctuation in valuation. PCA is also becoming widely used in VaR (see Chapter 9) to reduce substantially the number of risk factors. However, like all statistical techniques, the analysis is of course very dependent upon the historic data being a good representation of the future. An over-reliance on such findings to hedge against unstable market movements is extremely unwise, as many practitioners have found to their cost.

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Imperfect Risk Management

OBJECTIVE This chapter discusses the more recent risk management techniques that have been developed over the past decade, recognizing that markets are not perfect and that hedges sometimes have to be constructed from less-than-perfect instruments. Value-at-risk approaches are becoming standard in many banks, and the chapter will explore some of the practical issues that arise from its implementation. The chapter starts with a very simple transaction, and demonstrates how one-factor VaR may be calculated in various ways. The example is then extended to a two-factor model and eventually multifactor models. The last section here also looks at the relationship between delta hedging, minimum VaR hedging and hedge effectiveness. Some background work is then discussed, such as the selection of risk factors, alternative ways of implementing VaR in practice, and in particular practical problems of estimating volatility and correlation. A small FX option portfolio is then introduced, and its delta VaR calculated. The chapter proceeds to examine both historic and Monte Carlo simulation, and applies these techniques to this portfolio, comparing the various results. There are however various difficulties in implementing the simulation methods, and ways of improving and speeding them up are discussed, such as extreme value theory, weighted historic simulation and sampling strategies. The chapter finishes with three final topics. First, as trading spreads is becoming more common, especially within Europe, the calculation of spread VaR from market data is demonstrated. The calculation of equity VaR with and without the use of indices is then discussed. Finally, possible strategies for stress testing are described.

9.1

INTRODUCTION

The techniques described in the previous chapter are often called traditional or "desk-top" risk management. The latter name refers to the fact that individual trading desks usually only enter into a limited range of risks, and the more traditional risk measurement methods are perfectly adequate in this situation. However we made some quite extreme correlation assumptions about interest rate movements, namely that either the curve exhibited zero or perfect correlation. Obviously in practice, the truth lies somewhere in between. But these assumptions are frequently acceptable provided that the movement in interest rates is relatively small. Related to this statement, although not identical, is that the risk management time horizon is relatively short. Yield curve risk management used the bond market to hedge a swap portfolio, substituting, as was commented at the time, basis risk for absolute risk. It was assumed that this risk was zero, i.e. that the swap spread remained constant. Again, in practice, this is not true.

380

Swaps and Other Derivatives

In the mid 1980s, concern at the levels of unregulated risks banks were entering into as a result of the exponentially expanding off-balance sheet derivatives markets led bank regulators to introduce the fundamental concept of obliging banks to allocate a certain amount of capital against some overall measures of risk. The regulations started by defining the amount of capital required to support a measured amount of credit exposure. By the early 1990s, attention had moved on to market risk exposure, especially in the trading activities. The important point is that both banks and their regulators realized that these capitalbased approaches required measurement techniques that spanned a range of different activities, and not just a single trading desk. After conducting a number of simulations, a very crude "static framework" methodology was suggested by the regulators, first for credit risk and then for market risk. This basically meant that a bank would enter very high level summaries of their activities into a black box, and the output was the amount of regulatory capital required to sustain those activities. Intuitively, banks felt that if they could develop more precise methods of defining their risks, and persuade the regulators of this increased precision, then the levels of required capital would be reduced to their obvious benefit. The measurement of market risk was generally deemed to be an order of magnitude (at least!!) easier than modelling credit risk. By 1995 the regulators had been persuaded to permit "internal", i.e. individual bank-developed, market risk measurement models which would then be used to derive the capital required. The permission was granted under strict approval conditions1. To overcome the various shortcomings described above, a family of statistical techniques has been developed. Probably the best known are the "Value-at-Risk" (VaR) techniques, which have become the standard approaches for the modelling of bankwide market risk, and increasingly credit risk as well. This chapter will describe and contrast the major methods for calculating market risk VaR, and will finish with a brief discussion on credit-related topics. Because VaR is designed to measure market risks arising from a wide range of activities, more complex portfolios will be used as examples than in previous chapters. Is VaR a completely novel approach, or is it a natural development out of the desk-top approaches previously described? This is a question that will be discussed during this chapter.

9.2 A VERY SIMPLE EXAMPLE It is Sam on 4 January 2000. A bank dealer has just received a call from a customer to borrow USD 100 million for 12 months. He quoted 6^| which was accepted, and he confirms: $100 million to be paid by the bank on 6 January $100 million plus interest = $100m x (1 + 6.59375% x (8 Jan 2001 - 6 Jan 2000)/360) = $106,740,278 to be received from the customer on 8 January 2001 1

Credit risk regulatory capital, at the time of writing, cannot be determined using internal models, although many banks are using such models for their own purposes. The new Accord, proposed for 2004, does go some way towards permitting internal credit models.

Imperfect Risk Management

381

A bank would routinely value each transaction to the current market rates. The current mark-to-market value of the money to be received in 1 year's time would be calculated as: $106,740,278/(1 +6.59375% x 1.022) = $100 million which of course matches the money paid out — see column [1] of Worksheet 9.1. (Bid-offer spreads have been ignored in this discussion for ease of explanation.) The bank now has a market risk. If interest rates rise, it would lose value. The dealer can easily estimate the impact of moving rates by calculating the sensitivity of the transaction. For example, the PVBP is estimated by increasing the rate by 1 bp and revaluing, we get: $106,740,278/0 + 6.60375% x 1.022) = $99,990,424 i.e. a loss in value of $9,576, as shown in column [2]. So much for traditional measures, but it does not address an important question, namely "how much might rates move in one day?" It is common to analyse rate movements on a "return" or percentage basis, i.e. what was the percentage movement from one day to the next, instead of an absolute basis measured by simply taking the difference in rates. The reason for this is that a 50 bp absolute shift when rates are 2% is far less likely than the same shift when rates are 10%. Percentage shifts on the other hand are by and large independent of the current level of rates, and therefore consistently comparable. There are also some pragmatic theoretical reasons, such as the widely used assumption of returns being normally distributed in option pricing. The daily behaviour over the last 2 years (500 business days) of this cash rate is shown in Figure 9.1. The graph shows that there have been some large movements; one day experienced over a 10% rise which would have caused a loss of over six hundred thousand dollars!! However such days are very infrequent, as can be seen from the histogram Figure 9.2. The histogram was created by bucketing the movements into 50 bp bins. Consider one bucket, for example the one centred on 2%. Increases in the range from 1.75% to 2.25% occurred 16 times in the past 500 days or 3.2% of the time. Applying this average movement to the current level of rate, i.e. from 6.59375% to 6.59375%*(1 + 2%) = 6.72563%, would result in a loss of $126,134—see column [3] for confirmation. Of course a bigger movement may occur. For example, we find that on 40 days (or 8% of the time), there was an increase of 1.75% or larger over one day. Note that now we do not use the centre of the bucket, but the lower edge to indicate the size of movement, because we are interested in a particular increase or worse. This movement would result in a loss of $110,384 — as confirmed in column [4], On the assumption that these events are a good indication of the future, a VaR statement would be: "There is an 8% chance that the bank could lose $110,384 or more in one day due to market movements on the current deal."

This statement would define the 92% VaR for the transaction, i.e. conventionally quoted as (1 —probability). It must contain three elements: 1. the size of potential loss

382

Swaps and Other Derivatives

Worksheet 9.1 1. Simple sensitivity calculations Basic market data 4-Jan-00 6-Jan-00 13-Jan-00 7-Feb-OO 6-Apr-00 6-Jul-00 8-Jan-0l

7 32 91 182 368

0.02 0.09 0.25 0.51 1.02

5.53125% 5.81250% 6.03125% 6.21875% 6.59375%

Calculations for Introduction to VaR

[1] r=

[2]

[3]

[4]

[5]

6.59375%

6.59375%

6.59375%

6.59375%

6.59375%

P(mUSD)= d= CF(mUSD)= shift r' =

100 1.022 106.740278 0% 6.59375%

100 1.022 106.740278 1 bp 6.60375%

100 1.022 106.740278 2% 6.72563%

100 1.022 106.740278 1.75% 6.70914%

100 1.022 106.740278 10% 7.25313%

PV(mUSD)=

100

99.990424

99.873866

99.889616

99.372497

110,384

- 627.503

delta (m USD) =

-95.767

perturb (USD) =

-9,576

shift (bp) = PVBP approximation (USD) delta approx (USD) gamma (m USD) = delta/gamma approx (USD)

–126,134

–

13.19

11.54

-126,281

–110.496

–126,293

–110.506

-126,134

-110,384

183.427

383

Imperfect Risk Management

Time

Figure 9.1

Daily percentage movement in 12mo. USD cash rate

45 40 35 30 25 20 15 10 5 0 -

7

-

6

-

5

-

4

-

3

-

2

Figure 9.2

-

1

0

1

2

3

4

Histogram of daily movements

5

9

10

384

Swaps and Other Derivatives

2. the probability of this loss or worse 3. the time horizon and is incomplete if it is missing any element. Note the difference between traditional sensitivity measures and VaR statements. The former assume a predefined movement in the rate, in this case 1 bp, whereas the latter have additionally attempted to make a statistical statement about the likely size of movement. It is important to recognize that VaR extends the PVBP analysis, and does not supplant it. In calculating the VaR above, we used a "full valuation" approach, i.e. we shifted the rate by 1.75% and then revalued the trade. An alternative is to use a PVBP approximation as discussed in the previous chapter: Avalue

PVBP x shift in rates (bp)

For example, the 1.75% movement in rates was equivalent to: Ar = 6.59375% x(l + 1.75%) - 6.59375% =6.59375% x 1.75% = 0.0011539 or 11.54bp Therefore: Avalue

$9,576 x 11.54 bp = $110,496

The difference between the exact amount of $110,384 and the first-order approximation of $110,496 is due to second-order effects. The above expressions may be summarised as: Avalue ~ 10,000 x $9,576 x 6.59375% x 1.75% = 10,000 x PVBP x r x (% change in r) Based on the historic return data, we find: daily mean Qi) = 0.019% and daily standard deviation (a) = 1.290% The mean is often called the "drift" or "trend", whilst the standard deviation is of course simply the daily volatility of the rate. The percentage movement in the rate of 1.75% may be rewritten in terms of multiples of the volatility, i.e. 1.75% = 1.357 x a. The number 1.357 is called the "multiplier" (we will use the symbol k to represent it). Putting all this together, the PVBP VaR (also known as "parametric VaR") may be estimated as: VaR = 10,000 x PVBP x r x k x a = 10,000 x $9,576 x 6.59375% x 1.357 x 1.290% = $110,496 This result has been replicated more formally in Box 9.1. Interest rates have trended slightly upwards over this period, which will adversely affect the deal if this continued. However short-term trends are notoriously unstable, and it is common practice to ignore them, as we have above, when calculating VaR over short-time horizons. The trend is of course reflected in the 8% probability, as it is inherent in the historic data used to derive the histogram. We estimated there was an 8% probability of suffering this loss or worse. Obviously, as the multiplier is increased, the VaR also increases but the probability decreases. The relationship between the size of the multiplier and the probability is defined by the numerical histogram, and we have made no other distributional assumptions. Let us now assume that the returns follow a normal distribution. Using the delta approximation, this means that Avalue is also normally distributed. As we know the

385

Imperfect Risk Management

Box 9,1

Proof of one-factor delta VaR

To formalize what we have done so far, assume that there is a change Ar in the rate r. The change in value is approximately given by Taylor's theorem: Avalue ~ <9value/<9r x Ar = dvalue/dr x r x (Ar/r) where (Ar/r) is a return. As [<9value/<9r x r] is currently known: st devfAvalue} % dvalue/dr x r x st dev(Ar/r) = dva\ue/dr x r x & Therefore, if we characterize VaR by k multiples of st dev{Avalue}, we can write: VaR = <9value/$r x r x k x a The value of the transaction may be written as: value = $106,740,278/(1 + rd) where r is the rate and d the daycount fraction (365/360). Differentiating, we get: <9value/<9r = -$106,740,278 x d/(l + rd)2 =–$95,767,243 Note that this analytic delta is approximately 10,000 x PVBP. The delta VaR is therefore: delta VaR = -$95,767,243 x 6.59375% x 1.357 x 1.290% = $110,506 i.e. very similar to the previous estimate. standard deviation of the distribution2, the relationship between the multiplier and the probability is defined theoretically as follows: Multiplier 1 1.28 1.65 2 2.33 3

Probability (%) 15.9 10.0 5.0 2.3 1.0 0.1

Most practitioners start with the probability — 5% and 1% are the most popular — which then defines the multiplier. For example, to calculate the 95% VaR: VaR = 10,000 x $9,576 x 6.59375% x 1.65 x 1.290% = $134,440 per day Notice that strictly speaking this is negative as the PVBP is negative for an increase in rate. But conventionally VaRs are always quoted as a positive number even though they refer to a potential loss. Normality is a very common assumption, but frequently criticized. In particular, it has been observed that in practice the probability of large movements is considerably underestimated by a normal distribution; this is the phenomenon known as "fat tails". As "We have assumed the trend to be zero.

386

Swaps and Other Derivatives 50 40 30 20

-7

-5

-3

-1

Figure 9.3 Comparing the numerical results with a normal distribution

we can see from Figure 9.3, not only do the numerical results have longer, i.e. fatter tails, than the normal distribution, but also more peaked as well (ie. it is leptokurtic). If we felt that the normal was inappropriate, then we could work with the numeric distribution itself, or with a different theoretical distribution which may fit the data better. Student-t is a common choice, but it does have an additional parameter which has to be fitted. The 95% VaR may be estimated from the numerical results by, for example, linearly interpolating the historic cumulative frequency. A percentage movement of 2.219% corresponds to a 5% probability, which in turn gives a 95% VaR of $140,093. This is higher than the normal estimate, as expected.

9.3 A VERY SIMPLE EXAMPLE EXTENDED The above example calculated the VaR for a simple USD transaction. Let us assume that we are now a German bank, implying that all transactions have to be valued in DMs. This introduces the additional complication of movement in the exchange rate. The spot rate on 4 January 2000 was 1.8 DM per USD with a daily volatility of 2%. The value of the transaction in DMs is: value = {$106,740,278/0 + rd)} x S = DM180 million where S is the spot exchange rate. There are now two risk factors, r and 5, and we can calculate the 95% VaR with respect to each separately as above: 1. PVBPr = -DM17,236 (this is the USD PVBPxS) per 1 USD bp PVBPs =$10,000 per 1 bp shift in S (This PVBP is defined as A value/ AS which has dimensions of DM/(DM/USD) = USD) 2. VaR r = -10,000 x 17,236 x 6.59375% x 1.65 x 1.29% = -DM241.992 VaR s = 10,000 x 10,000 x 1.8 x 1.65 x 2% = DM5,940,000 VaRs is much higher than VaR r , despite PVBPr being greater than PVBP s . In part this is because the spot volatility is higher than the rate volatility, but the main reason is because PVBP works in terms of absolute movements, whereas VaR works in terms of percentage movements. A 1 % move in 1.8 is a much greater absolute movement than a 1 % shift in the

387

Imperfect Risk Management

rate. Note that the negative sign on VaR r has been retained, although obviously for reporting purposes it would be ignored. VaR was introduced as a risk measurement technique that can span a range of different activities, in contrast to the simpler PVBP approaches. Note that the units of the PVBPs are different, and therefore not combinable, where the units of the VaRs are consistently money (in this case, DM). However, we need to develop a method of combining together the individual VaRs in some appropriate fashion into a single risk measure. The standard deviation of the overall value of the transaction may be approximated using the result in Box 9.2. Before we do the calculation, what do we expect intuitively? If the interest rate increases, the trade loses value, whilst if the spot rate increases the trade gains in value. If USD interest rates increase then we would expect the dollar to strengthen in the short term, i.e. S to increase. Therefore we anticipate a positive correlation between changes in r and S, and that this correlation should reduce the overall risk. Let us assume prS = 0.5: {VaRTotal}2 = {-241,992}2 + {5,940,000}2 + 2{-241,992} x {5,940,000} x 0.5

Box 9.2

Proof of the two-factor delta VaR

Suppose there is a simultaneous shift Ar and AS in both r and S respectively. As a first-order approximation, the change in value is: Avalue % <9value/<9r x Ar + dvalue/dS x AS = dvalue/dr x r x (Ar/r)+ dvalue/dS x S x (AS/S) where the terms in parentheses are returns. The standard deviation of value may easily be calculated as follows: 1. var(iance)(Avalue) = E{Avalue2} — E{Avalue}2 2. but E{ Avalue} = c?value/<9r x E{ Ar} + <9value/dS x E{ AS} = 0 as we have assumed that all trends are zero, or ignorable 3. E{Avalue2} = [<9value/<9r x r]2 x E{(Ar/r) 2 } + [<9value/dS x S]2 x E{(AS/S)2} +2[<9value/dr x r] x [dvalue/<9S x S] x E{(Ar/r)(AS/S)} 4. var( Avalue) = [<9value/<9r xrx cr,.]2 + [<9value/dS x S x
More generally, if C represents the 2x2 correlation matrix Pr.S

1

then the

above expression may be written as VaR'-C-VaR where VaR is the vector of individual VaRs. This expression gives rise to yet another name for delta VaR, namely VCV.

