Solutions Manual - A Primer for the Mathematics of Financial Engineering

Contents Preface

IX

Acknowledgments

Xl

o.

1 1 11 11

Mathematical preliminaries 0.1 Solutions to Chapter 0 Exercises. . . . . . . . . . . . . . . .. 0.2 Supplemental Exercises. . . . . . . . . . . . . . . . . . . . .. 0.3 Solutions to Supplemental Exercises. . . . . . . . . . . . . ..

1 Calculus review. Plain vanilla options. 17 1. 1 Solutions to Chapter 1 Exercises. . . . . . . . . . . . . . . .. 17 1. 2 Supplemental Exercises. . . . . . . . . . . . . . . . . . . . .. 32 1.3 Solutions to Supplemental Exercises . . . . . . . . . . . . . .. 33 2 Numerical integration. Interest Rates. Bonds. 45 2.1 Solutions to Chapter 2 Exercises. . . . . . . . . . . . . . . .. 45 2.2 Supplemental Exercises. . . . . . . . . . . . . . . . . . . . .. 57 2.3 Solutions to Supplemental Exercises. . . . . . . . . . . . . .. 58 3 Probability concepts. Black-Scholes formula. Hedging. 3.1 Solutions to Chapter 3 Exercises. . . . . . . . 3.2 Supplemental Exercises. . . . . . . . . . . . . 3.3 Solutions to Supplemental Exercises. . . . . .

Greeks and . . . . . . . .. . . . . . . . .. . . . . . . . ..

63 63 82 83

4 Lognormal variables. Risk-neutral pricing. 91 4.1 Solutions to Chapter 4 Exercises. . . . . . . . . . . . . . . .. 91 4.2 Supplemental Exercises. . . . . . . . . . . . . . . . . . . . . . 106 4.3 Solutions to Supplemental Exercises. . . . . . . . . . . . . . . 107 Vll

CONTENTS

Vl1l

7

8

Taylor series. Solutions to Chapter 5 Exercises. . Supplemental Exercises. . . . Solutions to Supplemental Exercises.

吨E - 4

5.1 5.2 5.3 6

咱E 4 4 1 4 4 l 4

Taylor ￥ formula.

1i1 土门，中。，"

5

3346

Finite Differences. Black-Scholes PDE. 135 6.1 Solutions to Chapter 6 Exercises. . . . . . . . . . . . . . . . . 135 6.2 Supplemental Exercises . . . . . . . . . . . . . . . . . . . . . . 150 6.3 Solutions to Supplemental Exercises. . . . . . . . . . . . . . . 151 孔1ultivariable calculus: chain rule , integration tion , and extrema. 7.1 Solutions to Chapter 7 Exercises. . . . . . . . 7.2 Supplemental Exercises. . . . . . . . . . . . . 7.3 Solutions to Supplemental Exercises. . . . . .

Lagrange multipliers. Newton's method. ity. Bootstrapping. 8.1 Solutions to Chapter 8 Exercises. . . . . . 8.2 Supplemental Exercises. . . . . . . . . . . 8.3 Solutions to Supplemental Exercises. . . .

Bibliography

by substitu161 . . . . . . . . . 161 . . . . . . . . . 171 . . . . . . . . . 173

Implied volatil179 . . . . . . . . . . . 179 . . . . . . . . . . . 197 . . . . . . . . . . . 198 203

Preface The addition of this Solutions Ivlanual to "A Primer for the Mathematics of Financial Engineering" offers the reader the opportunity to undertake rigorous self-study of the mathematical topics presented in the Ivlath Primer , with the goal of achieving a deeper understanding of the financial applications therein. Every exercise from the Math Primer is solved in detail in the Solutions Manual. Over 50 new exercises are included , and complete solutions to these supplemental exercises are provided. 如1any of these exercises are quite challenging and offer insight that promises to be most useful in further financial engineering studies as well as job interviews. Using the Solution Manual as a companion to the Ivlath Primer , the reader will be able to not only bridge any gaps in knowledge but will also glean a more advanced perspective on financial applications by studying the supplemental exercises and their solutions. The Solutions Ivlanual will be an important resource for prospective financial engineering graduate students. Studying the material from the Math Primer in tandem with the Solutions Manual would provide the solid mathematical background required for successful graduate studies. The author has been the Director of the Baruch College MFE Program 1 since its inception in 2002. Over 90 percent of the graduates of the Baruch IvIFE Program are currently employed i口 the financial industry. "A Primer for the Ivlathematics of Financial Engineering" and this Solutions Manual are the first books to appear in the Financial Engineering Advanced Backgrou日d Series. Books on Numerical Linear Algebra , on Probability, and on Differential Equations for financial engineering applications are forthcommg. Dan Stefanica New York , 2008 lBaruch MFE Program web page: http://www.baruch.cuny.edu/math/master 山tml QuantNetwork student forum web page: httr旷/ www.quantnet.org/forum/index 抖lp IX

Acknowledgments "A Primer for the Mathematics of Financial Engineering" published in April 2008 , is based on material covered in a mathematics refresher course I taught to students entering the Financial Engineering Masters Program at Baruch College. Using the book as the primary text in the July 2008 refresher course was a challenging but exceptionally rewarding experience. The students from the 2008 cohort of the Baruch WIFE Program who took that course were driven to master the material and creatively incorporate ideas at an even deeper level than that of the WIath Primer book. WIany of the supplemental questions in the Solutions WIanual came about as a result of the remarkable interaction I had with 也is very talented group. I am grateful to all of them for their impressive efforts. Special thanks go to those who supported the proofreading effort: Barnett Feingold , Chuan Yuan-Huang , Aditya Chitral , Hao He , Weidong Huang , Eugene Krel , Shlomo Ben Shoshan , Mark Su , Shuwen Zhao , and Stefan Zota. The art for the book cover was again designed by Max Rumyantsev , and Andy Nguyen continued to lend his tremendous support to the fepress.org website and on QuantNet.org. I am indebted to them for all their help. This book is dedicated to my wonderful family. You give sense to my work and make everything worthwhile. Dan Stefanica New York , 2008

Xl

Xll

ACKN01iVLEDGJVIENTS

Chapter 0 Mathematical preliminaries. 0.1

Solutions to Chapter 0 Exercises

Problem 1: Let f : IR • IR be an odd function. (i) Show that xf(x) is a丑 even function and x 2f (x) is an odd function. (ii) Show that the function gl : IR • IR given by gl(X) = f(x22 is an even function and that the function g2 : IR • IR given by 如何) = f(x :J) is an odd function. (iii) Let h : IR • IR be de主ned as h(x) 工 xtf(x J ) ， where i and j are positive integers. When is h(x) an odd function? Solutio衍 Since

f(x) is an odd function , it follows that f( -x) = - f(x) ,

(i) Let h (叫 = xf(x) and

f2 (x) = x 2f(x).

\j

x

εIR.

(1)

Using (1) , we 自nd that

h(-x) = -xf(-x) = xf(x) = h(x) ,

\j

x

εIR;

(2)

f2 (-x) = (… X)2 f( -x) = - x 2f(x) = - f2 (x) , \j x εIR. (3) We co卫clude from (2) that h (x) is an even function , and , from (3) , f2 (x) is an odd function. (ii) From (1) , it follows that

gl(-X) = f ((_X)2) = f(x 2) = gl(X) , \j x εIR; g2(-X) = f ((_x)3) = f(-x 3) = - f(x 3) = - g2(X) ,

(4) \j

x εIR. (5)

We conclude from (4) that gl(X) is an even function , and , from (5) , that 如何) is an odd function. (iii) If j is a positive integer , it follows from (1) that

f ((- x )J)

=

(-1 )J f (x) , \j x εIR.

(6)

2

JVIATHE1VIATICAL PRELIMINARIES

Then , usi吨 (6) ， we 在nd that

h(-x) = (-x)if((-x)j) = (-1)ν ((-l)jf(x j )) 工( - 1)i+ jxi f (x j ) = (-1)忡jh(x) ， V x εJR. TTIerefore7if t +j is aIIeven iMeger7the fmctionfz(z)is an eveIl function , and , if i 十 j is an odd integer , the function h(x) is a~ ~dd function. 口?

- 吐 (T(n ， 1, x)) ,

3

( 2 一 (η 十 1)3ηx n 一 1 十 (2η2 十 2η-1)(η 十 2)(η 十 l)x n \ 2 n十 ) l\ -n (η+ 3)(η + 2)x 1 6(1 - x) 1 日1

、，

/一 (η 十 1)3η(η - 1)x n - 2 十 (2η2 十 2η -1)(n+2)( η 十 1)η x n 一 1\

Ii\ -n 2 (η 十 3)(η 十 2)(η 十 1)zn6/ 1日1

、 ，

(一加川一川仇 2η 一阳仙 O 一η2(η 十 3)(η 十 2 勾)(η 十 1)

Problem 2: Let S(n , 2) = I:~=1 k 2 and S(η ， 3) = I:~=1 k 3 . (i) Let T(n , 2, x) = I:~=1 k 2 xk . Use the recursion formula T(n , 2 , x)

0.1. SOLUTIONS TO CHAPTER 0 EXERCISES

6 η(η+ 1) (一 (η 十 1)2(η- 1) + (2η2 十 2η- 1)(η 十 2) … η(η+ 3)(η+ 2))

(7)

6 η (n+1)(2η+

and the fact that

1)

6 T(η ， 1 ，

x)

Z 一 (η 十 1)x n + 1 十 nx n + 2

(1 - X)2

(8)

Therefore , S(η ， 2)

=

η(η+ 1)(2η+

1)

to show that T(η ， 2 ，

x)

2 X +x 一 (η +

1)2 x n+1 十 (2η2 十 2η - 1)x n + 2 _ n 2x n + 3 (1 - x)3

(ii) Note that S(η， 2) = T(η ， 2" l);"Ys~ yHopital's rule to evaluate T(η ， 2 ， 1) ， and conclude that S (川)= 7加1i(2叫)

\lit-/ -qJ 、飞，，/

QU

η

咱t4

\飞 11I/

MZH /Il--\+.3 .qb

才tA

吨3·-A

η

十

/Ill--飞飞

+

(9)

for i = 4 , given that

=η(ηf1);

~.aiution: (i) ~he res~lt follows from (7) and (8) by usi鸣 Quotient Rule to differentiate T (η ， 1 ， x).

(ii) It is easy to see that T(η ， 2 ， 1) = I:~=1 炉工 S(η ， 2). By using I'Hopital's rule we 盘时 that T(η ， 2 ， x) is equal to lim Z 十 x 2 一 (η 十 1)2 x 饥十 1 十 (2η2 十 2η - 1)x n + 2 一 η 2 X n十3 z• 1 (1-x)3 lim 1 +2x 一 (η 十 1)3 xn 十 (2η2 十 2η- 1)(η 十 2)x n十 1 _ n2(η +3)x n + 2 -3(1 - x)2

.付 tu

n

一一

and conclude that S(n , 3) = (呼叫"

JFi 飞

(iv) Note that S(η ， 3) = T(η ， 3 ， 1). U~e I'Hopital's rule to evaluate T(η ， 3 ， 1) ，

cυ

川J)24mμ))

Problem 3: Compute S(η ， 4) = I:~=1 k4 using the recursion formula

1川 -

(iii) Compute T(η ， 3 ， x) = I:~=1 k 3 x k using the recursion formula

(iii) Finding the value of T(凡 3 ， x) requires usi吨 Quotient Rule to differen x). (iv) The solution follows similarly to that from part (ii) , albeit with more complicated computations. 口

t杜iate T(η ， 2 ，

=

S(川)

(叮 γ

Solution: For i = 4 , the recursion formula (9) becomes

仰， 4)

=

H价十户一言 0)

s(n ,

j))

n(η+ 1)(2η 十 1)

6

'

4

l\!fATHENIATICAL PRELIMINARIES

~(川)5 _ 1 -

S(n , 0)

一臼川一则川)一叫川))

η(η 十 1)(6η3 十 9η2 十 η-1)

5

0.1. SOLUTIONS TO CHAPTER 0 EXERCISES (ii) We substitute η 十 1 for n in (12) and obtain that

口

X n +3 -

2x n 十 2 - Xn+l 十 2(η+ 1) + 3.

(14)

By subtracti吨 (12) from (14) , we find that X n +3 =

ηin

,i 4

2X n +l -

Xn

+

2η+3 ，

V η;三

1，

(12)

with Xl 二 1 and X2 = 5. (ii) Similarly, show that 3X n +2 - 3X n +l

+ X n 十 2 ， V η;三

with Xl = 1, X2 = 5, and 句= 14. (iii) Prove that the sequence (x n ) 吃 1 satisfies the linear X n +4 - 4X n +3

+ 6X n +2 -

1，

(13)

Solve this recursion and show that η(η Xn

P(z)

二=

十 1)(2η+ 1)

V η>

+ Xn

-

0, V n ~三1.

(16)

= 泸 -4♂ + 6z 2 - 4z 十 1 =

(z-1)4.

The polynomial P (z) has root λ= 1 with multiplicity 4. We conclude that the there exist constants Ci , i = 1 : 4 , such that = 01 十 O2η 十 03η2 十 04n3 ， V η;三1.

Since Xl = 1, X2 = 5, X3 = 14 , X4 = 30 , it follows that 0 1, O2 , 0 3 and 04 satisfy the following linear system (1 11 2 14 1 3 9 1 4 16

rec旧slOn

0, V n 二三 1.

4xη+1 十 X n

/iE\vhu

The characteristic polynomial associated to the recursion (16) is

Xn

X n十 3

V n 二三 1.

By subtracting (13) from (15) , we find that

、飞，，/

/'''飞、、

+ (η 十 2?

亏1品

X n十 1

3x n十 1 一 3X n +2 十 Xn+l 十 2 ，

X n +4 - 4X n +3 十 6X n +2 - 4X n +l

Subtract (10) from (11) to find that X n +2

X n +4 =

(10) , obtain that

X n +2 -

V n ~三1.

、I I J

+ 1 for

3X n +l 十 X n 十 2 ，

(iii) We substitute η 十 1 for n in (13) and obtain that

nu 、、、，，，，，

n

41· 南

X n 十 (η +1? ， V η 三 1 ，

X n十 1 -

with Xl = 1. (i) By s由stituti吨

/Ftt\

-

才'·-A

Problem 4: It is easy to see that the sequence (xη) 吃 1 given by X n = I::~=l k2 satisfies the recursion

3x n十 2

1 8 ) \ C I O1 5 2 \ ) - /f 1 27 I I 0 3 I - I 14 6 4 \ 04 \ 30

We obtain that 0 1 = 0 , O2 = ~， α=i aBdq=i and therefore 2η3η(η + 1)(2n 十 1) n - …十一十一= ,

1.

V n;三1.口

Conclude that S(η ， 2)

三二 k

2

η(η+ 1)(2η+

6

1)

Vη

> 1.

Problem 5: Find the general form of the sequence (x n)n 2: 0 satisfyi吨 the linear recursion X n +3 = 2X n +l 十 X n , V n 二三 0, with

Solution: From (11) , we obtain that the first terms of the sequence (xn)n三 l are Xl = 1, X2 = 5, X3 = 14 , X4 = 30. (i) The recursion (12) follows immediately by subtracting (10) from (11).

Xo

= 1, Xl = -1 , and

X2

= 1.

First Solution: By direct computation , we obtain X3 = -1 , X4 = 1 ，圳工 -1 ， = 1. It is nat 盯al to conjecture that X n = (-l)n for a町 positive integer η. This can be easily checked by induction. X6

1VIATHE1VIATICAL PRELIMINARIES

6

Second

Sol州on: Alternati 叫y，

the linear recursion X n 十 3 P(z)

-

=

we note that the seque丑ce (x n ) 吃o satis且es 0 , with characteristic polynomial

+1 =

(z 十 1)(z2 -

Z -

The roots of P(z) are 一 -1 ，与E 眨， a时导L§. Th 阳 e臼re£ 白 f扣ore， 出t h 阳 坠e盯re 盯 exist ∞ c臼∞ on I丑n归st S阳 t 严叩\

/

=

01 (-1) η 十 O2

I \

严-\

~

11 - vi月 1 03 ←」二 I U\ 2 I

+

I

2

/

η

1 斗、 15 \ 卜」二 I

=

(-1)η ，

Vn 三 o.

X n +1

, Vn

主 O.

口

3X n + 2, V η;三 0 ，

4x n十 1 -

with Xo = 1 and Xl = 5. (ii) Find the general forml由 ηi丑 (ο17η) ，

-

(18)

=

z2-4z 十 3

=

(z-1)(z-3) ,

which has roots 1 and 3. Thus , X n 二 01

4X n+1 - 3x n 十 1 ，

4xη+2 - 3xη十 1

(19)

5X n+2 - 7X n十 1

=

z3 - 5z2

+ 7z -

V η;三 o.

V n 二三

o.

(22)

+ 1,

V η;三 o.

(23)

+ 3x n, V n

~三 o.

(24)

3 = (z - 1) 2 (z - 3).

Therefore , there exist constants 0 1 , O2 , 0 3 such that Xn

=

01 3η 十 O2η

+ 0 3,

V η 二三

o.

Since Xo = 1 ，问= 5, and X2 = 18 , we find that 0 1 = ~， O2 = -~， α=-i We conclude that 3η+2 -

+ 023n ,

(21)

(ii) The characteristic polynomial of the linear recursion (24) is P (x)

~三 o.

3X n +1 十 η+3 ， V η 三 o.

Subtract (22) from (23) to find that

Sl郎ti扰t1巾

(ii) The characteristic polynomial of the linear rec旧sion (19) is P(z)

Xn+2 -

X n十 3

0, V n

V η 二兰三 0,

Substitute η+ 1 for n in (22) to obtain that

(17) from (18) and obtain that

+ 3x n

+ 3x 阳 r1 ，

with Xo = 1, Xl = 5, and X2 = 18. (ii) Find the general formula for X n , n 主 o.

Xn+3 -

Xn+2 - 4xη十1

(20)

+2 ， V n 二三 0,

5xη+2 一 7X n+1

-

Xn+2 -

it follows that

Sl削ract

η

Subtract (20) from (21) to find that

3x n , V η 二三 0,

3x n十 1 十 2.

3x n 十

By substituting n + 1 for ηi口 (20) we obtain that

for X n ， η 主 o.

Zη+2 -

We

Zη+3

(17)

Solution: (i) Let n = 0 in (17) to find that Xl = 5. By

-

Solution: (i) The first three terms of the sequence can be computed from (20) and are Xo = 1, Xl = ?' and X~_~ 18.

with Xo = 1. (i) Show that the sequence (x n ) 吨。 satisfies the linear recursion X n十 2 -

口

with Xo = 1. (i) Show that the sequence (Xn)n:::::O satisfies the 且 1i丑 near 陀 r ec盯S

Problem 6: The sequence (x n )吃o satisfies the recursion X n十 1

1, V η 二三 O.

-

Problem 7: The sequence (x n ) 吨。 satisfies the recursion

η

By solving the 3 x 3 linear system for 0 1 , O2 , and 0 3 obtained by requiring that Xo = 1, Xl = -1 , and X2 = 1 we find that 0 1 = 1, O2 = 0, and 0 3 = o. We conclude that Xn

= 2· 3n

Xn

1).

C1 , O2 , and 0 3 such that Xn

Since Xo = 1 and Xl = 5, we obtain that C 1 = -1 and C 2 = 2 and therefore

2xη刊 - X n =

z3 - 2z

7

0.1. SOLUTIONS TO CHAPTER 0 EXERCISES

Xn -

2η

4

~ Vη> o. 口

JVIATHEJVIATICAL PRELIMINARIES

8

Problem 8: Let P(z) = 2:立。 αiZi be the characteristic polynomial corresponding to the linear recursion

2二 α向十i

°

0, V n ~

=

(25)

Ih(x) + h(x) I /' 1~ YY> """J h(x) I J.. lim ~;mn Ih(x) I <' IiTftplznl:三 liryhγ| 十 brylznl< n and therefore、 by definition , h(x) + h(x) = O(x ) as x • 0. Let gl(X) 二可♂) and g2(X) = o(俨 )asx • 0. The凡

• 01 x

I

均) C η λ P( λ

1. 2

lims叩 μ17;1< ∞

t 一」 沪H 俨1

叫 ωtλ 汇2二 ω

n一→。C

and that

+ C λη肿叫+礼1 P'(从川 材) λ

10

~~_1 k 2 一手

lim sup:::='凡-J. n 一今汉二

In other words , the sequence (Yn)n?:.O satisfies the rec旧sion (25).

o(x n ) + o(x n )

nZ?ZM 。(的，

= 。但n) and

lims叫 h(x) II < ∞ DI 一一一 00 I

俨

I

-

。(♂)，

as

x

•

0;

o(x n ) , as x • 0.

h(x) =

O(俨) as

and

p limsup

x • 0. Then ,

Ih(川

1 一一~I x→0ι| 俨 i

<∞-

Similarly, prove that

口

> 0. Show that

。(♂) +O(♂)

OJ

'也，

as

n →∞;

Z 十州， asη →∞

(26) (27) Solution: Recall that

2二 k

2

中俨十 1) 删去 k 3

一一

?十州， asη →∞? ~n

•

n →∞;

as

n川口

汇2二 α川十 C队λ沪

ηω+ 礼1

η

aT

2-z

-LZn

-z

Cη 队 λn

m•

/飞

。(的，

i.e. , show that

h(x)

川叫 -n →

-2

骂问

nZT 汇 M

2二 α以η+ i)λ叫 Cωη2 汇二 α川+刊4 十 C2 汇 二二iω 叫 α向iλ 俨 η肿十

Sol州on: Let

lil--io !li--io iil-lili--QU n3-

Problem 10: Prove that

anyη;三 0 ，

Problem 9: Let n

o I x ,. I

十仪

P'(z) = 汇 ωii- 1

汇 αiYn+i

OL 一但

寸

、飞 Bt/

1i

ld

O

k

lLZl

-znE

/吨 -Z\/

l--iili--nud

m n叫

9 一让

\l/-e

<一 ~v

/飞一

+LρU

• --nq z-B z-d g-W oa--1

x•

川 …J - n =

M-m 叶风

Z

blzpk mmm d h n t

Sol毗on: Note thatλis a root of multiplicity 2 of P(z) if and only if P(λ)=0 and P'(λ) = 0 , where

Then , for

°

叫四 1=0 and 叫 92(~) I = n

刃臼

where C is an arbitrary consta风 satisfies the recursion (25).

It is easy to see that

、 JJλu

Assume that λis a root of multiplicity 2 of P(z). Show that the seque丑ce (Yn)吨。 given by Yn = cηλ飞 η;三 0 ，

9

0.1. SOLUTIONS TO CHAPTER 0 EXERCISES

10

MATHEMATICAL PRELI]l.ifINARIES Then ,

0.2 lim n→∞

k=l k -L:~=1

2

ηU

""""η

7A2

η3

im L...tk=l 气一 τ n→∞

η"

""",, n 1认 3 limι~k=l 内

二

n→∞

ητ

lim 1日1

L:~卢 η4 k=l 一 τ

n→∞

ηu

1

~… η(η+ 1)(2η+ 1)

η→∞ 唱

6η3

3

3η2 十 η

<

..…二『

η→∞

6η2

limd(η 十 1)2 η→∞ ,.

l

4

2η 十 1

1 2

..…-

n→∞

4η

<

∞;

Supplemental Exercises α>

1. Let

0 be a positive number. Compute

Vα 十卢+~

∞;

<

一

<

11

0.2. SUPPLEMENTAL EXERCISES

∞;

2. Let

α> 0

be a positive number. Compute 1

00.

α 十一一--，

α 十注工

。(的?

3. (i) Find x > 0 such that as

ZZ

η →∞;

号 + O(n2 ) ,

as

as

T 十 O(η3) ，

η →∞;

as

η 一→ 00.

=

2.

(ii) Find the largest possible value of x > 0 with such that there exists a number b > 0 with

η →∞;

Z

。(的 7

Z

♂

ηZT 汇 ?γUMnZM

We conclude that

z

b.

Also , what is the largest possible value of b? 口

0.3

Solutions to Supplemental Exercises

Problem 1: Let

α>

0 be a positive number. Compute

户十卢十日τ 灿t剖tω ion 仇 仰仰 η

t出 ha 肮t

limit by l , ther口1

挝 1t

follows that l must satisfy

jα +jα+ 币二=

d丰 l ，

(28)

which can be solved for l to obtain that 1+ VI+石 2

(29)

12

!\/fATHEMATICAL

PRELIMINARIES

.-.

We now show that , for anyα> 0, the limit of 飞/α+ 、/α +vaτ-. does e于风 whick i?quimieM to provi丑g that the 叫uen 配阳 C臼e (X ωηρ)η 乌 哈三沪o 坦 is ∞ c on口阴飞

0.3. SOLUTIONS TO SUPPLE1VIENTAL EXERCISES

Solution: If we know that the continuous fraction (30) has a limit l, then l must satisfy

l= α+; 白 l2 - al -

whe 臼re 均 X 0= 飞v αand

va百二，

X n +1 -

We will how that the

\I η 主 O.

and therefore

l丑 creasm 丑g.

Let l be given by (羽)， i.e. , let l = 与严

va

X n +1

va:+l = l ,

The 自rst

since l is the positive 叫叫O∞ n of (闵 剑 28 创)， 肌a削丑叫 8 d 山 t he 臼时 re 盯 伽 eflf inductior 瓦 1 ， we 且 fiII卫ld tl扭lat Xη < l for all η> O.

Xo

(ii) The sequence (x r山三 o is increasi 吨·

since

Xn

<

X叫字=} Xn

<

vaτX n 仁丰 zi-zn 一

< land

Xn

> 0 for all n 三 O. vVe conclude that

η→∞川

α2 十 1

Xl

a3 + 2αα4 十 3α2+1

= 一一一一，

X2

= 一一一一-:

XQ = ，1， 3α3 十 2α

α2α2 十 l'

1 十飞11 十 4α ι 2

Xn

X2

< l <

X3

<

Xl·

- α 十一一-一一=

α+ 县 lin

(α2 十 l)Yn + α

αYn 十 1

?

V n>07

(33)

with Yo = α. Note that Yn = X2n for all η 主 O. Assume that Yn < l , where l is given by (32). Recall from (31) that

l2 -.al - 1 = 0

(34)

and that

h川 一

t 2 一 αt

1

α + a~...

Yn+1

< X n十 1 for all

Problem 2: Let α> 0 be a positive number. Compute α 十一一一寸­

<

To show this , let (y山

We showed that the seque口ce (xn)n~O is increasi吨 and bounded from above. We conclude that the seque口ce is convergent. Therefore、 the limit l=li问→00 Xn satisfies the equation l = vaτI an-d is given by (29) , i. e. , ,..,

n

Based on this observation , we co时 ecture that the subsequence (X2n)吃O made of the even terms of the sequence (xn)n~O is increasi 吨 and has limit equal to l , and that the subseq旧lce (X2附 1) 吃o made of the odd terms of (X n ) 吨。 is decreasing and has limit equal to l.

(xn _ 1+ V;+石) (x n - l-f石)

lim x ，..， 1m X

= α，

Xo

川) (xn 十盯了一 1) since Xn n> O.

α+17V n 三 O. X

few terms of the seque丑ce (xn)n三o are

α< 0 ,

> O. Note that 2:-zn 一 α

-

We note that the terms of the sequence are alternatively larger and smaller than the value of l given by (32) , i. e. ,

It is easy to see that Xn

(31)

l= α+va可2.

(i) The sequence (xn )吃o is bounded from above bv l. Note that Xo = < l. If we assume that Xn < l ， 白en va丰百 <

1 = 0,

(32) 2 To show that the continuous fraction (30) does have a limit , we must prove that the sequence (xn)n主o is converge 风 where Xo = αand

seq阳

X n +1 -

13

(30)

1

(t- a+vt+

4

)

(t 一 α-F) (35)

14

1VIATHE1VIATICAL PRELI1VIINARIES

We will show that , for allη;三 0 ， Yn十1 >νnand Yn+1 < l. Note that , by de且nition (33) , Yn > 0 for all η 三 O. Then , from (33) , it is easy to see that

Yn+1 > Yn 仁斗 (α2 十 I)Yn 十 α>αd+Uη 仁斗 α(ν; 一 αYn - 1) < O. (36) From (35) , and using the assumptio且 that Yn < l , it follows that

Yn+1 <

with Zo = 年It is easy to see that Zn = X2叫 for all n 三 0 As expec'ted , the sequence (zn)n三o is decreasing and h部 limit equal to l. The proof follows by induction: assuming that Zn > l , we show that Zn+1 < Zn and Zn礼 > l. This proof is very similar to that given above for the seque丑ce (Yr山三o and is left to the reader as an exercise. We conclude that lim X2n十1 = l.

/ d可4-a \

Eom(36)and(37)pwe comiude tht7if yn
>

(37) Um for any

the last equality can be derived as follows: 2

α2 - al 十 1 = l 仁中 l 一 α=α2l - al 十 l 仁斗 α (l2 - al - 1) = 0 , where ttle last equality is tfle same as(34). We conclude from (38) that , if Yn < l ,' then Yn+1 < l for all η 主 O. III other words7we showed by induction that the seque口ce (Yn)吨。 giv by the recursion (33) with Yo = αis increasi 吨 and bounded from above by l. Therefore , the seque口ce (Yn)吃o IS co盯ergent. Denote by h = limn→∞ Yn the limit of the sequence (Yn)n '2 o. From (33)-and using (35)v, ;;e obtai~ th~t

J 一 (α2 十 l)h 十 α 《 1 一 αh 十 l 仁斗 α (lt 一 αII - 1) = 0

白叫It - I) (It +咛 -a) =0 Since h > 0 , it follows that h = l , i.e. , that limn→∞ Yn = l. Recall that Yn 二 X2n for allη;三 O. We showed that the subsequence made of the even terms of (x 阳 仇 7η 1 by (32) , i.e. , that 一 lim X2n 二 l. (39) n一→ α2

lim im Xn Xn -

一 (α2 + l)zn + α αZn 十 1

η>0 ' V , .v:::-v ,

l -

α+ v'歹τz

一一一一一

口

Problem 3: (i) Find x > 0 such that

XX"

2.

(41)

(ii) Find the largest possible value of x > 0 with such that there exists a number b > 0 with XXX' = b. (42) Also , what is the largest possible value of b?

Solution: (i) If there exists x such that (41) holds true , then x 2 = 2 缸 aI丑l therefore x = y'2. We are left with proving that ~V2'

-/2 Y";'

= 2.

Consider the sequence (x n )n '2 0 with Xo = y'2 and satisfying the following recurSIOn: 队+1 - -/2x n _ γrz/27V 71 主 O. It is easy to see by induction that the sequence is increasing and bounded from above by 2 , since X n 十 1 > Xn 牛二中 2Xn/2 > 2 xn - I/ 2 牛工:> Xn > X n 一 1 ;

Xn+1 < 2 伫二? 2xn / 2 < 2 宇::} x n /2 < 1 仁::} Xn < 2. We conclude that the seque口ce (x n)n'2 0 is convergent. If l = limn--t∞ X n ， then

Similarly, we define the sequence (zn)n '2 o by the recursion 十

(40)

From (39) and (40) , we find that

l 白 (α2 + I)Yn 十 α<αlYn 十 J 白 Yn < α217L1 = l; (指)

l- α

15

n一→汉二

Y~ 一 αun-1=(un-J)(r2)

0.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

l =2l/27 which is equivalent to ll/l = 21/ 2 .

(43)

16

lVIATHElVIATICAL

PRELIMINARIES

Then

f' (t) = ~卜勺丁 77 一77?γt;J 俨!?于1叫(t)

Chapter 1

ex

~?te that the function f (t) is i肌reasi吨 for t < e and decreasing for t > e. Therefore?there will be two values of t such ttlat(43)is satis五ed， i.e. , s时l that tl/t=21/27OIIe value being equal t027and the other OIle greater tham Since zn<2for a!1n 主 o and l = limn→∞ x n ， we co口clude that l = 2 , 缸 a口n1 therefore that x = 飞/2 is the solution to (41). (ii) If there exists a number b > 0 such that xxx·

-

Calculus review. Plain vanilla options. 1.1

Solutions to Chapter 1 Exercises

Problem 1: Compute

J ln(x) dx.

Solution: Using integration by parts , we find that

b

for a given x > 0 , then x b = b and the时·ore x = b叭 Recall from part (i) that the functiozlfO)=tl/t has &Il absolute maximum at t =e.We cOBelude that Z 二 b 1 /b 三 maxt1 / t _ e 1 /e 自 1 .4447，

JIn仪

户何叫川 d伽x f

-

川 x 闻阳峭 Inn叫巾刷(仪例 叫 Z叫) 一 f μ叭Z呻巾喇 户 倒川 (In 以 Inn叫(

川忡 Z ln( 巾叫 叫巾(仪仲 n Z

t>O

and that the largest value of b such that the limit (42) exists is b = e.

口

Problem 2: Compute

J x击x) dx by using the substitution u = ln( x)

Solution: Let u = ln(x). Then du = 警 and therefore

JXl~(X)

xln(x) dx -- =

JJ ~

du =

ln(lull

=川x)l) + C 口

Problem 3: Show that (ta 阳 缸川 a nlX 叫)'=1ν/(巾C∞ Oωsx 叫)2 a丑

J1: 丁训dx l+x

-

町肌ctω 阳叫 阳 taa缸I

Solution: Using the Quotient Rule , we find that

(tan x)'

(:25)f=(sinx阳 x - sin x (cos x)' 、、，，，，

45i

才tA

r''' 、1

(COSX)2 (COSX)2 + ( s i n x ) 2 l (COSX)2 (cosx)2

CHAPTER 1. CALCULUS REVIEW. PLAIN ~生NILLA OPTIONS.

18

To prove that

J 1;'x dx = 2

arctan( x) , we will show that

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

Problem 4: Use I'H6pital's rule to show that the following two Taylor approximations hold when x is close to 0:

(arctan(训 =Jτ

Jτ王旬 l+j;

i 十 x~

Let f(叫= tan x. Then arcta叫x) = f-I(x). Recall that

(f- I (x))1 =

♂自 1 + x 十二

J

•

In other words , show that the following limits exist and are constant:

and note that l' (x) 二 (tanx)' =讨x)2; cf. (1. 1). Therefore , (arcta口 (x))' = (COS(f-I(X))? = (cos(arctan(x)))2.

mJ百三(1 +~)and liJ-(lfz呼)

(1. 2)

X

Let α= arctan(x). Then tan(α)=ι It is easy to see that

•o

x'::;

X

•o

x

Solution: The numerator and denominator of each limit are differentiated until a finite limit is computed. L' H6pital's rule can then be applied sequentially to obtain the value of the initialli皿it:

x 2 十 1=__1

(cos(α))2'

since (sin(α))2 十 (cos(α))2 = 1. Thus , (cos(arcta口 (x)))2 = (co巾)?

19

1;~ dτ王一 (1 +~)_ 1;…对古 -i

l

汇;6

(1. 3)

x2 十 1

1 8

lim m• 0

..…一

♂巧。

x2

2x

vVe conclude that

From (1. 2) and (1. 3) , we conclude that

dτzz l+i+0(泸)，

(arcta叫 x))'

as

x

•

0

Similarly,

and therefore that

f 市 dx 二 arctan(x)

eX - (1 十 Z 十豆}

+ C

lim

•o

X

We note that tke antiderivative of a rational fuzlctiOII is often computed using the Sl阳titution x = tan (~)

则:12Eample?to compm j 市 dx using the substitution x = 阳(~)，

dxx -= -;~ ((tan tan ((~\飞~ ) ) dz dz \-\2)) 山

eX 一 (1 十 Z 十二)

+

1

x

吊'山

1IZ

v

;

X

•o

h

十二十 O(内部户。口

1 l品

m说

-

(∞S(~))2 1 (sin(α))2+(COS(α))2-2(cos(5))2u =

~X

lim 二­

Problem 5: Use the definition e = limx-tχ(1 + ~)X of e to show that /IIiI\ 1-z \1llI/ 1-e

11+ 山~))2 2(∞J(5))2dz

I~ 命 dz

x

lim eX - 1 m1日1 • o 一一一一­ ox

/ ~ \ '" . . . dz.

Then

f

lim eX 一 (1 + x) z• 6 3x 2

、/

=可a叫n叫巾(归例 Z叫) 十

C

口

Solution: Note that 1

1 一­ Z

x-I Z

1

1

z

1+ 击

x-I

l

6'

20

CHAPTER 1. CALCULUS REVIEW. PLAIN VANILLA OPTIONS.

The口?

n∞

一十)十

n∞

町

l I Z -口 -1

♂

hhkh

才Z A

· 唱E A

/ J I I飞 -、

m切

1i3

\Iiil/ l-z

Solutio 衍 Let F (x) = J f (x) dx be the antiderivative of f (x). From the Fundamental Theorem of Calculus , it follows that Z-z

IH

1 e

fα +h F(α + h) - F( α - h) f(x) dx = n~_ I 2fz 2h Jα-h Using I'H6pital's rule and the fact that F'(x) = f(x) , we find that

Sl日 ce

h•

1 lim1+__. _ 1 z→∞

n~ 62h

fα+h

I J α -h

FII! h• 6

f (, x), dx =

-

x-I

lim

h•

and

J马 (1 +击r一 =J414♂

e.

口

~左 i七言 l(X) 叫协刷(仰例♂叫V)

=

since

f (x)

Note: Let F(t) = F'( α) is

J~ f(x) dx.

F'(α)

F( α +I \'""

h) -

h)

2

F( α -

.L

I~)

I' α 十 h

= i2h二 I Jα

\飞 IlI-/

们uu

、飞 4 ，/

Problem 8: Let

f : lR •lR

、

ρ ' ' '门 '凶 '''

'of--

/IIl-飞、、

d一 'ω

f(b(x))b'(x) - f( α (x ))α '(x).

\'""

h)

I~)

+ 0

(h

2

)

(1. 4)

,

-h

f(x) dx + O(h 2 ).

口

given by

的) = LCi e呐?

Therefore ,

-Lp 一鸣垣 . J/ ( 川一 υ

川-y12在叭叭一←厂/

Solution: Note that

e

?-b

一

~

U·t , 一一八，/

I'

yti )'

\11/

2σ 2T

( e-

positive constants. Compute

一

(In (去)十 (r 十号 )T 门内

\

are

and ti , i = 1 :叽

-e

xcrVT

Ci

、、E，， J

(一

where

''ι

.L

(b(x))2 飞

们uu

eXD

(

/lt 飞

飞12 7f T

凸-O-~

tHρU

1

1

J ' a E飞 、 、 / ， ， ‘ ‘ 、 、

而

V

d 一句 d 一句

g'(x)

-

h)

as h • o (if F( 剖 (t) = fll(t) is contin∞叫. Since F'( α) = f( α) ， formula (1. 4) can be written as

Recall that Vd UU

o

2h

1

rJ

2h α h) + f( α f( +

The central finite difference approximation of

.L

=

f(α) ftm 31j

F( α -

is a continuous function.

阳 wh 1

Solut刮4切 on:

h) -

F( α +

f( α) ，

Problem 6: Let K , T ， σand r be positive constants , and define the function g : lR •lR as

g(x)

21

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

NUAFU

-tte-uta; =t?e-Uti

The丑 7 n

Problem 7: Let f(x) be a contin∞ us function. Show that

出去 ffz 川

f'(y) = - L Citi e一队; i=l n

f"(y) = LCi 中 -yti i=l

口

f' (y)

and

f" (y).

22

CHAPTER 1. CALCULUS REVIEW. PLAIN VANILLA OPTIONS.

Note:The fmction f(u)represents the price ofa bond with caskHOWS Ci paid at time tt as a function of the yield Uof tfle bond.WheIlscaled appropztate1377 the 丑rst and second derivatives of f (y) 吼叫1 respect to y give the duration and convexity of the bond , respectively. Problem 9: Let f : JR3 x = (Xl , X2 , X3).

•

(ii) W~ substitute the 叫ues from (1. 6) , (1. 7) and (1. 8) for f( α) ， ，Dt(α) ~nd D 2f(α) ， respectively, in the expression f (α) + Df(α) (x 一 α) + (x 一

1

α)t D 2 f(α) (x 一 α)

3

(i) Compute the gradient and Hessian of the fur时ion f (x) at the point α= (1 , -1 , 0) , i.e. , compute Df(l , -1 , 0) and D2f(1， 一 1 ， 0). (ii) Show that

D 2f(α)

(x 一 α)

/ Xl - 1 \ X2 + 1 J

I

(5 , -1 , -3)

I

X3

/

4

飞

o

…

0 飞/

1

Xl -

1 飞

寸仙一 1 ， X2 十 LZ3)i-103ilh+1) J

3 -2

飞

J

X3

3 + (5Xl - X2 - 3X3 6) + (2xi - 5Xl - XIX2 + X2 十 3X2X3 + 3X3 - x~ 十 3) …

Here，叽 α ， and x 一 αare 3 x 1 column vectors , i. e. , 吁t牛才 l··-

,,"

f(x) ，

\飞 Il--1/

一十均

ZZ 1in

α

Z

一一

α

τiqhMηο

/JIl-飞\

一一

=

2xi - x向 +3x仇 -zi //iz--\

//11111\ \Illi--/ -i1inu

ZZZ

+

飞

刷工川的 +Df(仲

Z

and obtain that

f(α) +Df(α) (x 一 α)+1(z 一 α)t 2

JR given by f(x) = 2xr - XIX2 十 3X2均 - x~ ， where

\It--/''

23

1. 1- SOLUTIONS TO CHAPTER 1 EXERCISES

口

Problem 10: Let Note: Formul~ (1. 5)_ is the quadratic Taylor approximation of f(x) around the pointαSin~e_!(x) is a second order polynomial , the quadr~tic Taylor approximation of f(x) is exact. ,

Solution: (i) Recall that

Df(x)

飞/4 时'

for t

n u q 气u n L

才i n U Q d

I=

A吐 1 i n u

//11111\

\11111/

Note: This exercise shows that the function u(x , t) is a solution of the heat equation. In fact , u(x , t) is the fundamental solution of the heat equation , and is used in the PDE deri飞ration of the Black-Scholes formula for pricing European plain vanilla options. Also , note that u(x , t) is the same as the density function of a normal variable with mean 0 and variance 2t. θu

Df(α)

D 2 f(α)

ξJR.

絮 and g主， and show that

Solution: By direct computation and using the

Then , f(α)

> 0, x

θtθz2·

- Xl + 3X3 , 3X2 - 2X3) ;

在 (x) 忐 (x) 忐 (x) 品 (x) ~(x) 蔬 (x)

Compute

2

~-已一古，

u(x , t)

θuθ2也

(θfθfθf) 瓦 (x) 瓦 (x) 瓦(叮 (4X l - X2 ,

D 2 f(x)

I

,

f(l , -1 , 0) = 3 D f(l , ~1 ， 0) = (5 , - 1 , - 3);

(1. 6) (1. 7)

的(1， -1 ←(一;-i e-;)

(1. 8)

一一

at

=

一

1 2

'J问

_t-u/~ V

1

一-=

y4王

e

.r 2

"4τ+

x2

.r 2

1

一一=

d召 1

Product 丑ule ， 2

.r e- 4i

.r 2

=一一一=e 右十一·一 =e 一石; 4t 2 y4言: 2t y4巧 θu

一一一

x

ω

ax

-

-…-叩】

1

.--…-一

2t v41r t

ρ ~

.r 2 4t·

,

(

x2

(

we find that 1\\

I 一一. I 一 ~II 2

\4\

t ))

(1. 9)

CHAPTER 1. CALCULUS REVIE1iV. PLAIN VANILLA OPTIONS. nu••

4t4

From (1. 9) and (1. 10) , we ∞ c onc 叫 tcll时 e 由 i比 t ha创t θ2 U

θu

θx 2

θ t'

A

Note: This is a but臼rgy pead tMer mks a 10吨 Pωi阳1 in a b飞阳rfly pread if the price of the underWIngmset at maturity is expected to be h the K1 三 S(T) 三 K3 range. SoJTLtiOTZJA butterny spread is &Il optiOIls portfolio made of a long positiOII IIe call option with strike kh a long position iIla call option with strike khazld a short position intwo calls with strike equal to the average of tlm strikes K 1 and K 3 , i.e. , with strike K2 二且寿县 ; all options have the same maturity and have the same underlying asse( The payof at maturity of a butterdy spread is aiways11OIIIlegativtand it is POSItive if ttle price of the mdeElying asset at maturity is betweeiltbe strikes K 1 and K 3 , i.e. , if K 1 < S(T) < K;. For our particular example?tfle values of the three call options at maturity e , respectively,

C2 (T) C3 (T)

max(S(T) - K 1 , 0) 皿ax(S(T) - K 2 , 0) max(S(T) - K 3 , 0)

max(S(T) - 30 , 0); max(S(T) - 35 , 0); max(S(T) - 40 , 0)

and the value of the portfolio at maturity is V(T)

= C 1 (T)

S(T)

,,,,,, E

mm 黯

25

< 30 30 < S(T) < 35 35 < S T) < 40 40 < S(T)

。

S(T) - 30

。

。

O O

O S(T) - 30

S(T - 30 S(T - 35

S(T) - 30

。

sjTj35 SrT) - 40

40 - S(T)

。

L....J

Problem 11: Co斗sider a portfolio with the following positions: • long one call option with strike K 1 = 30; • short two call options with strike K 2 = 35; • long one call option with strike K 3 = 40. All options are on the same underlyi吨 asset and have maturity T. Draw the payof dlagram at maturity of tile portfolio?i.e-7plot the value of the portfolio V (T ) at matu y ity as a funct ion of S (T )7t he price of t he m d e臼r创l坊刷 抖 yrm J 臼se S 创t 肮 at time T.

C1 (T)

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

、、、

才1 4

/ t t‘ 、

δx 2

1二 币

θ2 U

1二 二 制

24

2C2 (T)

+ C 3 (T).

Depend,ing o,n t~~， ~~lu~s_ of the spot S(T) of the underlying asset at maturity, the value V(T) of the portfolio at t'i~e T is given beiow~

Problem 12: Draw the payoff diagram at maturity of a bull spread with a long position in a call with strike 30 and short a call with strike 35 , and of a bear spread with long a put of strike 20 and short a put of strike 15. Solution: The payoff of the bull spread at maturity T is 只 (T)

= max(S(T)

30 , 0) -

max(S(T) - 35 , 0).

on the value of the spot price S(T) , the value of the bull spread at maturity Tis

Depe口ding

I S(T) < 30 I 30 < S(T) < 35 I 35 < S(T) I 0 I S(T) 30 I 5 I vi (T) I The value of the bear spread at maturity T is 巧 (T)

= max(20 - S(T) , 0) - max(15 - S(T) , 0) ,

which can be written

i口 terms

of the value of S(T) as

A trader takes a long position in a bull spread if the underlying asset is expected to appreciate in value , and takes a long position in a bear spread if the value of the underlying asset is expected to depreciate. 口 Problem 13: Which of the following two portfolios would you rather hold: • Portfolio 1: Long one call option with strike K = X - 5 and long one call option with strike K = X 十 5; • Portfolio 2: Long two call options with strike K = X? (All options are on the same asset and have the same maturity.) Solution: Note that being long Portfolio 1 and short Portfolio 2 is equivalent to bei吨 long a butte

26

CHAPTER 1. CALCULUS REVIE1iV. PLAIN VANILLA OPTIONS.

rather aomegative)payof M maturity-Ttmrefore7if you are to α ss包 uη阳 ω n7 POSItion ln either om of th portfolios (口 ot to purchase the portfolios) you , are better of owniIlg Portfolio 17since its payofat maturity wiH always be at least as big as the payoff of Portfolio 2. More precisely, note that

V(T)

问 (T)

-

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

当〈几

VeT) O

K 1 < SeT) < K 2 K 2 < S(T ) < K 3 K 3 < SeT)

页

- 5) ， 0) 十 max(S(T) 一 (X + 5) , 0) - 2max(S(T) - X , O).

The value of the portfolio at time

T

is detailed below:

V(T) S(Tf<X - 5 O X - 5 < S(T) < X S(T) 一{~- 5) X < S(T) < X + 5 (X + 5) S(T X_+ 5 < S(T) 。 --

Z1K1

X1 C 1(0)

十 X2 C2(0) 十 X3 C3(0)

At maturity T , the value of the portfolio will be

X1 C1(T) + X2 C2(T) 十 X3 C3(T) Xl max(S(T) - K 1, 0) + X2 max(S(T) - K 2, 0) + X3 四 ax(S(T) - K 3, 0) ,

respectively. Dependi鸣 on the value S(T) of the underl}也 g asset at maturity the , value V(T) of the portfolio is as follows:

Xl K 十 K x2)K (Xl 3 X1 1 X2 2 Xl + X2 十 X3

X2K2 X2 K 2 - X3K3

。;

0; 0; 0.

For C 1 (0) = 8 , C2(0) = 5 , C3(0) = 3 and K 1 = 100 , K 2 = 120 , K 3 = 130 , the problem becomes finding Xl 主 0 ， and X2 and X3 such that

8X1

+ 5X2 + 3X3 30X1 + 10x2 Xl 十 X2 + X3

〈〉一>一

SoJTLUOTZJFor marbitrage opPOEtunity to be preseIlt7there must be a portm folio made of the three options with nonnegative payoff at maturity and with a negative cost of setting up. Let K 1 = 100 < K 2 = 120 < K 3 = 130 be the strikes of the options. Denote ,by x~ ， X2 ,!3 ..!~e options positions (which can be either negative or positive) at time 0. Then , at time 0 , the portfolio is worth

+ X2C2(0) 十 X3 C3(0)

〈〉一〉一>一

Problem 14: Call options with strikes 100 , 120 , an 丑 d 130 on the sam 丑le undern 呐it由 yi凶 g ass们E时 w 出 盯 ht 缸 肮 机 刷 切 h e sa 丑 me lema t I u 吨 江哟 l牛 (there is no bid-ask spread). Is there an arbitrage opportunity prese时? If yes , how can you make a riskless pro缸?

V(T)

TiT-

Note that VeT) is no 日 negative whe 丑 SeT) 三 K2 only if a long position is taken in the option with strike K 1, i.e. , if Xl 三 0. The payoff VeT) decreases when K 2 < SeT) < K 3, accounting for the short position in the two call options with strike K 2, and then increases when SeT) 三 K3 · We conclude that VeT) 三° for any value of SeT) if and ∞ly if Xl 主 0 ， if the value of the portfolio when S (T) = K3 is nonnegative , i.e. , if (Xl 十 x2)K3 - X1 K 1 - X2 K 2 ~三 0 ， and if Xl 十 b 十均主 0. Thus , an arbitrage exists if and only if the values C 1(0) , C2(0) , C3(0) are such that we can find Xl , X2 , and X3 with the following properties:

Xl C1 (0)

=

1δS

--Z X1K1 x2)S(T) 一 + 3Z22S x 1K 1 (Xl 十 X2 + x3)S(T) (Xl

\i2 (T)

max(S(T) 一 (X

V(O)

27

。;

(1. 11) (1. 12) (1. 13)

0; 0.

(For these option prices , arbitrage will be possible since the middle option is overpriced relative to the other two options.) The easiest way to find values of Xl , X2 , and X3 satisfying the constraints above is to note that arbitrage can occur for a portfolio with long positions in the options with lowest and highest strikes , and with a short positio日出 the option with middle strike (note the similarity to butterfly spreads). Then , choosi 吨 X3 = -Xl - X2 would be optimal; ef. (1. 13). The constraints (1. 11) and (1. 12) become

5X1 + 2X2 < 3X1 + X2 >

nunu

These constraints are satisfied , e.g. , for Xl corresponds to X3 = 2.

1 and X2 -

-3 , which

28

CHAPTER 1. CALCULUS REVIEW. PLAIN VANILLA OPTIONS.

Buying one option with strike 100 , selling three options with strike 120 , and buying two options with strike 130 will generate a positive cash flow of $1 , and will result in a portfolio that will not lose money, regardless of the value of the underlying asset at the maturity of the options. 口

more shares of the underlying asset and usi吨 the casg proceeds to make the dividend payments. Then , the short position in e- q1 shares at time 0 will become a short position in one share 1 at time T. The value of the portfolio at maturity is

V(T) Problem 15: A stock with spot price 40 pays dividends continuously at a rate of 3%. The four months at-the-money put and call options on this asset are trading at $2 and $4 , respectively. The risk-free rate is constant and equal to 5% for all times. Show that the Put-Call parity is 丑at satis丑ed and explain how would you take advantage of this arbitrage opportunity.

Solution: The followi吨 values are give口 : S = 40; K = 40; T = 1/3; r = 0.05;

P(T) + S(T) - C(T)

-

C -

39.5821

> 39.3389 - Ke-

= max(K - S(T) , 0) + S(T) - max(S(T) - K , 0) = K ,

regardless of the value S(T) of the underlying asset at maturity. Therefore ,

V(T) = 一 (P(T) 十 S(T) - C(T)) + 39.5821erT - - K + 39.5821e rT - - 40 + 40.2473 -

The Put-Call parity is not satisfied , since

P + Se-

= - P(T) - S(T) + C(T) + 39.5821erT .

Recall from the proof of the Put-Call parity that

q = 0.03; P = 2; C = 4. qT

rT

.

(1. 14)

Therefore , a riskless profit can be obtained by "buying low and selling by selling the portfolio a丑 the left hand side of (1.14) and buying the portfolio on the right hand side of (1.14) (which is cash only). The riskless profit at maturity will be the future value at time T of the mispricing from the Put-Call parity, i.e. ,

且igh飞 i.e. ，

(39.5821 - 39.3389) 产= 0.2473

(1. 15)

To show this , start with no money and sell one put option , short e- qT shares , and buy one call option. This will generate the following cash amount:

29

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

0.2473.

This value represents the risk-free profit made by exploiting the discrepancy from the Put-Call parity, and is the same as the future value at time T of the misprici吨 from the Put-Call parity; d. (1. 15). 口 Problem 16: The bid and ask prices for a six months European call option with strike 40 on a non-dividend-paying stock with spot price 42 are $5 and $5.5 , respectively. The bid and ask prices for a six months European put option with strike 40 on the same underlying asset are $2.75 and $3.25 , respectively. Assume that the risk free rate is equal to O. Is there an arbitrage opportunity present?

V(O) = 一 P(O) - S(O)e- qT 十 C(O) + 39.5821 = O.

Solution: For r = 0 , the Put-Call parity becomes P + S - C = K , which in this case can be written as C - P = 2. Thus , an arbitrage occurs if C - P can be "bought" for less than $2 (i. e. , if a call option is bought and a put option is sold for less than $2) , or if C - P can be "sold" for more than $2 (i.e. , if a call option can be sold and a put option can be bought for more than $2). From the bid and ask prices , we 直nd that the call can be bought for $5.5 and the put can be sold for $2.75. Then , C - P can be "bought" for $5.ι $2.75=$2.75 , which is more than $2. Therefore , no risk-free profit can be achieved this way. Also , a call can be sold for $5 and a put can be bought for $3.25. Therefore , C - P can be "sold" for $5-$3.25=$ 1. 75 , which is less than $2. Again , no risk-free profit can be achieved. 口

Note that by shorting the shares you are responsible for paying the accrued dividends. Assume that the dividend p句rments are financed by shorting

lThis is similar to converting a long pm;ition in c qT shares at time 0 into a long position share at time T: thro吨h continωus purchases of (fractions of) shares usi吨 the dividend payments: which is a more intuitive process.

P + Se- qT

-

C

39.5821 ,

since shorting the shares means that e- qT shares are borrowed and sold on the market for cash. (The short will be closed at maturity T by b可i吨 shares a丑 the market and returning them to the borrower; see below for more details.) At time 0 , the portfolio consists of the following positions: • short one put option with strike K and maturity T; • short e- qT shares; • long one call option with strike K and maturity T; • cash: +$39.582 1. The initial value of the portfolio is zero , since no money were invested:

i丑 one

30

CHAPTER 1. CALCULUS REVIEVV. PLAIN VANILLA OPTIONS.

Problem 17: You expect that an asset with spot price $35 will trade in the $40一$45 range in one year. One year at-the-money calls on the asset can be bought for $4. To act on the expected stock price appreciation , you decide to either buy the asset , or to buy ATM calls. Which strategy is better , depending on where the asset price will be in a year?

Solution: For every $1000 invested , the payoff i口 one year of the first strategy, i.e. , of buying the asset , is

叫T) = 1~~0 S(T) , w且ere

S(T) is the spot price of the asset in one year. For every $1000 invested , the payoff in one year of the second strategy, i.e. , of investing everything in buying call options , is

1I2 (T)

= 半 max(S(T) - 35， 0) 工(平叫)-35)iijzjZZJ

It is easy to see that , if S(T) is less than $35 , than the calls expire worthless and the speculative strategy of investing everything in call options will lose all the money invested in it , while the first strategy of buying the asset will not lose all its value. However , investing everything in the call options is very pro自table if the asset appreciates in value , i.e. , is S(T) is signi五cantly larger than $35. The breakeven point of the two strategies , i.e. , the spot price at maturity of the underlying asset where both strategies have the same payoff is $39.5161 , since 1000 _ ._, 35

1000

一:~ S(T) = 一一 (S(T)

- 35) 仁斗 S(T)

=

39.516 1.

If the price of the asset will , indeed , be i口 the $40一$45 range in one year , then buying the call options is the more profitable strategy. 口 Problem 18: The risk free rate is 8% compounded continuously and the dividend yield of a stock index is 3%. The index is at 12 ,000 and the futures price of a contract deliverable in three months is 12 ,100. Is there an arbitrage opportunity, and how do you take advantage of it?

Solution: The arbitrage-free futures price of the futures contract is 12000er - q)T = 12000e(0.08一0.03)/4

12150.94 > 12100.

Therefore , the futures contract is underpriced and should be bought while hedged statically by shorting e- qT = 0.9925 units of index for each futures contract that is sold.

1.1. SOLUTIONS TO CHAPTER 1 EXERCISES

31

At maturity, the asset is bought for 12100 and the short is closed (the dividends paid on the short position increase the size of the short position to 1 unit of the index). The realized gain is the interest accrued on the cash resulting from the short position minus 12100 , i.e. , eO.08 / 4 (e- O.03

32

CHAPTER 1. CALCULUS REVIEW. PLAIN VANILLA OPTIONS.

1.2

Supplemental Exercises

1. 3. SOLUTIONS TO SUPPLENIENTAL EXERCISES 8. Create a portfolio with the following payoff at time T:

1. Compute

户口(x) dx 2. Compute

V(T) =

xne x dx

(I n ( x) )n dx

5. Let

f(x)

z+ \11I/ 1-z

(

]

V Z

= 一古=exp(σ 〉习作

>-

才t4

<

寸'····-

e

吨E在

/IIlt\

<

十

寸土

+

\飞 llI/

/it\ l-zZ

0 三 S(T) < 20; 20 三 S(T) < 40; 40 三 S(T) ，

9. Call options on the same underlying asset and with the same maturity, with strikes K 1 < K 2 < K 3 , are trading for C1 , C2 and C3 , respectively (no Bid-Ask spread) , with C1 > C2 > C3 . Find necessary and sufficie时 conditions on the prices C 1 , C 2 and C 3 such that no-arbitrage exists corresponding to a portfolio made of positions in the three options.

J

4. Show that

2S(T) , if ( < 60 - S(T) , if l S(T) - 20 , if

where S(T) is the spot price at time T of a given asset. Use plain vanilla options with maturity T as well as cash positions and positions in the asset itself. Assume , for simplicity, that the asset does not pay dividends and that interest rates are zero.

J

3. Compute

33

C bid and C，αsk ， and by 1毛id and ~αsk ， respectively, the bid and ask prices for a plain vanilla European call and for a plain vanilla European put option , both with the same strike K and maturity T , and 0日出e same underlying asset with spot price S and paying dividends continuously at rate q. Assume that the risk-free interest rates are constant equal to r. Find necessary and sufficient no-arbitrage conditions for Cbid , Cask , Pbid , and 凡sk·

10. Denote by

(x 一 μ)2\ 2σ2

\

i

)

Assume that g : JR • JR is a continuous function which is uniformly bounded , i.e. , there exists a constant C such that jg( x) I 三 C for all x E JR. Then , show that

M: 削川x

=

g(μ)

1.3

Solutions to Supplemental Exercises

Problem 1: Compute

J

n x ln(x) dx

6. Let

内)工艺

Ci

Solution: If n 乒 -1 ， we use integration by parts and find that

Compute g' (y).

f n + l l f zn+1·-dz x nz ln(x) dx = 一-Mz)-j η + 1 η十 1)

7. A derivative security pays a cash amount C if the spot price of the underlying asset at maturity is between K 1 and K 2 , where 0 < K 1 < K 2 , and expires worthless otherwise. How do you synthesize this derivative security (i.e. , how do you recreate its payoff almost exactly) using plain vanilla call options?

n十

x 11n (吟 For n

= 一1，

x

n十 1

、A

+ C.

we obtain that

rn~x) 一 WJ dx

= (ln(x)?

+

C.

口

34

CHAPTER 1. CALCULUS REVIEW. PLAIN VANILLA OPTIONS.

c

vv n >-

414

口

♂

pu

十

'h 川h

吁i

η

门

一-

z

\飞 lll-/

nZM Lf-M

/III-11\

n

(1. 16)

(1\1 < In I 1 + -=. I < 一、 Vx> 1.

Let

x)

:一咛 +D;

f(x)

x

一

(1. 17)

咔+~)一击

g(x)

The丑 7

= eX , the followi吨 general forml 一­

,J

Vv Z >-

才i

1

Z 十 1\

儿(x) = j ♂eX dx = xneX - n j ♂-1川 = xne x 一吼一 1 协 Vn;:::l fIJ

4t-A

Note that (1. 16) is equivalent to

By using integration by parts , it is easy to see that

,2 z neZ G

吨'-4

Sol仰 on:

一二一-

PIt-

<

pu

as

儿 (x) 工 j xne x dx

Since fi。但)

Z+ \111/ 1-z

/I111\

<

十

fn(x)

z i

For every ir由ger n 主 0 ， define the function

t\liI/ al-Z

十

Solution:

民才

/♂♂ dx

σ/Iil\

h

QU

U

''EA

A且 I

P VAO 'D em

Problem 2: Compute

35

1.3. SOLUTIONS TO SUPPLE1VIENTAL EXERCISES

j' (x)

g'(x)

1

一一一+. 2

x

I

X

1

一

(x + 1) -

L一十

x(x 十 1)

I

1

(x + 1)2

x2 (x + 1) <~ 0:-, -

1< 0

x(x + 1)2

We conclude that bo也 f (x) and g( x) are decreasing functions. Since Problem 3: Compute

lim

j(1n(x))η dx

X 一→汉二

it follows that

Solution:

f(x) =

lim

X-→。c

f(x) > 0 and g(x) > 0 for

For every integerη 三 0 ， let

g(x) = 0 ,

all

x > 0, and

therefore

:>咛+:)〉 ZLI7VZ>07

儿 (x) 工 j (In (x) )n dx

which is what we wanted to show; cf. (1. 17).