388

Swaps and Other Derivatives 6.30 6.20 6.10 E, 6.00 | 5.90 * 5.80 *~ 5.70

5.60 -1

0.5

-0.5

Correlation

Figure 9.4 Impact of correlation

VaRTotal = DM5,822,777 per day The total VaR has indeed been reduced from VaRs but notice that the sign of VaRr must be retained in the calculations. Figure 9.4 shows the impact of changing correlation. Some special cases: (a) zero correlation: (b) positive perfect correlation:

{VaRTotal}2 = {VaRr}2 + {VaRs}2 VaRTotal = VaRr + VaRs

A number of banks believe that correlations are too unstable to include in their VaR calculations, and often combine the individual VaRs using one of the above expressions. It is important to appreciate that, like it or not, a correlation assumption is being tacitly made. In summary, what can we conclude? Traditional risk management revolves around sensitivity analysis, namely the potential to lose money for a predefined movement in one or more risk factors. Delta VaR combines sensitivity with volatility, which is a measure of the likely size of movement over a given time horizon, to produce a risk measure that is defined in monetary terms. It then goes further, to associate a multiplier of the volatility with a probability by assuming some underlying (analytic or numeric) distribution. Finally, because the dimension of the risk measures is consistent across all risk factors, they may be approximately combined together using correlation. So VaR started off with a traditional approach, but rapidly extended risk measurement into new directions.

9.4 MULTIFACTOR DELTA VaR The above result may easily be generalized to encompass N factors. If the value of a portfolio P depends upon the current levels of the risk factors xi , i = 1, . . ., N, then as the factors change: A value % £],<5, x AJC, = £,.£,- x jc, x {Ax,/*,} and

where <5, = dvalue/d.v,

389

Imperfect Risk Management

var(Avalue) =

x

Pi / x [&j x xt

x a

/\

Hence {VaRp}2 = VaR' C VaR as before. To illustrate, we will use the USD swap portfolio that was used as a running example in the previous chapter. Amongst other analyses, we calculated its gridpoint sensitivities, i.e. the PVBP with respect to each of the individual market rates used to construct the discount curve, and also the amount of a bond required to delta hedge the portfolio against a parallel shift in the market curve. The size of the delta hedge was 11.82 million nominal principal. The first step is to calculate the gridpoint sensitivities of both the swap portfolio (which we have already done) and the hedging bond, as shown below:

Market rates 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap Syr swap 7yr swap l0yr swap

Market rate sensitivity Bond Swap hedge portfolio (1m nominal) (USD m) 17.33 -0.41 30.92 -1.01 -22.99 2.21 -271.77 1.75 -2,237.29 2.69 -44.96 -2,778.03 -345.37 1,213.15 0.00 8,601.22 0.00 0.00

Each market rate was increased by 1 bp independently, and the change in the value of the portfolio and bond recorded. The results are shown in columns [1] and [2] of the valuation worksheet (not printed but on the CD) and again in the main VaR worksheet for this example. The second stage, given the volatilities and correlations of the individual market rates as shown in Worksheet 9.2, is to calculate the individual VaRs using the usual delta formula. The results are shown in column [5] for the swap portfolio and [6] for 1 million nominal of the bond of Worksheet 9.3. The 95% delta VaRs are then calculated for each, using the correlation matrix, to be $52,289 and $4,002 respectively. The impact of a hedge may be calculated by first constructing new individual VaR values as follows: VaR(i)portfolio & Hedge = VaR(i)Portfoli + hedge ratio X VaR(i) Im Nominal Hedge

for the ith market rate — see column [7] which is using the delta hedge ratio of 11.82 million of the bond. The total VaR has been reduced from $52,289 to $50,554, i.e. only just over 3%. Delta hedging the portfolio has not in fact reduced the overall VaR risk very significantly. We could see if the VaR hedge could be improved. The total VaR is given by: {VaRTotal}2 = {VaRPortfolio +n H VaR Bond } / -C-{VaR Portfolio +n H VaR Bond } for some hedge ratio nH. If we differentiate with respect to nH, and set the result to zero, we get the minimum VaR hedge ratio: nH = —{VaRPortfolio.C-VaRBond}/{VaRBond.C.VaRBond}

Worksheet 9.2 Rate vol (pa) 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap l0yr swap

Rate vol (pd)

20.00% 19.00% 18.00% 17.00% 16.00% 15.00% 14.00% 12.00% 9.00%

.26% .20% .14% .08% .01% 0.95% 0.89% 0.76% 0.57%

Correlation matrix 3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap l0yr swap

3m cash 1 0.7 0.6 0.5 0.4 0.35 0.3 0.2 0.05

6m cash 0.7 1 0.75 0.65 0.55 0.5 0.45 0.35 0.2

12m cash 0.6 0.75 1 0.8 0.7 0.65 0.6 0.5 0.35

2yr swap 0.5 0.65 0.8 1 0.85 0.8 0.75 0.65 0.5

3yr swap 0.4 0.55 0.7 0.85 1 0.9 0.85 0.75 0.6

4yr swap 0.35 0.5 0.65 0.8 0.9 1 0.9 0.8 0.65

5yr swap 0.3 0.45 0.6 0.75 0.85 0.9 1 0.85 0.7

7yr swap 0.2 0.35 0.5 0.65 0.75 0.8 0.85 1 0.8

l0yr swap 0.05 0.2 0.35 0.5 0.6 0.65 0.7 0.8 1

C/5

63 O.

O Er

391

Imperfect Risk Management 80,000 70,000 60,000 .ro 50,000 ~& 40,000 H

30,000 20,000 10,000 6

8

10

12

14

16

18

20

Bond hedge

Figure 9.5 VaR hedging

Column [8] calculates the VaR for this hedge ratio, which results in a VaR reduction of 12.8%. The optimal hedge cannot reduce the total VaR to zero, as we can see in Figure 9.5. The expected reduction in VaR may be calculated by substituting back, i.e.: VaRTotal = VaRPortfolio X y {1 — Pe

where VaR Portfolio is the VaR of the unhedged portfolio and paveraee is defined by ±{VaRPortfolio.C.VaRBond}/{VaRPortfolio x VaRBond}. The worksheet calculates the average correlation to be 0.489, which means that the total VaR can only be reduced to (1 - 0.4892)1/2 = 87.2% of the unhedged portfolio. Perfect hedging can only be achieved in the exceptional circumstance that paverage — 1, i.e. with a perfect correlation matrix as shown in column [9]. Before finishing this section, it may be worth noting as an aside that a closely related style of hedging, usually called "minimum variance hedging", has been widely used in many physical futures markets for a long time. For example, you wish to hedge the price of Norwegian crude oil. There is no directly relevant futures contract, but there are Brent oil futures. By collecting price histories of Norwegian crude and Brent futures, the correlation may be measured and the anticipated effectiveness of the hedge assessed. One of the techniques employed in these markets, which could also be used in the VaR hedging above, is the concept of "big step" correlation. Consider a historic price series; divide it into two series of small movements and large movements by defining some filter. Then take a second series and subdivide that into two; obviously in this case we do not apply the filter again, but simply ensure that the movements on the corresponding dates are included. Because movements in market prices are due to the interaction of many factors, the correlation between the two series of small movements is very often extremely low. However a big movement may have been caused by a single large exogenous event likely to affect both series; the result is that the correlation between the series of large movements is usually very much higher. One approach therefore to the creation of a hedge is to base it solely on the big step correlations, on the grounds that this would provide protection against large movements and accept that

392

Swaps and Other Derivatives

Worksheet 9.3 VaR calculations of portfolio and hedge

Rate

3m cash 6m cash 12m cash 2yr swap 3yr swap 4yr swap 5yr swap 7yr swap l0yr swap

Sensitivity of portfolio (USD)

[1]

17.33 30.92 -22.99 -271.77 -2,237.29 -2,778.03 1,213.15 8,601.22 0.00

Sensitivity of bond (USD/m nominal)

[2]

-0.41 -1.01 2.21 1.75 2.69 -44.96 -345.37 0.00 0.00

VaR multiplier =

1.65

Daily vol (% pa)

Current level of rate

[3] 1.265% 1.202% 1.138% 1.075% 1.012% 0.949% 0.885% 0.759% 0.569%

[4] 6.03125% 6.03125% 6.59375% 6.895% 7.025% 7.085% 7.135% 7.225% 7.335%

Total VaR

VaR reduction Average correlation Anticipated residual risk

393

Imperfect Risk Management

VaR of portfolio

VaR of bond (1m nominal)

218.11 369.73 -284.74 -3,324.29 -26,242.37 -30,809.26 12,645.94 77,820.35 0.00

[6] -5.18 -12.02 27.35 21.44 31.51 -498.57 -3,600.17 0.00 0.00

52,318.68

4,001.60

[5]

Delta hedge 11.820 m of bond to buy

Optimal hedge 6.3894 m of bond to buy

Hedge 7.531 m of bond to buy

VaR of portfolio & bond using delta hedge

VaR of portfolio & bond using optimal hedge

VaR of portfolio & bond using perfect correlation

[7]

[9]

[8]

156.87 227.64 38.49 -3,070.87 -25,869.93 -36,702.17 -29.906.82 77,820.35 0.00

185.01 292.92 –110.01 -3,187.30 -26,041.04 -33,994.81 –10,356.95 77,820.35 0.00

179.09 279.19 -78.78 -3,162.82 -26,005.06 –34,564.11 –14,467.86 77.820.35 0.00

50,554.08

45,645.72

0.00

3.4%

12.8% 0.489

87.2%

394

Swaps and Other Derivatives

the hedge will be ineffective against small movements. The level of the filter is probably best determined by simulation3.

9.5 CHOICE OF RISK FACTORS AND CASHFLOW MAPPING The value of a portfolio is likely to depend upon a variety of different underlying risk factors. There tends to be little debate about some factors. For example, if the valuation requires the conversion from a number of different currencies, then the various spot rates would be used as factors. If the portfolio had an equity or commodity proportion, then unless it was very heavily weighted towards a very small number of stocks or commodities, indices are usually used as surrogates — see Section 9.12 for further details. A more conscious choice has to be made for interest rates. The examples above have all used market rates. It would however have been perfectly feasible to use forward or zero coupon rates as alternative frameworks. Obviously the volatilities and correlations would have to be calculated for these rates. Because these rates are not directly observable but are themselves constructed from the market data, there may be differences in the derived data which would in turn lead to differences in the estimated VaRs. One popular choice4 is discount bond prices. A discount bond is a zero coupon bond paying 1 at maturity. For a bond of maturity T, its price today pT = 1 x DFT = DFT, i.e. the prices are equivalent to discount factors. Obviously they are very artificial and suffer from the various estimation problems discussed in the earlier chapter on curve building. They are also closely related to zero coupon rates, e.g.: DFT = exp{—zTTr} or DFT = (1+ zT/n)-nT i.e. continuously or discretely compounded. However they do possess one major advantage. Consider the swap portfolio used in the previous example; its value may be written as: value = Y^, CF, x DF, Obviously this expression is linear in the DFs, and hence has zero second-order effects. The delta method for calculating VaR should therefore provide identical results to any full valuation approach. Differentiating with respect to one of the DFs, say DF,, the derivative is equal to CFs. This method is often discussed in terms of "cashflows", without the explicit recognition that the cashflow is simply the discount bond price derivative, and as such follows the earlier VaR derivation. The formula we had above for the individual VaRs may be simplified: VaR s = dVsldps x px x k x as = CFS x ps x k x as — PVs x k x as i.e. the present value of each cashflow multiplied by the volatility of the appropriate bond price. Because many traditional risk management systems work in terms of zero coupon rates, simple transformations can be applied to convert them into zero coupon price terms. For 3

The author used this technique to hedge a portfolio of AS Eurobonds being traded in London with the AS bond futures contract traded in Sydney. Obviously the markets are related, but only loosely. He took the price histories of some typical old Eurobonds, regressed them against the bond futures, estimated optimal filters, and developed decision rules for creating a hedge against the actual trading book. 4 Encouraged by Risk Metrics and the publications of JP Morgan.

Imperfect Risk Management

395

example, rate volatility may be converted into price volatility by using the widely used approximation 5 : ap = ff, x (>•//?) x dp/dy Consider the artificial zero coupon yield curve below: Maturity 3 year 4 year 5 year 7 year

Yield (%) 8 9 10 12

Yield volatility (%) 2.78 2.53 2.42 2.25

Modified duration 2.778 3.670 4.545 6.250

Price volatility (%) 0.62 0.84 1.10 1.69

This uses the formula pT = (1 + y)-T and modified duration —(1/p) x dp/dy — T / ( 1 + y). Notice that the results are probably intuitive, namely that interest rates typically exhibit declining volatilities as maturity increases, whereas bond prices always have increasing volatilities with maturity. The earlier swap portfolio example had cashflows occurring on a variety of dates. However we only have knowledge, i.e. volatilities and correlations, about specific rates, often called "gridpoints". The portfolio was valued by using the following process: 1. a given set of market rates had to be completed, i.e. missing ones estimated, by some type of interpolation; 2. the discount curve calculated by bootstrapping the completed market curve; 3. the discount factors on the cashflow dates estimated again by some type of interpolation, and finally the cashflows discounted to produce the value. The estimation of the VaR followed the same process. The delta sensitivity was calculated by shifting a market rate, and recording the change in the value at the end of the process. The interpolation methods used for the VaR calculations are those used for the original valuation process; there is internal consistency. The process has mapped the portfolio sensitivities onto the gridpoints. We could of course adopt exactly the same process when using discount bond prices. Worksheet 9.4 shows the following example: • Box 1 contains some gridpoint data in terms of discount bond prices at years 1 and 2 (these are taken directly off the previous swap portfolio example) plus price volatilities and a price correlation; • Box 2 contains a single fixed cashflow approximately midway between the gridpoints. Generalizing, we have a cashflow at time T, and two gridpoints ti and ti+1 which bracket T. First we need to value this cashflow. Let us estimate DFT = I T (DF i , DFi+1) where / is some interpolation function. The worksheet uses continuously compounded interpolation, i.e. the zero coupon yields are calculated in column [1] of Box 3, linearly interpolated in "Often written as ap =• av x y x M D which stands for modified duration. This is derived from two approximations: 1. if p —f(y), then sp % df/dy x .y, where s is the standard deviation; 2. vr — (npY x [exp{(<7 p )"1} — 1] % (iipY x (ap) t where v is variance and np the expected value of p, and where;? is distributed log-normally.

Worksheet 9.4 Example spreadsheet demonstrating alternative cashflow mapping approaches

1'oday's date:

4-Jan-00

VaR multiplier =

1. Gridpoint data

8-Jan-0l 7-Jan-02 Correlation =>•

1.65

3. Derived data Time

DFs

Price Volatility

1.03 2.04

0.936853 0.873099 0.8

0.050% 0.086%

2. Cashflow data Time

CF

1.65

-1.57583

c-c Yield [1] 6.347% 6.656%

4. Derived results Linear interpolation Weights

Interpolated Yield

0.38187 0.61813

5. Allocated cashflows as sensitivities Sensitivities (scaled PVBPs) [2] -0.92711 -0.81172

1 2 First-order value -

[1] 0.936853 0.873099

–

1 .577282

Sensitivities (analytic) [3] -0.92713 -0.81175 – 1 .577320 Total VaR =

Interpolated DF

[3]

6.538% Present value =

DFs

Yield St dev [3] 0.049% 0.042%

[2]

[1]

21-Aug-0l

Yield Volatility [2] 0.767% 0.634%

95% VaR [4] -716.58 -1,005.70

1,636.45

0.897579 –

1 .4 1 443

6. Allocated cashflows by matching value and sensitivities DFs 1 2

Allocated CFs

Discounted CF

95% VaR

-0.60506 -0.97077

-0.56686 -0.84758

–467.66 –1,202.71

0.936853 0.873099

Sum=

-1.57583

–1.41443 Total VaR =

7. Direct calculation of cashflow VaR

Linearly interpolated Calculated

Yield Vol [1] 0.684%

Price Vol [2] 0.074%

95% VaR [3] –1,726.02

0.649%

0.070%

1,636.45

8. Allocated cashflows by matching value and VaR a = 0.00000030 16 b=- 0.00000079 12 c = 0.000000 1926 x = 0.271 5927722 1 2

Allocated PV -0.384150 -1.03028

Allocated CFs -0.41004 –1.18003 Total VaR =

95% VaR –316.92 –1,461.97 1,726.02

1,601.61

398

Swaps and Other Derivatives

column [2] of Box 4, and finally converted back into a bond price in column [3] of Box 4. The cashflow in Box 2 has a value of —1.41443 (million USD). The linear interpolation weights are shown separately in column [1] of Box 4 for future purposes. If VT = CFT x DFT, then we can estimate the two sensitivities OFr/dDF, = CFr x d!T/dDFj and <9Fr/dDF,+1 = CFr x d/r/(9DF;+1 either by perturbation or analytically as shown on the worksheet in columns [2] and [3] respectively in Box 5. Given the price volatility data, the individual VaRs may be calculated (using the analytic sensitivities, but it hardly matters which), and hence the total VaR of $1,636 using the correlation coefficient. This very much repeats the previous VaR method described above. An alternative approach is to allocate the cashflow CFT onto the two gridpoints. We have already seen one way of doing this (see Section 8.6) by defining the resulting cashflows as CF, and CFi+1 which are subject to the following constraints: preserve cashflow: CFT = CFi 4- CFi+1 preserve value: VT = CFi x DFi + CFi+1 x DF,i+l and solve for CFi and CFi+1 as shown in Box 6. The individual VaRs may then be calculated using the above expression, and finally the total VaR. The result is not the same as the previous total VaR, because the allocation is not consistent with the valuation structure in that sensitivity is not maintained. However it is not grossly different, as may be seen by changing the correlation. Another popular approach is first to estimate the VaR of the cashflow by interpolation, and then do the allocation whilst preserving this VaR. For example, the price volatilities in Box 1 are converted into yield volatilities (see column [2] of Box 2), linearly interpolated and converted back to price volatilities (see row "linearly interpolated"; columns [1] and [2] of Box 7). The VaR is now easily calculated to be VT x OT x k = $1,726. We now wish to calculate the allocated cashflows so that: • preserve value: VT = Vi + Vi+1 • preserve VaR: VaRT = VaR(CFi,

C

F

i

6

8 . The resulting VaR matches, of course, the VaR estimated by interpolating the volatilities. But this VaR is not the same as the VaR of $1,636 calculated originally. Furthermore, if the correlation shifts, we find that the original VaR changes but this new one does not! This is because the correlation was not used in the latter's calculation. The problem lies in the following: 1. yi and yi+1 were interpolated to get yT; 2. yield volatilities . 6

i.e. the VaR equation may be written as: (VT x <7r): = (
Substituting for V/^ and rearranging, we get: [07 + a~+i where .v, = VJ VT. This may easily be solved as a.\~ + b.\ + c = 0. The root is chosen so that, if possible, the signs of CF, and CF,_, match that of CFr.