口

By using integration by parts , it is easy to see that , for any η;三 1 ， Problem 5: Let

户口(x)t dx = 川)川 j (In(x))η→

仲) =忐叫一(工f)2)

and therefore

fn(X ) = x(I n ( x))η 一 η fn-1(♂)，

V η 主1.

Since fo(x) = 凯 the following general formula can be obtained by ir吐uction:

j (1卫 (x))n dx = fn(x) = x 主 L

(γk n !

\ .L ~nl

I f,;

(ln(x))k 十 0，

Vn 主1.口

Assume that g : R •ffi. is a continuous function which is uniformly bounded 2 , i.e. , there exists a constant 0 such that Ig(x)1 三 o for all x ε ffi.. Then , show that

叫:削g(x) dx = 仲)

2The uniform boundedness condition was chosen for simplicity: and it can be l'吐axed: e.g.: to functions which have polynomial growth at infinity.

CHAPTER 1. CALCULUS REVIE 1iV. PLAIN VANILLA OPTIONS.

36

Solution: Using the change of variables y = 乎?附在nd that

汇 f(x)g(x) dx

志汇 g(x)

exp

(气;)2)

1

J 一∞

Ig(μ) - g(μ+ 句)1 e一专 dy

J 62(£)

<

2Cε

= 一安

告 (1

Citi

+ y)t i

十1

(1. 23)

<-ss <

K

门，，"

(1. 25)

ιmwmw

Vy ε[-62(E)， 62(ε)]，

气/〈<一

< 飞

ρ·?ip?ip?· 晶·

g(μ+σy)1

that

nucnu

阿凡 ι"

we 自nd

/ B E E f - -飞

62(ε) 一

(1. 24)

V T

一-

I(μ+ 句)一 μ1=σIyl <61(ε) 一一::; 61 (ε) ， V y ε[-62(ε) ， 62(E)].

(1. 21) and (1. 24)

Problem 7: A derivative security pays a cash amount C if the spot price of the underlying asset at maturity is between K 1 and K 2 , where 0 < K 1 < K 2 , and expires worthless otherwise. How do you synthesize this derivative security (i.e. , how do you recreate its payoff almost exactly) usi吨 plai川anilla call options?

Solution: The payoff of the derivative security is

Iyl

Ig(μ)

'(y)

JR., it follows from (1. 22) that

It is 叫 to s叫风 if () <去87then

The凡 from

ZJ

Compute g' (y).

J 62(£)

+τ: (∞ Ig(μ) - g(μ 叫)1 汁 dy v ζπ

们u u

v L,1f

I

e

，α

r- 62(£)

1

g(y)

Solution:

(1. 22)

σ

J 一∞飞 !"L，1f

十

「LfdM)e4dy 十气: (∞片 dy < ε

2ML2

uu

Problem 6: Let

(1. 21 )

Using the fact that the integral (1. 19) exists and is finite , we obtain that , anyε> 0 , there exists 62 (ε) > 0 such that

η3

V Ix 一 μ1< 61(ε) .

μ'

Ig(μ )-g(x)1 < 飞

(1. 26)

叫:f(z)g(Mzg(μ) 口

(1. 20)

Our goal is to show that the right hand side of (1. 20) goes to 0 as σ\、 O. Since g( x) is a continuous function , it follows that , for 创lY ε> 0 , there exists 61 (ε) > 0 such that

nu

μ'

/

We conclude , by definition , that

沉沉 L陀广 nil← -」

r∞

-

→头=

Z

1一一份仰

1

f(x)g(x) dx = 一) (g(μ) - g(μ 十 σy)) e- 号 dy. j一∞ d订一∞

E

7α

f∞

Since Ig( x) I 三 C for all x

z nud z

J-OO

g(μ) - /

V L/Tr

rlu

<一〈

flL

(1. 19)

since，吗， the function 在e-15is tue p时ability density function of the standard normal variable. From (1. 18) and (1. 19) we obtain that

for

Ig(μ) - g(μ+σν)\ e一号 dy < ε-

From (1. 20) , (1. 23) , and (1. 26) , it follows that , for anyξ> 0 , there exist 61(E) > 0 and 62(E) > 0 such that , if σ<:237then ∞∞

kI/I

ο

= 1,

r 62(ε)

I

v L,1f J -62(£)

门时 M

V

→是=

dx

μ'

去/∞ λ dy

37

and therefore

vi言汇 g(μ+ 仙句 Recall that

1. 3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

Since V(T) is 出 d isc ∞ on 挝t挝 叫inuo 盹 i让t cannot be rep 抖lica 创ted exactly us岳i吨 call 丑lopt址10口 whose p a;句~offs are contir丑lUaus.

38

CHAPTER 1. CALCULUS REVIE 1iV. PLAIN

v:生NILLA

OPTIONS.

We approxi口时 e the payoff V (T) of the derivati飞肘ec盯ity by the followi吨 payoff 0,

(

I ~(T)

= <

c(S(T) 一 (K1

I

l

c,

-

if S(T) < K1 一 ε; E))/ε ， if K1 一 ε 三 S(T) 三 K1 ; if K 1 < S(T) < K 2 ; if K2 三 S(T) 三 K2 + ε; if K2 十 ε < S(T).

c - c(S(T) - K2 )/ε ， 0,

(1. 27)

< K1

~(T) 一 ε

。

K21

15εS二三十

KK1

K 2 ~二 K2 + ε < S(T)

ε

(T ;(S(T)

c-

•

(fKC(KS12

;(S(T)

+ 60 -

33(T)

= 2S(T) - 3(3(T) - 20).

This is equivalent to a long position in two units of the underlying asset and a short position in three calls with strike 20. To replicate the payoff S(T) - 20 of the portfolio when 40 三 S(T) ， note that S(T)-20

=

60-3(T) 十 2S(T)-80

= 2S(T) - 3(3(T)-20) + 2(S(T)-40).

This is equivalent to a long position in two units of the underlying asset , a short position in three calls with strike 20 , and a long position in two calls with strike 40. Summarizing , the replicating portfolio is made of • long two units of the asset; • short 3 call options with strike K = 20 on the asset; • long 2 call options with strike K = 40 on the asset. We check that the payoff of this portfolio at maturity, i.e. ,

vi (T) = 23(T) - 3 max(S(T) - 20 , 0) + 2 max(S(T) - 40 , 0)

(1S)K(T20))() 2K+1))=C )EQ+()S)f(((SK)E T ε))·;;0 (T K

39

One way is to use the underlying asset , calls with strike 20 , and calls with strike 40. First of all , a portfolio with a long position in two units of the underlying asset has value 2S(T) at maturity, when S(T) < 20. To replicate the payoff 60 - S(T) of the portfolio when 20 三 3(T) < 40 , note that 60 - 3(T) = 2S(T)

Note that V(T) = ~(T) unless the value S(T) of the underlyi吨 asset at maturity is either between K1 一 ε and K 1 , or between K 2 and K2 十 ε. The payoff ~(T) can be realized by goi吨 long c/εbull spreads with strikes K1 一 ε and K 1 , and shorti吨 c/εbull spreads with strikes K 2 创 an K2 十 ε已. In other words , the payoff V(T) of the given derivative security can be synthesized by taking the following positions: • long c/εcalls with strike K1 一 ε; • short c/εcalls with strike K 1 ; • short c/εcalls with strike K 2 ; • 10吨 c/εcalls with strike K2 十 εIt is easy to see that the payoff ~(T) is the same as in (1. 27): S(T)

1. 3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

(1. 29)

is the same as the payoff from (1.28): 同 (T)

23(T) 3(T) S 20 2S( T)=3(S(T) - 20) = 60 3(T) 20 < S(T 否){ 三 40 60 - S(T) + 2(S(T) - 40) = S(T) - 20 40 < S(T)

Problem 8: Create a portfolio with the following payoff at time T: V(T)

r 23(T) , if = < 60 - S(T) , if l S(T) - 20 , if

0 三 3(T) < 20; 20 三 S(T) < 40; 40 三 S(T) ，

(1. 28)

where S(T) is the spot price at time T of a given asset. Use plai丑 vanilla options with maturity T as well as cash positions and positions in the asset itself. Assume , for simplicity, that the asset does not pay dividends and that interest rates are zero.

Solution: Using plain vanilla options , cash , and the underlying asset the payoff V(T) can be replicated in different ways.

As a consequence of the Put-Call parity, it follows that the payoff V(T) from (1. 28) can also be sy毗lesized using put options. If the asset does 口ot pay dividends and if interest rates are zero , then , from the Put-Call parity, it follows that C = P+S-K. Denote by C20 and P20 , and by 0 40 and 凡0 ， the values of the call and put options with strikes 20 and 40 , respectively. Then , the replicating portfolio with payoff at maturity given by (1. 29) can be written as V - 23 - 3020 + 2040 . (1. 30)

40

CHAPTER 1. CALCULUS REVIE1iV. PLAIN VANILLA OPTIONS. To synthesize a short position in three calls with strike 20 , note that

1.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

41

which is equivalent to taking a short position in three units of the underlying asset , taking a short position in three put options with strike 20 , and being a long $60. Similarly, to synthesize a long position in two calls with strike 40 , note that 2040 2凡。十 28 - 80 , (1. 32)

• short 3 call options with strike K = 20 on the asset; • long 2 call options with strike K = 40 on the asset. The second replicating portfolio will be made of the following positions: • long e- qT units of the asset; • short $20e- rT cash; • short 3 put options with strike K = 20 on the asset; • long 2 put options with strike K = 40 on the asset. Note that any piecewise linear payoff of a single asset can be synthesized , in theory, by using plai丑 vanilla options , cash and asset positions. 口

which is equivalent to a borrowing $80 , taking a long position in two units of the underlying asset , and taking a long position in two put options with strike 40. Usi吨(1. 31) and (1. 32) , we obtain that the payoff at maturity given by (1. 29) can be replicated usi吨 the followi吨 portfolio consisti吨。f put options , cash , and the underlying asset:

Problem 9: Call options on the same underlying asset and with the same maturity, with strikes K 1 < K 2 < K :3' are trading for 0 1, O2 and 0 3, respecti飞吨T (∞ Bid-Ask spread) , with C1 > O2 > 0 3 Find necessary and sufficient conditions on the prices 0 1 , O2 and 0 :3 such that no-arbitrage exists corresponding to a portfolio made of positions in the three options.

一 3020

V

-

- 3P20 - 38 + 60 ,

(1. 31)

3020 十 2040

二

28 28 -

=

8-3马。十 2凡。一 20.

3马。一 38 +60 十 2凡。+

28 - 80

(1. 33)

The positions of the replicati吨 portfolio (1. 33) can be summarized as follows: • long one unit of the asset; • short $20 cash; • short 3 put options with strike K = 20 on the asset; • long 2 put options with strike K = 40 0丑 the asset. We check that the payoff of this portfolio at maturity, i.e. , 巧 (T)

= 8(T) - 20 - 3 max(20 - 8(T) , 0) + 2 max(40 - 8(T) , 0)

is the same as the payoff from (1. 28):

2日

8(T 页) ~ 20 B(T) 3(20 8(T) - 20 十 20 < 8(T) 三 40 40 < 8(T)

页{4SC到L--Tf芦S8(T)) T-)邪 T2川日 o=--S(T))~ 28(T) 60 - 8(T)

2(40

If the asset pays dividends continuously at rate q and if interest rates are constant and equal to 飞 in order to obtain the same payoffs at maturity, the asset positions in the two portfolios must be adjusted as follows: The first replicating portfolio will be made of the following positions: • long 2e- qT units of the asset;

,

Solution: An arbitrage exists if and only if a no-cost portfolio can be set up with non-negative payoff at maturity regardless of the price of the underlying asset at maturity, and such that the probability of a strictly positive payoff is greater than O. Consider a portfolio made of positions in the three options with value 0 at inception , and let Xi > 0 be the size of the portfolio position in the option with strike K i , for i = 1 : 3. Let 8 = 8(T) be the value of the unde臼rl 创坊 yi凶 n asset at maturity. For no-arbitrage to occur , there are three possibilities: Portfolio 1:Long the K 1-option , short the K2 一 option， long the K 3-option. Arbitrage exists if we can find Xi > 0 , i = 1 : 3 , such that

X1 0 1 - X2 02 十 X303 = 0;

(1. 34)

x1(8 - K 1) - x2(8 - K2 ) 十句 (8 - K3 ) 三 0 ， V 8 主 O. (1. 35) We note that (1. 35) holds if and only if the following two conditions are satisfied: Xl - X2 十 Z32三 0; (1. 36) x1(K3 - K 1) - x2(K3 - K 2) ~ O. (1. 37) vVe solve (1. 34) for X3 and obtain

Since we assumed that X3

O2

01

03

.."'03

(1. 38)

> 0, the following condition must also be satisfied: X2

01

一，一

句

O2

(1. 39)

42

CHAPTER 1. CALCULUS REVIE1V. PLAIN VANILLA OPTIONS.

Recall that C1 > C2 > C 3 . Using the value of that (1. 36) and (1. 37) hold true if and 0日ly if C1 -

X2

、

Xl

… C2 -

X3

from (1. 38) , it follows

C3

(1. 40)

、-

C3 '

X 2 K 3 - K1 Xl

-

K3

-

(1. 41)

K2

Also , note that if (1 .40) holds true , then (1. 39) is satisfied as well , since

C1 - C3 C2 - C3

C1 ­ C

二

-

C1 、

C3 C2 -C3 -

(1 .4 2)

Portfolio 2:Long the K 1-option , short the K 2-option , short the Arbitrage exists if we can 且口d Xi > 0 , i = 1 : 3 , such that 的 C1 - X2C2 - X3C3 Xl (8

0;

- K 1 ) - x2(8 - K2 ) 一句 (8 - K3 ) 三 0 ， V 8 三 O.

K3 一option.

(1. 43) (1. 44)

The ineql叫ity (1. 44) holds if and only if the following two conditions are satisfied: Xl - X2 - X3 主 0; (1. 45) Xl (K3 -

K 1) -

x2(K3 -

K2 ) 三 O.

(1. 46)

However , (1. 43) and (1. 45) cannot be simultaneously satisfied. Since C 1 > C 2 > C3 , it is easy to see that 1

=

C? C

C'}. C <-

X2 一二 -X'}. 一二 --v

1

1

43

x1(8 - K 1 ) + x2(8 - K2 ) 一句 (8 - K3 ) 三 0 ， V 8 三 O. The inequality (1 .48) holds if and only if

+ X2

(1. 48)

- X3 主 O.

(1 .49) It is easy to see that (1. 47) and (1. 49) cannot be simultaneously satisfied: C, C? 3 Xl 一二 +X2 一二 > X1 +X C3 C3 Xl

~

~

~

since C1 > C2 > C3 . Therefore , no arbitrage can be obtained by being long the options with strikes K 1 and K 2 and short the optio日 with strike K 3 . We conclude that (1. 42) , i.e. ,

2

We conclude that arbitrage happens if and only if we can find Xl > 0 and X2 > 0 such tl时(1.40) and (1. 41) are simultaneously satis丑ed. Therefore , no-arbitrage exists if and only if K 3 -K1 K 3 -K2

1.3. SOLUTIONS TO SUPPLE1VIENTAL EXERCISES

In other words , no arbitrage can be obtained by being long the option with strike K 1 and short the options with strikes K 2 and K 3 . Portfolio 3:Long the K 1-option , long the K2 一 option， short the K 3-option. Arbitrage exists if we can find Xi > 0 , i = 1 : 3 , such that 0;

C1 一 α

'二

K3 -K2 is the only condition required for

、

C2 -

C3

no-arbitrage.

口

Problem 10: Denote by Cbid and Cask , and by 凡id and 凡sk ， respectively, the bid and ask prices for a plain vanilla European call and for a plain vanilla European put option , both with the same strike K and maturity T , and on the same underlying asset with spot price 8 and paying dividends continuously at rate q. Assume that the risk-free interest rates are constant equal to 7'. Find necessary and sufficient no-arbitrage conditions for Cbid , Cαsk ， Pbid , and 凡sk·

Solution: Recall the Put-Call parity C-

8e- qT -

P

Ke- rT ,

where the right hand represents the value of a forward contract on the underlying asset with strike K. An arbitrage would exist • either if the purchase price of a long call short put portfolio , i.e. , Cα砾，一 Pbid were less than the value 8e- qT - K e- rT of the forward contract , i.e. , if Cαsk 一凡id<Se-qT-kfrT;

X') 十 X3· --.,

X1 C1 十 X2C2 - X3C3 -

K3 -K1

(1. 47)

• or if the selling price of a long call short put portfolio , i.e. , Cbid - Pask were greater than the value 8e- qT - Ke- rT of the forward contract , i.e. , if Cbid 一凡sk

We conclude that there Call parity if and only if

>

Se-

qT

- K e- rT

is 丑。-arbitrage directly

Cαsk …凡id 三 8e- qT

-

K

following from the Put-

e- r1、三 Cbid 一凡 sk·

口

Chapter 2 Improper integrals. Numerical integration. Interest rates. Bonds. 2.1

Solutions to Chapter 2 Exercises

Problem 1: Compute the integral of the function f(x , y) = x 2 - 2y on the region bounded by the parabola y = (x + I? and the line y = 5x - 1. Solution: We first identify the integration domain D. Note that (x + I? = 5x -1 if and only if x = 1 and x =2 , and that (x + 1)2 三 5x - 1 if 1 < x < 2.

Therefore , D = {(x , y) 11 三 Z 三 2 and (x+ 1)2 三 ν 三 5x - I}.

Then ,

fin 川白

[(汇 ιιi;:(归户 Z [

((怡均牛 2与U 一 泸的州叫叫)川川将巾 U|tcc U汇口口 口 :ζ:Uω; !: 斗 L 二二 J;斗ω户)2)讪 ) dx

I 泸 (5x

-

1 一一- (归仙川 Z叶川+札1呐一((仰一 1)2 一归仙川+刊1阳Z

1，'(σ5x 一 1 一川附一仰一 1忏叫十刊(川 俨(←一泸仇川十叶川 l 3xZ卜川一 -2川Z叫)川d白Z户←=←一; 口 Problem 2: Let

f : (0 ，∞)→lR denote the Gamma functio孔

的)

χ x

=

飞 俨1 V俨Zα←叫~一→ l 俨俨 45

i.e. ，

le t.

CHAPTER 2. NUMERICAL INTEGRATION. BONDS.

46

(i) Show that f(α) is well defined for any α> 0 , i.e. , show that both

[xα-1

e- x dx

[ ' x"

•

1 e- x

zα-le-z <

_ 时12α-1 e一♂ dx

dx

出 [xα一1

dx

exist and are finite. (ii) Prove ，旧吨 i的egration by parts , that f (α) = (α- 1) f( α 一 1) for any α> 1. Show that f(l) = 1 and conclude that , for any η 三 1 positive integer , f(n) = (η-I)!.

Solution: (i) Let α> 0. Intuitively, note that , as x \、 the order of x o:- 1 , since limx\", 0 e- x = 1. Since

~tIl}

(1 xα-1 t\", 0 Jt

0，

the functio丑 Zα-l e -x is 0丑

°

Choose n(ε) = max(m(ε ， N)). From (2.3) and (2 .4) we obtain that

t\、oαIt

xα一1

e-x dx -

lim I t\o Jt

X

dx

to show that , for any s > η(ε) ,

Z

smce lim x o:- 1 e- x

limxα-1

z\0

2\J

lim t o:- 1 e- t t → οc

-

0,

ια-1

lim 二--:-

t• x

eL

= 0.

for α> 1;

e

Z

,G

m 曲

q&

气 BEEt--liId

αz

才t 4

ffl'o

飞/

划已

e

1··A

伽

FI--00/ 飞

JZα

1lιtn

(2.2)

卜

I∞川-X dx < 飞 Vs> η(ε)

K

°

『f n n U 吃i U

exists and is finite. l\IIaking these intuitive arguments precise is somewhat more subtle. We include a mathematically rigorous arguments for , e.g. , showing that the integral in (2.1) exists and is finite. By de丑nition， we need to prove that , for a町 ε> 0 , there existsη(ε) > such that

•ijd

Z Z7 ufbfqh 飞 lI/{

(2.1)

m叭 卜

e- x dx

一 -11J

it follows that

咄咄-f

f(α)

=卡

1,

"~UV

fzMe曰 :Etlzα-1

= 1

Assume that α> 1. By integration by parts , we find that ]αef''

JiITl (1 - e- ) -

m)=fev=JI叫tfzdz= 出 (_e-t + 1) -z

t

which is what we wanted to show; d. (2.2). (ii) It is easy to see that

斗{旷日仁川

e- dx -

(2.6)

2e- s !2 < 2e- n (f. )!2 < 飞 xα-1 e- x

- 3川

I

Jl

and

<J坦/，' ~-x切=出(_2e- t !2 十川)

α ! 「U扒 L 1 1 χii 况一-

"~UV

< ε(2.6)

J豆豆/，' X"-l 川Z

X"-l 川Z

fl'ok 扫人 wh

Jim

(2.5)

We can then use

α

exists and is finite. In a similar intuitive way, note that , as x →∞， the function x O:一 l e -x is 0日出e order of e- x , since the exponential function dominates any power function at infi丑ity. Since x

(2.5)

zα-l e-z <e-z/27V Z 〉 η(ε);

it follows that

I

x

Also , since limx →。c e- x !2 = 0 , it follows that , for any ε> 0 , there exists m(E) > such that 2e- m (f. )!2 <已 (2 .4)

/,

l

(2.3)

1m x- - e_-x/2 -,- =

2e- n (f. )!2

dx 工 lim 主 |zllim(l-tα)

Jo

e-z/27 Vx> N ,

Iiα -1 X-•

e- x

°such that

Note that there exists N >

Sl日ce

and

47

2.1. SOLUTIONS TO CHAPTER 2 EXERCISES

CHAPTER

48

2.

NUMERICAL INTEGRATION. BONDS.

For any positive integerη> 1, we 五nd that f(n) = f(l) = 1, it follows by induction that f(η) = (η-1)!

(η -l)f(η-1). 口

2.1. SOLUTIONS TO CHAPTER 2 EXERCISES No. Intervals NIidpoint Rule 0.17715737 4 0.17867774 8 0.17904866 16 0.17914062 32 0.17916354 64 0.17916926 128 0.17917070 256 0.17917105 512

Since

3

Problem 3: Compute an approximate value of J1 -IX e-xdx using the NIidpoint rule , the Trapezoidal rule , and Simpson's rule. Start with η= 4 intervals , and double the number of intervals until two consecutive approximations are within 10- 6 of each other.

Solution: The approximate values of the integral found using the NIidpoint , Trapezoidal , and Simpson's rules can be found in the table below: No. Intervals NIidpoint Rule Trapezoidal Rule Simpson's Rule 4 0 .40715731 0 .4 1075744 0 .40835735 8 0 .40895737 0 .40807542 0 .40836940 16 0 .40829709 0 .40851639 0 .40837019 32 0 .40835199 0 .40840674 0.40837024 64 0 .40837937 0 .40837024 0 .40836569 128 0 .40836911 0 .4 0837253 0 .40837082 256 0 .40836996 512 0 .40837018 0 .4 0837039 0 .40837028 1024 0 .40837023

(ii) Without computi吨 f(4)(X)， note tl时 the denominator 1 + x 2 of f (x) is bounded away from 0 , and that the fourth derivative of the numerator of f(x) is on the o-rder of X- 3 / 2 , which is not defined at 0 , and is unbounded in the limit as x 、 o. (iii) Usi吨 Simpsor内 rule ， the following approximate values of the integral are obtained: No. Intervals 4 8 16 32 64 128 256

The approximate value of the integral is 0 .408370 , and is obtained for a 256 intervals partition using the NIidpoint rule , for a 512 intervals partition using the τ'rapezoidal rule , and for a 16 intervals partition using Simpson's rule.

49

Simpson's Rule 0.179155099725056 0.179169815603871 0.179171055067087 0.179171162051226 0.179171171372681 0.179171172188741 0.179171172260393

口

Problem 4: Let f : 卜 R→ --+R 问问 gi忖阳 V (ωi) Use NIidpoint rule with tal = 10- 6 to compute an approximation of (1

(1

I

币 5/2

= JoI f (x) dx = JI ~一一亨 o 1+ OJ

'--I

----

The approximate value of the integral is 0.17917117 , and is obtained for a partition of the interval [0 , 1] using 64 intervals. 口 Problem 5: Let K , T ， g :R • R as

σand r

x~

(ii) Show that f(4) (x) is not bounded on the interval (0 , 1). (iii) Apply Si皿psor内 rule with n = 2飞 k = 2 : 8, intervals to compute the integral I. Conclude that Simpson's rule converges.

Solution: (i) The approximate value of the integral is 0.179171 , and is obtained for a partition of the interval [0 , 1] using 512 intervals:

g(x) where b(x) =

(In

be positive constants. Define the function

= 在 rhu?

(圣)十(叫去) T) / (σ-IT). Compl.叫 (x)

Solution: Recall that

irf叫 = f川。)

CHAPTER 2. NUMERICAL INTEGRATION. BONDS.

50 Therefore ,

2.1. SOLUTIONS TO CHAPTER 2 EXERCISES

51

it follows that

g"(t) = h(t) ,

V左 i七言b印(归ω 阳 Z功)

g'(x)

which is what we wanted to show.

…(

l

xO"-J2百…y

(In

(圣)十 (γ 十号) T)2 \ 2σ2T

\

n

Problem 7: The continuously compounded 6-month , 12-month , 18-month , and 24-month zero rates are 5% , 5.25% , 5.35% , and 5.5% , respectively. Find the price of a two year semiannual coupon bond with coupon rate 5%.

~

J'

Solution: The value B of the semiannual coupon bond is Problem 6: Let 1巾) be a continuous fun创0口 such that J之o Ixh(x)ldx exists. Define g(t) by

以t)

=

[>O(川)h(x) = h(t) .口

,Tb

α

,Tb

、 、E E，J '

、飞}/

ι'LU

rlu

i'U

/ ' ' E飞 、

'hu

α

、、BJ/

,Tb

JZaE 飞

。，"

For our problem , α (t)

= t and f(x , t) = (x - t)h(x) ,

(2.8)

where h(x) is conti丑∞us. Then , ~{ (x , t) = -h(x) is continuous. Note that f(α (t) ， t)

= f(t , t) = (t - t)h(t) = O.

Since

= 二(

[' (x - t)h(x) dX)

口

(2.9)

Solution: Par yield is the coupon rate C that makes the value of the bond equal to its face value. For a 2-year semiannual coupon bond , the par yield can be found by solving 100 -

~

+

= - [\(X) dx T 28U

二( 1: 川x) = 一川川7

98.940623.

Problem 8: The continuously compounded 6-month , 12-month , 18-month, and 24-month zero rates are 5% , 5.25% , 5.35% , and 5.5% , respectively. vVhat is the par yield for a 2-year semiannual coupon bond?

From (2.7-2.9) , we conell由 that

g' (t)

= [2.5 2.5 2.5 102.5] .

时i

ι'U

、飞 l'/

iTb

v一cas且且ow

disc = [0.97530991 0.94885432 0.92288560 0.89583414], and the price of the bond is B =

i'U

α

rld

α

z

/ ，1 1 飞

ιtu

Z

7α

W

Mυ 一衍

PIllα

∞仙

一一

YG Z

\飞 liI/

、、 lt/

z

ι'U

rld

w

ft'11α

∞仙

/Ill-\

= [0.5 1 1. 5 2];

The discount factors are

A similar result can be derived for improper integrals , i.e. ,

d-a

r叫咐咐仰(仰队阳 O矶ω 川 ;• l叫 μ e-川川 ，2

where C 工 0ω.β05 ， and r(0 , 0.5) = 0.05 ， γ(0 ， 1) = 0.0525 , r(0 ,1. 5) = 0.0535 , γ(0 ， 2) = 0.055. The data below refers to the pseudocode from Table 2.5 of [2] for ∞ c om putir丑19 the bond price given the zero rate curve. Input: n = 4 Lcash且ow

vhu ,Tb

/'1 、白

+ rld

2

飞}/

Z ,?-U

7α

、iJ

Mυ-a

-4Lj

M叫凡 V PI''1α

一一

ι'LU

VG Z

、、 l'/

z

V

PI''1'α

ν叭 凡

\Ill-/ rlu

2

十 (10∞0 十

Solution: Recall that , ifα (t) and b(t) are differentiable functions and if f (x , t) is a continuous function such that ?It (x , t) exists and is continuous , then /-11iI\ d-a

=至 100 e- r (O，O.5)O.5 十空 100 e- r (即)十三 100 e- r (阳)1.5

d

and show that gil ( t)

B

100 e- r (O ,O.5)O.5 (100

+

+ ~ 1叫

C

~

100 e- r (O ,1)

2

户

+ 二 100 e- r (O ，1. 肌5

2

e- r (O，2)

QU

C

2(1 - e- r (O ,2)2) e- r (O ， O.5)O.5 十 e- r (O ， l) 十 e- r (O ，1. 5)L5 十 e- r (O ， 2)2

•

CHAPTER

52

2.

NUNIERICAL INTEGRATION. BONDS.

For the zero rates given in this problem , the corresponding value of the par yield is C = 0.05566075 , i.e. , 5.566075%. 口

2.1. SOLUTIONS TO CHAPTER 2 EXERCISES 十

(

~ ~

I 100

2.5 disc(l) Problem 9: Assume that the continuously compounded instantaneous interest rate curve has the form r(t) = 0.05 十 0.0051泣。 + t) , V t 三 O.

(i) Find the correspondi吨 zero rate curve; (ii) Compute the 6-month , 12-mo时h ， 18-month , and 24-month discount factors; (iii) Find the price of a two year semiannual coupon bond with coupon rate 5%.

+

99.267508.

Solution: The price , duration , and convexity of the bond can be obtained from the yield y of the bond as follows:

B D

= 兰 3 仪p(一到十川p(补 1 f ι3i

一 I )

: :-

B\ 仨t 2

1 (~9i

l丑 (ο1

+ t))

e- r (O ,O.5)O.5

_

e- r (O ,l)

0.94939392;

-

e-r(O ，1. 5) 1. 5 已 -r(O， 2)2 -

0.97478242;

二 0.92408277 ;

0.89899376.

(iii) The price of the two year semiannual coupon bond with 5% coupon rate IS

β:

0.05 ~({\ (\ ,,\{\ " 0.05 刁-100e-r(O?05)05 十一~V 2

4

(i \ 5 (5\ 飞 + 103 ~exp I -~y I I 飞 2 ~ ~ 2 \ 2'-' } J

I -~y I

1\ 100 e-一 r~I{\ (O ,I)

(2.10) ,, l

‘\

41·A

07

}

(i \ 2 5 (5\1 exp (一 ~y) + 103 -A~ exp (一 ~y ) I . 飞 2 4 \ 2'" } J 07

}

'

-

-

(2.12)

-

v一cash_自ow

= [3 3 3 3 103] .

Output: bond price B = 92.983915 , bond duration D = 2.352418 , and bond convexity C = 5.736739. 口

(ii) The 6咄onth， 12-month , 18-month , and 24-month discount factors are , respectively, disc(l ) disc(2) disc(3) disc(4)

VA"

exp

Lcash_flow = [0.5 1 1. 5 2 2.5];

0.005(1 十 t)

102.5 disc(4)

The data below refers to the pseudocode from Table 2.7 of [2] for computing the price , duration and convexity of a bond given the yield of the bond. Input:η= 5; y = 0.09;

;(0 阳+ 0ω0∞0 阳 +

+

Problem 10: The yield of a semiannual coupon bond with 6% coupon rate and 30 months to maturity is 9%. What are the price , duration and convexity of the bond?

) ~ B\ 仨t

庐t、 f 才 h扣 0ω05 十 0ωO∞0臼5盯阳川叫 l协丑叫(川 n( )

0.045

2.5 disc(3)

、、 1'/

= ;

+

呼14

叫OJ )

2.5 disc(2)

口

=一 I

Then ,

~({\ t)飞叮

e 一 I \V忡忡

门，中

ι'b

、、 Bt/

7α

俨'

TLV

一一

γ

JI 飞飞

Solution: (i) Recall that the zero rate curve r(O , t) can be obtained from the instantaneous interest rate curve r(t) as follows: l--tp''Ifo vv > nu nU , T T

0.05. __\ I

十丁一 IUU

53

0.05

+ 丁一 100 e-咐1. 5) 1. 5

Problem 11: The yield of a 14 months quarterly coupo且 bond with 8% coupon rate is 7%. Compute the price , duration , and convexity of the bond. Solution: The quarterly bond will pay a cash flow of 1. 75 in 2, 5, 8, and 11 months , and will pay 10 1. 75 at maturity in 14 months. The formulas for the price , duration , and convexity of the bond in terms of the yield y of the bond are similar to those from (2.10-2.12). For example , the price of the bond can be computed as follows:

B = 1. 75 exp

(

2 \ ( 5 \ / 民\ + 1. 75 exp I飞 - ~-"y I + 1. 75 exp I - ~-"y I 12'-' J \ 12'-' J

I - ~-"y I 飞

+1. 75 exp

12'-' J

( 1 1 \ ( 14\ + 10 1. 75 exp I - -,,- y I

I 一 ~~y I 飞

12'" J

飞/

CHAPTER 2. NUMERICAL INTEGRATION. BONDS.

54

The data below refers to the pseudocode from Table 2.7 of [2] for computing the price , duration and convexity of a bond given the yield of the bond. Input: n = 5; y = 0.07; I 2 5 8 11 141 t_c础且ow = 11-2 1~2 1~2 ~; ~;

I;

v_cash_flow = [2 2 2 2 叫

Output: bond price B = 10 1. 704888 , bond duration D = 1. 118911 , and bond convexity C = 1. 285705. 口 Problem 12: Compute the price , duration and convexity of a two year semiannual coupon bond with face value 100 and coupon rate 8% , if the zero rate curve is given by r(O , t) = 0.05 + O.Olln (1 + ~) Solution: The data below refers to the pseudocode from Table 2.5 of [2] for computing the price of a bond given the zero rate curve. Input:η= 4; zero rateγ (0 ， t) = 0.05 + O.Olln (1 十 i); t_cash一丑ow

= [0.5 1 1. 5 2] ; v_cash且ow = [4 4 4 104] .

Discount factors: disc = [0.97422235 0.94738033 0.91998838 0.89238025]. Output: Bond price B = 104.17391 1. Note: To compute the duration and convexity of the bond , the yield would have to be known. The yield can be computed , e.g. , by using Newton's method , which is discussed in Chapter 8. We obtain that the yield of the bond is 0.056792 , i.e. , 5.6792% , and the duration and convexity of the bond are D = 1. 8901 and C = 3.6895 , respectively. 口

2.1. SOLUTIONS TO CHAPTER 2 EXERCISES

55

ac ∞ ou 叩 po ∞ np 归 ay尼 r江即 rr I

payment.

口

Problem 14: By how much would the price of a ten year zero-coupon bond change if the yield increases by ten basis points? (One perce时 age poi时 IS equal to 100 basis points. Thus , 10 basis points is equal to 0.00 1.) Solution: The duration of a zero-coupo丑 bond is equal to the maturity of the bond , i.e. , D = T = 10. For small changes !:1 y in the yield , the percentage change in the value of a bond can be estimated as follows:

!:1 B

王一目

-!:1y

D =

…

= - 0.0 1.

0.001 . 10

We conclude that the price of the bond decreases by 1%.

口

Problem 15: A five year bond with duration 3~ years is worth 102. Find an approximate price of the bond if the yield decreases by fifty basis points. Solution: Note that , since the yield of the bond decreases , the value of the bond must increase. Recall that the percentage change in the price of the bond can be approximated by the duration of the bond multiplied by the parallel shift in the yield curve , with opposite sign , i.e. ,

!:1 B

万一句

- !:1 y

D

For B = 102 , D = 3.5 and !:1 y = -0.005 (since 1% = 100 bp) , we 五nd that !:1B 用

-

!:1 y D B = 1. 785.

The new value of the bond is Problem 13: If the coupon rate of a bond goes up , what can be said about the value of the bond and its duration? Give a financial argument. Check your answer mathematically, i.e. , by cor叩灿gggand gg7and ShhO 叭仰飞 咄 w ν these 旬 fUl丑 n囚 c ctior丑IS are either always positive or always 丑 1lega 肮，ti忖 ve.

-

B +!:1 B -

103.75.

口

Problem 16: Establish the following relationship between duration and convexity: 叮

C _ D :t.

Solution: Recall that

l-DU B-vu

a

n 以

nσ-rc

D

一­

Solution: If the coupon rate goes up , the coupon payments increase and therefore the value of the bond increases. The duration of the bond is the time weighted average of the cash flows , discounted with respect to the yield of the bond. If the coupo日 rate increases , the duration of the bond decreases. This is due to the fact that the earlier cash flows equal to the coupo丑 payments become a higher fraction of the payment made at maturity, which is equal to the face value of the bond plus

Bnew

-

θD

-=:.一

oy

,d

C

1θ2B

B

θy2

CHAPTER

56 Therefore ,

2.

NUMERICAL INTEGRATION. BONDS.

2.2

θB 一一

= -DB.

θν

(2.13)

B

\、飞 1111/''

no-rC

一

一一

ω一 句ω γ

一 -B

一切

切一句 ny

一十

BD

一一

ω一 句θ B

一-

叫σ τ 。

Using Product Rl由 to differentiate (2.13) with respect to y , we 直nd that :-uu B-2 D DB DB -D-hu

We conclude that

C

1θ2B

B

2.2. SUPPLEMENTAL EXERCISES

θν2

D2_ ~D. θu

0

57

Supplemental Exercises

1. Assume that the continuously compounded instantaneous rate curve γ (t) is given by γ (t) = 0.05 exp( 一 (1

+ t)2) .

(i) Use Simpsor内 Rule to compute the 1-year and 2-year discount factors with six decimal digits accuracy, and compute the 3-year discount factor with eight decimal digits accuracy. (ii) Find the value of a three year yearly coupon bond with coupon rate 5% (and face value 100). 2. Consider a six months plain vanilla European put option with strike 50 on a lognormally distributed underlying asset paying dividends continuously at 2%. Assume that interest rates are constant at 4%. Use risk-neutral valuation to write the value of the put as an integral over a finite interval. Find the value of the put option with six decimal digits accuracy using the NIidpoint Rule and using Simpson's Rule. Also , compute the 丑lack-Scholes value PBS of the put and report the approximation errors of the numerical integration approximations at each step. 3. The prices of three call options with strikes 45 , 50 , and 55 , on the same underlying asset and with the same maturity, are $4 , $6 , and $9 , respectively. Create a butterfly spread by going long a 45-call and a 55-call , and shorting two 50-calls. What are the payoff and the P &L at maturity of the butterfly spread? When would the butterfly spread be profitable? Assume , for simplicity, that interest rates are zero. 4. Dollar duration is defined as

D电=一旦 ωθu and measures by how much the value of a bond portfolio changes for a small parallel shift in the yield curve. Similarly, dollar convexity is defined as

C虫二工 θ2B

eθy2·

58

CHAPTER 2. NUMERICAL INTEGRATION. BONDS.

Note that , unlike classical duration and convexity, which can only be computed for individual bonds , dollar duration and dollar convexity can be estimated for any bond portfolio , assuming all bond yields change by the same amount. In particular , for a bond with value B , duration D , and convexity C , the dollar duration and the dollar convexity can be computed as D$ = BD and G$ = BG. You invest $1 million in a bond with duration 3.2 and convexity 16 and $2.5 million in a bond with duration 4 and convexity 24. (i) What are the dollar

d旧ation

and dollar convexity of your portfolio?

(ii) If the yield goes up by ten basis points，五叫 new approximate values for each of the bonds. What is the new value of the portfolio? (iii) You can buy or sell two other bo时s ， one with d旧时ion 1. 6 and convexity 12 and another one with duration 3.2 and convexity 20. What positions could you take in these bonds to immunize your portfolio (i. e. , to obtain a portfolio with zero dollar duration and dollar co盯exity)?

2.3. SOLUTIONS TO SUPPLEJYIENTAL EXERCISES

59

(ii) The value of the three year yearly coupon bond is B

= 5 disc(l) + 5 disc(2) + 105 disc(3) =

100.236424.

口

Problem 2: Consider a six months plai丑 vanilla European put option with strike 50 on a lognormally distributed underlying asset paying dividends co坠 tinuouslyat 2%. Assume that interest rates are constant at 4%. Use risk-neutral valuation to write the value of the put as an integral over a 且nite interval. Find the value of the put option with six decimal digits accuracy using the Midpoint Rule and using Simpson's Rule. Also , compute the Black-Scholes value PBS of the put and report the approximation errors of the numerical integration approximations at each step. Solution: If the underlying asset follows a lognormal distribution , the value S(T) of the underlying asset at maturity is a lognormal variable given by In(S(T))

= In(S(O)) +

/σ2\ 广(γ - q - v ) T + σ 飞ITZ，

2

where σis the volatility of the underlying asset. Then , the probability density function h(ν) of S(T) is

2.3

/

Solutions to Supplemental Exercises

h(y)

Problem 1: Assume that the continuously compounded instantaneous rate curve r(t) is given by

γ (t) 二

0.05

xp(一 (1

+ t)2) .

Solution: (i) Recall that the discount factor corresponding to time t is

叫-fa'州市 Using Simpson's Rule , we obtain that the 1-year, 2-year , and 3-year discount factors are

= 0.956595;

disc(2)

= 0.910128;

disc(3)

~呵!一\时

= 0.86574100.

LL "

(2.14)

if y > 0 , and h(y) = 0 if y 三 O. Using risk-neutral valuation , we find that the value of the put is given bγ P -

Use Simpson's Rule to compute the 1-year and 2-year discount factors with six decimal digits accuracy, and compute the 3-year discount factor with eight decimal digits accuracy. (ii) Find the value of a three year yearly coupon bond with coupon rate 5% (and face value 100).

disc(l)

=

(l叼一附(0))一 (r - q 一手) T)2 i

e-rTER川 max(K - S(T) , 0)] I' K

= e- rT / (K - y)h(ν) dy ,

(2.15)

JQ

where h(y) is given by (2.14). The Black-Scholes value of the put is PBS = 4.863603. To compute a numerical appr()ximation of the integral (2.15) , we start with a partition of the interval [0 , K] into 4 intervals , and double the numbers of -intervals up to 8192 intervals. We report the Midpoint Rule and Simpson's Rule appro均nations to (2.15) and the correspondi吨 approximation errors to the Black-Scholes value PBS in the table below: We 丑rst note that the approximation error does not go below 6 . 10-6 . This is due to the fact that the Black-Scholes value of the put , which is given by PBS Ke-r(T-t) N( -d2 ) - Se-q(T-t) N( -d1) ,

60

CHAPTER 2. NUIVIERICAL INTEGRATION. BONDS. No. Intervals Midpoint Rule 4 5.075832 8 4.922961 16 4.878220 32 4.867248 64 4.864518 128 4.863837 256 4.863666 512 4.863624 1024 4.863613 2048 4.863610 4096 4.863610 8192 4.863610

Error Simpson's Rule 0.212228 4.855908 0.059357 4.863955 0.014616 4.863631 0.003644 4.863611 0.000914 4.863610 0.000233 4.863610 0.000020 4.863609 0.000009 4.863609 0.000006 4.863609 0.000006 4.863609 0.000006 4.863609 0.000006 4.863609

Error 0.007696 0.000351 0.000027 0.000007 0.000006 0.000006 0.000006 0.000006 0.000006 0.000006 0.000006 0.000006

is computed usi吨 numerical approximations to estimate the terms N( -d1 ) and N( -d2 )工 The approximation error of these approximations is on the order of 10- f • Using numerical integration , the real value of the put option is computed , but the error of the Black-Scholes value will propagate to the approximation errors of the numerical integration. If we consider that convergence is achieved when the error is less than 10-5 , then convergence is achieved for 512 intervals for the Midpoint Rule and for 32 intervals for Simpson's Rule. This was to be expected given the quadratic convergence of the Midpoint Rule and the fourth order convergence of Simpson's Rule. 口 Problem 3: The prices of three call options with strikes 45 , 50 , and 55 , 0日 the same underlying asset and with the same maturity, are $4 , $6 , and $9 , respectively. Create a butterfly spread by going long a 45-call and a 55-call , and shorting two 50-calls. What are the payoff and the P&L at maturity of the butterfly spread? When would the butterfly spread be profitable? Assume , for simplicity, that interest rates are zero.

Solution: The payoff V(T) of the butterfly spread at maturity is if S(T) 三 45; 45 , if 45 < S(T) 三 50; V(T)=i55-S(TL if50<S(T)< 吮 l 0, if 55 三 S(T). (

0,

J S(T) -

The cost to set up the butterfly spread is $4 - $12 + $9 = $1.

2.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

61

The P&L at maturity is equal to the payoff V(T) minus the future value of $1 , the setup cost. Since interest rates are zero , the future value of $1 is $1 , and we conclude that ( -1 , if S(T) 三 45;

PU(T)={:jTJfii;:35; 二::; l

-1 ,

if 55 三 S(T)

The b时terfly spread will be pro主table if 46 < S(T) < 54 , i.e. , if the spot price at maturity of the underlying asset will be between $46 and $54. If r 并 0 ， it follows similarly that the butterfly spread is pr。在table if 45

+ erT < S(T) < 55 -

erT

口

Problem 4: You invest $1 million in a bond with duration 3.2 and convexity 16 and $2.5 million in a bond with duration 4 and convexity 24. (i) What are the dollar duration and dollar convexity of your portfolio? (ii) If the yield goes up by ten basis points ，五nd new approximate values for each of the bonds. What is the new value of the portfolio? (iii) You can buy or sell two other bonds , one with duration 1. 6 and convexity 12 and another one with duration 3.2 and convexity 20. What positions could you take in these bonds to immur山e your portfolio (i.e. , to obtain a portfolio with zero dollar duration and dollar co盯exity)?

Solution: Recall that the dollar duration and the dollar convexity of a position of size B in a bond with duration D and convexity Care D$ = BD (i) The value , duration and

and

co盯exity

C$ = BC.

of the two bond positions are

B 1 = 1, 000 , 000; D 1 = 3.2; C 1 = 16; B 2 = 2, 500 , 000; D 2 = 4; C1 = 24. Denote by B 工 B 1 + B 2 the value of the bond portfolio. The dollar duration and dollar convexity of the portfolio are

D虫=一坐立一旦一 θB2

ψθuθuθU

B 1D 1 C虫 =θ2B

+

B 2 D 2 - $13 , 200 , 000. θ2B+θ2β

川一一一­

eθν2θν2θν2

B 1C 1

+

B 2C2

-

$76 , 000 , 000.

62

CHAPTER

NUMERICAL INTEGRATION. BONDS.

2.

(ii) Using dollar duration and dollar co盯exity， the approximate formula i:::.. B 7 倡一- D i:::.. y

1

+ §C( i:::.. y?

for the change in the value of a bond ca丑 be written as

i: :. B 臼 -D向十 ;C$(AU)2

Chapter 3 (2.16)

Formula (2.16) also holds for bond portfolios , since the dollar duration and the dollar convexity of a bond portfolio are equal to the sum of the dollar duratioIIs azld of th dollar convexities of the bOIlds making up the portfolio7 respectively. Using (2.16) , we 五nd that the new value of the bond portfolio is Bne ω = B 十 i:::.. B 臼 $3 ， 500 ， 000 一 $13 ， 200 十 $38 二 $3 ， 486 ， 838.

(iii) Let B 3 and B 4 be the value of the positions taken i口 the bond with duration D 3 = 1. 6 and convexity C3 - 12 and in the bond with duration D 4 = 3.2 and convexity C4 = 20 , respectively. If II = B + B3 十 B4 denotes the value of the new portfolio , the丑

D$(II) = D$(B) + D$(B3 ) + D$(B4 ) $13.2mil + D 3 B 3 十 D 4 B4 ; C$(口) = D$(B) + D$(B3 ) + D$(B4 ) Then , D$(口)

=

°

=

3.1

Solutions to Chapter 3 Exercises

Problem 1: Let k be a positive integer with 2 三 k 三 12. You throw two fair dice. If the sum of the dice is k , you win w (k ), or lose 1 otherwise. Find the smallest value of ω (k) thats makes the game worth playing.

Solution: Consider the probability space S of all possible outcomes of throwing of the two dice , i.e. , S =

{(x , y) I x

°

if and only if

(mml 十1. 6川 2B4 $76mil + 12B3

-

0;

+ 20马- 0,

二 1

: 6, y

=

1 : 6}.

Here , x and y denote the outcomes of the first and second die , respectively. Since the dice are assumed to be fair and the tosses are assumed to be independent of each other , every outcome (x , y) 且as probability 去 of occurring Formally, the discrete probability function P : S • [0 , 1] is

$76mil + C3 B 3 十 C4 B4 .

and C$ (II)

Probability concepts. Black-Scholes formula. Greeks and Hedging.

P(x , y) (2.17)

The system (2.17) has solution B 3 = $3.25mil and B 4 = -5.75mil. We conclude that , to imm飞lnize your portfolio , one should buy $3.25 million worth of the bond with duration 1. 6 and convexity 12 and sell $5.75 million worth of the bond with duration 3.2 创ld convexity 20. 口

= 二 ， V(x ， y) εS 36

Let k be a fixed positive integer with 2 ~二 k ~二 12. The value X of your winning (or losses) is the random variable X : S →~ given by

J w(k) , if x 十 y = k; X(川)巳 i-17else If 2 三 k ~二 7 ，

then x

+ y = k if and only if

(x ， y) ε{(1 ， k - 1) , (2 , k - 2) ,…

In other words , x S. Then ,

E[X]

十y

, (k - 1, 1)}.

= k for exactly k - 1 of the total of 36 outcomes from

= 汇 P(x ， y)X(x , y) = 去 L (X ,Y)ES

(x ， y) εS

63

X(x , y)

64

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FOR l\/f ULA. k-l

=丁百一 ω (k) 十

36 一 (k ~~

- 1) ~/

(-1) 一_

ω (k)(k ~\'U/\'U

- 1) - 37 + k ;~

VI

I

1~(3.1)

The game is worth playing if E[X] 三 O. From (3.1) , it follows 出at the game should be played if

~二 k 三 12 ，

then

x十y

for 2 ::; k ::; 7.

E[X] Recall that

一-

EX

1-P

内，，却­

E [X] = p' 1 + (1 - p) . (1 十 E[X]) = 1 + (1- p)E[X].

n 一

Sol毗on: If the 自rst coin toss is heads (which happens with probability p) , then X = 1. If the 自rst coin toss is tails (which happens with probability 1 - p) , then the coin tossing process resets and the number of steps before the coin lands heads will be 1 plus the expected number of coin tosses until the coin lands heads. In other words ,

(3.4)

that

(η 十川 _ p)n+l

I

n(1 _ p)n+2

(3.5)

"ι

公

p

1- p

1

~ lim) ~k(l-p)日:一一 ·-r= 一. -pn→χ 仨7 1 - p y p

Similarly,

口

Problem 2: A coin lands heads with probability p and tails with probability 1 - p. Let X be the number of times you lllUSt flip the coin until it lands heads. What are E[X] and var(X)?

…门，，，"

The values of w(k) for k = 2 : 12 are as follows:

= 8;ω(6) =ω(8) = 6.2;ω(7) = 5.

咱i ω i f

E[X] =

E[X 2 ]

L 对 (1 _ p)k-l p 二 1 ~?')nl~~Lk2(I-p)k 1 - pn→ χ 仨?

Since T(η ， 2 ，

x)

三二 k

2

X +x 2

x

k

一

(n 十 1)2 x n+l

+ (2η2 十 2η (1

l)x n + 2

一

η 2 X n十3

x)3

we find that n

E[X 2]

.. p _ lim ~ k 2(1- p)k = ~ lim T(n , 2, 1 - p) 仨71-pn →∞

pl 1- P

…

(3.3)

+z-n-

From (3.3) and (3.5) and since 0 < 1 - p < 1, we conclude that

for 8 三 k < 12.

ω(2) =ω(12) = 35;ω(3) =ω(11)=17;ω(4) =ω(10) = 11;

17P

= τ了一

十一

k

k(l-p) fC

i口 (3 .4)， we 直nd

1i-41i

肿 一 、Z

From (3.2) , it follows that the game is worth playi吨 if E[X] 三 0 ， i.e. , if

23+ k ω (k) 三一一" 13 - k

立

(3.2)

36

/,,‘\-

M川 一 ­

By letting x = 1 - p

(x ，y) εS

ω (k)(13-k)-23-k

x)

pn→ χ 仨?

η 一

= 去汇 X(x， y) = 与生ω (k) 十 36-T-k)( 一 1)

We conclude that

品

Z 一

T(η ， 1 ，

~?') 1~ L k(1 - p)k

1

十一(

, (k - 6, 6)}.

= k exactly 13 - k times. Then ,

ω(5) =ω(9)

k (1 - p)k-l p =

n汇 M

E[X]

十y

L

= k if and only if

(x ， y) ε {(6 ， k-6) ， (5 , k-5) , …

In other words , x

Another way of computing E[X] is as follows: The coin will first land heads in the k-th toss , which corresponds to X = k , for a coin toss sequence of T T. . .T H , i. e. , the first k - 1 tosses are tails , followed by heads once. This c。如 toss sequence occurs with probability P(T)川 P(H) = (1 - p)k-l p . The凡

37 - k k-l

ω (k) 兰一一 oj'

If 8

65

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

p + (1 - p)2 p3

2-p p2

66

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORNIULA.

Therefore , var(X) = E[X 2 ] 一 (E[X])2

2- p

1

p2

p2

匕主口

(i) Show that the function f(x) is indeed a density function. It is clear that f(x) 三 0 ， for any x εJR. Prove that

汇 f川=

P'"

Problem 3: Over each of three consecutive time i时 ervals ofle丑gth 7 = 1/12 , the price of a stock with spot price 8 0 = 40 at time t = 0 will either go up by a factor u = 1. 05 with probability p = 0.6 , or down by a factor d = 0.96 with probability 1 - p = 0 .4. Compute the expected value and the variance of the stock price at time T = 37 , i.e. , compute E[8T ] and var(8T ).

Solution: The probability space 8 is the set of all different paths that the stock could follow three consecutive time intervals , i.e. ,

i

r 1- e 一αx if x > 0 F(x) = γ0770阳飞Nis (iv) Show that

M 三 t)

Sol础。即 (i)

E[(8T)2]

Jxe叫z

1

=

一 αx

e~一 α x

xe --

~_嘈

Uw

dx

αα2

r

'

一生主+~ xe一ω dx ααl

x2 e一αx

2xe 一 αx

2e 一 αz

αα2α3

fl04-g

z-d pu

z

1-d \飞 lil/

十

叫 de-d

α/Ill\

α

h以 优 - t

z--iii etnu \111/

一

一

一 -M 叫

出

ω=

二

α

-e 一α

/It--\

z-

一 -L

句

zi

znχ

/ l飞

α11

出 k ‘→

α

rJ

J J产 o -d

T 'ME

0;

)、77if Z 二。

r

I e-

ααl

f 泸川Z

PI-←

=

1

xe 一 α x

一一一一一+…

χ

J a e一 αzif Z >

f(x)

出(1 - e一叫

Z

(ii) By integration by parts we find that

一-

Problem 4: The density function of the exponential random variable X with parameterα> 0 is

fα 川z=J叫tα 川Z 出 (-rz)lZ

n 叫 W 凹

var(8T )

80u3 . p3 十 380 饥 2d.p2(1_p)+ 380 叫 2 . p(l _ p)2 十 80 d3 . (1 _ p)3 4 1. 7036; 但OU 3 )2 . p3 + 3(80u 2d)2. p2(1 - p) + 3(80ud2)2. p(l _ p)2 十 (80 d 3 )2 . (1 - p)3 = 1749.0762; E[(8T )2] 一 (E[8T])2 工 9.8835. 口

f 川x = 川 Vt 主 O

It is easy to see that

汇 f川

We conclude that

E[8T]

=

Note: this result is used to show that the exponential variable is memoryless , i.e. , P(X 三 t 十 s I X 三 t) = P(X 三 s ).

Note that

P(UUU) = p3; P(DDD) = (1 _ p)3; P(UUD) = P(UDU) = P(DUU) = p2(1_ p); P(UDD) = P(DUD) = P(DDU) = p(l- p?

1

(ii) Show tl时 the expected value and the variance of the expo阳 rando'm variable X are E[X] = and var(X) =去 (iii) Show that the cumulative de丑sity of X is

8 = {UUU, UUD , UDU, UDD , DUU, DUD , DDU, DDD} , where U represents an "up" move and D represents a "down" move. The value 8T of the stock at time T is a random variable de五日ed on 8 , and is given by 8T(UUU) 二 S旷 ; 8 T (DDD) = 8 0d3; 8 T (UUD) = 8 T (UDU) = 8T(DUU) = 8 0u 2d; 8 T (UDD) = 8 T (DUD) = 8 T (DDU) = 8 0ud2.

67

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

68

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORJYIULA.

E[X 2]

LW)dz=due-α♂ dx 工 αEtjtpzdz

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES for any two continuous functions

Solution: Let

叫一午一平 _ 2~3ax) I:

αεJR

be an arbitrary real number. Note that

叫ω

l

α21 g2(x)dx + 叫 仲州川 )g 川

1 α2

拄 i i垃i) 1f x < 0 , then F(x) = J~ f(s) ds = O. (仙 ∞ 1f x 主 0 ， the口 1-e

一αX

'O /II--\

\IlI-

〈一

z

、1 ，/

nL

7α

门u u

z

PI--α

、、，，/

Z

，α

门，，"

rId

\飞 11I/

F--11α

A哇

,GZ

Z

nu

,/

JU

Z

/It--\

Thu

od

Z

z

口

z

，α

rldn,,"

\飞 liI/

<-

内， h

,2 G

fI'11α

Z

\飞 llI/

Z

、、‘，，，，，

(3.6)

rIdJ'i\

ptt''1α

I~

-o /III-\

'D η3

一

(-e ω)

ny Z

''51 飞

一l'

αω巳一叫 = 1

z

which is equivalent to PI''1α

l

Z叫州)川d λρ 扣f只(阳

LU /Il--\

\Ill-/ rId

/'4飞、、

1- P(X < t) = 1 一

叮，"

,/lil\

then P(X 主 t)

VαεR

if and only if ，。

= (-e- α s) I~

l 9" 川+叫bf(Z)g(Z)dz+ff2 川三 0 ， b

。中

1f t 三 0 ，

fa" αf叫

α2

ftttIα

(iv)

=

V αεJR.

Recall that a quadratic polynomial P(x) = Ax 2 十 Bx + C is nonnegative for all real values of x if and only if P(x) has at most one real double root , which happens if and only if B 2 - 4AC 三 O. For our problem , it follows that

Therefore E[X X 判] 一 (但啊[阿冈附

b

b

~ 0,

咐:汇川s

JR.

ρb 飞U(υf(叫川+忖α 以仰州 巾Z )川)

αJ坦(一午 -25ff-23+ 主)

X var a町r刘叫州 叫叫 伊帆(仪问 刻) = 州

f , g : JR •

69

Recall that the conditional probability of A given B is

=

P(AIB)

P(A n B) P(B)

(3.7)

Let s , t 三 O. The凡 from (3.6) and (3.7) , we find that P(X 主 t+s I X 三 t)

P((X 三 t+s)n(x 主 t)) P(X 三 t)

e 一 α (t+ s) _,""

If

ε'-'0

工

e

P(X 三 t

+ s)

Problem 6: Use the Black-Scholes formula to price both a put and a call option wit且 strike 45 expiring in six months on an underlying asset with spot price 50 and volatility 20% paying dividends continuously at 2% , if interest rates are constant at 6%.

Solution: Input for the Black-Scholes formula:

P(X 三 t)

S = 50; K = 45; T - t =

0.5;σ=

0.2; q = 0.02; r = 0.06.

一讪-甲-

l f(x)g(川三 (l 尸 (x) (l dx )'

The Black-Scholes price of the call is C = 6.508363 and the price of the put is P = 0.675920. 口

g2 川)\

Problem 7: What is the value of a European Put option with strike K = 07 is the value of a European Call option with strike K = 07 How do you hedge a short position in such a call option7

毛iV hat

70

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORNIULA.

Solution: A put option with strike 0 will never be exercised , since it would mean selling the underlying asset for the price K = O. The price of the put option is O. A call with strike 0 will always be exercised , since it gives the right to buy one u日it of the underlying asset at zero cost. The value of the call at maturity is V(T) 工 8(T) ， and therefore V(O) 工 e- qT 8(0). This can be seen by building a portfolio with a long position on the call option and a short position of e- qT shares , or by using risk-neutral pricing: V(O)

e- rT E

RN [8(T)]

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

e.g. , for pricing volatility swaps. These two Greeks are called volga and vanna and are de如led as follows: θ(vega(V))θ(vega(V))

volga(V) =θσand vanna(V) =θS

It is easy to see that θ2V

volga(V) =万万

e- rT . e(r一伊 8(0) 二 e- qT 8(0).

A short position i口 the call optior丑1 祀 1 s hedged (s剖ta 创tic 伪 址11与吵 a y 坊圳 7→) by buying ∞ 0l口1 share of the underlying asset. 口

= - K(T - t)e-r(T-t) N(-d 2 ).

Solution: Recall that θCθP

p(C) = a~

p(P) = 万;

and

and

θ2V

vanna(V) -否瓦

The name volga is the short for "volatility gamma". Also , vanna can be interpreted as the rate of change of the Delta with respect to the volatility of the underlying asset , i.e. ,

Problem 8: Use formula p( C) = K(T - t)e-r(T-t) N(也) for p( C) and the Put-Call parity to show that p(P)

71

θ (~(V))

vanna(V)

θσ

(i) Compute the volga and vanna for a plai丑 vanilla European call option on an asset paying dividends continuously at the rate q. (ii) Use the Put-Call parity to compute the volga and vanna for a plain vanilla European put option.

Solution: (i) Recall that

By differentiating the Put-Call parity formula P

with respect

to 飞 we

+

8e- q(T-t) -

C

vega(C)

K e-r(T-t)

~(C)

find that

p(P) -

p(C) =

-K(T-t)e-r(T-t).

叶 ) + (γ -q 吟) (T -

d1

p(P) = ρ (C) - K(T - t)e-r(T-t) K(T - t)e-r(T-t) N(d 2 ) - K(T - t)e-r(T-t)

In (去)

+ (r

…

q)(T - t)

σ VT丁 t

tσ VTτz

2

Then ,

口

volga(C) Problem 9: The sensitivity of the vega of a portfolio with respect to volatility and to the price of the underlying asset are often important to estimate ,

t)

VT丁Z

-K(T - t)e-r(T-t) (1 - N(d 2 ))

since 1 - N(d 2 ) = N(-d 2 ).

飞/27了'

e-q(T-t) N(d 1 ) ,

where

Therefore ,

- K(T - t)e-r(T-t) N( -d2 ) ,

8e-q (T-t)叮叮 -LJ:

vanna(C)

θ(吵(C)) _ - 8e- q (T-t哺可-L Uσ

飞/2 7r

dJf11; -

dσ'

θ(~(C))_ e-q(T-t) N'(d1 ) 纽 - e- q 叫↓ λ ad 1 θσ

〉三作

θσ

72

CHAPTER 3. PROBABILITY. BLACK-SCHOLES

FOR~在ULA.