+

1

)

A

Imperfect Risk Management

399

But ar is a property of yT, and therefore we should be able to construct an expression for it directly without interpolation. Define:

where COT and (1 — a>T) are the interpolation weights given in Box 4. Then: K») = (a> r ) 2 v(j>/) + (1 - « r ) 2 v(>vi) + 2<w r (l - w7Mv^>( >',-+i )p/.,+i where v(-) and s(.) are variance and standard deviation respectively. The approximation relating v and a (see footnote 6) may then be used to estimate aT. The end results are shown in row "calculated", column [1] of Box 7: • the new o> is estimated to be 0.649% compared to the linearly interpolated value of 0.684%; • the total VaR is now identical to the original value. Furthermore, as the correlation shifts, OT changes accordingly and the VaRs calculated using the original sensitivity approach and this latest approach remain identical. To summarize, the cashflow approach has some attractions in that it permits a portfolio to be represented solely by cashflows accumulated at gridpoints. It is therefore simple to add new transactions into the portfolio, as these easily modify the cashflows. However, there are some difficulties, and simple interpolation is not sufficient to implement this method. The sensitivity approach uses allocation methods that must already exist within, and be consistent with, valuation methods, easily allowing a range of different interest rate risk factors, and requiring little additional work to implement. This latter approach will be used throughout the remainder of this chapter.

9.6 ESTIMATION OF VOLATILITY AND CORRELATIONS This was discussed at some length in Chapter 7; everything that was ascribed to volatility also applies to correlation. For relatively short time horizons, say under 1 month, weighted or GARCH estimation schemes are more popular than unweighted, unconditional estimation. Implied volatilities are seldom if ever used. This is for consistency. VaR is specifically designed to measure the risk over a range of different activities, possibly taking place in different geographical locations, and in different time zones. A large number of risk factors may be used to capture this risk. It is highly unlikely that there will be liquid options traded on all the risk factors. So whilst it may be feasible to obtain implied volatilities on some factors, it would not be possible to get them on all factors, and therefore it would be necessary to mix historic and implied volatilities — hardly a good idea! Calculating the estimates is fraught with practical difficulties. A time series of consistent, cleansed data is required for each risk factor. Non-business days will leave gaps in this time series, and many data providers plug the gaps by simply repeating the previous business day's value. The problem is exacerbated when trying to estimate the correlation between two risk factors from different countries as the non-business days are unlikely to match. One alternative is simply to omit all data referring to a day which is a non-business day in some country, but that may require the rejection of substantial amounts of data.

400

Swaps and Other Derivatives

This is hardly adequate for the calculation of accurate estimates, and complex statistical algorithms have been constructed to bridge the gaps more appropriately7. A similar issue is asynchronous data, i.e. data that are available at different points in time. For example, using closing prices in Tokyo and in New York means about a 14-hour gap. But the markets will not have remained constant, so the estimated correlations are likely to be biased downwards. Again algorithms have to be developed to make adjustments. These issues do not simply result in slight mis-estimates. A correlation matrix is by definition positive semidefinite8: if it were not, then it would be possible to find ourselves having to take the square root of a negative number when calculating the parametric VaR. If all the correlations are calculated consistently, then there are no problems. But the data issues described above, unless tackled carefully, can result in an infeasible correlation matrix9. Consider a typical international bank operating in, say, 20 currencies. If we assume two yield curves per currency (bond and swap), each represented by 10 data points (the regulatory minimum is six per curve), then this suggests:

IR: 20 x 2 x 10 = 400 FX: = 19 i.e. 419 volatilities and 87,571 correlations! A lot of data to be gathered, cleansed and estimated, even if it is only once a quarter. Luckily there are practical ways around this. Most transactions that the bank carries out are likely to be single currency, and the main risk for many of them is a change in the absolute level of the curve. As we have seen in the Appendix to Chapter 8, market rates are quite highly correlated. If these are the risk factors, then only two or at most three points would be required to capture the curve movement adequately for risk management purposes. Unless the bank is doing a lot of spread trades, correlation between the two curves in each currency is not very important, and could be reduced to the correlation between two indicative points, one on each curve. Furthermore, unless the bank is doing a lot of cross-currency trades, such as hedging Yen bonds with USD instruments, then the cross correlations can also be reduced to that between two indicative points. This would reduce the IR gridpoints down to about 120, but more importantly the correlations required to just over 1,50010. So it is feasible to reduce the data requirements quite significantly without seriously jeopardizing the effectiveness of the risk management. The estimation of volatility and correlation are crucial to the estimation of VaR, and the above discussion only outlines some of the issues. See, for example, Best (Footnote 9) for a much more detailed discussion. 7

See, for example, the EM algorithm developed by JP Morgan, Risk Metrics Technical Document, 4th Edn, 1996. Chapter 8. 8 i.e. the value x'-C-x ^ 0 for any vector x. 9 See also the discussion in Chapter 3 of Best, Implementing VaR, Wiley, 1998. 10 (a) Three points per curve = 40 x 3 = 120 intra-curve correlations. (b) Single point inter-curve correlation = 0.5 x 40 x 39 = 780. (c) FX correlations = 0.5 x 19 x 18 = 171. (d) FX/IR correlations = (max)0.5 x 40 x 19 = 380.

Imperfect Risk Management

9.7 A RUNNING EXAMPLE The swap portfolio was used above to illustrate how to estimate the VaR of a portfolio, and to compare it with a delta hedge. However we wish to discuss other VaR issues, and the swap portfolio lacks various required properties. Hence the portfolio below will be used as a running example throughout the rest of this chapter. S-Euro FX option portfolio (all options sold out of 24 September 1999)

1 2 3 4 5 6

9 month 15 month 3 month 21 month 7 month 18 month

USD USD USD USD USD USD

call on 10m USD put on 20m USD call on 25m USD put on 15m USD call on 20m USD put on 15m USD

at strike at strike at strike at strike at strike at strike

1.06 1.04 1.05 1.55 1.065 1.02

The current market data is shown below: Current market levels

Interest rates

$-Euro spot rate: 1.0432

6mo cash 12mo cash 2yr swap

Volatility of spot FX: 20% pa

USD 5.938% 6.031% 6.205%

Euro 3.125% 3.313% 3.850%

it has been reduced so that each curve only consists of three points. There are therefore only seven risk factors, six interest rates plus the spot rate. This is hardly realistic, but adequate for illustrative purposes. In particular, the example does not include the volatility of the spot rate as a risk factor, which is becoming increasingly common. There may however be some additional data estimation difficulties, as both the volatility of the volatility and the correlation with the other factors would then be required. The first stage is to value the portfolio as shown in Worksheet 9.5. The discount curves are built, and then zero coupon interpolated11. The appropriate zero coupon rates for each option are calculated in columns [7] and [8]. Finally the option prices, quoted in terms of Euros, are calculated by means of a macro and shown in column [10]. The value of the total portfolio is — 13,671,321; negative as all options have been sold and the premia taken up front. The second stage is to calculate the portfolio sensitivities. Changes in the risk factors (bp) may be entered in the shift area of Worksheet 9.5. The resulting PVBPs are given in the box. The annualized volatility and correlation data for the calculation of the VaR are shown in Worksheet 9.6. We wish to estimate 10-day 99% VaR, so each volatility will be scaled by A/( 10/250) as discussed above. The individual VaRs are calculated in column [4] of Worksheet 9.7 using the normal formula VaRx = 10,000 x PVBPx × x × k × ax and finally the total VaR. The worksheet shows that the portfolio has a delta VaR of €522,437. 11

The zeroth point is determined by backward extrapolation.

402

Swaps and Other Derivatives

Worksheet 9.5 Estimate risk factor sensitivity for a portfolio of FX options Today's date

24-Sep-99 Market data

Shifts (bp)

USD

Euro

6m 12m 2y

5.938% 6.031% 6.205%

3.125% 3.313% 3.850%

Euro/S

1.0432

USD 0 0 0

9 month 15 month 3 month 21 month 7 month 18 month

USD USD USD USD USD USD

call on 10m USD at strike put on 20m USD at strike call on 25m USD at strike put on 15m USD at strike call on 20m USD at strike put on 15m USD at strike

.06 .04 .05 .55 .065 .02

Maturity date [1] 24-Jun-00 24-Dec-00 24-Dec-99 24-Jun-01 24-Apr-00 24-Mar-01

Generating the discount curves 24-Sep-99 24-Mar-OO 24-Sep-00 24-Sep-01

USD DF

Euro DF

0.506 1.017 1.014

0.970857 0.942225 0.884890

0.984447 0.967420 0.925988

USD 5.928% 5.931% 5.935% 6.106%

gradient 0.000069 0.000069 0.001713

Euro 2.986% 3.144% 3.303% 3.839%

Zero coupon rates time 0.001 0.499 1.003 2.003

0 0 0

0

FX option portfolio (all options sold) 1 2 3 4 5 6

Euro

gradient 0.003165 0.003165 0.005363

Length of time [2]

0.751 1.252 0.249 1.751 0.584 1.499

Imperfect Risk Management

403

Shifted market data 6m 12m 2yr 6m 12m 2yr Spot

USD USD

Euro

5.938% 6.031% 6.205%

3.125% 3.313% 3.850%

Euro

Sensitivities (PVBPs) 1,023.27 – 1,745.90 -2,495.98 -941.82 2,455.03 3,724.28 433.62

1 0437

1 . Wf_J *i.

Principal m USD [3] 10 20 25 15 20 15

Type

Spot (E/S)

[4] call put call put call put

[5] 1.0432 1.0432 1.0432 1.0432 1.0432 1.0432

Strike (E/S) [6]

1.06 1.04 1.05 1.55 1.065 1.02

Domestic Foreign Z-c rate Z-c rate [7] [8] 3.223% 5.933% 3.437% 5.978% 3.065% 5.930% 3.704% 6.063% 3.171% 5.932% 3.569% 6.020%

Volatility Pa [9] 20% 20% 20% 20% 20% 20%

Price (Euro)

[10] 532.432 2,046.515 863,500 7,812,391 908,527 1,507,957

Value =

– 13,671,321.85

Original value =

-13,671,321.85

Worksheet 9.6 Market volatility and correlation information

USD

Euro

6m

12m 2yr 6m 12m 2yr Spot

Volatility Correlations — > pa 6m

12m

1 0.9 0.8 0.3 0.2 0.2 -0.4

0.9 1 0.95 0.3 0.3 0.2 -0.35

17% 15% 13% 20% 18% 16% 20%

2yr

6m

12m

2yr

0.8 1 0.2 0.3 0.25 -0.3

0.3 0.3 0.2 1 0.9 0.75 0.4

0.2 0.3 0.3 0.9 1 0.9 0.35

0.2 0.2 0.25 0.75 0.9 1 0.3

0.95

Spot

-0.4 -0.35 -0.3

0.4 0.35

0.3 1

Matrix for Monte Carlo simulation Decomposed correlation matrix A USD

Euro

6m

12m 2yr 6m 12m 2yr Spot

6m

12m

2yr

6m

12m

2yr

Spot

1 0.9 0.8 0.3 0.2 0.2 -0.4

0 0.436 0.528 0.069 0.275 0.046 0.023

0 0 0.286 0.267 0.018 0.230 0.028

0 0 0 0.913 0.894 0.820 0.576

0 0 0 0 0.292 0.408 -0.310

0 0 0 0 0 0.258 0.106

0 0 0 0 0 0 0.632

405

Imperfect Risk Management Worksheet 9.7 Spreadsheet to calculate the VCV VaR of an FX option portfolio VaR multiplier = 2.33 Sensitivities (per bp)

Levels

USD

Euro

[2] 1,023.27 – 1,745.90 -2,495.98 -941.82 2,455.03 3,724.28 433.62

[1] 5.938% 6.031% 6.205% 3.125% 3.313% 3.850% 1.0432

6m 12m 2yr 6m 12m 2yr Spot

VaR

Volatility [3] 17% 15% 13% 20% 18% 16% 20%

[4] 48,131.26 -73,604.36 -93,823.63 -27,430.57 68,213.55 106,907.55 421,593.91

522,437.21

Total VaR =

9.8 SIMULATION METHODS In Section 9.2, we briefly discussed two approaches; the delta approach which we subsequently expanded, and a full valuation method. This latter approach involved shifting a rate by an amount determined using a historical distribution, and then revaluing the transaction. We found that the results for the two were similar, with the valuation method generating a slightly higher VaR as expected. This valuation approach is a simple example of a simulation method. There are two alternative simulation approaches widely used, namely historic and Monte Carlo. The former takes changes in risk factors that have actually happened in the past, applies them to the current level of factors, and revalues. The latter randomly generates the changes in the risk factors according to some volatility/correlation structure. In this section, we will measure the VaR of the FX option portfolio using these two approaches. For historic simulation, 500 days (approximately 2 years of business days) of 10-day percentage change vectors have been collected; these are shown in Box 1 of Worksheet 9.8. To be precise, these are the simple percentage changes from dayi to dayi+10, dayi+1 to day i+11 , etc. This does involve some autocorrelation as the sampling periods obviously overlap, but this is normal practice. It would be statistically more accurate to use dayi to dayi + 10, dayi+11 to dayi+20, etc., but this would require samples from 5,000 days which may lack relevance. The calculated annualized volatility of each factor from this data set is shown on the worksheet; these compare closely with the long-run volatilities used in the previous section. Each change is applied to the current market rate, i.e. rnew = rcurrent × (1 + % rchange), to create new factors as shown below:

Current rates First change vector Resulting new curves

USD 6m 5.938% 6.138% 6.302%

12m 6.031% 5.340% 6.353%

2y

6.205% 4.762% 6.500%

Euro 6m 3.125% 0.236% 3.132%

Euro/$ 12m 3.313% 0.459% 3.328%

2y

3.850% 1.0432 0.868% -5.105% 3.883% 0.990

ON

Worksheet 9.8 Spreadsheet to perform historic simulation VaR for a portfolio of FX options Today's date Time horizon = VaR

24-Sep-99 10 days = 99% 95%

0.04 years

FX option portfolio (all options sold) 1 2 3 4 5 6

9 month USD call on 10m USD at strike 1.06 15 month USD put on 20 USD at strike 1.04 3 month USD call on 25m USD at strike 1.05 21 month USD put on 15m USD at strike 1.55 7 month USD call on 20m USD at strike 1.065 18 month USD put on 15m USD at strike 1.02

Maturity date

Length of time

Principal m USD

24-Jun-OO 24-Dec-OO 24-Dec-99 24-Jun-01 24-Apr-OO 24-Mar-01

0.751 1.252 0.249 1.751 0.584 1.499

10 20 25 15 20 15

Type call

put call put call put

Strike (E/S)

Volatility

pa 20% 20% 20% 20% 20% 20%

1.06 1.04 1.05 1.55 1.065 1.02

Market data

Rates Volatility (pa)

USD 6m 5.938%

12m 6.031%

2y 6.205%

Euro 6m 3.125%

12m 3.313%

3.850%

1.0432

17.19%

15.53%

13.68%

19.16%

17.26%

15.34%

19.89%

Euro/$

2y

Simulation 1. Historic percentage shifts 2. Shifted market data USD Euro Euro/$ USD 12m 6m 12m 2y 6m 6m 12m 2y 2y 0 6.138% 5.340% 4.762% 0.236% 0.459% 0.868% -5.105% 6.302% 6.353% 6 500% 2.059% 6.533% 5.769% 5.926% 6 214% 1 -2.830% -1.738% 0.145% -2.359% -0.206% 2 0.997% -0.913% -2.110% 1.530% -0.948% -1.290% -4.608% 5.997% 5.976% 6 074% 3 1.243% 1.398% 0.531% 2.552% 0.539% -0.655% 0.426% 6.011% 6.116% 6 238% 0.274% -4.213% 5.987% 6.113% 6 274% 4 0.831% 1.352% 1.113% 0.580% 1.771% 5 2.457% 4.560% 3.067% 4.146% 3.804% -0.124% 0.263% 6.083% 6.306% 6 395% 3.095% 8.148% 5.797% 5.944% 6 106% 6 -2.363% -1.441% -1.602% 5.507% 4.426% 7 1.815% 0.378% -0.695% 0.901% -2.657% -2.050% -0.990% 6.045% 6.054% 6 162%

Euro

6m 3.132% 3.051% 3.173% 3.205% 3.143% 3.255% 3.297% 3.153%

12m 3.328% 3.306% 3.281% 3.330% 3.371% 3.439% 3.459% 3.224%

Euro/$ 2y 3.883% 0.990 3.929% 1 . 1 1 1 3.800% 0.995 3.825% 1.048 3.861% 0.999 3.845% 1.046 3.969% 1.128 3.771% 1.033

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Swaps and Other Derivatives

Given the new curve, the worksheet calculates: • the new discount curves (Box 3 but not printed); • the zero coupon rates (Box 4 but not printed) and interpolated for the relevant dates of each option (Box 5 but not printed)12; • the new value of each option (Box 6). Hence the change in the value of the portfolio is calculated as a result of this change vector. The output from this process provides a vector of 500 observations of "changes in value" — see column [1]. There are various ways of proceeding to estimate the VaR. For example, to estimate the 99% VaR, we could: 1. select the fifth worst loss out of the 500 — see cell BG21; 2. calculate the 1% percentile — see cell BG22; 3. create a histogram, and interpolate that. Suppose we wish to create a histogram of 20 buckets, in units of 10,000. The minimum and maximum changes in value are -1,932,167 and 173,447 respectively, therefore a bucket is (max-min)/20 = 110,000 roughly in size. The histogram shown in Figure 9.6 is created. The tail of the histogram may be interpolated, either linear or negative exponential are probably most common — see cell BG23. The results are summarized below: Results from historic simulation Worst VaR Percentile VaR Bucketed VaR=»

99% -1,628,981 -1,619,060 -1,638,834

95% -1,031,286 -1,007,287 -1,030,167

All the 99% VaR estimates are considerably different to the delta VaR result, which was just over €500,000. The main cause of the difference is of course the negative gamma caused by the options being sold, which is ignored by the earlier approach and yet is highly significant — notice the long left-hand tail of the histogram. The portfolio consisted of three calls and three puts, in other words very much like a portfolio of straddles which would typically have low delta (and hence low delta VaR) and yet significant losses irrespective of the rates moving up or down. The long tail of possible losses indicates the (virtually) unlimited downside when selling options, whilst the very limited upside indicates that at best the option will not be exercised and the premium gained for nothing. Historic simulation is relatively simple to implement; all that is required is a historic data set to generate the change vectors which are then applied in turn to the current levels of the risk factors. Monte Carlo (MC) simulation generates the change vectors randomly using the statistical properties of the risk factors. The usual approach is to assume a generation process such as: dXi(t) = X i (t + dt) - X i (t) = //,{t, Xi(t)}dt + £, 0 it (t, X i (t)}dz k (t) i.e. the change in the ith risk factor at time t over an infinitesimally short period of time dt is due to a trend term (which itself may depend upon both time and the current level) plus 12

The first row in each box is the relevant times from today in years.

-1.93

-1.71

1.49

-1.27

-1.05

-0.83

-0.61

-0.39

Change in value

Figure 9.6 Results from historic simulation for FX option portfolio

-0.17

0.05

0.27

410

Swaps and Other Derivatives

a number of intercorrelated stochastic movements where usually dzi ~ N {0, 1} and E{dzi, dzj} = pij. As a special case:

with a constant drift and single constant volatility describes the usual lognormal process for Black & Scholes options. It can easily be shown that13: X i (T) = X i (0) exp{(//,. - Jo?) T+ ffj This means that the continuously compounded return over the period

T

1s

To demonstrate the sampling process, a simple worksheet has been created (Worksheet 9.9). It is divided into three boxes. Box 1 takes a single rate, X(0) = 8%, assumes a drift j/, volatility a and a weekly time horizon T=(1/52) years. It then generates a N(0,l) random number for dz, and calculates X(1). Given X(l), a new random variable is sampled to calculate X(2). This is repeated until X(52), i.e. we now have a path of weekly observations going out for 1 year. The graph below shows a variety of paths that may result from the single starting point. Obviously as expected, as we move more into the future, the paths widen out demonstrating how the volatility increases with time. Usually however we wish to generate an entire curve rather than an isolated interest rate. Box 2 of the worksheet has taken a simple curve of three points and evolved each rate completely separately. An example is shown in the graph. To interpret, the new threepoint curve at any time segment is a vertical slice through the three evolutions. Notice that the 10% and 12% gridpoints have risen, converged and crossed, whilst the 8% gridpoint has behaved more independently. But this sampling ignores the fact that interest rates along a curve are typically highly correlated. Let us assume the following high positive correlation structure C:

1 0.9 0.8

0.9 1 0.7

0.8 0.7 1

and decompose it into AA', where A is a lower triangular matrix, i.e.

1

0

0.9 0.8

0.436 -0.046

0 0 0.598

This is the Cholesky decomposition14—see Box 3. If we take the random vector of normal samples x as generated in Box 2, calculate y = ji + xA', and then y~N(n,C). The result is still a vector of random numbers, but with the correlation structure superimposed. The resulting graph shows each of the curves now moving much more in parallel, as expected. 13

By defining Y = ln(X), applying Ito's lemma and integrating from 0 to T. See for example Hull (2000) pp. 230–

231. 14 This decomposition is sometimes called taking the "square root" of a matrix. It can only be applied to positive semidefinite symmetric matrices which, as discussed above, a correlation matrix should be. There are other approaches that may be used, especially when faced with real-world ill-conditioned correlation matrices.

411

Imperfect Risk Management Worksheet 9.9 1. Simple demonstration of Monte Carlo sampling Current level Drift Volatility Time horizon No. of segments Time segment

10% pa 15% pa 1 year 52 0.019 years Multiple time paths of single factor

0

10

20

30

40

50

60

Time segments

Single factor uncorrelated sampling

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

8% 8.17% 8.23% 7.91% 8.13% 8.28% 8.41% 8.37% 8.35% 8.84% 8.85% 8.72% 8.76% 8.55% 8.44% 8.70% 9.08% 9.14% 9.20% 9.25% 9.31%

8% 8.06% 7.95% 7.72% 7.58% 7.47% 7.81% 7.69% 7.55% 7.59% 7.86% 7.99% 8.07% 8.07% 8.09% 8.19% 8.68% 8.81% 8.96% 9.06% 8.95%

8% 7.88% 7.80% 7.71% 7.54% 7.55% 7.37% 7.33% 7.46% 7.46% 7.48% 7.60% 7.36% 7.30% 7.11% 6.93% 6.96% 6.89% 6.77% 6.85% 6.85%

8% 7.92% 7.87% 8.13% 8.02% 8.33% 8.38% 8.26% 8.26% 8.04% 8.19% 8.09% 8.29% 8.31%, 8.36% 8.30% 8.58% 8.67% 8.30% 8.13% 8.03%

8% 8.06% 7.95% 7.81% 7.85% 7.91% 7.59% 7.52% 7.62% 7.56% 7.33% 7.48% 7.50% 7.47% 7.36% 7.30% 7.46% 7.65% 7.74% 7.59% 7.63%

8% 8.08% 7.97% 7.68% 8.19% 8.20% 7.99% 8.18% 8.18% 8.11% 8.02% 8.18% 7.77% 7.96% 8.25% 8.29% 8.09% 8.21% 8.40% 8.45% 8.48%

7.94% 7.86% 7.92% 7.95% 7.82% 8.11% 8.31% 8.58% 8.23% 8.16% 7.91% 7.91% 7.98% 8.13% 8.13% 8.05% 7.96%, 7.89%, 7.99% 8.06%,

(Continued)

412

Swaps and Other Derivatives

Worksheet 9.9 continued

2. Uncorrelated sampling of a simple 3-point curve

Current level Drift Volatility Time horizon Time segment

Variable 1

Variable 2

Variable 3

8% 10% 15% 1 0.019

10% 5%

12% 8% pa 12% pa 1 year 0.019 years

14% 1

0.019

Uncorrelated sampling of a 3-point curve 16 14 12 10

8 6 10

20

30

40

50

60

Time segments Uncorrelated variables

Uncorrelated samples

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-0.9110 -0.8189 -0.8128 0.5722 -0.3483 -0.3982 -0.5161 -0.1283 -1.8192 0.7630 1.1273 -2.4603 -0.0913 1.2813 1.2978 0.7829 1.3814 0.9017 1.6348 -0.2721

-0.4313 -0.2717 -0.0180 1.8620 -0.6500 0.6673 1.4890 -0.0515 -0.0960 1.2428 -0.2122 -0.2856 0.3434 0.4187 0.6966 0.2372 -0.1960 -1.7404 1.2561 -0.0464

-1.3320 1.0352 -0.8298 -1.3816 0.8761 -0.2154 -0.3931 1.5958 -0.9095 1.8963 0.3016 -0.3742 0.8731 -0.4909 0.0320 0.0324 -0.7143 0.9078 1.0316 0.2001

8% 7.864% 7.745% 7.629% 7.734% 7.693% 7.644% 7.577% 7.571% 7.299% 7.429% 7.618% 7.243% 7.243% 7.450% 7.665% 7.805% 8.044% 8.210% 8.505% 8.473%

10% 9.926% 9.883% 9.889% 10.256% 10.137% 10.278% 10.585% 10.584% 10.575% 10.840% 10.806% 10.756% 10.838% 10.937% 11.095% 11.157% 11.125% 10.760% 11.033% 11.033%

12% 11.751% 11.971% 11.823% 11.568% 11.753% 11.728% 11.668% 11.995% 11.831% 12.221% 12.300% 12.241% 12.437% 12.353% 12.377% 12.402% 12.272% 12.476% 12.708% 12.768%

413

Imperfect Risk Management

3. Correlated sampling of a simple 3-point curve Cholesky decomposition (A)

Correlation 1 0.9 0.8

0.9 1 0.7

1 0.9 0.8

0.8 0.7 I

0 0.436 -0.046

0 0 0.598

Correlated sampling of a 3-point curve ^f1 """"^

10

^

^

^ ^^*^~

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0

\

\

i

i

i

10

20

30

40

50

60

Time segments Correlated samples 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

-0.9110 -0.8189 -0.8128 0.5722 -0.3483 -0.3982 -0.5161 -0.1283 -1.8192 0.7630 1.1273 -2.4603 -0.0913 1.2813 1.2978 0.7829 1.3814 0.9017 1.6348 -0.2721

Correlated variables -1.0079 -0.8555 -0.7394 1.3266 -0.5968 -0.0675 0.1845 -0.1379 -1.6791 1.2285 0.9221 -2.3387 0.0676 1.3357 1.4716 0.8080 1.1579 0.0529 2.0189 -0.2651

-1.5058 -0.0234 -1.1458 -0.4542 0.2753 -0.4780 -0.7164 0.8544 -1.9951 1.6878 1.0921 -2.1790 0.4336 0.7121 1.0254 0.6348 0.6868 1.3443 1.8674 -0.0959

8% 7.864% 7.745% 7.629% 7.734% 7.693% 7.644% 7.577% 7.571% 7.299% 7.429% 7.618% 7.243% 7.243% 7.450% 7.665% 7.805% 8.044% 8.210% 8.505% 8.473%

10% 9.814% 9.660% 9.531% 9.786% 9.682% 9.678% 9.722% 9.706% 9.339% 9.632% 9.813% 9.377% 9.399% 9.651% 9.936% 10.102% 10.339% 10.359% 10.775% 10.730%

12% 11.717% 11.729% 11.522% 11.452% 11.521% 11.446% 11.326% 11.503% 11.138% 11.467% 11.692% 11.285% 11.382% 11.534% 11.747% 11.888% 12.041% 12.328% 12.729% 12.727%

414

Swaps and Other Derivatives

The above approach has been used to estimate the VaR of the FX option portfolio. First the correlation matrix has been decomposed; see Worksheet 9.6. Worksheet 9.10 (not printed) then uses a table function to generate 500 random scenarios, and to construct the histogram as before. It operates in the following steps: 1. generate a random vector of seven normal distributed variables, each drawn from a distribution with zero mean and appropriate standard deviation (volatility); 2. apply the decomposed correlation matrix to generate the correlated random vector; 3. calculate the new levels of each risk factor using the above formula with a time horizon of (10/250) years; 4. revalue the option portfolio (as in the historic simulation); 5. repeat this process as often as required — the worksheet does 500 samples. Example extract from MC worksheet Current Levels: Ann Vols:

USD Eur 6m 12m 2y 6m 12m 2y 5.938% 6.031% 6.205% 3.125% 3.313% 3.850% 17% 15% 13% 20% 18% 16%

1A. Normally distributed iid random variables (zero mean, and using the ann vols) -0.3 -0.6 -0.8 -0.5 -1.0 0.3 1B. Correlated random numbers -0.317 New Levels:

5.870%

1.0432 20% 1.1

-0.568

-0.812

-0.377

-0.962

-1.015

0.838

5.927

6.073%

3.076%

3.198%

3.725%

1.0779

Figure 9.7 shows the resulting histogram, which is very similar to that generated by historic simulation. The VaR estimates are as follows (but will obviously change for each simulation): Results from Monte Carlo simulation 99% 95% Worst VaR -1,621,110 -1,005.018 Percentile VaR -1,522,137 -931,547 Bucketed VaR -1,611,391 -971.391

In comparison to the delta method, simulation approaches are deemed to be more accurate but inherently slower because they require a full revaluation of the portfolio for each scenario. Intra-day simulation approaches are therefore unlikely despite increases in computer power, but overnight simulations are common.

9.9 SHORTCOMINGS AND EXTENSIONS TO SIMULATION METHODS Historic simulation is very simple to apply in practice, although there are often data integrity issues to be resolved during the collection of data from disparate sources and time zones. Figure 9.8 shows the progressive calculation of the 99% VaR as the number of scenarios is increased up to 500.

250

200-•

150

100"

50-

2.61

-2.31

-2.01 Figure 9.7

-1.71

-1.41

-1.11

-0.81

-0.51

Results from Monte Carlo simulation for FX option portfolio

-0.21

0.09

0.39

416

Swaps and Other Derivatives

The early part of the chart is extremely characteristic, with large upward jumps followed by slow declines. This is easily explained. Suppose 100 simulations have been done so far; the 99% VaR estimate will be determined at most by the two largest losses so far recorded. The impact of the 101st simulation now has to be included: either (a) the new change in value shows a loss greater than the current VaR estimate; the new VaR estimate is therefore increased, and possibly substantially; or (b) the new change in value shows a gain or a loss less than the current VaR estimate; the new VaR estimate is reduced but typically by a small amount as there are already 99 observations on the right-hand side of the distribution. Thus VaR estimates from a small number of scenarios are often described as unreliable or "choppy". To overcome this, a large number of scenarios are required until the estimate smooths out, as shown by the latter part of the chart. This introduces a second issue, namely that of "relevance"; just how relevant are changes that occurred some 2 years ago in estimating the VaR going forward from today? It is often argued that, especially for short-term VaR, what has happened most recently is of most relevance. This issue is sometimes referred to as "stationarity", i.e. does the historic data possess significant trends? Unfortunately the immediate solutions to these two shortcomings are in conflict. One can increase the reliability of the VaR estimate by increasing the number of scenarios, but that in turn reduces the relevance. Notice that this problem does not arise with MC simulation as the number of randomly generated relevant scenarios can be increased without limit15. The reason for the unreliability of the VaR estimates is, for example, we only use five scenarios out of 500 to estimate 99% VaR. Any change in one of those five may change the VaR dramatically, whilst changes in the other 495 are of no consequence. One way around this would be to fit a theoretical distribution to the histogram. This distribution would use all 500 observations, and therefore would be quite robust to one more observation. Unfortunately the fitting would be concentrated on the central mass of the observations, and would be likely to model the tail relatively inaccurately — precisely what is not required because VaR is a property of the tail. A ready compromise therefore is to select some threshold, such as — 1,000,000, and model (in this case) the 27 observations in the tail; this approach is called "peaks over threshold" (POT). The model could be either some simple regression on the frequency or cumulative distributions or by some more complex means. For example, increasing use is being made of extreme value theory (EVT), which effectively fits a statistical distribution simply to the tail. In the discussion that follows, the histogram will be flipped over, so that we are analysing its positive right-hand tail — this follows the usual convention in statistics. A common assumption is that the tail follows a negative exponential distribution: g(y) = (1//?)exp{— y / f i } for y ^ 0 where y = change in value — threshold 15

It is possible to combine random sampling with historic simulation in what is called "bootstrapping". Normal historic simulation effectively samples from the historic data without replacement, and where the observations are drawn in their order of occurrence. The bootstrap method is to sample from the data with replacement, i.e. it is possible to use the same change vector a number of times. But this method is critically dependent on the assumption of stationarity, i.e. no trends.