In (圣)

+

(γ - q)(T - t) ，~丁I 一二

σ2VT士t

θσ

C -

2

叫) + (γ -q 一手) (T - t)

Ke-q(T-t) N(d 1 ) Ke- r(T-t)N(-d

P -

σ2 飞厅可

-

Ke-r(T-t) N(出); Ke- q(T-t)N(-d

2) -

1 ).

where

d2

叶子气)旧 and 叫于一~)旧

σ

We conclude that

Then

volga(C)

Se-旷-tL.jT丁zi-e千生生

(3.8)

vanna(C)

-e-q(T-t) 」-e-4d2

(3.9)

川

νd开

d石

σ7

σ

(ii) By differentiating the Put-Call parity P 十 Se-q(T-t) - C = K e-r(T-t) with respect to 风 we find that vega(P) -

θP

否歹

C 三 P

仁=?

e-q(T-t) N(d I) - e-r(T-t) N(d2 )

三 e-r(T-t)N( -d 2 ) -

中=争 e一旷一气 N(d 1 ) + N( -d 1 )) 三 e-r(T一气N(d2 ) 牛=辛> e-q(T-t) > e-r(T-t) 牛=学

since N(d 1 )

γ 二三

+ N( -d2 ))

q,

+ N( -d1 ) =

N(d 2 )

+ N( -d2 ) =

1.口

Problem 11: (i) Show that the Theta of a plain vanilla E旧opean call option on a non-dividend-paying asset is always negative. (ii) For long dated (i.e. , with T - t large) ATJVI calls on an underlying asset paying dividends continuously at a rate equal to the constant risk-free rate , i.e. , with q = r , show that the Theta may be positive.

θ(vega(P))θ(vega(C))

= z Volga(C);

θσθσ

θ(vega(P))θ(vega(C))

Solution:

vama(P)===vanm(C)? θSθs

where volga(C) and van叫 C) are given by (3.8) and (3.9) , respectively.

口

(i) 丑ecall

that

Sσe-q(T-t)

~

@ ( C ) = - e 2 十 qSe-q(T-t)N(d 1 ) 2 、/2π(T -

-

t)

r K e-r(T-t) N(d 2 ).

Problem 10: Show that an ATM call on an underlying asset paying dividends continuously at rate q is worth more than an ATM put with the same maturity if and only if q 三飞 where r is the constant risk free rate. Use the Put-Call parity, and then use the Black-Scholes formula to prove this result.

For a non-dividend-paying asset , i.e. , for q = 0 , we find that

Solution: For at-th e-money options , i.e. , with S = K , the Put-Call parity can be written as

(ii) If q = r , the Theta of an ATM call (i.e. , with S = K) is

C- P =

e-q(T-t) N( -d1 )

δC

歹歹工 vega(C).

Therefore , Volga(P)z

73

Alternatively, the Black-Scholes formulas for at-the-money options can be written as

Note that θd 1

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

Se-q(T-t) - K e-r(T-t) _ K e-q(T-t) - K e-r(T-t) K e-r(T-t) (e(r-q)(T-t) -

1)

Therefore , C ~三 P if and only if e(r-q)(T-t) ;三 1 ， which is equivalent to r 主 q.

8(C) =

-Sσe-4-Tke-r(T-t)N(d2)<0. 2 飞/2作 (T - t)

Kae-r(T-t)

-~

@ ( C ) = - e 2 十 r K e-r(T-t) N( d 1 ) -

2 飞/2作 (T

- t)

r K e-r(T-t) N( d 2 )

Ke-r(T-t) (r(N(d 1 ) - N(d 2 )) 一 σJ) \2 飞/2汀 (T -

tr )

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORlvIULA.

74

75

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

since d 1 = d 2 十 σ VTτt and therefore

where

θd 1

d1 =

θ d2

θKθK

Note that lim

(T-t)→∞

Then ,

li~

(T-t)→∞

d1 = ∞

N(d 1 )

and

lim

(T-t)→∞

仇=一∞.

，~li~

= 1 and

(T-t) →∞

N(d 2 )

By

with respect to K , we find that

空气工 - e-r(T-t) N'(d2 ) 纽工 _ e-r(T-t) τLe手 252

=0

δK

8K

v 三作

θK

(3 叫

Note that

and therefore (T-t)→∞

differentiati吨 (3.10)

!γ(N(d1 ) - N(d2 )) 一 σe-4i \2y2作 (T - t) )

In

∞

z

d2

JT-t

(γ (N(d 1 )

-

飞

N(出))一

JT-t

σJ}

2y2作 (T -

t)

Then

)

口

'

σVT丁I

θd 2

1

and , from (3.11) and (3.12) , we conclude that θ2C

Problem 12: Show that the price of a plain vanilla European call option is a convex function of the strike of the option , i.e. , show that θ2C

一一=-=δ K2

> O. 一

Se-q(T-t) N'(d 1 )

-

Ke-r(T-t) N'(d 2 ).

By differentiating the Black-Scholes formula Se-q(T一忖T(d 1 ) -

Ke-r(T-t) N(d 2 )

w山w

Se-q(T一叫鞋一 Ke-r(T一川也)在一 e-r(T一切(d2 ) _e-r(T-t) N(d 2 ) ,

(坐 -252} 飞 θKθK

}

-

σK j2作 (T

- t)

e-r(T-t) e二Ji>O

口

Solution: The input in the Black-Scholes formula for the Gamma of the call is S = K = 50 ， σ= 0.3 , r = 0.05 , q = O. For T = 1/24 (assuming a 30 days per month count) , T = 1/4 , and T = 1, the following values of the Gamma of the ATM call are obtained:

with respect to K , we obtain that

Se-q(T-t) N'(d 1 )

θ K2

Problem 13: Compute the Gamma of AT l\!I call options with maturities of fifteen days , three months , and one year , respectively, on a non-dividendpaying underlying asset with spot price 50 and volatility 30%. Assume that interest rates are constant at 5%. What can you infer about the hedging of AT l\!I options with different maturities?

Solution: Recall that

C -

(3.12)

θKσKJT-t'

will be positive. We note that the positive value of 8( C) is nonetheless small , si丑ce lim(T-t)→∞ 8(C) = O.

州十 (γ -q 一手) (T - t)

In(K)

We conclude that , for T - t large enough , 8(C) = Ke- r 叫

(圣) + (γ -q-~)(T-t)

e-T(T一切(出) (3.10)

r(15days) -

0.057664; f(3months) = 0.052530; r(1year) = 0.025296.

We note that Gamma decreases as the maturity of the options increases. This can be seen by plotting the Delta of a call option as a function of spot price , and noticing that the slope of the Delta around the at-the-money point is steeper for shorter maturities. The cost of Delta-hedging AT l\!I options

76

CHAPTER

3.

PROBABILITY. BLACK-SCHOLES FOR1vIULA.

(iii) For clarity, let

= T - t. For r = q = 0 , we obtain from (3.13) that

7

vega(C) =

豆坦豆f VL1 汀 「

Problem 14: (i) The vega of a plain vanilla European call or put is positive , smce vega(C) - vega(P) - Se-q(T一明习气Le一手 (3.13)

since , for an ATJVI option with r = q = 0,

、、 lg/­

…

、、l/-

S 一旦;z 2)2 'Tr 7 ~

-

dS〉宇 -z;工 8~石 U

忐 (1 一宁)e牛 For σ= 0.3 and for time to maturity less than one year , i.e. , for 7 三 1 ， that h7

we 丑nd

>- nuQd

巧i 町i

巾 w he叫 = 巳F Cωom川附e 山 t he 仰 de叩pe丑毗丑ce of 吨叩&叫圳州 阴 (ρ C) on川 ti让ime 江凶I T 一 t ， i.e. , θ(vega( C)) θ(T - t) ,

/'1\-

C 一

飞/三'Tr

nu-v-e

g 一θ

SVT习气LJ7

2

By direct computation , we find that ，， lt 、-

=

(γ -q 十号)7 _aft σJ于一

i

β-T

吨a(C) 工吨a(P)

= In (主)十

d1

VL/Tr

Can you give a financial explanation for this? (ii) Compute the vega of ATM Call options with maturities of 自fteen days , three months , and one year , respectively, on a non-dividend-paying underlying asset with spot price 50 and volatility 30%. For simplicity, assume zero interest rates , i.e. ， γ=0. (iii) If r = q = 0 , the vega of ATJVI call and put options is

77

M二而

may be higher for short dated options , since small changes i口 the price of the underlying asset lead to higher changes in the Delta of the option , and therefore may require more often hedge rebalancing. 口

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

414

and therefore

vhu

θ(vega(C))

θ;三 0. T

and explain the results from part (ii) of the problem.

Solution: (i) The fact the vega of a plai口 vanilla European call or put is positive means that , all other things being equal , options on underlying assets with higher volatility are more valuable (or more expensive , dependi吨 0日 whether you have a long or short options position). This could be understood as follows: the higher the volatility of the underlying asset , the higher the risk associated with writing options 0日 the asset. Therefore , the premium charged for selling the option will be higher. If you have a long position in either put or call options you are essentially "10日g volatility" . (ii) The i即ut in the Black-Scholes formula for the Gamma of the call is S = K = 50 ， σ= 0.3 ， γ = q = 0. For T = 1/24 , T = 1/4, and T = 1, the following values of the vega of the ATJVI call are obtained: vega(15days) vega(3months) vega(lyear)

4.069779; 9.945546; 19.723967.

We conclude that , for options with moderately large time to maturity, the vega IS mcreasmg as time to maturity increωes. Therefore we expect that vega( 1year) > vega(3months) > vega( 15days), which is what we previously obtained by direct computation.

口

Problem 15: Assume that interest rates are constant and equal toγ. Show tflat7unless tfle price C of a ca11optiOIlwith strike k amd mat11rity T OIl a non-dividend paying asset with spot price S satisfies the inequality Se- qT -

K e- rT 三 C 三 SfqT7

(3.14)

arbitrage opportunities arise. Show that the value P of the corresponding put option must satisfy the following no-arbitrage condition:

K e- rT

- Se- qT

S P < K e- rT .

(3.15)

78

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORMULA.

Solution: One way to prove these bounds on the prices of European options is by using the Put-Call parity, i.e , P 十 Se- qT - C 二 Ke- rT . To establish the bounds (3.14) on the price of the call , no

C - Se- qT -

Ke- rT 十 P.

(3.16)

The payoff of the put at time T is 即以 (K - S(T) , 0) which is less th叩 the strike K. The value P of the put at time 0 cannot be more than K e- r1 , the present value at time 0 of K at time T. Also , the value P of the put option lllust be greater than O. Thus , O 三 P 三 K e- rT ,

(3.17)

and , from (3.16) and (3.17) , we obtain that Se- qT - Ke- rT < Se- qT - Ke- rT

+

P -

C < Se-qT.

To establish the bounds (3.15) on the price of the put , note that P

Ke- rT - Se- qT 十 C.

(3.18)

The payoff of the call at time T is max(S(T) - K , 0) which is l~s than S(T). The value C of the call at time 0 cannot be more than Se- q1 , the present value at time 0 of one unit of the underlying asset at time T , if the dividends paid by the asset at rate q are continuously reinvested in the asset. Also , the value C of the call option must be greater than O. Thus , 。三 C 三 Se-qT.

(3.19)

<

P

<

Ke- rT .

A more insightful way to prove these bounds is to use arbitrage arguments and the Law of One Price. Consider a portfolio made of a short position in one call option with strike K and maturity T and a long position in e- qT units of the underlying asset. The value of at time 0 of this portfolio is V(O) = Se- qT - C.

If the dividends received on the long asset position are invested continuously in buying more units of the underlying asset , the size of the asset position at time T will be 1 unit of the asset. Thus , V(T)

= S(T) - C(T) = S(T) - max(S(T) - K , 0)

since , if S(T) > K , then V(T) = S(T) 一 (S(T) - K) = K , and , if S(T) 三 K ， then V(T) = S(T) 三 K. From the Generalized Law of One Price we conclude that V(O) = Se- qT - C 三 Ke-y-T? and therefore Se- qT - K e- rT 三 C ， which is the left inequality from (3.14). All the other inequalities can be proved similarly: • To establish that C ~二 Se- qT ， show that the payoff at maturity T of a portfolio made of a long position i丑 e- qT units of the underlying asset at time o and a short position in the call option is nonnegative for any possible values of S(T); • To establish that K e- rT - Se- qT ~二 P ， show that the payoff at maturity T of a portfolio made of a long position in e- qT units of the underlying asset at time 0 and a long position in the put option is greater than K for any possible values of S(T); • To establish that P ~二 K e- rT , show that the payoff at maturity T of a portfolio made of a short position in the put option and a long cash position 口 of Ke- rT at time 0 is nonnegative for any possible values of S(T). Problem 16: A portfolio containing derivative securities on only one asset has Delta 5000 and Gamma -200. A call on the asset with 6.(C) = 0 .4 and f(C) = 0.05 , and a put on the same asset , with 6. (P) - -0.5 and f(P) = 0.07 are currently traded. How do you make the portfolio Delt aneutral and Gamma-neutral?

Solution: Take positions of size Xl and X2 , respectively, in the call and put options speci五ed above. The value II of the new portfolio is II = V + Xl C + 归 P ， where V is the value of the original portfolio. This portfolio will be Delta- and Gamma-neutral , provided that Xl and X2 are chosen such that

From (3.18) and (3.19) it follows that Ke- rT _ Se- qT

79

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

三 K，

6. (II)

=

6. (V)

+ xl 6.( C) + x2 6. (P)

=

5000 十 O .4Xl - 0.5X2

f(II) = f(V) 十 xlf(C) 十 X2f(P) = - 200 十 0. 05X l 十 0.07X2 The solution of this linear system is 250 , 000 ._. ~ ~_ 1= 一一一ι...::....::.. = -4716.98 53 - . - - - - and .---.-

0;

o.

330 , 000

x') = 6226 .4 2. -- "' = 一」一-=-=53

To make the initial portfolio as close to Delt a- and Gamma-neutral as possible by only trading in the given call and put options , 4717 calls must be sold and 6226 put options must be bought. The Delta and Gamma of the new portfolio are 6.(江) = f(口)

=

6. (V) - 4717 6.( C) 十 6226 6. (P) = 0.2; f(V)+4717f(C)+6226f(P) = 一 0.03.

80

CHAPTER

3.

PROBABILITY. BLACK-SCHOLES FOR1VIULA.

To understand how well balanced the hedged portfolio II is , recall that the initial portfolio had ~(V) = 5000 and r(V) = -200. 口

3.1. SOLUTIONS TO CHAPTER 3 EXERCISES

81

(iii) The new spot price and maturity of the option are 8 2 = 98 and T2 =

12~ /252 (there are 252 trading days in one year). The value of the call option

is $8 .4 53134 and the value of the por毛folio is Problem 17: You are long 1000 call options with strike 90 and three months to maturity. Assume that the underlying asset has a lognormal distribution with drift μ= 0.08 and volatility σ= 0.2 , and that the spot price of the asset is 92. The risk-free rate is l' = 0.05. vVhat Delta-hedging position do you need to take?

1000C2

units of the underlying asset , where d1 = In 二一

(圣) + (1' - q 吟)T

-σ 飞IT

with 8 = 92 , K = 90 , T = 1/4 ， σ= 0.2 ， γ= 0.05 , q = O. Note that , for Delta-hedging purposes , it is not necessary to know the driftμof the underlying asset , since ~ (C) does 口ot depend on μ. 口 Problem 18: You buy 1000 six months ATM Call options on a nondividend-paying asset with spot price 100 , following a lognormal process with volatility 30%. Assume the interest rates are constant at 5%. (i) How much money do you pay for the options? (ii) What Delta-hedgi吨 position do you have to take? (iii) On the next trading day, the asset opens at 98. What is the value of your position (the option and shares positio时? (iv) Had you not Delt a-hedged , how m旧h would you have lost due to the increωe in the price of the asset?

Solution: (i) Usi吨 the Black-Scholes forml出 with input 8 1 = K = 100 , T = 1/2 ， σ= 0.3 , l' = 0.05 , q = 0 , we 自nd tl时 the value of one call option is 0 1 = 9.634870. Therefore , $9 ,634.87 must be paid for 1000 calls. (ii) The Delta-hedgi吨 position for long 1000 calls is short 1000~(C)

= 1000e- qT N( d 1 ) = 588.59

units of the underlying. Therefore , 589 units of the underlying must be shorted.

= - 49268.87.

(iv) If the long call position is not Delta-hedged , the loss incl decrease in the spot price of the underlying asset is 1000(C2

Solutioη:

A long call position is Delta-hedged by a short position in the underlying asset. Delta-hedging the long position in 1000 calls is done by shorting 1000~(C) = 1000e- qT N( d1 ) = 653.50

58982

-

-

cd

=一 $118 1. 74.

For the Delta-hedged portfolio , the loss incurred is (100002

-

58982 ) 一 (1000C1 - 58981 ) =一 $3.74.

As expected , this loss is much smaller than the loss incurred if the options positions is not hedged ("naked"). 口

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORNIULA.

82

3.2

Supplemental Exercises

1. What is the expected number of coin tosses of a fair coin in order to get two heads in a row? What if the coin is biased and the probability of getting heads is p?

3.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

83

8. You hold a portfolio with 6.(口) = 300 , f(II) = 100 , and vega(II) You can trade in the underlying asset , in a call option with

6. (C) = 0.2;

f(C) = 0.1;

= 89.

vega(C) = 0.1 ,

and in a put option with 2. What is the expected number of tosses in order to get k heads in a row for a biased coin with probability of getting heads equal to p?

6. (P) = -0.8;

vega(P) = 0.2.

What trades do you make to obtain a 6.-, f -, and vega-neutral portfolio?

3. Calculate the mean and variance of the uniform distribution on the interval [a ,b] . 4. Let X be a normally distributed random variable with standard deviation σ> O. Compute E[ IXI ] and E[X 2].

f(P) = 0.3;

meanμand

5. Compute the expected value and variance of the Poisson distribution , i.e. , of a random variable X taking only positive integer values with probabilities

3.3

Solutions to Supplemental Exercises

Problem 1: What is the expected number of coin tosses of a fair coin in order to get two heads in a row? What if the coin is biased and the probability of getting heads is p?

n 一入飞 k

P(X=k) = 气卡， V k 三 0 ， whereλ>

0 is a fixed positive number.

6. Show that the values of a plain vanilla put option and of a plai且 vanilla call option with the same maturity and strike , and on the same underlying asset , are equal if and only if the strike is equal to the forward pnce. 7. You hold a portfolio made of a long position in 1000 put options with strike price 25 and maturity of six months , on a non-dividend-paying stock with lognormal distribution with volatility 30% , a long position in 400 shares of the same stock , which has spot price $20 , and $10 ,000 如 cash. Assume that the risk自free rate is constant at 4%. (i) How much is the portfolio worth? (ii) How do you adjust the stock position to make the portfolio Deltaneutral? (iii) A month later , the spot price of the underlying asset is $24. Wr时 is new value of your portfolio , and how do you adjust the stock position to make the portfolio Delta-neutral?

Solution: If p is the probability of the coin toss resulting in heads , then the probability of the coin toss resulting in tails is 1 - p. The outcomes of the first two tosses are as follows: • If the first toss is tails , which happens with probability 1 - p , then the process resets and the expected number of tosses increases by 1. • If the 趾st toss is he告ds ， and if the second toss is also heads , which happens with probability p气 then two consecutive heads were obtained after two tosses. ·If the first toss is heads?and if the second toss is tails?which happens with probability p(l - p) , then the process resets and the expected number of tosses increases by 2.

If E[X] denotes the expected number of tosses in order to get two heads in a row , we conclude that E[X]

= (1- p)(l + E[X]) 十 2p 2 十 p(l

p)(2 + E[X]).

(3.20)

We solve (3.20) for E[X] and obtain that 1 牛竹

E[X] = 寸主. p"

For an u~biase_d coir:, i.e. , for p = ~， we find that E[X] = 6, and therefore the expected number of coin tosses to- obtain two heads i~ a row is 6. 口

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORIvIULA.

84

Problem 2: What is the expected number of tosses in order to get n heads in a row for a biased coin with probability of getting heads equal to p?

3.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES var(U)

= E [eu - E[U])2] = E [ (U 一引 j

Solution: The probability that the first n throws are all heads is pn. If the k throws are heads and the (k + l)-th throw is tails , which happen wi让t吐h probability pk(οl 一 p) ， then the process resets after the k 十 1 steps; here , k = 0: (η-1). Then , if x(η) denotes the expected number of tosses in order to get ηheads in a row , it follows that

二 f(z-132)dz= 击 ~(X_b;a)' 一

且rst

'。，hLr

/飞

PA

n 飞、 J牛 'K

z n

二、 FE-­

+

吁1-

PA

向

八川V\Ill--/

十十

dA4

PA

、4 元

「

一

川川

忏川 γμM

V八

kkZK

-iJSStt-

-hT

UA

Pι11

k

二、 FH 一一­ n 飞 4k 门V

l 气 do

ηη

十+

PP

ηη

一一

η

Z

'hh

UA

Problem 4: Let X be a normally distributed random variable with and standard deviation σ> O. Compute E[ IXI ] and E[X 2]. Solution: We compute E[

叫ZM

Recall that

p

1- pn l-p7

p-

2二 kpk

(1 - p)2

η，p n 十 1-pη

1 一 (η 十 l)pn 十 npn+1

1-p

-一一十 x(η)(1

1-p

and therefore

+

x(n)(l _ pn)

E[IX 门

τ~ (一μ/σ 一(μ 十的) e一句+气L

I

e一 T

dz

ν 2i J一 μ/σ-，.

f

J 一 μ/σ

(μ + o- z) e一手dz

ft7

+ ,

一r.::= 1

~在 j一μ/σ

一 Td

山

It is easy to see that

1-du

JUJu

，α ，α

flγ，叩

υ

rb xj(x) ~" • 1 rb b+ α Ij dx = 一~ I xdx = 一­ α b一 α j α 2

+μ 一言=

土而

Solution: The probability density function of the uniform distribution U on the interval [a , b] is the c∞stant functi∞ j(x) = b~a' for all x E [a , b]. Then ,

V L/Jr

J-x

-tl:Vdz 一元zlJNJdz

1-pη

(η) =一一一-

E[U] =

vi言汇 lμ+ 的Ie一手 dz v 马丁II

Problem 3: Calculate the mean and variance of the uniform distribution on the interval [价].

IXI ] in terms of the cumulative dist由ution

of the standard normal variable Z. Note that X = μ+σ Z. Then ,

_ pn) ,

pn (1- p)' We conclude that the expected number of tosses in order to get n headsin a row for a biased coin with p时 ability of getti昭leads 叩al to pis 元57 If the ∞i丑 were unbia时， i.e. , for p = ~， the expected number of tosses in order to get n heads in a row is 2η+1 - 2. 口

meanμ

N(t) 工 vi言 L 巾z

npn 十 (η - l)pn十1

Then , we find that x(η)

85

N(-~) = 1-N(~);

(-e一手) I二 f= 一叫一￡); 在[~" e-'; dy N (;) ; (-e-~) J:二/σ= 叫一￡)

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FORJVIULA.

86

Then , it follows that

We conclude that

一 μ (l-N(;)) 十元叫一出

E[\X\]

二气 )

十叫;) +在叫一￡) μ (2N (;) -1)

+岳叫一出

One way to compute E[X 2] would be to compute the E[X 2] =

J-OO

While this would provide the correct result , an easier way is to recall that

=σ2 ，

we concl 时 e that

气!'

Vk 三 0 ，

where λ> 0 is a fixed positive numbe r.

et

=z;

上…

Recall that the Taylor expansion of the function eX is

∞汇M

∞ ZM

∞ ZM

P X

川丁

Solution: We show that E[X] = λand var(X) =λ By definition ,

eλ

(3.23)

。c

E[X 2]

= 汇 P(X

e一λ 已 ..

=

k)· k2 =

(k - 1)λk

台

(k-1)!

λ.

~一 λ 飞 k

汇气产

-' -~一 λ

e 兰川

k2

λk

亏 ~但

..一

x

飞 k-2

飞 k-l

十 λ7

where (3.22) and (3.23) were used for the last equality. We conclude that

n 一入飞 k

=

(3.22)

From (3.21) and (3.22) , we 丑 nd that E[X] = Similarly,

λ2

Problem 5: Compute the expected value and variance of the Poisson distribution , i.e. , of a random variable X taking only positive integer values with probabilities

= k)

、

e…

d2EtrA 罢工一

E[X 2] = var(X) + (E[X])2 = μ2 十 σ2

P(X

-

ZJ二-:-; -

x

2 var(X) = E[X ] 一 (E[X]?

Since E[X] = μand var(X)

λ k-l

一一一一一一一

followi吨 integral:

E[(μ+σ Z)2] = 仨/∞ (μ 十 M)2Jdz V'<""

EX

87

3.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

var(X) = E[X 2] - (E[X])2 = λ

口

Problem 6: Show that the values of a plain vanilla put option and of a plai 丑 vanilla call option with the same maturity and strike , and on the same underlying asset , are equal if and only if the strike is equal to the forward pnce.

Solution: Recall that the forward price is F = Be 一 (r-q)T From the Put-Call parity, we know that

c-

P

Be- qT

-

Ke- rT .

(3.24)

If a call and a put with the same strike K have the same value , i.e. , if C = P in (3.24) , then Be- qT = Ke- rT . Thu K -

Be 一 (r-q)T7

i.e. , the strike of the options is equal to the forward

price.

口

88

CHAPTER 3. PROBABILITY. BLACK-SCHOLES FOR1VIULA.

Problem 7: You hold a portfolio made of a long position in 1000 put options with strike price 25 and maturity of six months , on a non-dividend-paying stock with lognormal distribution with volatility 30% , a long position in 400 shares of the same stock , which has spot price $20 , and $10 , 000 in cash. Assume that the risk-free rate is constant at 4%. (i) How m时1 is the portfolio worth? (ii) How do you adjust the stock position to make the portfolio Delta-neutral? (iii) A month later , the spot price of the underlying asset is $24. What is new value of your portfolio , and how do you adjust the stock position to make the portfolio Delta-neutral? Solution: (i) The value of the portfolio is

1000P(0)

+

-803 + 400 =

( 1 \ ( 1\

I + 8038 I

飞 12 )

~-_ I

飞 12

}

十 1946

= 23400.

The new Delta of the portfolio is

-1000N( -d l )

+ 803 = 292.

To make the portfolio Delta-neutral , you should sell 292

shares.

口

Problem 8: You hold a portfolio with i:::.. (II) = 300 , r(II) 工 100 and vega(II) = 89. You can trade in the underlyi吨 asset， in a call option with i:::.. (C) 工 0.2;

r(C) = 0.1;

vega(C) = 0.1 ,

89

and in a put option with i:::..

(P) = -0.8;

r(p) = 0.3;

What trades do you make to obtain a

i:::..-,

vega(P) = 0.2.

r -, and vega-neutral portfolio?

Soltdi07ZJ You can make the portfolio T-and Vega-neutral by takiIlg positions in the call and put optiOI17respectively.By trading iII the underlying asset7the r aad Vega of tfle portfolio would not change7and the portfolio can be made i:::..-neutral. Formally7let z17Z27and hbe the positions i2the underlying asset7the Cali option?and the put option7respectively.The value of the zlew portfolio is II旧w=II 十 x 1 8 + X2 C + 句 P and therefore i:::..(丑nω) = i:::.. (II) 十 Xl + x 2 i:::.. (C) 十 x 3 i:::.. (P); r(II nω ) = r(口) + X2 (C) + X3 (P); vega(口new) - vega(II) 十 x2veg a (C) 十 x3 ve g a (P).

r

r

Then , i:::..(丑nω) = r(丑ηω) = vega(II n 叫 = 0 if and only if

•

(们 0.2X2- 0.8

- 403.

To obtain a Delta-neutral portfolio , 403 shares must be purchased for $8 ,060. The Delta-neutral portfolio will be made of a long position in 1000 put options a long position in 803 shares of the underlying stock , and $1 ,940 in cash. (iii) A month later , the spot price of the und 凹lying asset is S ( 古剖) = 24 a丑 the put opt悦ions have 直 fi飞ve months lef 丘t ur时 l垃t挝 削i让1 maturity. The Black-Scholes value of the put option is P (古) = 2.1818. The cash position has accrued interest and its current value is 1940 exp (呼) = 1946. The portfolio is wo巾 ~-_

SOLUTIONS TO SUPPLEMENTAL EXERCISES

4008(0) 十 10000 = 22927 ,

where 8(0) 二 20 is the spot price of the underlyi吨 asset and the value P(O) = 4.9273 of the put option is obtained using the Black-Scholes formula. (ii) The Delta of the put option position is -1000N( -d l ) = -803. (Here and in the rest of the problem , the values of Delta are rounded to the nearest integer.) The Delta of the portfolio is

1000P I

3.3.

0. lx 2 十 0. 3X 3 0. lx 2 十 0. 2X 3

-300;

= 一 100; -

-89.

The solution (rounded to the nearest integer)is Zl=-254向饥= -670、 均 =-1101n othr words , to make the portfolio A-7T-and Vega一丑eut叫: one must short 254units ofthe underlying asset and sell 670call options &创 an 口1 110 put

options.

口

Chapter 4 Lognormal random variables. Risk-neutral prIcIng. 4.1

Solutions to Chapter 4 Exercises

Problem 1: Let Xl = Z and X 2 = -Z be two independent random variables , where Z is the standard normal variable. Show that Xl + X 2 is a normal variable of mean 0 and variance 2, i.e. , Xl 十 X2 = V2 Z. Solution: 丑ecall that if Xl and X 2 are independent normal random variables with mean and variance μ1 and σ?7aMμ2 and σi ， respectively, then Xl + X 2 is a normal variable with mean μ1+μ2 and varia肌eσr+σ27and

Xl + X 2 =

f.t 1

+

f.t 2

+

For Xl = Z and X 2 = -Z , it follows that conclude that E[X]= μ1 十问= 0

and therefore X

and

;;;f.丐 z μ1= 问=

0 and

σ1= 句=

1. vVe

var(X) =σ?+σ~ = 2,

= Xl +X2 = V2Z. 口

Problem 2: Assume that the normal random variables Xl , X 2 , … , X n of meanμand variance σ2 a陀 uncorrelated， i.飞 cov(Xi ， Xj) - 0 , for all l 三 t 并 j 三 η. (This happens , e.g. , if Xl , X 2 , … , X n are independent.) If Sn = ~ L:~=1 Xi is the average of the variables Xi , i = 1 :风 show that

E[Sn] Solution: Recall that , for

= μand Ci ε~ ，

X几均4斗] 叫 E斗[~← q呐凡斗] ~吝辛←← Ci剧刷 i E町耶啊啊[仄问阳 =

91

var 问

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

92

刊主Ci

L crvar(Xi) +

4. 1. SOLUTIONS TO CHAPTER 4 EXERCISES

Problem 4: If the price S(t) of a ∞n-dividend paying asset has lognor口时 distribution with drift r and volatility 风 show that

2 ~二呐∞，v(Xi ， X j )

ER 川S(t

Therefore ,

E[Sn] var(Sn)

+ c5t)]

ERN [S2(t 十 c5t)]

E

[~字]

:土贝Xi]

咔字) 主 2二 var(Xi)

1

一 ·ημ

=

=μ;

η

主 zmE间十二三二-nCOVM) 1

c)

σ

(pu + (1 - p)d) S(t); (pu 2 十 (1 - p) d2) S2 (t ).

var(Y)

E[tft)j

(4.1)

(4.2)

space{U7D}ofttle possible moves ofthe price ofthe asset fromtimet to time t .L+ 8t ~ndo~ed with- the risk-neutral probability function P : {U , D} • [0 , 1] with P(U) = p and P(D) = 1 - p. Then S(t + 8t) is given by

= S(t)u;

S(t 十们)(D)

S(t)d.