Imperfect Risk Management

417

U X

418

Swaps and Other Derivatives

1.0

1.5

2.0

2.5

3.0

3.5

4.0

Figure 9.9 Negative exponential

The Appendix shows how to estimate an optimal ft from a set of tail observations, and also how to calculate the VaR using the formula: VaR P = t - P ln{(N/n t )(1 - P)} where nt is the number of observations above some threshold t, N is the total number of observations, and P the VaR probability (typically 95% or 99%). The results are very similar to the earlier historic simulation, suggesting that the simulation had stabilized by 500 observations — as we could see from Figure 9.8. VaR using a negative exponential model

0 = 5361,145 VaR confidence level VaR-NE expected loss

95% 1,027,794 1,388,939

99% 1,609,035 1,970,180

Now we have a model of the tail (Worksheet 9.11), we can explore other properties. One extremely useful one is the "mean excess function" defined as: e y (v) = E{y - v | y > v}

In words, it is the expected value of a loss, given that the loss will exceed a certain level. For example, if we set v = VaR, we find for a negative exponential model that: ey(VaR) = VaR + ft i.e. how much on average might you lose, given that the VaR level has been exceeded. This is probably a much more interesting statistic than the VaR number itself. Unfortunately, whilst the negative exponential distribution is very simple, it only possesses one parameter which is really for location. In practice, this is seldom sufficient, especially as the output from a simulation may have fat tails which decline more slowly than a negative exponential. A distribution frequently used in practice is the Generalized Pareto, whose density function is expressed as follows:

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419

Worksheet 9.11 Spreadsheet to demonstrate the construction of POT negative exponential tail Beta =

361,145

MLF 369.72 Threshold 1,000,000 N 500 Nu 27 V

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

[1] 932,167 832,365 789,202 653,515 628,981 618,960 598,201 564,571 543,290 511,751 501,751 481,645 350,199 272,720 267,192 192,339 176,533 169,793 127,156 118,161 111,934 107,783 81,723 77,002 31,286 6,024 4,671

sum

9,750,916

VaR confidence level VaR-EVT expected loss

95%

99%

1,027,794 1,388,939

1,609,035 1,970,180 when VaR is exceeded

420

Swaps and Other Derivatives

0.5

1.0

Figure 9.10

2.5

3.0

3.5

4.0

Negative exponential compared to GPD

for y > 0

+ x × y/

This has an additional parameter x controlling the tail. The GP reduces to the negative exponential if x is zero, but if x > 0, this corresponds to a tail that is fatter than that of a negative exponential, and if x < 0, then a thinner tail16 — see Figure 9.10. The Appendix describes how these parameters may be estimated using maximum likelihood techniques. Worksheet 9.12 calculates x whilst holding /? equal to the negative exponential value:

VaR using a generalized Pareto model

0 = $361,145 Z VaR confidence level VaR-GP expected loss

= –0.210 95% 99% 1,027,571 1,512,903 1,321,274 1,722,401

Notice that x is negative, suggesting that the historic simulation actually generated a thinner tail than the negative exponential. The VaR estimates are therefore slightly lower and hence, if one wished to be pessimistic, the negative exponential would be adequate. The modelling may also be approached in a different fashion, which provides some additional insights. Take the 500 observations and divide them into (say) 20 blocks, each containing 25 contiguous observations. Record the largest percentage loss Zi suffered within each block i; these are shown on the far right-hand side of Worksheet 9.8. The Appendix discusses the Generalized Extreme Value distribution that the Zs might, at least in theory, be drawn from. This distribution is given by:

where x = (Z — m)/s, m and 5 are location and dispersion parameters, and a models the tail. The GEV is given different names depending on the value of a; if a > 0, then this is called a Frechet distribution and has fat tails. 16

In this case, the region of y is restricted to ensure that [1 + x × y/fS\ > 0.

421

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Worksheet 9.12 Spreadsheet to calculate the parameters of a Pareto distribution from the results of a historic simulation To be minimized 1,000,000

Threshold Observations No. below

500 27

VaR confidence level VaR-EVT expected loss when VaR is exceeded

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27

MLF Xi = Beta-

26.2106 371.73 -0.210 361,145

95%

99%

1,027,571 1,321,274

1,512,903 1.722,401

y(i)

In(l+Xi*y/beta)

932,167 832,365 789,202 653,515 628,981 618,960 598,201 564,571 543,290 511,751 501,751 481,645 350,199 272,720 267,192 192,339 176,533 169,793 127,156 118,161 111,934 107,783 81,723 77,002 31,286 6,024 4,671

-0.7805 -0.6613 -0.6138 -0.4778 -0.4551 -0.4459 -0.4273 -0.3977 -0.3795 -0.3531 -0.3448 -0.3285 -0.2276 -0.1726 -0.1688 -0.1186 -0.1083 -0.1039 -0.0768 -0.0712 -0.0673 -0.0647 -0.0487 -0.0458 -0.0184 -0.0035 -0.0027

9,750,916

-6.9640

422

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The Frechet worksheet (Worksheet 9.13) takes the extreme block losses in column [1], reorders them in column [2] and calculates estimators for a in column [4] as described in the Appendix. Values for m and s are then estimated by maximizing a likelihood function in column [7]. A number of the estimators for a were used, and the one that gave the highest overall likelihood selected. The results from the worksheet are shown in the box below:

Percentiles 99% 95% 27.74% 77.78%

VaR 95% 99%

VaR -21.454% -16.565% -7.661% -12.050%

VaR 2,933,105 2,264,674 1,047,377 1,647,344

As discussed in the Appendix, a percentile from a GEV distribution describes a property of an extreme. For example, based on the results above there is a 5% chance that the worst loss out of a block of 25 observations will exceed 16.565% of the current value, i.e. €2,264,674. To estimate the level such that there is a 5% probability that any observation will exceed it, this corresponds to 95%25 = 27.74% percentile. Thus one can see that the VaR numbers provided by this method are again very similar to those estimated by the historic simulation. However, this form of modelling has another very interesting use, namely in the estimation of worst case scenarios. For example, the interpretation of the 99% percentile could be "there is a 1% probability that the worst loss on any day in a month (say, 25 business days) will exceed €2.93 million". The reason for discussing these two EVT approaches is that they extend historic simulation in two ways: • because they make use of more observations than simply the few in the far end of the tail, the VaRs are less choppy and hence more reliable; • modelling the tail enables statistical statements about the tail that could not have been addressed using the more conventional analysis, and in particular about the likely size of loss given that the loss has exceeded the VaR level. Both methods however assume stationarity, i.e. the analysis does not depend upon the temporal ordering of observations. Figure 9.11 shows the extreme losses from each block ordered back through time. Crudely classifying market conditions into high, medium and low volatility, subjectively it would appear that the market is currently experiencing medium volatility, and that it had much higher volatility earlier in the year and lower volatility over a year ago. The high volatility periods would of course have a much bigger impact on the VaR estimates than the other periods. Is this sensible, or should the current period of medium volatility carry the greatest weight? Risk managers are often divided on this point, some prefer to use recent market behaviour as the best predictor for future behaviour, and some would rather look back across the entire range of information available. If we wanted to modify the historic simulations to be more weighted to recent market behaviour, then a simple way would be to scale each of the changes as follows:

Worksheet 9.13 Spreadsheet to model extreme losses using a Frechel distribution Parameter estimation m s alpha

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Percentiles 8.315% 2.641% 3.36%

VaR (percentage) -21.454% -16.565% -7.661% -12.050%

99% 95% 27.74%, 77.78%

Maximum losses Zi

Reordered

Z[j]

ln(Z[j])

[1] -8.179% -11.690% -8.606% -13.087% -12.095% -10.985% -8.103% -7.543% -13.403% -14.133% -11.842% -11.289% –11.444% -4.001% -8.245% -6.689% -6.624% -6.854% -11.915% -7.359%

[2] -14.133% -13.403% -13.087% -12.095% -11.915% -11.842% -11.690% -11.444% -11.289% -10.985% -8.606% -8.245% -8.179% -8.103% -7.543% -7.359% -6.854% -6.689% -6.624% -4.001%

[3] -1.9566582 -2.0096929 -2.0335308 -2.1123967 -2.1273457 -2.1335166 -2.1464216 -2.1676888 -2.1813841 -2.2086686 -2.4527286 -2.4956026 -2.5036149 -2.5129394 -2.584494 -2.6092941 -2.6803604 -2.7047153 -2.7145387 -3.2186637

VaR (USD) 2,933,105.92 2,264,674.25 1,047,377.02 1.647,344.82

Hill's estimator [4]

2.65% 3.36% 8.43% 7.94% 7.13% 7.22% 8.18% 8.49% 10.09% 31.36% 32.68% 30.91% 29.56% 34.27% 34.45% 39.12% 39.24% 38.11% 84.10%

x = (Z -m)/s

f(x)/10

ln[f(x)]

[5] -0.05 1.28 0.11 1.81 1.43 1.01 -0.08 -0.29 1.93 2.20 1.34 1.13 1.18 -1.63 -0.03 -0.62 -0.64 -0.55 1.36 -0.36

[6] 1.393276 0.780374 1.379582 0.519623 0.697389 0.936054 1.391882 1.341081 0.471190 0.373661 0.748589 0.867662 0.833333 0.100526 1.393500 1.117822 1.093965 1.174097 0.733514 1.307704

[7] 0.33 -0.25 0.32 -0.65 -0.36 -0.07 0.33 0.29 -0.75 -0.98 -0.29 -0.14 -0.18 -2.30 0.33 0.11 0.09 0.16 -0.31 0.27

Maximum likelihood estimation = —4.05

424

Swaps and Other Derivatives

-2 -4 -6 g -8 -10 -12 -14 -16

V \ 1

2

3

4

5

6

7 8

9 10 11 12 13 14 15

16 17 18 19

20

Time back from today Figure 9.11

Maximum losses per block

• let VN and CN be the current volatility vector and correlation matrix, and VH and CH the historic volatility vector and correlation matrix respectively; • let ArH be a historic change vector; • estimate ArN = (V N /V H )×Ar H for adjusting for shifts in volatilities; • ArN = C 1/2 -C 1/2 -Ar H where the square root matrices are given by a Cholesky decomposition for shifts in correlations. A more sophisticated approach17 modifies the historic simulation so that a weight, related to the length of historic period, is attached to each observation. Observations corresponding to recent changes in the risk factors carry higher weights than observations from more distant periods. For example (see Worksheet 9.14): • let i = 0, 1, 2, 3, . . ., K represent the time of historic information (where 0 is the most recent) (column [1]); • let APi, be the percentage change in value of the portfolio due to the ith set of returns (column [2]); • define weights wi = [(1 — /)/(! — /*)]/'"' where £V w, = 1 and where /. is some number < 1. The worksheet shows the weights in column [3] using / = 0.95 and K = 500; • place APi in ascending order, together with the associated weights (columns [4] and [5]); • accumulate the weights, see column [6]; • to obtain q% VaR, locate (1 — q%) in the accumulated weights column and read off the associated &Pq. This may require interpolation to find the exact point; finally, calculate VaRq = current value xAP q . The following results are calculated in the worksheet: VaR 99% 95% 17

Required Interpolated weight return -8.24% 1% -5.82% 5%

Absolute

VaR -1,126,680 -795,207

See J. Boudoukh et al., The best of both worlds. Risk, 11(5), 1998, 64–67.

425

Imperfect Risk Management

Worksheet 9.14 Spreadsheet to estimate a weighted HS VaR Required weight 1% 99% 95% Lambda 5% 95% 5% Scales Returns

[1] 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

[2] -4.678% -1.338% -3.266% -0.078% -3.034% -0.382% -2.544% -0.689% -1.596% -1.412% -0.011% -8.179% -4.675% —0.388% -2.504% 0.057% -0.962% 0.623% -0.202% -1.914% -5.442% 0.453% -5.411% -1.957% -0.133% 0.683% –1.698% 0.320% 0.250% -0.170% -6.949% 0.192% -0.951% -3.962% 0.745%

Ordered Returns

Weighting

[3] 5.0000% 4.7500% 4.5125% 4.2869% 4.0725% 3.8689% 3.6755% 3.4917% 3.3171% 3.1512% 2.9937% 2.8440% 2.7018% 2.5667% 2.4384% 2.3165% 2.2006% 2.0906% 1.9861% 1.8868% 1.7924% 1.7028% 1.6177% 1.5368% 1.4599% 1.3869% 1.3176% 1.2517% 1.1891% 1.1297% 1.0732% 1.0195% 0.9686% 0.9201% 0.8741% 0.8304%

[4] 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35

-14.133% -13.403% -13.087% -12.095% -11.915% –11.842% -11.690% –11.444% -11.289% –11.058% -10.985% -10.838% -9.876% -9.309% -9.269% -8.721% -8.606% -8.557% -8.245% -8.179% -8.133% -8.103% -7.912% -7.878% -7.543% -7.359% -7.349% -7.294% -6.949% -6.854% -6.752% -6.689% -6.624% -6.479% -6.471%

Weighting

[5] 0.00003% 0.00008% 0.04697% 0.01520% 0.00000% 0.00001% 0.44872% 0.00000% 0.00000% 0.01061% 0.00280% 0.00000% 0.00000% 0.00252% 0.00000% 0.00004% 0.32985% 0.00000% 0.00000% 2.70180% 0.08258% 0.00086% 0.00001% 0.00005% 0.00022% 0.00000% 0.00144% 0.00000% 1.01953% 0.00000% 0.00000% 0.00000% 0.00000% 0.00000% 0.10138%

Interpolated return -8.2412% -5.8166% Accumulative Weighting

[6] 0.0000% 0.0001% 0.0471% 0.0623% 0.0623% 0.0623% 0.5110% 0.5110% 0.5110% 0.5216% 0.5244% 0.5244% 0.5244% 0.5269% 0.5269% 0.5270% 0.8568% 0.8568% 0.8568% 3.5586% 3.6412% 3.6421% 3.6421% 3.6421% 3.6424% 3.6424% 3.6438% 3.6438% 4.6633% 4.6633% 4.6633% 4.6633% 4.6633% 4.6633% 4.7647%

Absolute VaR -1,126,680 -795,207

426

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As expected, the VaRs are lower than those calculated using unweighted HS. The accuracy for most parameter estimation from a Monte Carlo simulation is proportional to 1 n where n is the number of scenarios. A typical number in practice, especially for pricing, is n= 10,000 which will give an error of 1%. Most simulations however are run with a stopping rule, that will stop the simulation if the parameter being estimated is not changing by more than a defined amount per set of scenarios. This accuracy is on average, and it is feasible for the simulation to have to run for many more scenarios before stopping. Given that each scenario requires a number of random variables, Monte Carlo is both computationally intense and also critically dependent upon the quality of the random number generator. Strictly speaking, these numbers are only pseudo-random, i.e. they have been generated by an algorithm using deterministic rules with no random components. These rules take an initial value (called a "seed") and generate a series of numbers which should pass the standard tests for randomness. Eventually the generated number will be equal to the seed, so that the generator will now cycle producing the same series. The cycle length must be much greater than the number of random numbers required for the simulation; if not, the apparent accuracy is spurious. The worksheet simulated changes in the risk factors using the expression: ,. - iff?) T+ as shown in Figure 9.12. The reason why so many samples have to be taken is to ensure that the random samples cover the whole of the 0–1 line, and do not cluster. But this idea gives rise to "quasirandom" sampling, i.e. if you want to ensure that there is good coverage along the line. why not:

-0.35 Figure 9.12

-0.25

-0.15

-0.05

0.05

0.15

0.25

0.35

Showing uniform samples on LH axis being transformed into normal samples

Imperfect Risk Management

• • • •

427

divide the line up into a number of segments (say, N) use the mid-point of the first segment as the uniform random number and transform that to a normal sample the next time a random number is required to be drawn from the same distribution, use the mid-point of the second segment

and so on. No random sampling at all! The error for an approach such as this is proportional to l/n. Unfortunately it will not work in practice because the simulation cannot stop until all segments have been sampled, i.e. there is no stopping rule. Hence the number of simulations that have to be run is NK where K is the number of risk factors. However there are techniques which pick the segments quasi-randomly, and can therefore reduce the number of samples required significantly, with an error of the order ln(n) K-1 /n. Unfortunately the methods do tend to break down with high K—sometimes called the "curse of dimensionality". Stein's algorithm is one that has proved to be remarkably robust, and appears to work well even for high K. There are a range of other variance reduction techniques, many of which are based on different ways of increasing the coverage. Anthetic sampling is a very old and simple method: if r is the uniformly distributed random number, then use ( 1 — r ) in the next scenario18!

9.10 DELTA-GAMMA AND OTHER METHODS In summary, delta approaches are fast but likely to be inaccurate in the presence of significant second-order (and higher) effects; simulation methods however are much more accurate but slow. Are there any compromises that might produce fast accurate results? One approach is to introduce the second term explicitly into the VaR calculations. This may be done either by extending the delta approach or by simulation. For a particular portfolio, the change in value for a single risk factor x is approximately given by Taylor's theorem: A value % (dPV/dx)Ax + ±(d 2 PV/dx 2 )x 2 (Ax) 2 For an option, especially if it is not too close to maturity, the approximation is quite good. Figure 9.13 shows the effectiveness of the approximation for an ATM option with 1 year and 0.25 years to maturity respectively. The changes in the underlying have to be quite large before the approximation becomes ineffective. Write the above equation as: Avalue % (6PV/dx)x(Ax/x) +

(d2PV/dx2)x2(Ax/x)2

If it is assumed that (Ax/x) is distributed normally, then (Ax/x)2 is chi-squared, and unfortunately Avalue has a distribution which cannot be analytically defined. The standard deviation of the distribution can be estimated relatively easily, but the relationship between the multiplier and probability can only be numerically calculated. Most methods use approximations to estimate the final distribution, although it is possible

A good broad introduction to sampling is Boyle et al., "Monte Carlo methods for securities pricing". Journal of Economic Dynamics & Control, 21, 1997, pp. 1267–1321. See also Stein, "Large scale properties of simulations using Latin hypercube sampling", Technometrics, 29(2), 1987, pp. 143 151.

428

Swaps and Other Derivatives 50403020-

10o80

90

100 110 120 130 140

(a)

-10-20 70

H 80

1 90

1 h H 1 100 110 120 130 140

(b)

Figure 9.13 ATM call option: (a) 0.25 year; (b) 1 year to derive an exact expression19. A numeric method was applied to the FX option portfolio, with the following results:

Portfolio

delta 2,452

99% VaR delta delta-gamma 521,618 1,602,522

The approach took roughly three times longer than the delta method. An alternative approach is to modify the simulation methods. For multiple risk factors, the above equation may be written as: Avalue % 6-Ajc +1 Ajc-F-Ajc where AJC is the vector of changes in the risk factors, and 8 and T the delta vector and gamma matrix for the portfolio. Notice that this expression has no knowledge of individual transactions within the portfolio. If a particular change AJC is observed, Avalue can be calculated extremely quickly. The delta vector (strictly a delta + vector) in Worksheet 9.15 had already been calculated when we estimated the VCV VaR. The gamma matrix elements were estimated by: (a) for the ith risk factor • yii = <5+ — dj (as in the previous chapter) (b) for a pair of different risk factors i and j • assume that Axi = Axj = 1 bp and that all other Axs are zero • therefore, Avalue % <5, + <5y + \(y ij + yji) = 6, + <5, + y,y as y/y = y,7 • i.e. yij = Avalue — ((5, + <5y). The full delta-gamma data are shown in the worksheet. In practice, many risk management systems can not calculate a complete gamma matrix with all the cross pairs, but only a "full gamma" matrix which consists of the leading diagonal with no crosses. There may even be systems that can only calculate gamma with respect to the 19 Which is outside the scope of this book. See Cardenas et al., "VaR: one step beyond". Risk, 10(10), 1997, pp. 72-75.

Worksheet 9.15 Spreadsheet to perform historic simulation VaR for a portfolio of FX options using a range of delta-gamma approximations 1. Sensitivity information of FX option portfolio Delta_up

USD Euro

6m 12m 2yr 6m 12m 2yr Spot

1023.27 -1745.90 -2495.98 -941.82 2455.03 3724.28 433.62

Gamma matrix USD 6m

12m

2yr

Euro 6m

12m

2yr

Spot

-0.380 -0.014 0.000 0.291 0.010 0.000 0.776

-0.014 -0.163 0.011 0.009 0.459 0.249 0.317

0.000 0.011 -0.020 0.000 0.256 0.272 0.108

0.291 0.009 0.000 -0.209 -0.006 0.000 -0.688

0.010 0.459 0.256 -0.006 -0.879 -0.644 -0.501

0.000 0.249 0.272 0.000 -0.644 -0.673 -0.352

0.776 0.317 0.108 -0.688 -0.501 -0.352 -2.219

0.000 0.000 -0.020 0.000 0.000 0.000 0.000

0.000 0.000 0.000 -0.209 0.000 0.000 0.000

0.000 0.000 0.000 0.000 -0.879 0.000 0.000

0.000 0.000 0.000 0.000 0.000 -0.673 0.000

0.000 0.000 0.000 0.000 0.000 0.000 -2.219

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 -2.219

Full (diagonals only) gamma matrix -0.380 0.000 0.000 0.000 0.000 0.000 0.000

0.000 -0.163 0.000 0.000 0.000 0.000 0.000

FX gamma only 0.000 0.000 0.000 0.000 0.000 0.000 0.000

0.000 0.000 0.000 0.000 0.000 0.000 0.000

rftntinupfi

Worksheet 9.15 (continued) Today's date 24-Sep-99 Time horizon = 10 days = VaR 99% FX 1 2 3 4 5 6

0.04 years Maturity date 24-Jun-OO 24-Dec-OO 24-Dec-99 24-Jun-01 24-Apr-00 24-Mar-01

option portfolio (all options sold) 9 month USD call on 10m USD at strike 1.06 15 month USD put on 20m USD at strike 1.04 3 month USD call on 25m USD at strike 1.05 21 month USD put on 15m USD at strike 1.55 7 month USD call on 20m USD at strike 1.065 18 month USD put on 15m USD at strike 1.02 Market data

Rates

USD 6m 5.938%

12m 6.031%

2y 6.205%

Euro 6m 3.125%

0

2y

3.850%

1

2 3 4 5

6 7 8 9 10 11 12 13 14 15 16 17 18 19 20

Euro

12m 5.340% -1.738% -0.913% 1.398% 1.352% 4.560% -1.441% 0.378% 2.603% 1.494% -3.473% 4.417% -6.051% 7.721% -3.348% -1.257% -1.460% 0.595% -2.177% 3.626% 5.348%

2y

6m

Principal m USD 10 20 25 15 20 15

Strike Type (E/$) call 1.06 put 1.04 call 1.05 put 1.55 call 1.065 put 1.02

Volatility pa 20% 20% 20% 20% 20% 20%

Euro/$ 12m 3.313%

2. Historic percentage shifts

USD 6m 6.138% -2.830% 0.997% 1.243% 0.831% 2.457% -2.363% 1.815% 4.028% 1.547% -3.265% 3.579% -7.633% 9.394% -5.357% -0.905% -3.055% 1.319% -0.480% 2.243% 5.654%

Length of time 0.751 1.252 0.249 1.751 0.584 1.499

1.0432 3. Shifted market data — defined as USD Euro 6m 12m 2y 6m -5.105% 36.44 0.74 32.21 29.55 6.533% -16.80 -10.48 0.90 -7.37 -4.608% 5.92 -5.51 -13.09 4.78 0.426% 7.38 8.43 3.29 7.98 -4.213% 4.93 8.16 6.90 1.81 0.263% 14.59 27.50 19.03 12.96 8.148% -8.69 -9.94 17.21 -14.03 -0.990% 10.78 2.28 -4.31 2.81 23.91 -2.570% 15.70 17.68 0.93 -1.730% 9.19 9.01 18.79 -17.74 -0.739% -19.38 -20.95 -20.81 -8.61 -7.634% 21.25 26.64 20.69 5.33 9.890% -45.32 -36.49 -27.95 1.23 -0.401% 55.78 46.56 41.69 23.20 8.202% -31.81 -20.19 -10.60 3.58 3.891% -5.38 0.94 -7.58 -0.95 -18.14 -8.80 -4.79 5.603% 4.82 1 .402% 3.59 4.31 4.20 7.83 -1.927% -13.13 -13.71 -2.85 4.23 -2.789% 13.32 21.87 18.41 7.19 -5.699% 33.57 32.25 24.78 4.74 Euro/$

12m

2y

4.762% 0.236% 0.459% 0.868% 0.145% -2.359% -0.206% 2.059% -2.110% 1.530% -0.948% -1.290% 0.531% 2.552% 0.539% -0.655% 1.113% 0.580% 1.771% 0.274% 3.067% 4.146% 3.804% -0.124% -1.602% 5.507% 4.426% 3.095% -0.695% 0.901% -2.657% -2.050% 2.849% 0.299% 0.717% 1.787% 3.029% -5.675% -2.976% -0.845% -3.353% -2.755% -2.048% -1.566% 3.335% 1.706% 1.884% -0.334% -4.504% 0.394% 0.339% 0.233% 6.720% 7.425% 5.436% 5.171% -1.708% 1.146% 3.297% 4.026% -0.153% 0.302% 0.120% 0.389% -0.772% 1.542% 0.714% -0.757% 0.695% 1.345% 2.160% 2.869% -2.210% 1.354% 1.313% 2.825% 2.966% 2.301% 2.558% 1.197% 3.994% 1.516% 0.489% 0.325%

absolute bp shifts Euro/$ 12m 2y 1.52 3.34 -532.51 -0.68 7.93 681.52 -3.14 -4.97 -480.74 1.78 -2.52 44.43 5.86 1.05 -439.54 12.60 -0.48 27.43 14.66 11.92 849.99 -8.80 -7.89 -103.30 2.38 6.88 -268.08 -9.86 -3.25 -180.45 -6.79 -6.03 -77.07 6.24 -1.29 -796.42 1.12 0.90 1031.74 18.01 19.91 -41.84 10.92 15.50 855.63 0.40 1.50 405.92 2.37 -2.92 584.56 7.15 11.05 146.30 4.35 10.88 -200.99 8.47 4.61 -290.96 1.62 1.25 -594.50

Imperfect Risk Management

O

O

431

432

Swaps and Other Derivatives

underlying alone (as commonly defined in option textbooks). The results below were also calculated for these situations. The worksheet uses the historic simulation data, and calculates the absolute change in basis points by 10,000 xrcurrent x %rchange as shown in Box 3. Four sets of calculations are done and shown in Box 4 (not printed): 1. 2. 3. 4.

using a delta approximation only, i.e. ignoring the gamma term; using delta plus the complete gamma matrix as above; using delta plus "full" gamma matrix; using delta plus FX gamma only.

The results are as follows—see also Figure 9.14 for the resulting histograms:

99% VaR

Delta approximation Delta-gamma approximation Delta-full gamma approximation Delta-FX gamma approximation

—548.001 —1,653,146 —1,636,911 —1,636,270

As expected, the delta approximation closely replicates the delta method whilst the three gamma methods produce VaRs similar to the full simulation. Indeed there is little different between the three which suggests that in many situations a full gamma approach is likely to be adequate. Obviously the individual options do not have to be revalued, and therefore this simulation is very much quicker than a full simulation, especially when applied to a much larger portfolio. In summary delta-gamma methods do appear to provide considerably increased accuracy over delta methods, and yet are much faster than full simulations. However they will only work when the third-order effects are relatively small. One approach that has been applied to overcome these effects is to use "gridpoint" approximations, i.e. instead of using a single delta and gamma for all changes, to use different deltas and gammas at different gridpoints20. However, for portfolios that contain significant third-order effects due to high gearing or discontinuous payouts, there is little substitute for full simulation. We have concentrated on the deltas and gammas of a portfolio. It was mentioned above that increasingly for option portfolios, volatility is being used as a risk factor in its own right. The delta-gamma approximations may easily be extended to include a vega term if necessary. Finally, there are a range of other approaches that are used to speed up the calculations. Usually the portfolio across an entire bank's operations is likely to consist of mainly "linear" transactions, especially if we worked in terms of discount bond prices, and only a relatively small residual will have significant gamma. For example, most banks would transact loans and deposits, buy bonds, enter into swaps and FX agreements, and only a very small proportion of business would possess non-linear or optionality properties. A delta approach would be sufficiently accurate for such linear transactions, and simulation 20

See, for example, M. Pritsker. Evaluating VaR methodologies. Chapter 27 in Understanding and Applying VaR. Risk Books, 1997.

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required only for the non-linear. The trick of course is to combine the VaR estimates together in some fashion, and usually the VaR-delta21 is used as a first-order estimate.

9.11 SPREAD VaR Credit-sensitive trading has significantly increased over the past few years: see, for example, the rapid growth in the Eurozone corporate bond market. With issuance in the governmental bond markets declining since 1998 across Western Europe and in the US, corporate bonds are now frequently quoted as spreads over some inter-bank curve, rather than spreads over governments. When calculating the hedges in the previous chapter, we explicitly assumed that the bond-swap spread remained constant. Of course in practice this isn't true, and the changing spreads represent a source of risk22. We will analyse a small (artificial) example, and show how the spread risk may be isolated: Details of bond portfolio Today's date:

24-Sep-99

Bond

Maturity

Ann Coupon

Dirty Price

Valuation Curve

1 2 3 4 5 6 7 8 9 10

29-Mar-04 12-Jun-02 10-Dec-Ol 13-Sep-02 1 7-Jun-03 19-Feb-Ol 5-Sep-03 8-Jul-02 25-May-04 10-Nov-03

5% 6.50% 8.25% 7% 4.50% 5.50% 6.75% 4.75% 9.50% 7%

98.61 100.49 104.23 93.48 99.68 102.98 98.49 100.60 99.38 104.90

Libor Libor Libor Libor Bond Libor Libor Bond Libor Libor

1 2 3 3 1 2 3 2

There are 10 bonds in the portfolio, valued off four different curves. The bond curve is the base curve, and the Libor curves are increasing spreads over it. The current levels of each curve are known, together with volatility and correlation histories: Current curves

iy

3v

5y

Bond 4.688% 4.731% 4.855%

Libor 1 5.438% 5.581% 5.805%

Libor 2 6.438% 6.781% 7.205%

Libor 3 8.438% 9.181% 10.205%

The total 1 day, 95% VaR of the portfolio may easily be calculated in the usual way as follows: 21

See M. Carman, Improving on VaR, Risk, 9(5), 1996, 61-63. This is termed "specific risk" in the banking capital regulations; it represents the risk that the credit quality of an issuer of a security may decline so that the market price is adversely affected. The Eastern European crash of 1998 highlighted specific risk as a source of market risk. 22

434

Swaps and Other Derivatives

• calculate the sensitivities of each bond with respect to its valuation curve; • sum the sensitivities for each curve to get the total sensitivity with respect to each curve; • apply the delta VaR approach in the usual way, using the data in Box 1 of Worksheet 9.17. Box 1 of Worksheet 9.16 shows the detailed calculation: total VaR = $59.168 million. But where did this VaR come from? Define Vi as the vector of individual VaRs from the ith curve, and Cii as the corresponding 3x3 correlation submatrix. We can calculate the VaR due to each one of the individual curves moving; for example, the VaR due to Libor 3 curve moving is {V 3 C 3 3 V 3 } 1 / 2 = $32.486 million. We can also calculate the contribution due to the interaction between the curves by using {2Vi-CijVj}: note that this number is not necessarily non-negative as Cij, is not a proper positive semidefinite correlation matrix, and therefore a square root cannot be taken. The results are shown in Box 2. The total contribution of the four curves may be calculated in a variety of ways; in the box, all the interaction VaR was allocated to the curve with the lower credit. The problem with this type of analysis is that we can say nothing about the source of the movement: was it due to the bond curve or to one of the spreads? The difficulty is that the Libor curve implicitly includes the bond and spread curves. In order to be able to break the VaR down into its fundamental components, we need to transform the world from {Bond—Libor} into {Bond—Spread} where the spreads are additive over the bond curve. In this world, the effects may easily be separated. Unlike before, the bond valued off the Libor 3 curve will now have sensitivities to the bond curve as well as the three spread curves. The spread properties must first be calculated: • a current spread S is simply the difference between two curves X and Y; • the spread volatilities can be calculated using var(S) = var(X) — 2 cov(X, Y) + var(Y): • the correlations can be calculated from cov(X,S) = cov(X,X)—cov(X,Y), etc. Box 2 of Worksheet 9.17 shows the precise details, and the resulting {Bond—Spread} data. First the total VaR can be recalculated; see Box 3 of Worksheet 9.16. This is of course exactly the same as before: no risk has been generated or removed. However the individual VaRs are quite different; for example, bond curve VaR is now much higher because, when that curve moves, it affects all the bonds. If we look at the components as shown in Box 4, we see that some of the interaction effects are negative, suggesting that there is a negative correlation between some of the components unlike in the {Bond—Libor} world. The percentage contributions are quite different as well, suggesting much greater potential losses if the bond curve moves adversely. But the major advantage to this approach is in a better understanding of the true risks being run. For example, suppose we have a portfolio which is long Libor 3 bonds and short government bonds. We could manipulate the amount of the government bonds so that the net VaR off the bond curve is very low. However this hedge will have no effect if any of the spreads shifted, so our VaR report should still show relatively high VaR for the spread risk.