Then , by definition ,

ER E RN[S2(t + 8t)]

P(U) . S(t 十们)(U) + P(D)· S(t 十 8t)(D) (pu + (1 - p)d) S(t); P(U) . (S(t 十们)(U)? 十 P(D) . (S(t 十们)(D))2 (pu 2 十 (1 - p) d2) S2 (t ) .口

优p (2μ + 0: 2 )

(i:

2 -

1)

盯=需在 th叫 =(γ 一手) 8t and 歹工 σv5i and 切向

vaT

(吐了))

叫 ((γ_ ~2) /it 十叩2) 仪p e

Solution: We can regard S(t 十们) as a random variable over the probability

S(t 十们)(U)

+ σ) 圳 8t叼 Z.

叫μ+ 三);

E[Y]

Show that

ERN[S2(t 十 6缸t均创)刀!

(4.4)

丑〈俨(了俨 S贝S(t (川 J = I 一一 J 8t 例 t均) ) \ /

口

Problem 3: Assume we have a one period binomial model for the evolution of the price of an underlying asset between time t and time t 十 8t: If S(t) is the price of the asset at time t? t~en the p:~ce S.(t 十 8t) of the asset at time t 十 8t will be either S(t)叽 with (risk-ne飞巾al) probability p, or S(t)d , with probability 1 - p. Assume that u > 1 and d < 1.

ER

(4.3)

e(2r+σ2)8t S2(t).

Recall tl时， if ln(Y) =μ 十歹Z is a lognormal random variable with parameters μand 歹， the expected value and variance of Yare

η

since COV(Xi , X j ) = 0 for all 1 三 t 泸 j 三 η.

er8t S(t);

Solution 1: If the price S (t) of the non-dividend payi吨 asset has lognormal distrib 叫0 川ith drift 川 nd 叫a创t址削 il 过lity 川 r given by

2

一τ ·nσ­ n血

93

2r8t

=eTSt;

(4.5)

(2 (γ_ ~2) 们+州)2) .(e(σ而)2- 1J4 . 6 ) (产 -1)

(4.7)

Note that ,

E[tft)] mr(tf))

」一 ·E[S(仆 8t)] S(t)

(4.8)

-L-Var (S(t+60)-

(4.9)

S2(t)

From (4.5) and (4.8) , and from (4.7) and (4.9) , respectively, we conclude that E[S(t 十 8t)]

var (S(t + 8t))

er8t S( t);

e叮产-

1) S2(t)

(4.10)

CHAPTER

94

4.

LOGNOR1VIAL VARIABLES.

RN

PRICING.

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES

Note that var (S(t 十们)) = E[S2(t

=

S2(t) 可言=

+ 8t)] 一 (E[S(t 十们)])2

E[S2(t 十 6t)]

V L-7r

- e2r6ts2(t).

(4.11)

fχ(.L£ J- _2£ exp 12γ们十 σ2 6t -

J-x

(X-2 o-V5t )2\

i

n

I

飞.<.

~

( (x - 2σ V5t)2 飞 Ifχ exp {一门)

V L,7r J -x

E[S2(t 十 6t)] = var (S(t 十 8t)) + e2r6t S2( t)

J-

I

/

S2(t)e(2r+ e> 2) 6t

From (4.10) and (4.11) it follows that

95

\

.<.

I

S2 (t) e (2r+e>2)6t ↓/沉 e-4ds

e 2r6t十σ 26t S2(t) ,

V L,7r J-x S2(t)e(2r+σ2)6t;

which is what we wanted to show.

Solution 2: Note that S(t 十归) can be written as a function of the standard

the substituti∞ S 可 -2σv5t was used above

口

normal variable Z as follows: I

S(t 十们)

=

I

σ2 飞

Problem 5: The results of the previous two exercises can be used to calibrate a binomial tree model to a lognormally distributed process. This means 在nding the up and down factors u and d , and the risk-neutral probability p (of goi鸣 up) such that the values of ERN[S(t 十 6t)] and ERN[S2(t+6t)] given by (4.1) and (4.2) coincide with the values (4.3) and (4 .4) for the lognormal model. In other words , we are looking for u , d , and p such that

r\

S(t) 仰~ ~ r 一言) 6t 十 σ训tZ)

Then ,

E[S(t + 6t)]

刁卖 i七言 fι:S附 川 (tω t均) m 左 ι 叫6们ι←一宁 t +o-V5t ♂卜斗一 ζx;)νdx μ1

pu 十 (1 - p)d

pu2

r∞/ (x 一 σVti? 飞 J

S(t) 矿Ot 苟言儿仪p \一

+

(1 _ p)d2 =

寸t-h

、、tIJ

向 AU

\I L,7r

S2(t) exp t 2r6t -

(4.15)

I

dx

/ rτ-

x-

\

嘈

o-:.:I 6t + 2σ yot x - 2) dx

J 一∞\/

1.

Show that the solution can be written as

E[S2忻州]=去 (00 S2(均叫2 (γ_ ~2) 6t + 2σvti x) e-~ n

/tt 飞

ud -

SimiLr1377mobtaizltbM

I

aTb

(ii) Derive the Cox一RoωS臼S卧εs-一-Rubi阳 ns时teir丑1 paramet 忧ri坦za 创tion for a binomial tree , by solving (4.14-4.15) with the additional condition that

1,

J 一∞\\

(4.14)

‘

(e r6t - d) (u - er6t )

since ÷eJ兰 is the density function of the standard normal variable.

I

(4.13)

mha

viz 汇巾ν-

去=

t "一 一 衍

p

where we used the substitution y = x 一 σvti and the fact that

f∞

e(2r十σ2)6t

Ml-ub d-r e-e

S(t 阶 t均)e旷T5"t?

1

(4.12)

Since there are two constraints and three unknowns , the solution will not be umque. (i) Show that (4.12-4.13) are equivale时 to

- 2-')α

T r 仰附)汩 )e旷沪

\I L,7r

= er6t ;

p

= 主二f; 也 = u 一 α

where

A =

A+

YA2才; d=A-VA可7

~ (川十 e(r+e>2)6t)

.

(4.16)

CHAPTER 4. LOGNORIvIAL VARIABLES. RN PRICING.

96

Solution: (i) Formula (4.14) can be obtained by solving the linear equation (4.12) for p. To obtain (4.15) , we 在rst square formula (4.12) to obtain

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES

the other solution of the quadratic equation (4.20) corresponds to the value of d, since

L二 =A 一州 -1 A2 _ 1

d = u1

e2r8t

p2 U 2 + 2p(1 - p)ud + (1- p)2d2 -

97

A 十飞!

v 哥…

-

口

and subtract this from (4.13). We 缸d that

p(l - p)u 2 - 2p(1 - p)ud 十 p(l - p)d2

Prabl 巳ill 6: Show that the series 1二;二l 击 is conve带风 while the series 2:%二1i and E二;二2 世而 are diverge风 i.e. ， equal to ∞

已 (2r十σ2)8t _ e2r 8t

Note: It is known that

which can be written as

p(l - p)(u - d)2 = e 2r 们 (eo-

2

8t

1).

_

Using formula (4.14) for p , it is easy to see that

p(l - p)

=

and

( e r 8t 一份 (u-e r 8t ) (u - d)2

(e r8t

d) (u - e

-

2 ) = e2r 8t (e o- 8t

-

1r

2

6

战(兰 ;-h(η))

(4.18) where

From (4.17) and (4.18) , we conclude that r们

汇丰

(4.17)

γ 自 0.57721

=

~

is called Euler's constant.

Solution: Since all the terms of the series 2:%二l 击 are positive , it is eno吨h to show that the partial sums ~ 1

1)

·一

(ii) By

mt山iplyi吨 out

ue r8t

(4.15) and

usi吨 the

fact that ud = 1, we obtain that

1 _ e 2r8t 十 de r 8t = e(2r十σ2) 8t

-

_ e 2叫

( 4.19)

\

/

U

\li/

u

〈一

U 十 1 一 ( e- r8t + e(r十σ2)叫 z

一η 一

咱i

which can be written as

jl 1-K

e(r十泸州 7

++ /lil\

-

汇丰

1 气/立

- e- r8t

are uniformly bounded , in order to conclude that the series is convergent. This can be seen as follows: L-41 14

After canceling out the term _e 2r8t , we divide (4.19) by e耐 and obtain U十d

仨 k2

1+ 各(kll) 工 1 十二击 -i

<

To show that the series 1二;二1

+1-2A=0; U

2,

V n ~三 2.

t is diverge风 we will prove that

4η4

d. (4.16) for the de自nition of A. In other words , u is a solution of the quadratic equation

u 2 - 2Au 十 1 = 0, which has two solutions , conclude that

A + VA亡landA-VA亡1. u=A 十 J万丁I;

i丑(η) 十二<

(4.20) Since u > 1 , we

)~二<

乞fk

In(n )

+ 1, V

n 三 1

(4.21)

The integral of the function f(x) - ~ over the interval [1 , n] can be approximated from above and below as follows: Note that

fjdzzzf+1:dz

CHAPTER 4. LOGNORNIAL VARIABLES. RN PRICING.

98

Since j (x) = ~ is a decreasing function , it is easy to see that -L
\-- /

-

Since

Z 巳 (k， k 十 1).

7α

ZfzLzdz

which is the same as

(4.24).

口

(4.22) Problem 7: Find the radius of convergence R of the power series

n 1-z p''''''1i 用·山

，α

三气

Zfl:dz

<

位 k ln(k)'

Zfidz (4.23)

and investigate what happens at the points x where Ixl = R. Solution: It is easy to see that

The inequality (4.21) follows from (4.22) and (4.23) , since

2二 x

f:dz=1日(η

川) -ln(叩门十 L <土 -L 内 (η) 仨 k ln(k)'

vn>2;

立 -L
…( 4.25)

1

klB(k)21 口 (2)'

(424)

and co肌lude that the series '2二;二2 时而 is divergent For example , (4.24) can be proved as follows: ~

rk+1

I-~-_-_dx-)'I

儿 xln(x)ω中台儿

1η~

rk+1

-一二~dx< 亨 l

xln仙

台儿

1

主~

1

一一一一7" dx = 予一一-一

k ln(k) 山仨 k ln(k)'

which is equivalent to I".'

二

' 一 … 一 ‘ '

儿土豆(幻川 I

k

2二 αJ?

(4.28)

with 句:时可， k 三 2. Note that

In a similar fashion , by considering the integral of x l~(百 over the interval [2 , n] , we can show that

1

xk

.二

zi= 一;十主;

川

(4.27)

坦(lB(n))-h(ln(2))+ 」一<安 1 nln(η) 仨 kln(k)'

Z

z 击= -1 十 zi

kd

In (l n(2)) ,

99

we conell陇 from (4.26) and (4.27) that

flJI 1-z >

/-l-dz=ln(ln(η)) x ln(x)

J2

Then , ZfV

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES

二

nln(n)\ 仨 kln(k)'

(4.26)

lim

k→∞

Iα川 =lim(」-YK=1(429) k→∞\ k ln(k))

Recall that , if li问→∞|时11k exists , the radius of co盯erge肌e of the series E二;二2αkXk is given by

(4.30) 一一」 - li叫→∞ |αk 1/k ' From (4.29) and (4.30) the radius of co盯ergence of the series (4.28) is R = 1 We co肌lude that the series is co盯ergent if Ixl < 1 , and not co盯ergent if Ixl > 1. If x=工 -1. the series becomesS ~∞ Since ~k=l ζ且 kln(k)' i:)!ll ce 'Lthe ne 'Lterms erms ζ且 kln(k) have alternating signs and decrease in absolute value to 0 , the series is co丑vergent. If x = 1, the series becomes '2二;二1 kl~(k)' which was show口 to be divergent in Problem 6 of this chapter. 口 R

一

1

Problem 8: Consider a put option with strike 55 and maturity 4 months on a non-dividend paying asset with spot price 60 which follows a lognormal

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

100

model with drift μ= 0.1 and volatility σ= 0.3. Assume that the risk-free rate is constant equal to 0.05. (i) Find the probability that the put will expire i口 the money (ii) Find the risk一时的ral probability that the put will expire in the money. (iii) Compute N( -d 2 ).

/JIll\

T

σ

q

\Ill-/

十

μ'

n

\111/

2 σ一 2

/111\

一一

T 一的 叹一别

Solution: (i) The probability that the put option will expire in the money is equal to the probability that the spot price at maturity is lower than the strike price , i.e. , to P(S(T) < K). Recall that

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES The凡 d 2

101

= 0.511983 , and N( -d 2 ) = 0.304331 ,

which is the same as the risk-neutral probability that the put option will expire in the money; cf. (4.31) To understand this result , note that PRN(S(T)

< K)

=

P

(z

< 一叫到 +(γ-q-4) 气

1σ VT

d z d. (4.31) and (4.32).

N(-d 2 );

=

J

口

Then , P(5(T)

< K)

P(;2< 忐)

=

P

(叫:2)< 叫忐))

P( (μ-q-DT+dZ< 叫录)) )

Problem 9: (i) Consider an at拍e-money call on a non-dividend paying asset; assume the Black-Scholes framewor k. Show that the Delta of the option is always greater than 0.5. (ii) If the underlyi吨 asset pays dividends at the continuous rate q , when is the Delta of an at-the-money call1ess than 0.5? Note: For most cases , the Delta of an at-the-money call option is close to 0.5. Sol毗 on:

(i) Recall that the Delta of a call option is given by

-qT 刷=严N(咄 +(γ -q+ 号)T} 1σ〉于

For 5 = 60 , K = 55 , T = 1/3 ， μ= 0.1 , q = 0 ， σ= 0.3 , and 7' = 0.05 , we obtain that the probability that the put will expire in the money is 0.271525 , i.e. , 27.1525%. (ii) The risk-neutral probability that the put option will expire in the money is obtained just like the probability that the put expires i丑 the money, by substituting the risk-free rate 7' for μ ， i.e. ,

PRN (5(T) < K)

P

(z

叫蒜)

- (γ -q 一手)寸

1σVT

1

For an at-th e-money call on a non-dividend paying asset , i.e. , for K = S and q = 0 , we find that

=N (

(r 十 j) ♂)三

(ii) If the underlyi吨 asset pays dividends at the ∞ c or巾 of ar口1 ATM call 恒 iS

J (4.31)

TN(d 1 )

=

e刊γ -q+ 叮

(iii) Recall that d2

叫到十 (7' - q 一主)T σ叮

(4.32)

For a fixed risk-free rateγand 自xed maturity T , we conclude that ~(C) < 0.5 if and only if the dividend yield q and the volatility σof the underlying asset

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

102 satis句T

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES

When pricing a plain vanilla call using ris k-neutrality, we proved that n-rT r ∞ II _2 \ 9 \ CBS(O) = 8(0) 二言= exp ( (γ 一二) T+ σdz-4:}dz

the following condition:

V 二7i

一 盯旷 Kee-一→Tr This happens , for example , if r = q and T is large enough , si丑ce lim N (

T•

00

\

(jV! 1 ~

103

1

and

J

lim

T•

00

I

J -d 2

~

L, )

\\

)

古 /ιL 仆;〉〉ef一勾手幻d

叮 S d 出1均) 一 Ke- T TN(μd 贝(0创)N(何 也 ω公2). …

0.5 严=∞口

In other words , we showed that

产去/∞ Adz=e一叩 (d2 );

(4.34)

V L,7i J -d2

Problem 10: Use risk-neutral pricing to price a supershare , i.e. , an option that pays (max(8(T) - K , 0))2 at the maturity of the option. In other words , compute V(O) = e- rT E RN [(max(8(T) - K , 0))2 ], where the expected value is computed with respect to the risk-neutral distribution of the price 8(T) of the underlyi吨 asset at maturity T , which is assumed to follow a lognormal process with drift r and volatility σ. Assume that the underlying asset pays no dividends , i.e. , q = O.

Solution: Recall that

σ

十

qr-2 T

d z

~ J-d2 叫

The integral from (4.36) is computed by co 皿 pleti吨 the square as follows:

的。)

e~-rT

V

/c\

r ∞/哎\

/叫( (2γ - (j 2)T + 2σdz-Z}dz

L,7i J -d2

∞ (x 的 O)e~ /r 侃p ((一 L,7f v 一出 飞

and note that

L, )

\

-rT

- 2σ VT)2

z

>

叫茄)一 (γ 一手 )

一们IT

T

z 一也

V L,7i

-d 2

8 2 (0) 户2)T ↓/∞

飞

工

「

2

VT? } I

叫(一白J dy \

L，

S 0创印)诈e 俨气泪栩(仰例 的州

ff7T'1

汀咽

e-一TTV 的阶 K2与乍 七言

左 i l兀;

2巧)泞 T+2川Z卜一ζ x;)νdx P( (ρ伪2衍T…一→σ巾 恬 lι:〉仪叫叫叫 均巾巾((←价

T 户(价

we used the substitution y = x - 2σ -IT and the fact that

巾Z

X V布i七言 l仆:〉e叫

一叫

2气协(仰0 的内 d +刊 咐S俨呻

(j

L，

飞!"l，7i J 一 (d 2 +2σ -IT)

dx

I

(x 一 2 的 0) 川 )T 牛/∞叫{一 J

Then ,

m

十 (2γ+σ 2)T 、1

ηL,

J

8(T) 三 K 白

(4.35)

From (4.33) , (4.34) , and (4.35) , we conclude that V(O) = K 2e- rT N(d 2) - 2K8(0)N(d1 ) 同一 rT ?\ r∞ / +8 2(0) (2 …协 2σ VTx - ~- dx. (4.36) )

\飞 111/

/ft--\ /III-\

\飞 111/

-

cu nu exDA

俨'

Cυ

T

5二汇叫 (r- ~2) T 十 σdz-2)dz= 川 d1 )

'rorαm Fr ￥

(4.33)

(4.36) and (4.37) , we conclude that

V(O恒 的-rT N(d2) -

2K8(0)N(d1 )

+ 8 2 (0) 户 )TN(d2 十 2σ 叮) .口

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

104

Problem 11: If the price of an 部set follows a normal process , i.e. , d8 = μdt+ σdX ， then

8(t2)

8(tl) 十 μ (t2 - t 1 ) + σd产石 Z， V 0 < tl < t2·

=

Assume that the risk free rate is constant and equal to r. (i) Use risk neutrality to 且nd the value of a call option with strike K and maturity T. (ii) Use the Put-Call parity to find the y~~ue of a put ~ption with s~rike K ~n'd maturity T , if the underlying asset follows a normal process as above. Sol仰on: (i) Using risk-neutral prici吗， it follows that

C(O) = e- rT ER川max(8(T) - K , 0)] ,

= 8(0) +γT 十 σVTZ.

(4.38)

> K iff Z

>

K - 8(0) - rT 一

= e- rT ERN [8(T) - K] = e- rT (8(0)

〉三1r

Jd

-

Ke- rT 去 f e-hz y '21r Jd

+

zm

，α

曲

， d

dl 一而

∞f f d d f I l d

tl挝lat

zc

J!!叫 1-vi言 f: e九 = tm

一切

=

it(-e 一句 I:

d2

e 2;

1- N(d) = N(-d)

where N(t) is the cumulative distribution of the standard normal variable. We conclude that rT rT 0(0) 工 (8(0) + rT)e- N(-d) - K e- N(-d) + m

e- rT 厅、 IT ~.L e 一 τ.

v

+ γT-K)，

since ER From (但4.3ω9创) and (任4 .4钊0创)， we obtain that

已-rTσ vT e +卢J 飞 /2 汀

since 1 - N( -d)

严 V在 i七言 [(卡归仰贝仰(仰仲 O rT「 严一 (问贝喇 O的如)忖十忖州叫 们巾町 rTTη 叩旷)诈 γγ 巳e-r -叮 ÷ L l∞ 汁 dx

Note

0(0) - P(O)

一

d2 τ

;[

- d

Then , 0(0)

(ii) Regardless of the model used for describi吨 the evolution of the price of the underlying asset , the Put-Call parity says that a portfolio made of a long position in a plain vanilla European call option and a short position in a plain vanilla European put option on the same asset and with the same strike and maturity as the call option has the same payoff at maturity as a long position in a unit of the underlying asset and a short cash position equal to the strike of the options. Using risk-neutral pricing , this can be written as

Ke-rTN(d) 一 (8(0)

Note that 8(T)

105

P(O) = 0(0) 一 (8(0) 十 γT)e- rT 十 Ke- rT K e -rT (1 - N ( -d)) 一 (8(0) 十 rT) ε -rT(l - N( -d))

where the expected value is computed with respect to 8(T) given by 8(T)

4.1. SOLUTIONS TO CHAPTER 4 EXERCISES

(4.39)

= N(d).

口

+ rT)e- rT N(d)

~-rT

十

b

_.

/币刀

二 V .L e一专

(4 .40)

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

106

4.2

4.3

Supplemental Exercises

=

Solutions to Supplemental Exercises

(言。- I日(η)

Xn

is convergent to a limit between 0 and 1. Note: The limit of this sequence is γ 臼 0.57721 ， the Euler's constant. 2. Assume that an asset with spot price 50 paying dividends continuously at rate q = 0.02 has lognormal distribution with mean μ= 0.08 and volatility σ= 0.3. Assume that the risk-free rates are constant and equal to r = 0.05.

ln(η)+j
4. Use risk-neutral pricing to find the value of an option on a 口。且一 dividend-paying asset with lognormal distribution if the payoff of the option at maturity is equal to max((S(T))α- K , O). Here ， α> 0 is a fixed constant. 5. Find a binomial tree parame~rization for a risk-neutral probability (of going up or dow时 equal to ~. In other words , find the up and down factors u and d such that σ

)

FλU

何，"

伊' 门， A

- p)d2

ρuρU

pu2 十 (1

\;/

η 三 1，

<

zn <

vv n

>

42···b

It is easy to see that

Xn+1 一句 :-L-ln(η+ η 十 l Therefore , X n十 1 <

Xn

1) +

ln(η) .

if and only if

(n+ 1\ tI
e

<

/η+1\η+1 I 一一-I = 飞

η/

(~1 \η+1 11 十一 i 飞

η/

which holds for anyη;三 1 ， from the definition of e. We showed that the sequence (x讪=1:∞ is decreasi吨 and bounded from below by 0 and from above by 1, The seque口ce is therefore convergent to a limit between 0 and 1. 口

T( -7十

(l-p)d

ιι "λU

+

1,

which can be written as 才i

3. If you play (American) 1 roulette 100 times , betting $100 on black each time , what is the probability of winning at least $1000 , and what is the probability of losing at least $10007

(剖- ln(η)

Solution: Recall from (4.21) that

1 一η

(ii) Find 95% and 99% risk-neutral con在dence intervals for the spot price of the asset in 15 days , 1 month , 2 months , 6 months , and 1 year , i. e. , assuming that the drift of the asset is equal to the risk-free rate.

=

is convergent to a limit between 0 and 1.

(i) Find 95% and 99% confidence intervals for the spot price of the asset in 15 days , 1 month , 2 months , 6 months , and 1 year.

pu

107

Problem 1: Show that the sequence

1. Show that the sequence Xn

4.3. SOLUTIONS TO SUPPLENIENTAL EXERCISES

'?L

ifp=i 1 American roulette has 18 red slots: 18 black slots: and two green slots (corresponding to 0 and 00). European roulette: also called Fr ench roulette: has only one green slot corresponding to O.

Problem 2: Assume that an asset with spot price 50 paying dividends continuously at rate q = 0.02 has lognormal distribution with mean μ= 0.08 and volatility σ= 0.3. Assume that the risk-free rates are constant and equal to r = 0.05. (i) Find 95% and 99% confidence intervals for the spot price of the asset in 15 days , 1 month , 2 months , 6 months , and 1 year.

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

108

(ii) Find 95% and 99% risk-neutral con且dence intervals for the spot price of in 15 days , 1 month , 2 months , 6 months , and 1 year , i.e. , assuming that the drift of the asset is equal to the risk-free rate. theωset

Solution: If the asset has lognormal distribution , then S(t) = =

S(O) 叫( (μ-q-f)t 十 σ0z }) 2 } 飞飞

TXT

50 exp(O.Ol 日十 0.30z).

< Z < 1. 95996) = 0.95; P( -2.57583 < Z < 2.57583) = 0.99.

Therefore , the 95% and 99% con直dence intervals for S (t) are [50 exp(O.Ol 日一 0.30 . 1. 95996) , 50 exp(O.Ol 日十 0.30· 1. 95996)]; [50 exp(O.Ol 日- 0.30 ·2.57583) , 50 exp(0.015t + 0.30 ·2.57583) ], respectively. The risk-neutrα， l confidence intervals for the spot price of the asset are found by substituting the risk-free rate r for μin (4 .4 1) to obtain SRN(t) = 50 exp( -0.015t + 0.30Z). Therefor飞 the

川-

r -100 ,

1.

100 ,

with probability 器; with probability 盖

Note that

E[ Wi]

μ =

10 ,

. _ _,

= 一 19\(-100) --~/

9. __ 100 +' 1~n 100 =一一 19--~ 19 ' 600\/10

σ= 削 (Wi) = E[(Wi )2] 一 (E[Wi ]? = 二口.

Let W = L:i~~ Wi be the total value of the winr山gs after betting 100 times. Since every bet is independent of any other bet , it follows that W is the sum of 100 independent identically distributed random variables. From the Central Limit Theorem we 自nd that W

自

100μ+ 10σZ

=

100006000\/10 ---~- ~Y 一 =-Z 19 19

一一一+

The probability of winning at least $1000 can be approximated as follows:

95% and 99% confidence intervals of SRN(t) are

[50 exp(-O.Ol 日一 0.30 . 1. 95996) , 50 exp(-O.Ol 日十 0.30 . 1. 95996)] [50 exp( -0.01日- 0.30 ·2.57583) , 50 exp( -0.01日十 0.30 . 2.57583月 7 respectively. For t ε{ 主 ι i ，~， 1} , we obtain the following confidence intervals: t 15 days 1 month 2 months 6 months 1 year

109

Solution: Recall that an American roulette has 18 red slots , 18 black slots , and two green slo~s. Therefore , every time you bet on black , you win $100 with probability 击 and lose $100 with probability 击 In other words , if V只 is the value of the winnings in the i-th round of playing , then

(4 .4 1)

Recall that the 95% and 99% con直dence intervals for the standard normal distribution Z are [-1. 95996 ,1. 95996] and [-2.57583 , 2.57583 ], i.e. , P( -1. 95996

4.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

95% CI S(t) 43.36 , 57.76 42.25 , 59.32 39 .43, 63.72 32.25 , 75.35 28.20 , 91. 38

99% CI S(t) 41. 45 , 60 .4 3 40.05 , 62.57 36.56 , 68.70 29.17 , 86.99 23 .44 ,109.90

95% CI SRN(t) 99% CI SRN(t) 43.28 , 57.66 41. 36 , 60.30 42.14 , 59.18 39.96 , 64 .4 1 39.23 , 63 .41 36.36 , 68.37 32.74 , 75.21 28.73 , 85.70 [27.36 , 88.68 22.75 ,106.65

Problem 3: If you play (American) roulette 100 times , betting $100 0丑 black each time , what is the probability of winning at least $1000 , and what is the probability of losing at least $10007

P(W>

1000) 臼

=

[ 100006000 飞/101 PI 一一一一十 v … Z> 1000 I \

19

19

.-

J

P(Z > 1. 5284) = 0.0632.

The probability of losing at least $1000 can be approximated as follows:

P(vV <

-1000)

目

[ 100006000 飞/101 PI 一一一一十 v 一 Z < -1000 I

=

P(Z < -0 .4743) = 0.3176.

\

19

19

J

We conclude that the probability of winning at least $1000 is approximately 6% , and the probability of losing at least $1000 is approximately 32%. 口 Problem 4: Use risk-neutral pricing to 且nd the value of an option on a non-dividend-paying asset with lognormal distribution if the payoff of the option at maturity is equal to max((S(T))α- K , O). Here ， α> 0 is a 在xed constant.

CHAPTER 4. LOGNORMAL VARIABLES. RN PRICING.

110

Solution: Using risk-neutral pricing , we find that the value of the option is

~~oblem 5: Find a binomia) tree parametrization for a risk-neutral proba旧rd 出s ， 直缸缸时 r1 bility (of going up) equal to ~. In other wo t包L 缸 aI丑ld d such that

ιι

σ

)

向 λU

5且:e句 = e-rTN归)

η4

Recall from (但4.34 刊 钊) 由 4 t ha 创t

去

2

-7 十

ρupu

+

(1 - p)d pu 十 (1 - p)d2 pu

v，，，，、、

亏琴;江汇/汇川川~(川川川(牛← 仙阳(侈问仰附附 μ S贝仰俐(仰阶附附 0创创)川)

'?心"'

=

口

σJT

δ

叩 V(O)

TN

α

The丑7

k

ρu

-a.

d

俨'

d

(吕 )+(γ-4)T

σ

Z>

α

牛::::::}

T N

十

S(T) 三 K 1 /α

In α

\lliI/\111l/ α

Note that (S(T))α 三 K is equivalent to S(T) 三 K /α. Using (4 .42) , we fin

俨'

where

1

十

叫

咱t 4

S(O叫 (γ-DT 十 σVTZ)

//I11\

α

=

//ll1\

ex

Dι

盯)

v nu - s nu

α

where

111

note that the change of variables y = x 一 ασ JT was used above. We conclude that

川 一2

V(O) = e- rT ERN[max((S(T))α - K , O) ],

4.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

ifp=i Sol州附It is

easy to see that , if p = ~， then U 十 d =2eTdt;

Therefore ,

叩) =嘿~1~ exp ((α- 1)ι 子~dz-Zhz -

u

2

+ d2

=

2e(2r 十 σ2)dt7

and therefore

Ke-rTN(α)

ud

( 叶 d)27(u2+d2)

- 2e2rM 一俨σ2)M

Note that u and d are the solutions of Z2 一(也十 d)z 十 ud = 0 , which is the same as Z2 _ 2erbt z 十 2e 2rM _ e(2r 十 σ 2)M _ 0. (4 .43)

V卖 i七言 1~ex优叫叫 均((←(归川 p( α

We solve (4 .4 3) and find that

优叫p(←归忻川川一斗斗机)沙俨 1)

叫叫(川川 α←川一」斗1) (γ 十 子)川 T才) 左 VIi七言 ιι1:+Q刊叩叮叫 αω叫 而叮而) 叫叫(-一5引) dν σd 叫(α 一刊+子) T) 在 1: 叫引 dy OUVT

叫川 (r 十字)T) N(a+QσVT);

u

d since d <

u.