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Worksheet 9.16

Spreadsheet to calculate VaR off different sets of curves

1. VaR analysis: Bond — Libor k=1.65 Bond

1y

3y 5y

Liborl

ly

3y 5y

Libor2

ly

3y 5y

Libor3

ly

3y 5y

Rate

Vol

Delta

VaR

4.6875% 4.7313% 4.8550% 5.4375% 5.5813% 5.8050% 6.4375% 6.7813% 7.2050% 8.4375% 9.1813% 10.2050%

18.0% 16.5% 14.0% 20.0% 18.5% 16.0% 24.0% 22.5% 20.0% 29.0% 27.5% 25.0%

-0.003722 -0.045606 -0.012703 -0.011066 -0.014313 -0.028942 -0.006256 -0.057762 -0.034639 -0.011365 -0.046256 -0.027921

-0.518 -5.874 -1.425 -1.986 -2.439 -4.435 -1.595 –14.542 -8.236 -4.589 -19.270 -11.754 59.168

Total VaR

2. Individual VaR off Libor curves VaR Bond Liborl Libor2 Libor3

7.705 8.342 23.640 32.486

VaR2 59.369 69.593 558.859 1,055.334

Interactions between curves Bond Bond Bond Liborl Liborl Libor2

and and and and and and

Liborl Libor2 Libor3 Libor2 Libor3 Libor3 Total VaR 59.168

Contribution Percentage 59.369 153.553 747.808 2,540.082 3,500.812

83.960 100.652 116.256 88.298 158.192 1,210.300 3,500.812

2% 4% 21% 73%

436

Swaps and Other Derivatives

3. VaR analysis: Bond — Spread

iy

Bond

3y 5y

iy

Spread 1

3y 5y

ly

Spread2

3y 5y

ly

Spreads

3y 5y

Rate

Vol

Delta

VaR

4.6875% 4.7313% 4.8550% 0.7500% 0.8500% 0.9500% 1.0000% 1.2000% 1.4000% 2.0000% 2.4000% 3.0000%

18.0% 16.5% 14.0% 104.1% 87.0% 79.3% 170.2% 134.5% 110.7% 82.2% 75.8% 58.4%

-0.032409 -0.163937 -0.104205 -0.028687 –0.118331 -0.091502 -0.017621 -0.104018 -0.062560 -0.011365 -0.046256 -0.027921

-4.512 -21.116 –11.687 -3.697 -14.441 –11.370 -4.949 -27.709 –16.004 -3.083 -13.885 -8.073 59.168

Total VaR

4. Individual VaR off spread curves Bond Spread 1 Spread2 Spread3

VaR

VaR~2

36.436 28.217 47.020 20.811

1,327.590 796.195 2,210.835 433.087

Interactions between curves Bond Bond Bond Spread 1 Spread 1 Spread2

and and and and and and

1,327.590 535.882 642.420 994.920 3,500.812

Spread 1 Spread2 Spread3 Spread2 Spread3 Spread3 Total VaR

Contribution Percentage

-260.313 -544.736 135.425 -1,023.679 250.488 175.920 59.168

3,500.812

38% 15% 18% 28%

Worksheet 9.17 Market volatility and correlation information 1. Market data between bond curve and three Libor curves Bond ly 3y 5y Libor 1 ly 3y 5y Libor2 ly 3y 5y Libor3 ly 3y 5y

Z-c rates

Volatilities

4.688% 4.731% 4.855% 5.438% 5.581% 5.805% 6.438% 6.781% 7.205% 8.438% 9.181% 10.205%

18.0% 16.5% 14.0% 20.0% 18.5% 16.0% 24.0% 22.5% 20.0% 29.0% 27.5% 25.0%

Volatilities correlation matrix between bond and 3 Libor curves

Bond pa

Liborl

Libor2

Libor3

ly

3y

5y

ly

3y

5y

ly

3y

5y

ly

3y

5y

Bond ly 3y 5v

18% 17% 14%

1 0.9 0.8

0.9 1 0.95

0.8 0.95 1

0.7 0.65 0.5

0.65 0.7 0.65

0.45 0.55 0.6

0.3 0.3 0.2

0.2 0.3 0.3

0.2 0.2 0.25

0.2 0.2 0.15

0.15 0.25 0.25

0.15 0.15 0.2

Liborl ly 3y 5y Libor2 ly 3y 5y

20% 19% 16% 24% 23% 20%

0.7 0.65 0.45 0.3 0.2 0.2

0.65 0.7 0.55

1 0.85 0.7 0.2 0.15 0.1

0.85 1 0.9

0.7 0.9 1

0.2 0.2 0.15

0.15 0.25 0.25

0.1 0.17 0.2

0.25 0.3 0.2

0.2 0.3 0.3

0.2 0.15 0.25

0.3 0.3 0.2

0.5 0.65 0.6 0.2 0.3 0.25

0.2 0.25 0.17

0.15 0.25 0.2

1 0.9 0.75

0.9 1 0.9

0.75 0.9 1

0.6 0.7 0.65

0.5 0.65 0.75

Libor3 ly 3y 5y

29% 28% 25%

0.2 0.15 0.15

0.2 0.25 0.15

0.15 0.25 0.2

0.25 0.2 0.2

0.3 0.3 0.15

0.2 0.3 0.25

0.75 0.6 0.5

0.85 0.7 0.65

0.85 0.65 0.75

0.75 0.85 0.85 1 0.65 0.65

0.65 1 0.75

0.65 0.75 1

Worksheet 9.17

Volatility and correlation information (continued)

2. Market data between bond curve and three spread curves Z-c rates

Volatilities

4.688% 4.731% 4.855% 0.750% 0.850% 0.950% 1.000% 1.200% 1.400% 2.000% 2.400% 3.000%

18.0% 16.5% 14.0% 104.1% 87.0% 79.3% 170.2% 134.5% 110.7% 82.2% 75.8% 58.4%

Bond ly

3y 5y Spread 1 ly

3y 5y Spread2 ly

3y 5y Spread3 ly 3y

5y

Correlation matrix between bond and 3 spread curves

Bl B2 B3

S211

Bl

B2

B3

S11

S12

S13

1 0.90 0.80

0.90 1 0.95

0.80 0.95 1

–0.11 –0.07 –0.17

–0.04 –0.08 –0.10

–0.17 –0.18 –0.16

-0.17 -0.14 -0.14

1 0.74 0.75

0.74 1 0.92

0.75 0.92 1

-0.45 -0.35 -0.26

S11 S12 S13

-0.11 -0.04 -0.17

-0.07 -0.08 -0.18

-0.17 -0.10 -0.16

S211 S212 S213

-0.17 -0.23 -0.08

-0.14 -0.16 -0.14

-0.14 -0.13 -0.13

-0.45 -0.31 -0.36

-0.35 -0.39 -0.38

S212

S213

S321

S322

S323

-0.23 -0.16 -0.13

-0.08 -0.14 -0.13

0.02 0.02 0.04

0.04 0.10 0.10

0.05 0.05 0.09

–0.31 -0.39 -0.30

-0.36 -0.38 -0.39

0.24 0.34 0.16

0.17 0.19 0.17

0.23 0.05 0.17

-0.26 -0.30 -0.39

1 0.91 0.76

0.91 1 0.89

0.76 0.89 1

0.04 0.23 0.43

-0.03 -0.01 0.01

-0.03 0.14 0.13

0.16 0.17 0.17

0.04 -0.03 -0.03

0.23 -0.01 0.14

0.43 0.01 0.13

1 0.21 0.26

0.21 1 0.60

0.26 0.60 1

C/3

73 (/> V

S321 S322 S323

0.02 0.04 0.05

0.02 0.10 0.05

0.04 0.10 0.09

0.24 0.17 0.23

0.34 0.19 0.05

0.

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Imperfect Risk Management

439

9.12 EQUITY VaR Finally, a brief look at calculating VaR when there are equities (or indeed commodities) in the portfolio. Equities may be handled quite simply by treating each one as a separate risk factor. For example, consider the following simple USD portfolio:

Holding 100,000 500,000 1 ,000.000

Stock 1 Stock 2 Index Equity forward Total (USD) Total (DEM) 3mo. USD Libor USD -DEM spot rate

Current price 10 4 5

Current value (USD) 1,000,000.00 2,000,000.00 5,000,000.00 -896,805.90 7,103,194.10 12,785.749.39

7% 1.8

It consists of two stocks, a holding of the index, and an equity forward contract to pay $12 per share on 500,000 shares of Stock 1 in 3 months' time. Assuming (quite simplistically) zero growth in the share price, the value of the forward is 500,000 x [10-12xDF3]. We wish to calculate 1 day, 95% VaR in DEM. We therefore have five risk factors: the two stock prices, the index price, 3mo. Libor and the spot rate. Given appropriate volatilities and correlations, it is straightforward to calculate VaR = $712,472 (see Worksheet 9.18). However using individual stocks may increase the data requirements significantly. Consider for example a single portfolio replicating the S&P 500: the number of crosscorrelations is in excess of 100,000! For a bank in which equity constitutes a significant proportion of activity, the accuracy provided by modelling the individual stocks may well warrant the time and cost of collecting and cleansing the data. But for many organizations, the effort is simply not worthwhile. Beta analysis of the equity market is very common, with beta defined in: rs = a,. + ßs r1 + ex

where rs, and r1 are the return on a share s and on the index / respectively; as is excess return on shares (in theory, this should be zero); ßs is the coefficient linking share performance to the index; ss is an error term, assumed to be uncorrelated with either the market or the other stocks. We can therefore write23: var(r s ) = (ß s ) 2 var(r 1 ) + var(s) If we can assume that var(es) is negligible, then cr(s) = ßs x a(I). We can therefore replace the individual stocks in the above example by the index, i.e.: VaR(equity) = {<5, x / + dl x Sl x ß1 + <52 x S2 x ß2,} x < r ( I ) x 1.65 21

This may easily be generalized for a portfolio: if rp = w r then: var(r p ) = w'-C,.-w = (w'-ß-ß-w) x var(r I ) + w' D, w

where D, is a diagonal matrix of var(e ).

Worksheet 9.18

Portfolio:

Equity VaR: treating stocks and index as separate factors

Stock 1 Stock 2 Index Equity forward

All stock in USD Holding 100,000 500,000 1,000,000

Total (USD) Total (DEM) Current FX

Current Price Current Value (USD) 1,000,000 10.00 2,000,000 4 5,000,000 5 -896,806 to receive 500,000 shares of stock 1 in 3 months' time, and pay 12 per share 7,103,194 12,785,749 7% 0.982801 1.8

3m Libor 3m DF DEM/USD

multiplier Factors Stock 1 Stock 2 Stock Index 3m Libor FX

Volatility (PD) 1.26% 2.53% 2.21% 0.95% 1.58%

Level 10 4 5 7% 1.8

Sensitivity (DEM) 1,080,000.00 per 1 unit increase 900,000.00 per 1 unit increase 1 ,800,000.00 per 1 unit increase 260.79 per increase of 1 bp 71,031.94 per 0.01 increase

1.650 delta VaR 225,407 150,271 328,719 2,858 333,565 712,472

Notes: Stocks 1 & 2 and the Index are all treated individually as separate factors, i.e. not using the betas. This means that we need the individual vols to calculate the individual VaR and also would require a correlation matrix for all factors.

I

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Imperfect Risk Management

where dr is the sensitivity with respect to the index, etc. Continuing the example, we have now reduced the portfolio down to effectively three factors—see Worksheet 9.19: Factors

Beta

Stock 1 Stock 2 Stock Index 3m Libor FX Total VaR

0.75 1.25

VaR individual stocks 225,407.15 150,271.43 328,718.76 2,857.51 333,564.74 712,472.06

VaR beta analysis 788,925.03 2,857.51 333,564.74 885,693.46

Depending upon a single factor to represent each stock is of course a gross approximation, but it is a trade-off between accuracy and data availability. It is feasible to extend the above analysis into using more factors24 to represent the behaviour of the individual stocks, but such methods are not so widely accepted.

9.13 STRESS TESTING VaR measures risk under "normal" market conditions. All the methods described ultimately use historic information, either directly through simulation or indirectly through the calculation of volatilities and correlations, and therefore are tacitly suggesting that the future will resemble the past. If the past period had low market volatility, then obviously the VaR estimates of the future would reflect that. There is a definite user and regulatory requirement for additional "stress" testing, i.e. revaluation under extreme movements of market factors. The requirement for this has become more evident rather than less over the past few years. This section will briefly describe some of the approaches used in stress testing, and then discuss its general effectiveness. Stress testing may be done in a number of ways. 1. By changing the current level of market factors. For example, shifting each factor by some (positive or negative) multiple of its volatility, i.e. fnew = f old exp{+ma} ^ fold(l + ma) where a is the time horizon adjusted volatility. An alternative is to make an absolute shift in the factor level, but this may result in negative factors. The selected portfolio could then be revalued using the shifted factors, and the change in value reported. The above, whilst sounding simple, presents difficulties in practice because the shifts completely ignore the existence of correlation. So a more realistic method is to subdivide the factors into two sets. The first small set consists of "important" mainly independent factors which may be shifted as above. The second, much larger, set consists of factors that will change as a result of the correlated relationships. For example, a factor in set 1 could be the 5yr swap rate; then all the other swap rates are put into set 2 as they are highly correlated with the 5yr. Their shift would then be the same as that of the 5yr rate. For example: let Fl and F2 be the two sets. Assume that the covariance matrix C can be partitioned into submatrices (C11 corresponding to F1, n x n; C22 corresponding to F2, 24 See, for example, The arbitrage pricing theory approach to strategic portfolio planning, Financial Analysts Journal, May/June, 1984, 14–26 or M. A. Berry et al.. Sorting out risks using known APT factors. Financial Analysts Journal, March/April, 1988, 29–42.

Worksheet 9.19

Portfolio:

Equity VaR: treating stocks and index as the same factor

Stock 1 Stock 2 Index Equity forward Total (USD) Total (DEM)

All stock in USD Holding 100,000 500,000 1,000,000

Current Price 10 4 5

Original value Current

FX

Current Value (USD) Beta 1,000,000 0.75 2,000,000 1.25 5,000,000 -896,806 7,103,194 12,785,749 12,785,749

7%

3m Libor 3m DF DEM/USD

0.982801 1.80 multiplier

Factors Stock 1 Stock 2 Stock Index 3m Libor

FX

Volatility (PD)

Level

10 4 2.21% 0.95% 1.58%

5 7% 1.8

Sensitivity (DEM) 1 ,080,000.00 per 1 unit increase 900,000.00 per 1 unit increase 1,800,000.00 per 1 unit increase 260.79 per increase of 1 bp 71,031.94 per 0.01 increase

1.650 delta VaR

788,925 2,858 333,565 885,693

C/5

65 g.

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443

m x m; C12 = C21 the cross matrices, n x m and m x n}. So we could: • generate the set of percentage changes R1 for set F1; • calculate the changes for F2 by R2 = C21 •C]-111-R\ • 2. Changing the levels of the factors is the main form of stress testing, primarily because these are events that can happen very rapidly. But it is also useful to stress test the volatilities and correlations in a consistent fashion. As they are averages, they react more slowly than levels, but can have significant effects. Shifting volatility is also equivalent to changing the time horizon, which is itself a surrogate for liquidity. It is probably sensible to use the same subdivision as above. Shift the volatilities in set 1, and then interpolate/extrapolate in some fashion to estimate the shift in the volatilities in set 2. 3. Shifting the factors or the volatilities by large amounts is likely to increase the risk measures but probably gain few other insights. The impact of stressing the correlation matrix on the other hand is far less intuitive, and may reveal a range of unexpected events. Unfortunately this form of stress testing is also the most complicated because we have to ensure that the resulting, stressed matrix is still a well-specified correlation matrix, i.e. it is still positive definite. For example, suppose that we partition the correlation matrix £7 as {£2aa, £^/>, Qha, Qhh} where the submatrix Qaa is to be stressed: • let Q?aa be the new stressed matrix; we assume it is itself well-specified; • it is perfectly feasible however to find that £ls = {£2saa, fin/,, ft/,,,, Q/,/,} is not wellspecified; • so that either £lsaa or the cross matrices fia/) = Qhll will have to be adjusted until ST2S is well-specified; • Kupiec25 has suggested a simple (albeit potentially computer-intensive) algorithm when Qs is not well-specified; o define Qsr = (1 - c)ttv + cQ, for 0 < c < 1; o where Qs° is not well-specified, whilst £2 vl is well-specified; o there is likely to exist a positive value of c so that £2SC is well-specified. Selecting the actual correlation shifts is less straightforward. Percentage shifts can only apply, and these must be bounded to ensure that the correlations lie between ±1 at all times. It may be better to consider shift strategies such as shifting all correlations closer to zero or to ±1, or even setting the correlations equal to the extremes. Many organizations perform scenario analysis under the name of stress testing. Scenarios are typically taken from the extreme movements in the past such as the Far Eastern crash of 1997 or the collapse of the Eastern European markets in 1998. Alternatively, as such crises will never repeat themselves exactly, hypothetical future events such as China defaulting or a US president being assassinated could be used. Scenario analysis is often combined with the historic simulation over a short period of time with extreme moves. However scenario analysis can be very time-consuming to create for a global bank, and severe doubts have been expressed as to whether it is really worthwhile26. 25

P. H. Kupiec, Stress testing in a VaR framework, Journal of Derivatives, 6(1), 1998, 7-24. "See, for example, Shaw, "Beyond VaR and stress testing", in Understanding and Applying VaR. Risk, 1997. pp.211–224.

444

Swaps and Other Derivatives

APPENDIX: EXTREME VALUE THEORY Suppose there is a set of observations zi-, i = 1, . . ., nt from a simulation such that zi ^ a defined threshold t27. Define yi = zi — t > 0.

Peaks Over Threshold: Negative Exponential The negative exponential distribution has a density function:

This can be interpreted as: g(y) is the likelihood of y actually being observed as a single sample drawn from a negative exponential distribution with parameter ft. The likelihood function is similarly defined as:

LF{y1, y2, y3 • • ., yn|ß} = g(y1)g(y2)g(y3) • • g(y n ) The process of maximum likelihood estimation is to find the value of ft that maximizes LF, i.e. the most likely distribution for the observed samples. The usual method is to find ft such that:

=0 Very often it is easier to work with ln(LF), as: dln(LF)/dß? = and therefore any value of ß that sets dLF/dß to zero will also set dln(LF)/dß. Since:

the (logarithmic) likelihood function is: ln(LF) = -nt, In ß–(1/ß)£,..>>, Differentiating with respect to ft and setting to zero, we get the optimal value of ft = (l/nt)$2/.yi. We could now calculate a VaR off the negative exponential by using the cumulative probability distribution: G(u) = prob{0 < y < u} = 1 — exp{— u/ß} for some u > 0 i.e. uq = —ß ln(q) where q is the probability of exceeding uq = 1 — G(uq). But we need to convert this result back into the z- world. We can rewrite prob{0 < y < u} as prob{t < z < u + t} given that z > t, by simply substituting for y. If we represent the cumulative z-distribution as F, this probability may also be written as: {F(u + t) - F(0)}/prob(z >t) = {F(u + t) - F(t)}/{1 – F(t)} Therefore: F(u + t = F(t) + G(u){1 - F(t)} 27

Note that we are dealing with the positive, right-hand tail of observations.

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F(t) is simply the probability that an observation will not be in the tail. Our best estimate for this is {1 — n t N } where N is the total number of observations in the simulation. Substituting, we get: F(z q ) = 1 - (nt/N) exp{-(zq -t

or )(l - F(zq))} = t -

Peaks Over Threshold: Generalized Pareto28

The generalized Pareto distribution has a cumulative density function as follows:

where X is a "tail" parameter. If X = 0, then the GP reduces to a negative exponential. If X > 0, the GP has a fatter tail than the negative exponential, if X < 0 then a thinner tail. So it simply allows a second parameter to fit the observations more accurately. The density function is:

The (logarithmic) likelihood function is: LF = -nt In ß- (1 + l/X)£/In(1 +ry/) where

X/ß.

By setting the two differentials, dLF/dß and dLF/d%, to zero, we get: ft = {!>/}/{«; - * £/««} =/iW X = (1/nt,)£, ln(l + TJ;,.) =/ 2 (t)

where M/ = y,/(\ + T>';)

(A9.1) (A9.2)

which may be solved iteratively. However the solution is not well behaved, and a pragmatic two-step alternative that works well in practice is to estimate ß using the negative exponential formula, and then improve the tail fit by calculating x by maximizing LF. Once x and ß have been estimated, the q% VaR estimate may be calculated from: VaRq - t + (ß/X){[(ni,/N)(l - q)]-y - 1} using the same argument as above. Other estimates may also be calculated. The mean excess function for estimating expected losses given a particular level L has been exceeded is given by ESL defined as: ESL = L + E{y - L|y > L} = {L + ß - Xt}/{1 – X} This formula applies when X — 0, i.e. for a negative exponential. Of particular interest is the expected loss when the VaR is exceeded, i.e.: ESq = { VaRq + 0 28 See "EVT for Risk Management", working paper by A. J. McNeil, ETH Zentrum, May 1999.

446

Swaps and Other Derivatives

Block Maxima This is based upon the statistics of extremes, also known as order statistics. Consider a distribution, say a normal one. Assume that you take k random samples from that distribution zi-, i = 1 , . . . , & and assume that they have been placed in order so that Z1 > Z[1]> z[2] > — We then repeat the whole process, generating the largest sample Z2 for the second set, and so on. We end up with a set of observations of maxima Zj where j is the sample set. Order statistics is the study of these maxima, and in particular what can be said that is independent of the original underlying distribution. It has been found that these maxima asymptotically follow a generalized extreme value distribution29:

where x = (Z — m)/s is a normalized variable, and a is a "shape" parameter. If: a ^ 0 the distribution is called a "Frechet" where 1 + a min{ yi} ^ 0 a < 0 the distribution is called a "Weibull" a = 0 the distribution is called a "Gumbel" simplifying to H(y) = e x p { e - y } and its density function is: h(x) = exp(-[l + ax]-1/3t)[l + ax] - ( 1 + l / x ) (l/s) Trying to estimate a likelihood function for all three parameters simultaneously is complicated by non-linearities, and a common approach is to adopt a semiparametric method, i.e. estimate a somewhat crudely first, and then fit m and s by regression or maximum likelihood methods. For example, given the set of extreme losses Zi, i = 1 , . . . , « from the n blocks, reorder them so that Z[j]> Z[j+1] A Hill estimator for a is given by {average[ln Z[1], . . ., In Z[p]] — In Z[p]} for some threshold p30. There is little guidance for the setting of p, but we can use increasing values until a stabilizes. The fitted distribution can provide information about the extreme of a set of data. For example, suppose we estimate prob{Z ^ L} for some cut-off L. Now Z is the extreme value drawn from a set of samples, i.e. {Z > z1,, Z ^ z2, . . ., Z ^ z<.} where zi, is the ith observation in the set. If we assume that the observations are independent, then prob {z>L} = prob{Z > L}k.

29

See, for example, Chapter 14 in Stuart et al., Kendall's Advanced Theory of Statistics, Griffin. 5th Edn. 1987. For their application in risk management, see the articles by Dowd on EVT in Financial Engineering News, issues 11 to 13, 1999 plus references. This is probably the most popular approach — see Hill. "A simple general approach to inference about the tail of a distribution". Annals of Statistics. 35. 1975. pp. 1163–1173.

Index Note: Page references in bold refer to Worksheets

accrual accounting 88 Actual/365 convention 12 American swaptions 319–20 amortizing 74 anthetic sampling 427 ARCH (AutoRegressive Conditional Heteroskedastic) modelling 274–5 asset packaging 95-106 10 year par asset swap with a bond trading below par 102-3 DP asset swap with a bond trading below par 98 par asset swap with a bond trading below par 99 simple discount asset swap with a bond trading below par 104–5, 107 AUD coupon 258 AutoRegressive Conditional Heteroskedastic (ARCH) modelling 273–4 average rate swaps 129–31 back-to-back loan market 2 backwardation 175 Bankers Trust 307 Barone-Adesi approximation 325 barrier caps 307 basis risk 48 basis swaps 137–42 reference rates 137 Bermudan swaptions 318–19 bid-ask spreads 12 bid-offer spreads 12 big step correlation 391 BIS Accord 89 Bjerksund approximation 325 Black and Scholes model Garman-Kohlhagen variant 319–24, 325 Black-Derman -Toy (BDT) 269–70

Black option pricing model 267, 268–70 Black's model for floorlet 287 blending 59–63 blipping 30, 336 bond calculating yield of par bond 44 convertible 165 Pfandbriefe 357 bond market, relationship between swap market and 43-50 bond repurchase (repo) market 47-8 bootstrapping 52, 416 call-put parity theorems 323–4 cap pricing model 269, 270 Capital Asset Pricing Model (CAPM) 119 capital gain 96 caplet 268 capped loan 300–1 caps 288–96, 290–3 barrier 307 chooser 308 curve 289 mid-curve 294 periodic 308 cash hedge 24 cash stub 25 cash to first futures (CTFF) 25 cashflow mapping 394–9 Cholesky decomposition 410 chooser caps 308 choppy scenarios 416 collar pricing 297 collars 288-96 commercial paper 137 commodity swaps 171–4 comparative advantage 2, 48 bond market and 48

448

comparative advantage (cont.) pricing through 38–41 compound swaps 129 constant maturity swap (CMS) 148-52, 150, 151, 153, 155, 158–9 with convexity adjustment 198–9, 200–1 contango 175 continuously compounded interpolation 25 convenience yield 174 convertible bonds 165 convexity (gamma) effect 30 approaches to 184–9 convexity bias in futures 199-203 general mismatch swap 189-95 measuring 184–203 of swaps 152–6 yield curve swaps 195–9 correlations, estimation of 399–400, 404, 405 CP-Libor rates 140–1 credit-adjusted (CA) DFs 111–14, 112-13 credit-adjusted swap pricing 121–8 credit default swap (CDS) 108, 120 credit risk management models 111 credit spread swaps 114 credit swaps 106–21 cross-currency basis swap (CCBS) 205-7, 212-13, 229-34 cross-currency equity swaps 258-9, 260–3 cross-currency swaps 2-3, 205–65 valuation 241–7, 242–5, 248–53 volume 6 cubic interpolation 14 currency-protected (quanto) swaps 171 curse of dimensionality 427 curve cap 289 curvelock swaps 175 customized average rate swap 132-3 cylinder 294 delta hedging 330, 362 delta-gamma 427–33, 429–31 delta-gamma curve hedge 352–3 delta-gamma hedging 362 delta-gamma-rotational curve hedge 354–5 delta VaR multi-factor 388–91 one-factor 385 desk-top risk management (traditional market risk management) 333–79 diff swaps 230–3,231 digicaplets 296 digital options 299–300 discount bond prices 394 Discount Corporation of New York Futures 372 discounting 11-16 doubles 12

Index D(irty) P(rice) 96–7 dual currency swaps 247–58, 254–7 EIB-TVAswap 39 embedded structures 300–7 embedded swaptions, structures with 317–21 EONIA swaps 134, 135 Equitable Life 299 equity swaps 165-71 equity VaR 435–41,440,442 equity warrants 165 equivalent portfolios 336–8 EU Capital Adequacy Directive 93 Euro swap curve 357 EuroClear 106 Eurodollar futures contracts 18 European Investment Bank-Tennessee Valley Authority (EIB-TVA) swap 3, 4, 10 European OverNight (EON) rate 134 evolution of swap market 8-10 extendibles 315, 316 extreme value theory (EVT) 416, 444–6 block maxima 446 peaks over threshold: generalized Pareto 445 peaks over threshold: negative exponential 444–5 fair breakeven coupon 258 fair price of a swap 20 fairness 13 fat tails 385 Fed funds 137 Fisher effect 165 fixed-fixed cross-currency swaps 234–41, 236–9 fixed-floating cross-currency swaps 220–9 fixed-floating inflation 162–4 fixed-for-floating commodity price swap 171 fixed notional equity swap 170, 171, 172, 173 floating-floating (cross-currency basis swap, CCBS) 205–7, 212–13, 229–34 discounting and 211–21 pricing and hedging 207–11 floating interest 3 floating price against floating interest swap 171 floors 288-96 foreign asset, synthetic, creating 222–4 forward inflation curve 163 forward rate agreements (FRAs) 16–20 forward rate sensitivity 339–40 forward rates 14–16 forward volatilities 277–88, 278–85 Frankfurt Indexed OverNight Average (FIONA) swap 134, 135 Frechet simulation 422, 423, 446 full gamma matrix 432 future valuing 31, 32–3 futures contracts 16–20

Index FX options 321–6 hedging 326–31, 328–9 vanilla 322

ISDA credit derivative documentation 110 revised 110 ISDA Master Agreement 106

gamma effect see convexity effect gamma spike 330 gap analysis 331 GARCH modelling 275, 399 Garman-Kohlhagen variant of Black & Scholes model 319–24, 325 general collateral (GC) rate 47 General Electric Capital Corp (GECC) 304 gilt discount curve 161 greek hedging 182, 363 gridpoint risk management forward rates 346–8 market rates 338 zero coupon rates 341–4 gridpoint sensitivity 347 gridpoints 395 gross market value 6 Gumbel 446

Lehmann Brothers 10 Libor forward curve 137–8 linear interpolation 14 log-linear interpolation 14 long-term FX forward contract (LTFX) 234–41

haircut 47 Heath-Jarrow-Morton (HIM) modelling 186, 264 historic simulation 405–8, 406–7, 421, 424–6, 425 IBM 5 imperfect risk management 379–446 implied forward (IF) method 71 in-arrears swap 128, 130, 190–1, 192 index amortizing 74 inflation (RPI) swaps 156–65, 166–7 hedging of 373-5, 374 inflation-Libor swaps 165 interest rate futures 16–20 interest option portfolios, risk management of 362-73 interest rate envelope modelling 122 interest rate OTC options 267-330 interest rate risk management 336–8 interest (plain vanilla) rate swaps, generic (IRS) 35-63 7 year generic US dollar swap, definition 37 building a blended curve 62–3, 60–1 definition 35–6 implying a discount function 50–62, 54, 58 pricing through comparative advantage 38–41 relationship between bond and swap markets 43–50 relative pricing 41–3 volume 6 interest rate volatility 271–6 interpolation 14

Margrabe spread option model 309 mean excess function 418 mid-curve cap 294 minimum variance hedging 391 modified following day convention 12 money market swap 20 construction of discount curve 28–9 pricing 22–3 pricing off a futures strip 26–7 pricing off a futures strip using future valuing 32 Monte Carlo (MC) simulation 405, 408-10, 410-14, 411-13, 415, 426 multiplier 384 muni curve 139 Muni-Libor swaps 143–5, 143–7 muni swap 146–7 municipal 137 Nelson-Seigel curve 111, 112, 162 New Market Data Worksheet 89 non-generic swaps, pricing and valuation of 65–93 5 year amortizing using hedging swaps 76 using IF 78 using NPA 77 alternative to discounting 85 complex example 75-85 pricing 65-72 using hedging swaps 67, 80–1 using NPA 69, 83, 90–1 swap valuation 85–93 using IF 70,84,86–7,92–3 using IF with reference rate method 73 Nordea Bank 10 notional principal amount (NPA) 37, 71 method 72, 82 off-market 207 one-factor delta VaR 385 order statistics 446 OTC derivative market, current state 7-8 overnight indexed swaps (OIS) 131–7 par non-generic swaps 65

450

par volatilities 277-88 parallel loan market 1 Pareto distribution 420, 421 participation yield curve swap 152 participations 295, 298 peak exposure limit (PEL) 125 peaks over threshold (POT) 416, 419 period date approach 72 periodic caps 308 perpetual swaps 207 Pfandbriefe bonds 359 plain goal programming 369 polynomial splining 160 potential future exposure (PFE) envelope 125 potential swap exposure, modelling 124 Prime swaps 137, 142 Procter & Gamble swap 307 property swaps 174 PV01 report 338 PVBP (present value of a basis point) 338 pyramiding 343 quanto diff swaps 211, 230–3, 232, 264–5 quanto swaps 171 quasi-random sampling 426 Rand Overnight Deposit (ROD) swap 135, 136 realized volatility forward contract 175 receiver's swaption 314 reference rates for basis swaps 137 method 65, 72 repo (bond repurchase) market 47-8 repo transaction 47 retractibles 315, 316 reverse bootstrapping 343 reverse repo 47 risk free DFs 111 risk-neutral pricing models 111 robust programming 370, 371 rollercoaster swap 72–5 RPI swaps 156–65, 166–7 hedging of 373–5,374 sale and buy back market 47 Salomon Brothers 5, 35 same day transaction 12 sensitivity calculations 380 short-term interest rate swaps 11–33 short-term swaps 20–30 simple mismatch swaps 128–9 simulation methods 405–14 shortcomings and extensions 414–27 smile effect 276 sneer effect 276 Solver 162 SONIA 135

Index specific risk 433 spread VaR 433–4,436–7,438–9 spreadlock swap 49, 50, 175 square-root rule 271, 272 static framework methodology 382 static hedging 373 stationarity 416 Stein's algorithm 427 Stensland approximation 325 step-down (amortizing) 74 step-up 72 Sterling OverNight (SON) rate 134 stress testing 441-3 swap curve analysis of 375-7 3-monthly forward curve 54 smoothing forwards 58 swap futures 356–8 swap markets, relationship between bond market and 43-50 swapnote futures 359, 360–1 swaptions 308–15.311,312–13 tailed hedge 30 tangential contracts 203 Taux Moyen Mansual du Marche Monetaire (TMMMM; T4M) 135 Taux Moyen Pondere (TMP) 134 Taylor's theorem 182. 345. 427 T-Bills 137 term repo 48 theta risk 360–2 time value of money 11–16 total return swap (TRS) 108 trading volatility 175 traditional market risk management (desk-top risk management) 333-7 Turbo swaps 193–5, 196–7 two-factor delta VaR 387 USD bond, swapping into floating Yen 222–4 USD Libor, swapping an SA bond into 226–8 USD-Yen FX forward contracts 320 USD-Yen LTFX 236–9 Value-at-Risk (VaR) techniques 382–6 calculation of portfolio and hedge 392–3 delta multi-factor 388-91 one-factor 385 equity 435–41.440.442 extended 386–8 spread 433–4,436–7,438-9 two-factor 387 vanilla FX options 322 variable maturity (VM) structures) 148 variable notional 171

Index

451 variance hedging 181, 183 variance swaps 176 volatility cones 286 volatility, estimation of 399–400, 402-3, 404, 405

volatility surface 276 volatilitv swaps 175-81 Weibull 446 Whaley approximation 325 World Bank 5 World Bank-IBM swap 4, 5, 9

Yen principal repayment 258 Yen swap curve, bootstrapping 209-10, 214–18 Yen-USD diff swap 231 Yen-USD quanto diff swap 232 • ,, •, 343–55, 347, ... . .*? „ yield curve swaps 142-52 zero cost collars 295 zero coupon bootstrapping 52 zero coupon rate sensitivity 342-3

352,