口

Mt 1 十 M_ 刷耐几 0- 2 6衍 = 旷♂沪叫 叮 气(← J山e 一

工矿 M

(1 一

ν e o- 2M

-

O 1) ,

c

ap er wo

hu

4EU

Taylor's formula and Taylor series. ATM approximation of Black-Scholes formulas. 5.1

Solutions to Chapter 5 Exercises

Problem 1: Show that the cubic Taylor approxi口lation of vfτx around 0 IS

"十 Z 自 1 +王一旦+主3 2 8 ' 16

Solution: Recall that the cubic Taylor approximation of the function f(x) around the pointα 工 o is f(x) 臼 f(α)

+

(x 一 α) 1' (α) 1(_.\

+

(x 一 α)2.£1/(_.\ \~

...2 -/ 'f"(α) " -- /

,

十

(x 一 α)3 p fi

f(O) 十 x1'(O) 十三f响)十三 f(川0). 2" 6 ,/

f(的 (α)

(5.1)

For f(x) = Vf石， we find that

1' (x) = 卢z ; f ff ( z h ar丑ld

U

Z ) 3/ 2 ;

尸内 )(x) (μ = ~

therefore

f(O) = 1; 1' (0) 寸;ff(0)=-j; 内)=;(52) From (5.1) and (5.2) we conclude

由前

dτZ 但十三2 -zf+21 口 8 ' 16 Problem 2: Use the Taylor series expansion of the function eX to find the value of eO. 25 with six decimal digits accuracy. 113

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

114

Solution: Recall that the Taylor series expansion of the function f(x) = ♂ around 0 converges to eX at all points ♂ ε JR.， i.e.

,

∞巾 k

♂=三示，

5.1. SOLUTIONS TO CHAPTER 5 EXERCISES l 1- x 2

L: x

2k

= 1 + x 2 + x 4 忖 6+... ， \i xE(-l ， l)

Problem 4: Let

ι(_l)k+l x k

l. e.

T(x) = ):

ι (0.25)k w

_

be the Taylor series expansion of f (x) = In (1 T(x) = f(x) for all x such that Ixl < 1.

, eO.

25

=

lim

xn

, where

xn

= 安哗fii7h 三0 "'.,

Let

Pn(x)

n→ 3二--

k=O

Note that the sequence {xn}n=O:∞ is increasing. It is then enough to compute Xo , Xl , x2 , …, until the first seven decimal digit~ _
X6

= 1. 28402540;

Xl X4 X7

= 1. 25; = 1. 28401698; = 1. 28402541 ,

X2

X5

= 1. 28125; = 1. 28402507;

=

已 (_l)k+l x k ) ~

be the Taylor polynomial of degree ηcorrespondi吨 to f(x). Since T(x) = f(x) = T(x) for all Ixl < 1 if and only if

limn→∞ Pn(x) ， it follows that

J兰在 If(x) 一凡 (x)1

= 0,

(5.3)

n …

,G

ι'U

i1

,

4'Lν-

f\iz z-m 一-刊

f(x) - Pn(x)

口

罔

eO. 25 自 1. 284025.

\i lxl< 1.

(i) Show that , for any x , 才 t4

and conclude that

十 x). Our goal is to show that

叶一 i? wj-6 、1

X3

= 1; = 1. 28385417;

Xo

口

JR.

\i xE

For x = 0.25 we find that 已0.25

115

(ii) Show that , for any 0 三 x < 1, Problem 3: Find the Taylor series expansion of the functions

If(x) - Pn(x)1

叶泸) and 击 around the point 0 , using the Taylor series expansions of In(l - x) and 占

If(x) 。c

.;:::.. . . x k

z,.

x2

x3

x4

1

1-x

L: x

k

= 1 + x + x2 + x 3 十

，

\i

In (1 - x 2 )

= 一了 。c zffz-z2-21-Ef-zf一仨t k

234

7V Z

ε(-1 ， 1);

Pn(x)1 三( -x)nlln(l 十 x)1

and conclude that (5.3) holds true for all x such that -1 < x < O.

Solution: (i) From the integral formula for the Taylor approximation error we know that

xE(-l , l)

By substituting x 2 for x in the Taylor expansions above , where Ixl < 1, we find that

n

and prove that (5.3) holds for all x such that 0 三 x< 1. (iii) Assume that -1 < x < O. Show tl时

Solution: Recall that In (1 - x) = 一〉 ;-z-z 一一一一一一一…， V x E [-1 , 1); 乞t k 2 3 4

:三 x In(l 十 x)

f(x) - Pn(x) Since f(x) = In (1

=

+ x) , we

I{X

(x - t)n An+l)

一石「 ftmj(t)dt.

(5 .4)

obtain by inductio旦出at the de巾atives of

f(x) are (k)

fwj(z)=

( -l)k十 l(k

- 1)! 』 (1 + x)k

V

k >1

(5.5)

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

116

(x - t) n(-1 )7川 n! J I{X 一一~ \ -/ . d Jo n! (1 十 t) 叫山 {X ( -1)η十2(X - t)n A Jo (1 + t)叫一

卫鬼 (-x)nlln(l

(5.6)

主二!< ι \7 0
一一

J豆豆 If(x) 一凡(x)1

f(EY T41dt 三 42Ifzdt

f(x) - Pn(x) =

巳 [0 ， 1).

0, V x

8

=

r-

-I-

(5.7)

s ( -1)η十2(-8

{-s ( -1?n+2(8 十 t)n J-IrJo (1 + t)叫…。

s

(8 十 t)n

I

\7

0
{S (8 - z)η{ I:i 8 n l z < I 一一… Jo (1 - z)卅1 厅 - Jo 1- z a

8

n

(一 l叫 1 - z))I;二;

-8叫n(1-8)

=

( -x)nlln(l 十 x)l;

8 ll叫 1

n

(5.9) (\(5.10\) (5.11 )

~ (川十川附

u

eCJVlt :

(5.12)

d

e 一σJ瓦

(5.13)

1 1 (T' σ\~ p - 一十一 i 一一~ I v8t. 2 ' 2\σ2/ . --

we find that

If(x) - Pn(x)1

V x E (-1 , 1) ,

Use Taylor expansions to show that , for a small time step 8t , u , d and p may be approximated by

By taking absolute values and using the fact that

主二三 <8‘ 1-z 一'

where

A =

(8 - z)n

{S

-

J

h

(1 十 t)叫…

Using the substitution t = -z , we obtain that =

- t)n

= 0,

u = A+VA亡I d = A-V万1· er8t - d p 一u-d'

(1 + t)叫山

I

f(x) - Pn(x)

(5.8)

Problem 5: In the Cox-Ross-Rubinstein parametrization for a binomial tree , the up and down factors u and d , and the risk-neutral probability p of the price going up during one time step are

-x. From (5.6) , it follows that

{X(_1)n+2(x - t)n J ~~ ~~-n+l -/ dt = I (1 + t) 肿1 ~V Jo

I。 \

ε(-1 ， 0).

and co叫ude t~at the Taylor series expansion of the function f(x) = ln(l 十 x) converges to f (x) for a叮 x with Ixl < 1.口

We conclude that

(iii) Assume that x 巳 (-1 ， 0) and let

0, V x

From (5.7) and (5.8) we obtain that Jl忠 If(x) 一凡 (x)1

lim If (x) - Pn (x) I

0

=

,

we obtain that lf(x) 一凡 (x)1

+ x)1

We conclude that

(ii) Let x ε[0 ， 1). By taking absolute values in (5.6) and using the fact that +t

117

recall that , by definition 8 = -x. Note that , for a叮 Z ε(-1 ， 0) ，

From (5 .4) and (5.5) it follows that

f(x) - Pn(x) =

5.1. SOLUTIONS TO CHAPTER 5 EXERCISES

- 8)1

J_.

dz 内

(5.14)

In other words , write the Taylor expansion for (5.9-5.11) and for (5.12-5.14) and show that they are identical if all the terms of order 0 (8t) and smaller are neglected.

Solution: We will show that the Taylor expansion for (5.9-5.11) and for (5.12-5.14) are identical if all the terms of order 0 ((8t) 3/2) and smaller are

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

118

5.1. SOLUTIONS TO CHAPTER 5 EXERCISES

neglected. In other words , we will show that

1 + a 05t +

U

eu!8i 十 0((6t)3/2);

(5.15)

d

e一σ!8i 十 0((6t)3/2).

(5.16)

1

1

a

p

2

,

1- a 05t

2σ2

(5.17)

日ι

l+x 十二十州7 川→ 0; 1+j 十 O(此 as x • 0; l-j+0( 向 as x • 0 l 十 ?+0川)

Then ,

e

σ v&t

~

(e-

r<5t

+ e(r+u

2)

+

6们t 宁 十 O叫仰州(仅仰阳(仔仰州 户= (1+ 平。((ot)'))' -

Therefore , (5.15) and (5.16) are established. Finally,

1 十 (r 忖协 0(( 们 )2))

1

1

+ 0(( 们 )2; d亡工工作26t + 0((6t)2) = σ 币、Ii十 0(6t)

0((6t)2)) 0((6t)2) I

σ v6t + 0(( 们 )3/2)

and

A 十 J万τ丁

1 十字 + 0((M)2)

er<5t - d u - d

σ yft;i + (γ- ~)们十 0(( 们)附)一 σ 十 (γ

σ26t

u -

工

(1 + r6t + 0(( 圳))一 (1 一 σ yft;i 十字 + 0((6t)3/2)) (1 + a yft;i 十字+ 0((们 )3/2) )一 (1 - a yft;i 十字 + 0((6t)3/升

1+

。但!l σ v6t .(1 ( 十一叫一 +

6缸t l+a v6t 十 字 +0叭咐州(仅 (θ仰川 阳 ω 1 一 d 十宁+叫川 u = eU V&t +0((M)3/2); d = e- u V&t +0((6t)3/2).

<5t)

;(1-dt+0(( 们 )2)

+ a v6t + 0((M)3

0((σ 05t )3)

we conclude that

p

A =

(σ yft;i)2 而 十一门一+

eUV&t

In particular , note that

飞/i 十 O(M)

+ 字+ O( 川

Since

Recall the following Taylor approximations: ex

A 一 J万丁I

d

2

一十一 (γ 一三 )v6t十 0((出)3/2) ，

缸 6t 宁 + 叫创仰州(付仰川

2σ yft;i + 0((6t)3/2) 1 , a2

一号 )yft;i + O(6t)

2σ 十 O( 们)

一十一 (γ 一巳 )v6t十 0((6t)3 j 2) ， 2

2σ2

wh Problem 6: (i) What is the approximate value ~凡αp附 prox 创t price money put option on a nor丑1 di忖飞 vider丑ld一paying underlying asset with 印 s po S = 60 , volatility σ= 0.25 , and maturity T = 1 year , if the constant risk-free interest rate is r = 07 (ii) Compute the Black-Scholes value PB 归S氏向仲仲如?卢 户 r俨伫r严叫咆气­ mate the relative approximate error

一

IPBS户闪=0 -~α即叫 r=O ， q=ol PBS， r=O ， 俨 O

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

120

(iii)Assume thatT=0.06and q=0.03Compute the approximate value Pα.pprox，r=0.06俨0.03 of an ATM put option and estimate the relative approxi日late

error

jPBS，r=0.06，俨0.03 一凡pprox，r=0.06俨0.031 PB S ,r=0.06 ,q=0.03 ' where PB S ,r=0.06 ,q=0.03 is the Black-Scholes value of the put option. u

(5.18)

Solution: (i) Using the approximate formula

Pα附町=O，q=O

_ /T 21r'

as

-

V

5.984134.

川丁

PBS，r=O ，俨O

approximate formula

IT ( (γ +q)T\ Pappr叫附 = aSy ;1r \ 1 一 j

we obtain that

-

QU 7

that

=

P 自 σ s f[;

(1JTlq)T

-

2

十

叮

S (e- qT - e- rT ) 2

(5.19) (5.20)

p-c 十川=

-2 S

rT

(寸 e- ) 十 Se- qT

_

Se- rT

kfrT7

since K = S for ATM options. (ii) Using the new approximation formula (5.20) , we obtain that -

4.86103 1.

The Black-Scholes value of the put option is PB S ,r=0.06 ,q=0.03 = 4.886985 , and therefore 1.4 761 %.口

0.014761

Problem 7: It is interesting to note that the approximate formulas

(rγ 旷 )T 卜一J价巳叮 υ C 臼 σS占 (ν1 γ丁 lγγγqω

Based on the linear Taylor expansion e- x 自 1 一凯 new approximation formulas for the price of AT lVI options which satisfy the Put-Call parity can be obtained by replacing rT and qT by 1 - e- rT and 1 - e- qT , respectively. The resulting formulas are

凡，pprox_new ， r=0.06 ， 俨 0.03

4.886985 ,

and therefore IPBS， r=0.06， 俨0.03 - ~αpprox ， r=0.06 ， q=0.031 .r BS,r=0.06 ,q=0.03

Ke- r

Solution: (i) From (5.19) and (5.20) , it is easy to see that

Pα.pprox，r=0.06，q=0.03 = 4.814848.

PBS ,r=0.06 ,q=0.03

S (e- qT 一 (γ - q)T) 并 Se- rT

(i) Show that the Put-Call parity is satisfied by the approximations (5.19) and (5.20). (ii) Estimate how good the new approximatio丑 (5.20) is , for an ATM put with S = 60 , q 二二 0.03 , σ = 0.25 , and T = 1, if r = 0.06 , by computing the corresponding relative approximate error. Compare this error with the relative approximate error (5.18) found in the previous exercise.

IPBS，r=O俨0 一凡pprox，r=O，q=O 1= 0.002604 = 0.26%.

自nd

C

jz

and therefore

From the Black-Scholes formula , we

-

P 自 σS1/.!.-e qT+ 2

PBS ,r=O ,q=O = 5.968592 ,

Usi吨 the

P 十 Se- qT

JZehedSM-frT)

(ii) From the Black-Scholes formula , we find that

(iii)

for ATM call and put options do not satis有T the Put-Call parity:

C 用 σ Sy ;1r

we obtain that Pα.pprox，r=O， q=O -

5.1. SOLUTIONS TO CHAPTER 5 EXERCISES

十

(r - q)TS:

IPBS ， 问 0.06 ，俨 0.03 - ~α.pprox_new ， r=0.06 俨 0.031 PBS ,r=0.06 ,q=0.03

工 0.005311

= 0.5311%.

Recall from Problem 7 that the approximation error corresponding to the original approximation formula is 1. 4761%. We conclude that , for this particular example , the new approximation formula is more accurate. 口

2 (γ - q)T

2

σ

Problem 8: Consider an ATM put option with strike 40 on a non-dividend paying asset with volatility 30% , and assume zero interest rates.

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

122

Compute the relative approximation error of the approximation

σ币汪叫叫…p川 1 ， 3, 5, 10 , and

P 臼

20 years

Solution: We expect the precision of the approximation formula for ATJVI options to decrease as the maturity of the option increases. This is , indeed , the case: Approximation Error T PBS Pαpprox 1 4.787307 4.769417 0.38% 1. 13% 3 8.291860 8.199509 1.88% 5 10.704745 10.507368 3.76% 10 15.138795 14.589748 7.55% 20 21 .409489 19.906608

Problem 9: A 在ve year bond worth 101 has duration 1. 5 years and convexity equal to 2.5. Use both the formula

(5.21)

which does not include any convexity adjustment , and the formula

+

~C(b:..y)2 ，

(5.22)

the price of the bond if the yield increases by ten basis points (i.e. , 0.001) , fifty basis points , one perce风 and two perce风 respectively.

to 自nd

Solution: Denote by B n 毗 D the approximate value given by formula (5.21) for the value of the bond corresponding to the new yield. Then , b:.. B = Bneω .D-B a叫 from (5.21) , it follows that Bnew ,D =

B

(1 - Db:.. y).

(5.23)

Similarly, let Bnew.D.C the approximate value for the value of the bond given by formula (5.22). We obtain that

,

Bn w ,D ,C

=

B

(1 - D

t>y 斗叫) .

Note that pz1017D =157and C =25i口 (5.23) and (5.24). The following approximate values are obtained for b:.. y ε{0.001 ， 0.005 , 0.01 , 0.02}: Bneω ，D

0.0010 100.8485 1B0η0ε.ω8，4D8，6G 0.0001% 0.0050 100.2425 100.2457 0.0031% 0.01 99 .4850 99 .4976 0.0127% 0.02 97.9700 98.0205 0.0515% b:.. y

Bne ω ，D ，G - Bne叽 D B neω ，D

IPBS 一凡pproxl [ PBS，r=O，q=。一

b:.. B E一目 - Db:.. y

123

The last column of the table represents the percent difference between the approximate value miIlg d11ratioa done?and ttle approximate value usiIIg both duration and convexity, i.e. ,

Here , the Approximation Error is the relative approximation error defined as

b:.. B 王一用一 Db:..y ，

5.1. SOLUTIONS TO CHAPTER 5 EXERCISES

(5.24)

〔 一

CHAPTER

124

5.2

TAYLOR'S FORMULA. TAYLOR SERIES.

5.

Supplemental Exercises

5.2. SUPPLEMENTAL EXERCISES 6. The goal of this exercise is to compute

11In(1- x) In例

1. (i) Let g(x) be an i凶nitely differentiable function. Find the linear and q~adrati~ Taylor approximations of eg(x) around the point O.

(ii) Use the result ~bove to compute the quadratic Taylor approximation around 0 of e(x+I)2 (iii) Compute the quadratic Taylor approxi可lation around 0 of by using Taylor approximations of eX and eX·. 2. Show that

e- X 一 -L=0(内，

as x

l 十Z

e(x+I)2

(i) Show that :地(同 - x) ln( ♂))

l'扑

+

十

4i

Z 1-z \飞 111/

已

才t - A

十

PAVAOve

//111\

[

仨IηJ叫O

In(l 一叫

1

立

(iv) Use that fact that

14

VV Z

>-

立主 =Z

41·&

ha

4L

4EU

才叶到 ln(x

5. (i) Find the radius of convergence of the series ~4

1 十

L

2!

、、 8

十 '

L

4!

~12

十二一+. '

6!

(5.25)

(ii) Show that the series from (5.25) is the Taylor series expansion of the function X2 ez2 十 e2

。，"

Vv Z >-

咱54

\飞 lif/

♂

吨E L

<

十

<

ρU

十

4ti

♂

/Iit\

/fiI\ 1-z+ l-2

2 作 -6

to obtain that + l-2 -b \1111/ 1-z

0,

(iii) Prove that

4. Recall that <

= x/I lim- (问 - x) ln(x)) =

μl1 1h 〉」 ln (μ Z 凶丑叫(1 一 功训巾叫 以坤口叫巾 川

around the point 0, and find its radius of convergence.

<

(5.26)

(ii) Use the Taylor series expansion of 1 叫 1- x) for jxj < 1 to show that

• o

叫;号~) z \111J/' 1-z

dx

and conclude that the integral (5.26) can be regarded as a definite integral.

3. Compute the Taylor series expansion of

/JIl--\

125

7. Consider an ATM put option with strike 40 on an asset with volatility 30% and paying 2% dividends continuously. Assume that the interest rates are constant at 4.5%. Compute the relative approximation error to the Black-Scholes value of the option of the approximate value

.IT ( 1 一

Pα附 oX ， r 手 O ， q 并 o -σ BV

;1T \

(7'十 q)T \(7' -

FJ

-

if the put option expires in 1 , 3, 5, 10 , and 20 years.

q)T

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

126

5.3

5.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES Problem 2: Show that

Solutions to Supplemental Exercises

e-x 一 -L=0(z2)7aS 1 十 Z

ρυ

1

Solution: (i) Let j(x) = eg(x). Then = g'(x)eg(x)

j"(x) = (g"(x)

and

1 十Z

+ (g' (x))2)e g(X).

1- x + x 2

+

O(x 2) ,

as

x

•

O.

e- x

j(O)

•• _,

x 2 _"

+ x j' (O) 十五 f气。) +

。

J

O(x ) ,

x

as

•

0,

1 一l+x

l~X

as

x • 0;

as

x

•

O.

as

x

are

。但2) ，

•

O.

一二十 O(x3 ) ，

In e g (川

I

x

•

1 (L (2x)2 3 1 + 2x 十一γ + O(x ) ,

•

0,

I

口

xJ

叫;52)=l中川川

O.

=

I

0

Solution: Note that the function

quadratic Taylor approximations

(2X)2 " f ~3\ J e""x = 1 + 2x + \-;一 + O(x ) , x2 e _ 1 + x 2 十 O(x气 as x

-t

around the point 0 , and find its radius of convergence.

as x • O. (ii) By letting g(x) = (x 十 I? in (5.27) , we find that as

x

(1 +纠 飞 1-

(5.27)

e(叫 )2 工 e + 2ex + 3ex 2 + 0 (川)，

as

Problem 3: Compute the Taylor series expansion of

becomes

Usi吨 the

十。但 3) ，

l+x

The quadratic Taylor approximation

eg(x) =

0

Note that we implicitly proved that

eg(x) = eg(O) 十 xe到 O)g'(O)

(iii)

I-x 十二十州，

e- x 一 -L= 一主十 O(泸)

= j(O) 十 x f' (O) + O(x 2) , as x • 0,

can be written as

j(x)

•

Therefore ,

The linear Taylor approximation j(x)

Z

Solution: The qu础 atic Taylor approximations of e- x and m 山

Problem 1: (i) Let g(x) be an infinitely differentiable function. Find the linear and quadratic Taylor approximations of eg(x) around the point O. (ii) Use the result above to compute the quadratic Taylor approximation around 0 of e(x+1? (iii) Compute the quadratic Taylor 斗pproximation around 0 of e(x札 )2 by using Taylor approximations of eX and eX*.

f' (x)

127

as

x

•

0;

is not defined for x = -lor x = 1. Therefore , the largest possible radius of convergence of its Taylor series expansion around 0 is 1. The Taylor series expansions of the functions In(l 十 x) and In(l - x) are In (1

+ x)

it follows that e(x十1)2=e -dz-et=e(l十 2x 十 2x 2 )(1 十 x 2 ) 十 O(x3 ) e 十 2ex + 3ex 2 + O(x 3 ) , as x → O. 口

In (1 - x)

立

fz2z3z4

(-OMJZZ 一一十一一一+... k 2 '3 4

军空气 x k

x2

x3

x4

, V ♂ ε(-1 ， 1];

一予:二 = -x 一一一一一一一... , V x ε[-1 ， 1) ， 乞fk 234

CHAPTER

128

TAYLOR'S FORMULA. TAYLOR SERIES.

5.

5.3. SOLUTIONS TO SUPPLEl\1ENTAL EXERCISES Fr。因 (5.31) ， we also find that /IIl-\

In(1 + x) 一 l叫 1 - x)

xk

=

•

, I.

,

1

2叶子+车十

\liI/ 1-z

\liI/ /II11\

n

-EA

414

+

=

1 十

至 271

，\I川 xEε 川)

X+ j

Z

Recall that

Sol州on:

1

1

3(2z+1)21-tF 一

1 十一) (1+~时

(1 十 ~r\ \l x?1

(5.29)

1+ 一」龟

12x(x 十 1)'

(]斗"\ 2 y 3 , 2ν5 In (飞 ~一立) = 2y 十一+一十... , \I y E (-1 , 1). 1 - y) 3 ' 5

(5.30)

\l x?1

(5.33)

x> 1.

\l

(5.34)

Using (5.34) , we 丑nd from (5.33) that

(1 十 ~r十二

Recall from (5.28) that

< e1吨

e<(l+:yl

Problem 4: Prove that

e <

1y 一」 + 1)2k

3(2x 十 1)2 位 (2x

and therefore that (5.28)

and has radius of ∞ cO ∞ I口1飞vergence equal tω0 1.口

(1+D 忖 <

1十

x k \λ 2x 2j + 1

t; ((一叫+讨 (

十

Z

1土 - q G

and have radius of convergence equal to 1. We conclude that the Taylor s~ries expansion of In (i~~) is

叫;占~)

129

e

卢

< e(1+:)? VG17

and we conclude that

-0'

(1 十 D 叫一百 < e ,

For any x 三 1 ， substitute y = 古 i口 (5.30) and obtain that

咔 +D =击+… 2

Jwn2

4 、民十

,

\I x

> 1,

which can also be written as

From (5.31) , we find th创 1

(5.35)

(ii) Show that the series from (5.35) is the Taylor series expansion of the function

(→)咔 +:)7VQ17

<

十

t-2 +

十

4t4

川一创

(5 出)

口

Problem 5: (i) Find the radius of converge肌e of the series 泸 -4

, \I x > 1.

The left inequality of (5.29) is therefore established.

十

(什)咔 +1)=1+ 。J 川十川一1 亏\d +. ., XI

\I x 二三 1.

ex2

+ e- x2 2

which is equivalent to

hv

z

w -n

一一

Z

α

The right inequality of (5.29) is therefore established.

T

句

I

α

X

一-

飞

Solution: (i) The series (5.35) can be written as a power series as follows:

(5.32)

l回

/ ...\ x+~ (1 十二 I , \ I x 二三1.

句

<

∞ ZH

e

Vv DA >-

130

CHAPTER

TAYLOR'S FORMULA. TAYLOR SERIES.

5.

__k!

1.

一一----，一

k→χ(~)κd 在

[

们 (k!)l / k ..…-

:地(问- x) In (x))

1 e

k

k 二二x

咱止

一门 L

P-n 川/

...."

Ll吨1

,,,,,,,, n4p

~

e1/2 lim

p→ x 飞/2p

(5.36)

=ε

,.

(

11日1

I

2p ，'~

\

,,,.

-

I

(2p)1/

(iv) Use that fact that

limp→∞ |α4p\1 / 4p

立去 z

~~，

ez=57Vzd?

to obtain that

Sol毗 on:

(i) First of all , note that

(5.38)

x/1

/lil\wv

叽→ U \ / 气l /

h 内 Ur-ww

1/fIi1\/JI--\

一

、飞 E1'' ，

UU

、 112J

n

-ai

/tt 飞、

、 、l /

/''飞飞

vhu

nuυ ηο

一;

Uu-f

一

ll

， ， ，E E 、 、 - 、 ，U J J

〓

一\jv.

/飞

1-uul

\Iil/n

一一

/ll\/i!\

口

∞

which is the same as the series (5.35).

m∞ h H俨

37+ZI+ 石「+

Jl\n

1+z4z8212

:与(l n(l 一仙 (x ))

恼，

k!

lim (同 - x) ln(x))

We compute the left hand side limit of (5.38) by changing it to a limit to infinity corresponding to y = ~ as follows: /i\Bh 1-U"1 且俨且俨一

2位

f工 x 4j - --但 (2j)!

=

m∞

-

z

\lil/

;倍 ;21+ 主 i二;11) 1 王三产+ (_1)k x 2k

1

才112(1 一叫) dx 工 2-Z ?V口 (1 - x) In(x))

it is easy to see that

2

立

T (x) is

which means that the series (5.35) is co盯ergent for all x E JR (ii) Using the Taylor series expansion

ez2 十 e- X2

(iii) Prove that

[ln(1 一叫) dx

l l lims叩k→况 |αkl 1/k

1 一…一一一τ;:;-

0

Therefore , the radius of convergence of the power series R 一

In

2

1/

p~叩\ ((2p)! )l /2p )

-

=

0,

才1ln川 h川

-

P叫 ((2p) !)1/ 4p

= lim (叫- x) In( x)) = x/1

that the integral (5.37) can be regarded as a de直nite integral. (ii) Use the Taylor series expansio口 of ln(l - x) for Ixl < 1 to show that

自nd tl川

J坦 lα年 111年

(5.37)

a时 co肌lude

25

lp

m明

Using (5.36) , we

In(1 一叫)

(i) Show that

It is then easy to see that

and therefore that

131

Problem 6: The goal of this exercise is to compute

From Stirling's formula we know that lim 1日1

5.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

132 丑ecall

5.3. SOLUTIONS TO SUPPLE1VIENTAL EXERCISES From (5 .4 1) and (5 .4 2) , it follows that

that

[忡

J41-j)=: and therefore

J坦咐 -D")

=

(5 .40)

-1 ln(ν)

1

li~ (ln (1-

x) ln(x)) =

0

1口 (1

1)

(η+

1)2

Vx 托叫 ε 叫( -11υ ， 均1)

一→

\III-/

一汇去 (5 .44)

Here , we used the fact that

立主 =Z

- x). Then ,

nl:(x) dx = - I I) ~ ~一一 )Il n In \....-

[信

and the telescoping series

(5 .4 1)

主 G 一击) zA 兰 G 一击)

-叩 m

-Zifznln(Z)dz

/IIlI\

lN

From (5 .4 3) and (5 .44) , we conclude that

(iii) Using integration by parts , it is easy to see that xn+I ln(x) xn+I znln(z)dzzd-r

1 一川

ln(1 - x) ln( x) [ln(1-X)ln(x)

L忑

η(η+

1日 才

absolutely ∞ c or盯erge旧 to

V「 ZK

一

汇

1.

x) , i. e. ,

(ii) The Taylor series expansion of ln(l 川lι M 一 Z叫) =

VO<x<

1

1 一η / i i l \

ln(l 一 x)l叫 x) ，

g(x) =

Therefore ,

∞汇叫卜

The integral (5.37) is equal to the de自缸 fin 丑旧 l让it怡e in 丑挝t怡egra 址1 be 创tweer丑1 0 and 1 of the

g(l) = 0;

1

η(η 十 1)η+1

111 n n+ 1 (η+1 )2

∞ C ∞ O I丑1 毗

g(O 创) =

1)

1

η(η 十 1)2 x/I

(5 .43)

Then , it is easy to see that

We conclude that

x) ln(x)) =

1

l n(η+

l地 (ln(1-z)lm(z))=J 马亏一= 0

炖(川 -

立

(iv) Note that

From (5.39) and (5 .40) it follows that

i坦s

133

1.

[I中叫l川

f

Then

1xnln仪 1

( x n + I l日(x) 一 η

+1

xn+ I

\[

(η 十 1 )2 )

Problem 7: Consider a口 ATIVI put option with strike 40 on an asset with volatility 30% and paying 2% dividends continuously. Assume that the interest rates are constant at 4.5%. Compute the relative approximation error of the approximation

10

L一一 ~~.~~(x 肿Il n (x) ) 1)2

(η+

dF

VGl

(5 .42)

凡，pprox ，呐

IT

(

(γ 十 q)T \(γ -

q)T

134

CHAPTER 5. TAYLOR'S FORMULA. TAYLOR SERIES.

if the put option expires in 1, 3, 5, 10 , and 20 years. Solution: The approximate option values and the corresponding approximation errors are given below:

T

凡.pprox

4.1317 3 5.9834 5 6 .4652 10 5.2187 20 -2.5067 l

Error PBS 4.1491 0 .4 2% 6.1577 2.83% 6.9714 7.26% 7.3398 28.90% 6.0595 NjA

While the approximation formula is still within 3% of the Black-Scholes value when the maturity is three years or less , it deteriorates for long dated options , and even produces a negative value for the 20-years option. 口

Chapter 6 Finite Differences. Black-Scholes PDE. 6.1

Solutions to Chapter 6 Exercises

Problem 1: A butterfly spread is made of a long position in a call option with strike K 一叽 a long position in a call option with strike K 十凯 and a short position i口 two calls with strike K. The options are on the same underlying asset and have the same maturities. (i) Show 也at the value of the butterfly spread is C(K 十 x)

- 2C(K)

+ C(K -

x) ,

where , e.g. , C(K + x) denotes the price of the call with strike K 十 x. (ii) Show that , in the limiting cωe when x goes to 0 , the value of a position in 主 b耐er组y spreads as above co盯erges to the second order partial derivative of the value of the option , C , with respect to strike K , i.e. , show that C(K 十 x)

-

2C(K)

+ C(K -

x2

2、6

x)

护C =一τ (K).

8K2

(iii) Show that , in the limiti吨 case when x • 0 , the payoff at maturity of a po副on in ~ butte吻 spreads as above is goi吨 to approximate the payoff of a derivative security that pays 1 if the underlying asset expires at K , and o otherwise. Note: A security that pays 1 in a certain state and 0 i丑 any other state is called an Arrow-Debreu security, and its price is called the Arrow-Debreu price of that state A position h butter23rspreads as above7with Z smll7is a synthetic way to construct a口 Arrow-Debreu security for the state S(T) = K.

i

Solution: (i) The value of a butterfly spread , i. e. , of a long position in a call option with strike K - x and value C(K - x) , a long position in a call option with strike K 十 x and value C(K + x) , and a short position i口 two calls with strike K and value -2C(K) is C(K

+ x) -

2C(K)

135

+ C(K - x).

136

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

(ii) The value of a position in 去 butterfly spreads is

毛.. (C(K 忖) x

2C(K)

+ C(K -

x)).

(6.1)

The value of a call option as a function of the strike of the option is infinitely many times differentiable (for any fixed point i丑 time except at maturity). The;efore the expression from (6.1) represents the central finite difference approximation of 在 (K) and we know 山t

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

Problem 2: A bull spread is made of a long position in a call option with strike K and a short position in a call option wi出的rike K + x , both options being on the same underlying asset and having the same maturities. Let C(K) and C(K + x) be the values (创 time t) of the call options with strikes KandK 十叽 respectively.

(i) The value of a position in ~ bull spreads is cmf(K刊)In the limiting case when x goes to 0 , show that

C(K 叫一 2C (K) +C(K - x) = 些 (K) + O(泸)

z\", 0

σ K2

C(K

+ x)

- 2C(K)

•

3J

a s n nuoo : 7 ; 刘

十

、z l

/』飞

cukυ

、 飞 『/ Q U

nu )d k a m i d …n u ihJTL

θC

一 (K) = 印

y

e

l-z

-p

注

4EU '+b

e

QU

li同 fx(8) \人j

= 0, V 8

并 K.

•

+ O(x) ,

O. We conclude that x

θC

=一一 (K). θK

(ii) The payoff at maturity of the bull spread is

(6.2)

1.

as x

2、o

Let 8 笋 Kbea 直xed value of the spot price of the underlyi吨 asset. Then 儿 (8) = 0 for any x such that 0 < x < IK - 81 , and therefore Z

一

C(K) - C(K 十 x)

Note that fx(K) = 1 for any x 并 0 ， and therefore Ii闷儿(K) = z \>U

C(K 十 x) - C(K)

θK

( 0, if 8 < K - x; j 仨竖立1 ， if K - x ::; 8 ::; K; {刑， -- ~-1 互年-8 ， if K::; 8 ::; K + x; I 0, if K + x < 8.

=

Solution: (i) Since the value C(K) of a call option as a function of the strike K of the option is infinitely many times differentiable , the first order forward 且nite difference approxim~tion ~f ~去 (K) is

川叫

ZSTMCUL 叫

S}<-rrH

吭-

川

ιd

Eff

一<一/\一<哑 KZSZ 削

〈一<一十口

、 、 ‘ ，/ ，υ ，v ' '

uU 十

阶I L

一 -b

一叫乱?

K

UZ?

江

句 KrJ

bu-hMP

臼

忱刘

矶

hm 巴幻

d-SKKKM

川

d-

的υ 的 f f f f g

M

』

eτhU

b

创 n 叫

刘 (lzzlur L-1 七十 15J

口 yrII\rr

」

ftt 〈 l 飞

US-f/KJJ

丑

mULXS

e

、 、i J

T=

fx(8)

θ2C =一τ (K).

8K2

'u DAa e-I7·1·1·1·IE

DTne

x)

+ C(K -

x2

C(K) - C(K 十 x) θC =一 ~;/(K) x θK

(ii) Show that , in the limiting case when x 0 , the payoff at maturity of a position in ; bull spreads as above is going to approximate the payoff of a derivative security that pays 1 if the price of the underlying asset at expiry is above K , and 0 otherwise. Note: A position in ~ bull spreads 部 above， with x small , is a synthetic way to construct a cash-or-nothing call maturing at time T.

Then , in the limit as x goes to 0 , we obtain that z\o

137

r 0, if 8 三 K; max(8 - K , O) - max(8 一 (K 十 x) ， O) ={ 8-K , if K<8 三 K+x; x, if K + x < 8.

,

If gx(8) denotes the payoff at maturity of a position in ~ bull spreads , then 如何)

(6.3)

From (6.2) and (6.3) , we conclude ~hat ， in the limiti吨 case when x • 0 , the payoff at maturity of a position in ; butterfly spreads as above is 1 if the underlying asset expires at K , and 0 otherwise. 口

If 8 三 K ，

r

=

0, if 8::; K; <与互， if K < 8 三 K+x; l 1, if K 十 x < 8.

then gx(8) = 0 for any x Ii同 gx(8) Z\。U

=

> 0 and therefore 0, V 8 三 K.

(6 .4)

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

138

If S > K , then gx(S) therefore

=

1 for a叮 x such that 0 < x < S - K , and

Ii同 gx(S) x \ ,U

=

1, V S

as h

→ O.

Problem 3: Find a second order finite difference approximation for f'( α) using f(α) ， f(α + h) , and f(α + 2h). Note: This type of approximation is needed , e.g. , when discretizing a PDE with boundary conditions i盯olvi吨 derivatives of the solution (also called 抖tions (c∞ O丑时叫 t悦lnuou Robin boundary conditi∞ s). For example , for Asian Op computed ave臼ra略ge rate call , to be more precise) , this type of finit~ _difference approximation is used to discretize the boundary conditio丑誓+嚣= 0 , for R=O.

Solution: To obtain a finite difference approximation for f'(α) in terms of f(α) ， f(α + h) , and f(α 十 2h) we use the cubic Taylor approximation of f (x) around the point x = 风i. e. ，

口

Problem 4: Find a centralfinite difference approximation for the fourth derivative of f at 风 i.e. ， for f(4)( α) ， usi吨 f(α - 2h) , f(α -h) ， f(α) ， f(α 十 h) ， and f(α+ 2h). What is the order of this 缸ite difference approximation?

Solution: We will use the following Taylor approximation of f(x) around the point x = α: f(x) = f(α) 十 (x 一 α) f' (α) 十

,

(6.6)

+ 0 ((μZ 一 α叫) 7) , as

f( α +2h)

h 2 _",' h 3 _1 飞 A f(α) + hf'(α)+Efff(α) + '~ f(3)(α) + 0 (的;

(6.7)

4h 3 川 A f(α) + 2hf'(α)+ 约2f"(α)+ 了 f(3)(α) + 0 (的， (6.8)

as h • O. We multiply (6.7) by 4 and subtracting the result from (6.8) to obtain f(α + 2h) - 4f(α +h) =

as h 缸lite

•

For symmetry reasons , and keeping in mind the form of the central diι ference approximation for f" (α) ， we use (6.10) to compute _IA\.

f(α +

2h)

+ f(α -

OJ

_fA\.

OJ

as h • O. We multiply (6.11) by 4 and subtract the result from (6.12). We solve for f(4)( α) and obtain the followi口g second order 丑nite difference approximation: (4)(~\ _ 2h) - 4f(α +h) _ f(α J \~ + -'UJ ~J \~ 'UJ 十 6f(α) ~Jh~

f~4) (α)

ash → O.

I

I

- 4f(α - h) 十 f(α - 2h) 十 O( 的?

I

口

2h -3f(α) - 2hf'(α) 十了 f(3)(α) + 0 (的， (6.9)

difference approximation

f'( α) ， we of f' (α):

obtain the following second order

一 f(α + 2h) + 4f(α +h)-3f(α)

2h

-f(α +

h4 , h6 h) = 2f(α) + h2f"(α) 十一 12 f(4)(α)+-f(川的 ,--/. 360 十 o (h 7 ) , as h • 0; (6.11) 4 6 4h , 8h α) 十一 f(6)( α) 2h) = 2f(α) + 4h 2f"(α) + "or: 3 f(4)(,--/. 45 7 十 o (h ) , (6.12)

f(α 十 h) 十 f(α -

3

O. By solving (6.9) for

f'( α)

(6.10)

x 一→ α-

as x → α. By letti吨 Z 工 α 十 h and x = α + 2h in (6.6) , we obtain that f( α 十 h)

(x 一 α)2 II(x 一 α)3(3) 2 -J f" (α) 十 6f()(α)

+(zff(4)(α) +(

f(α) 十 (x 一的f'(α) +(气川"(α) 十(气川但) (α) 十 o ((x 一 α)4)

139

(6.5)

> K.

From (6.4) and (6.5) , we conclude th~t ， in the Iimiti吨 case when x • 0, the payoff at maturity of a position in ~ bull spreads as above is 1 if the underlying asset expires above K , and 0 otherwise. 口

f(x) =

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

h 2 r (3)

十'; f(3)(α) + 0 (的

2h) + 4f(α 十 h) - 3f(α)2 2 h + O (的?

Problem 5: The goal of this exercise is to emphasize the importance of symmetry 坦白nite difference approximations. Recall that the central difference approximations for the first and second order derivatives are f'(α)

f"(α)

f(α +

h) -

2h

f(α -

f(α +h)-2f(α) ~~""J

h) + I

十 f(α J \""

2

(h ) ; h) '~J

+ 0 (h2) ,

140

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

as h • O. In other words , f'( α) and f"(α) are approximated to second order accuracy by using the value of f at the point αand at the pointsα- hand α + h that are symmetric with respect toα. We investigate what happens if symmetry is not required. (i) Find a second order finite difference approxi口lation of f' (α) usi吨 f(α) , f(α - h) and f(α + 2h). (ii) Find a first order finite difference approximation of f" (α) usi吨 f(α) , f(α - h) and f(α + 2h). Note that , in general , a second order finite difference approximation of f" (α) using f(α) ， f(α - h) and f(α + 2h) does 日ot exist. Let (3 <α<γsuch thatα- (3 = C(γ 一 α) ， where C is a constant. (iii) Find a finite differe肌e approximation of f' (α) 四吨 f(α) ， f( (3), and f(γ) which is second order in terms of b 一 αI ， i.e. , where the residual term isO (Iγ 一 α1 2 )

(iv) Find a 直nite difference approximation of f" (α) using f(α) ， f( (3), and f ('Y) which is first order in terms of b 一 αI. Show that , in general , a second order finite difference approximation of f" (α) using f(α) ， f ((3) and f ('Y) is not possible，叫esMzEFFie?mlessβandγaresym皿etric with .resp创 to α.

Solution: (i) and (ii). vVe use the cubic Taylor approximation of f(x) around the point x = 矶 i.e. ， f(x) =

f(α) 十 (x 一 α) 1' (α) + (x 一 α? f"(α) 十 (x 一 α)3 f(3)(α) + 0 ((x 一 α)4) ，

2 as x → α(6.13)

By letting x = α - h and x = α + 2h in (6.13) , we obtain that 2

;

(6.14)

4h3

f(α + 2h) = f(α) 十 2h1'(α) + 2h f"(α) 十了 f(3)(α) +0 (的， (6.15) 2

as h • O. We eliminate the terms ∞ cO丑川t 创口旧in 吨 g f"气 1 (μα) by ml咄i毕 pI趴 抖 yri鸣 (仰6.14 钊) 均 by 4 an 丑1 su 由 忱tractir吨 the resul忧t from (仰 b 6.1臼5盯叫). By solvi吨 for f'( α ), we obtain the following second order 自nite difference approximation of l' (α) : 一 f(α + 2h) + 3f(α) 4f(α … h)h 2 1'(α)-6h-Ef(川α) + 0 (h 3 ) f(α + 2h) + 3f(α) - 4f(α - h) 22 6h 十 o (h ) ,

as h • O.

141

Similarly, we eliminate the terms containi吨1'(α) by multiplying (6.14) by 2 and adding the result to (6.15). By solving for f"( α) ， we obtain the following 负rst order finite difference approximation of f" (α) :

f"(α) = f(μα +叶州叫 2孰h均圳川)卜一斗3 豆♂扩??俨俨)忡川十叶叫 2勾f附 f(归 α

+ 2h) 一 3f(归刨 叫) 十 2f(归 α α一 h 川) ? + O (h)7 3h 2

as h • O. (iii) a时 (iv). Denote γ 一 αby h , i.e. , let h = γ 一 α. Then ， α- (3 = Ch. We write the cubic Taylor approximation (6.13) of f(x) around the point α for x = γ = α + h and for x 二二(3= α - Ch and obtain

f( 'Y)

h 2 _"., h3 f(α) + h1' (α) + '; f勺) + '~ f(3)(α) + 0 (h 4 )

f( (3)

C h _"., h _(9飞 A f(α) - Ch1' (a) 十一f- fll(α) 一 -rf(3)(α) +0 (的， (6.17)

c

2 2

;

(6.16)

3 3

as h • O. By eliminating from (6.16) and (6.17) the terms containi吨 f"(α) and solving for 1' (α) we obtain the following finite difference approximation: C 2 f ('Y) - (C 2

-

1) f (α) - f( (3)

1'(α)1C(C 十 l)h+O (的

(6.18)

Similarly, we eliminate from (6.16) and (6.17) the terms contai时日g 1' (α) and solve for f" (α) to obtain the following finite difference approximation: f"( α)

3

h rill \ h f(α - h) = f(α) - h1'(α)+Eff(α)-lff(3)(α) + 0 (h 4 )

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

=

2

Cf(γ) 一 (C+1)f(α)

+ f( (3)

C(C 十 1)h 2

十 o

(h).

(6.19)

Note that , i口 general ， the finite difference approximation (6.18) of f'(α) is second order , while the finite difference approximation (6.19) of f"(α) is 且rst order. The finite difference approximation (6.19) of f"(α) would be second order , e.g. , if C = 1 or if f(3)(α) = O. Also , note that , for C = 1, i.e. , if (3 =α -h and γ=α 十 h are symmetric with respect to the point α ， then (6.18) becomes the central finite difference approximation of l' (α) ， i.e. , f(α +

h) -

f(α -

h)

1'(α)=+O (的 门飞 The same would not be true for (6.19) , which becomes f(α 十 h)

-

2f(α)

f"(α)=79

+ f(α - h)

十 o

(h) ,

(6.20)

142

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

Using the Taylor approximation In (1 +叫 :Z-4 十 O(X 3 ) we find that

instead of the central finite difference approximation f(α +h)-2f(α)

f协) =ω

+ f(α - h)

十 O(的

i (l n (1

I I

+ h) - h)

qA\l/

才t4

十

\飞 IlI/

\1llI/

门4

以 -2

B， ， ， ，

oh

and therefore \Ill-/

/l 飞

门，，"

一一

、 、t E J /

门，"

11

十

::; e ~~~ lei (ln(l+h)-h) -

一一

飞

vhM

。

'h-2

\飞 Ill-/

/It--1\ /III-\

门 ，4

。

门，"

/i

叮，"

e仇( ei (ln(l+h)

。

=

-mb

'hM

= 0 : n , we obtain that =

i 臼眈 jE 2盟g窍 野3 l卡e仇i h μ (←川沪 伊 e呻巾 i(队忡 (1( in

1.- 2

<e i22号 I-i亏 +O(的|

<已;十 O(h2 )

(1 + h)忡17

which is what we wanted to show. (ii) Let Y(x) = eX be the exact solution of the ODE. It is easy to see tl时 the approximation error ei can also be written as ei =的 - Y(Xi) = (1 + h)i - eih = ei1n (1+h) _ eih

4t4

O

vhM \, IJ

回 iq |

Solution: (i) Recall that yo = 1 and Yi+ 1 = (1 十 h)Yi ， for all i = 0 : (n - 1). It is easy to see by induction tl川的= (1 +}札 for all i = 0 :η: Initial condition: for i = 0, we know that yo = 1 = (1 十 h)o. Induction step: assume that Yi = (1 十 h)2. Then , =

:η ，

\11llI/ \1111/

Since eih < e for all i

n一→。C 2=U:η

+ h)Yi

十

-nb

=0

、 飞B， ， ， ， 、 、

for all i

/JIll-\ ,/Il--1\

町，血门，血

ih 三 1 。

be the approximation error of the finite difference solution at the node Xi , i = 0 :η. Show that this finite difference discretization is convergent , i.e. , that lim J;ll ftx Iei I = O.

since

-+ -o

纠/

0:η.

(ii) Note that y(x) = eX is the exact solution of the ODE. Let ei = Yi - Y(Xi) = (1 + h)i - eih

的+1 = (1

= O(h 2 ) is

\1111/\lll/

//II11\ /IlI-\ 几

=

以 -2

Vi

'hM7

with yo = 1. Show that

00

l e←

Jγ-nVA

十

difference

一

缸lite

1

:叽 where

，寸，什十

_ 1

P

臼

= 0

[ln(l 十h)-h]

十十仍

/lit\/iil\2-

ei

= (1 + f仰i ， V i = 0 : (η- 1) ,

+ f讥

:ηNote that the estimate iO(h 3 )

= 0

以一 2 以 -20

y' (x) = Y (x) , V x ε[0 ， 1]; ν(0) = 1.

(1

1.- 2

sharp , since , for i =叽 the product ih is equal to ih = nh = 1. From the Taylor approximation eX = 1 十 X + O(x 2 ) , it now follows that

Problem 6: Consider the following first order ODE:

的=

- h}

h2

s年ce ih 三 1 ， for all i

的+1

\ \

-i亏+ iO(的工-~亏+ O(时) ,

口

(i) Discretize the i且terval [0 , 1] using the nodes Xi = ih , i h= 去 Use forward finite differences to obtain the following discretization of the ODE:

1.- 2

i ( ( h 一亏十 O( 的)

of f"(α). This is due to the fact that , for C = 1, the coe自cient of h from o (h)' f~om (6.20) cancels out and the next term of order 0 (的 becomes relevant.

143

咐= o(~) We conclude that lim J;llftx Iei I

n一→。o

2=u:n

- 0,

and therefore that the finite difference discretization scheme of the ODE is convergent.

144

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

Note that we actually showed that the finite difference discretization is first order convergent , i. e. ,

233|eii

=

°G)

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

145

From the boundary conditions (6.22) , we find that Yo = 1 a时 Un=i (ii)For n=67tkeaIlite difemme dberetizatiOII(624)of tke ODE (621) can be written in matrix form as

口

AY

b,

where A is a tridiagonal 5 X 5 matrix given by Problem 7: Consider the following second order ODE: 2

3x y"(x) - xy'(x) + y(x) -

0 , \I x

ε[0 ， 1];

A(i , i-1) A(i , i 十 1)

(6.21)

-5

Y'(Xi)

y( 叫一 1)

2h

+

Problem 8: Show that the ODE

y"(X)

2y'(x) 十 x 2 y(X)

=

0

can be written as

f(x , Y(x)) ,

where

-

0,

(n-1) 叫

and

f(x ,Y(x))

Y

Z

‘

Solution: Note that y"(x) = 2y'(x) - x 2 y(x). Then ,

Y'(x) = 0, Vi = 1 :

(削)

\Iil/ 、、飞，，，，，

十的

Y(X)

of

/IIiI\

/''飞飞

UU

since Xi = ih , which can be written as

(3 叶)的一 1 一 (6i 2 -1 如 ( 3i' 一 DYi 十 1

口

=

内，，"

'hM

b

- i 。4

+ UU

气

μ-2

却」 h

q，旬

Uu-

十一

qο

'h

Y

Y'(X)

2

O( 的 7

into (6.23) , use the approxi血 a~e values yi for the exact values y(Xi) , for i = 0 :叽 and ignore the O( h'2) term. The followi鸣 second order finite difference discretization of (6.21) is obtained: q&

n叫U

2

h2+0(hzh

y(X i+ 1) -

25.5 -95 77.5

\illit--/

1iqAqOAUZWO It--BElt--/

ν "(Xi)

y( 向一 1)

0 0

(6.23)

We substitute the second order central difference approximations for Y"(Xi) and y'(均)， respectively, i.e. ,

y(X i+ 1) - 2Y(Xi) +

}qovhu

\飞

Vi=l:(η- 1).

,

/IEt--11111\ UUUUU

Solution: (i) Let Xi = 仇， i = 0: 凡 where h = ~. We look for Yo , y1 , . ..，仇 such that yi is an approximate value of y(Xi) , f6'r all i = 0 :ηBy writing the ODE (6.21) at each interior node Xi = ih , i = 1 : (η- 1) , we obtain nu

13 O O O

A

一。，"

(i) Partition the interval [0 , 1] intoηequal intervals , corresponding to ∞ des Write the finite difference discretization Xi = ih , i = 0 :凡 where h = 土. n of the ODE at each node Xi , i = 1 : (η … 1) ， usi吨 central 丑nite difference approximations for both y' (x) and y" (x). (ii) If n = 6 , we find , from the boundary conditions , that Yo = 1 and y6 = ~. The 直丑 ite difference discretization scheme presented above will have 自 ve equations can be written as a 5 x 5 linear system AY = b. Find A and b.

i = 1 : 5; i = 2: 5; i = 1 : 4,

OOO 钊」

,

64

I. e.

VVVVVV

1) ,

3i2-j7

。口引印。

(6.22)

川 2800

的 )=1;U(hj

3X;Y"(Xi) - xiy'(Xi) + y(Xi)

一 (6i 2 3i 2 十乞

A( i , i)

(UZj)z (勾 fdt2U(Z)) (工 2;)(38)=( 工 2

n

Y(x)

口

146

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

Problem 9: Consider a six months plain vanilla European call option with strike 18 on a non-dividend-paying underlying asset with spot price 20. Assume that the asset has lognormal distribution with volatility 20% and that interest rates are constant at 5%. (i) Compute the Greeks of the call optio凡i. e ， ~， f , P, vega , and 8. Use finite differences to 丑nd approximate values for the Greeks. Recall that

A

θCθ2CρθC f =一一· ρ 一一一 θ S' θ S2' t' θγ7

一一一: 一

vega

…一

δC

_

-τ一;。 。σ

θC θT

Denote by C (S , K , T， σ， r) the value of the call option obtained from the Black-Scholes formula. (ii) The forward and central difference approximations ~f and ~e for ~， and the central difference approximation f e for fare ~f -= C(S 十 dS， K， T， σ， r) - C(S， K， T， σ， r) f = dS

~e

- C(S - dS， K， T， σr); 2dS f = C(S+dS， K， T， σ， r) 一 2C(S， K， T， σ， r) + C(S - dS , K , T， σ， r) e (dS)2

=

一

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

147

Solution: (i) Usi吨 the formulas for the Greeks of a plain vanilla call option derived from the Black-Scholes formula , we find that ~

= 0.839523;

f

= 0.086191;

vega = 0.01501;

P = 7.045377;

8 = -1. 394068.

(ii) The Black-Scholes value of the call option is C = 2.699703. The corresponding forward and central difference approximation errors for the Delta and Gamma of the option are: dS 0.1 0.01 0.001 0.0001 0.00001 0.000001

6. f

6. e

0.843774 0.839952 0.839565 0.839526 0.839522 0.839522

0.839464 0.839521 0.839522 0.839522 0.839522 0.839522

fe 0.086199 0.086196 0.086195 0.086196 0.086207 0.090594

I~-~fl

I~-~el

If - fel

0.00425158 0.00042959 0.00004228 0.00000349 0.00000037 0.00000076

0.00005840 0.00000138 0.00000082 0.00000081 0.00000081 0.00000081

0.0000081~

0.0000042: 0.0000042l 0.0000042! 0.0000153 0.0044029

C(S+dS， K， T， 叽 γ)

一

-

nunununu

18 一 8fl_

181

呼BE--

reJ-

门《 U

a 一 σ。 一

0.1 0.01 0.001 0.0001 0.00001 0.000001

W一 乃

If - fel

一『

6. - ~el

a

1

二可

1

V 一

6. f 6. e f e 6. - 6. fl

σb­

The corresponding relative approximation errors are enL

Compute the approximation errors for the following values of dS:

dS

(iii) The 且由 e difference approximations of vega , P and 8 give川y (6.25-6.27) are vegaf 3 .448385;ρf = 7.045624; 8 f = - 1. 392998.

n

Ip- Pfl = p

一一一一一

0.000035;

nnn7C::巧内 …-

Problem 10: Show that the value of a plai口 vanilla European call option the Black-Scholes PDE. In other words , show that

satis直es

(iii) Let dσ= 0.0001 ，价= 0.0001 , and dT = 在， i. e. , one day. Find the following forward difference approximations for vega ， ρ ， and 8: vegaf Pf

8f

C(S， K， T， σ +dσ， r) - C(S , K , T， σ， r) dσ

C(S , K , T， σ7 俨十 dr) - C(S , K , T， σ， r) dr C(S , K , T + dT， σ， r) - C(S， K， T， σ， r) dT

(6.25) (6.26) (6.27)

θC 一 θt

+'

1 叮叮 θ2C δC ';a 2 S 2 一丁 +(γ - q)S一

2-

~

8υθS

-

rC -

0,

where C = C (S , t) is given by the Black-Scholes formula. Solution: Although direct computation can be used to show this result , we will use the version of the Black-Scholes PDE invol飞ring the Greeks , i.e. ,

@+jd价十 (γ 一仙一 rC

0,

148

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

substitute for .6., rand

.6.

=

e -

e the values

qSe 一矿-t) ρ -q(T-t)

r-L

e-q(T-t) N(d 1 );

Sσ 、/2作 (T

qSe-q(T-t) N( d1 )

- t)

e-T-

O. σ Se-q(T-t)

_

d~

~ v ~~G /~ ., e-~ ， 2 、/2作 (T … t)

and substitute for C the value given by the Black-Scholes formula , i.e. , C -

Se-旷-t) N(d 1 ) - Ke-r(T-t) N(d 2 ).

Then ,

@+jd付 + (r …仙一 γc qSe-q(T-t) N(d 1 ) 十 σ2s2

σ Se-q(T-t)

r Ke-r(T-t) N(d 2 )

-

-21

2 飞/2作(T-t)e2

一手

e-q(T-t)

t)

2 Sσ 、/2作 (T -

十 (r 一仙一 q(T-t) N(d 1 ) 一 r(仇旷 川一吁呗 v q创阳叫 (TTι 归叫一→卅t功)N(μd出 1廿 队.口 0

Problem 11: The value at time t of a forward contract struck at K and maturing at time T , on an underlying asset with spot price S paying dividends continuously at the rate q , is j(S , t) = Se-q(T-t) -

Ke-r(T-t).

Show that j(S, t) satis五es the Black-Scholes PDE , i. e. , show that θj 1 _202 θ2j 百 + ~O'~S:~a;2 I

+

δf (γ - q)S页 - rj

= O.

Solution: It is easy to see that

红 = aSe-q(T-t) - r K e-r(T-t) θt

Then ,

":1--

.---

AU-nAU rff-qa

T

口可 A

十

oa

e-q(T-t);

θS

θ2 j θ S2

O 一 θ cυ

2u-cυ

σ

1-2 nL

cυ

十

w-a

坐

，

_ rKe-r(T-t)

+

0

一 γ (Se-q(T-t) - Ke- r 叫)

d~

r K e-r(T-t) N( d2 )

-

6.1. SOLUTIONS TO CHAPTER 6 EXERCISES

fJ-S T rlu

A

-

口

+

(γ -

149 q)Se-q(T-t)

150

6.2

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

Supplemental Exercises

1. Let 1(x) = x 3 ♂ -6♂. Show that the central differe肌e approximation for l' (x) around the point 0 is a fourth order approximatio泣

2. Let

6.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

4. Consider a six months 5% in-the-money plain vanilla European call option with strike 30 on an underlying asset with volatility 20% , paying dividends continuously at a 2% rate. Assume 出at the interest rates are constant at 5%. (i) Use central differences to compute the finite difference approximations ~c and r c for ~ and r , respectively, i.e. ,

-

2C(K , T) + C(K - x , T) 1(S) = x2 where , e.g. , C(K , T) denotes the value of a plain vanilla call option with strike K and maturity T on an underlying asset with spot price S following a lognormal distribution. Show that , for any continuous function g : JR. •JR., C(K 十队 T)

~c

rc

户。c

Ii玛 I

Z\U I 可出~ J-OC

(J

2S2

r

2

8

0

connecting the r and the θof plain vanilla European options is exact if the underlying asset pays no dividends and if the risk-free interest rates are zero. In other words , for , e.g. , call options , show that , if r = q = 0, then

1 +σ2S2 r(C)

6.3

十 十 二 一

8(C)

2 (ii) If q = 0 but

r 并 0，

σ2S2 r(C) 1 十一一一·一一一

8(C)

1 1+

'T

N'(d2)

~ ~一-~μ 2r /T习 N(d2 )

(iii) Consider a six months plain vanilla European call option on an underlying asset with spot price 50 and volatility 30%. Assume that the interest rates are constant at 4%. If the asset pays no dividends , compute 1

+

Solutions to Supplemental Exercises

Problem 1: Let 1(x) = x 3 ex - 6e X • Show that the central difference approximation for 1'(x) around the point 0 is a fourth order approximation.

show that

2

- C(S - dS) 2dS ' C(S + dS) - 2C(S) + C(S - dS) (dS)2

(ii) Compute the Delta and Gamma of the call using the Black-Scholes ~c - ~ 1and Ire - r I. Note formula , and the approximation errors 1 that these approximation errors stop improving , or even worsen , as dS becomes too small. How do you explain this?

3. (i) Show that the approximate formula 十二一-.一目

C(S 十 dS)

for dS = 10-2 with i - I : 12 , where , e.g. , C(S + dS) = C(S 十 dS， K， T， σ， r) denotes the Black-Scholes value of the call option corresponding to a spot price S 十 dS of the underlying asset.

1(S)g(S) dS = g(K).

1

151

σ2S2 r(C) 一一一- .一一一

2

8(C)

if the options are at-the-money, 10% , 20% , 30% , and 50% in-themoney, and 10% , 20% , 30% , and 50% out-of-the-money, respectively. What happens if the asset pays dividends continuously at a 3% rate?

Solution: Recall that , in general , the central difference approximation of the first derivative is a second order approximation , i.e. ,

- f(-h) 1' (0) = f(h) J \'''/ 2~ \ '''/ + 0

2

(h 2 )

,

as h

•

O.

(6.28)

To see why, for the function f(x) = x 3 ex - 6e气 the central difference for l' (x) around the point 0 is a fourth order approximation , we investigate how the approxi日lation (6.28) is derived. The Taylor approximation of f (x) around the point 0 for n = 5 is appro均nation

f(x)

=

f(O)+x1'(O)+ 旦f"(O) 十 21f(川0) +马队)(0) +主严)(0) 2 6 24 120 6 + 0 (x ) , as x • O. (6.29) J

\ - /

'

J

\ - I

'

J

\ - I

'

152

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

We let x = h and x = -h in (6.29) and sum up the two resulting formulas. After solvi鸣 for 1' (0) we obtain

1' (0) =f(h) - f( -h) 一句句(0) - 1~:f(5)(0) + 2h

6"

\ /

120

0

(的，

(6.30)

as h • O. For f (x) = x 3eX - 6e x , we find that f(3) (x) = (x 3 + 9x 2 十 18x )e X, and thus that f(3)(0) = O. Also , f(5)(0) = 54 乒 o and (6.30) becomes

f(h) - f( -h)9h4 f(h) - f( -h) f f ( 0 ) = 2 h - 3 5 + O (问 :2h+O(的? as h • O. In other words , the central difference approximation for around the point 0 is a fourth order approximation. 口

f' (x)

where we used the s巾的itutions z = 8 一 (K -x) and ω =K+x­ two integrals above , respecti飞rely. Let z = xy a丑dω= xt. Then dz = x dy , dω = x dt , and we find that

1:

f(S)g(S) dS =

l'扑 1' yg

We let x \、、 o 阳 i n (仰6.3臼 勾) . Since the function 9 : ~ →~ is continuous , we 2 obtain that

炖时 fι: 贝阴川 S冈)归川 g 8

=

以9 州(仪川 K

which is what we wa时ed to show; ef. (6.31). For the sake of completeness , we provide rigorous proof of the fact that (6.32) becomes (6.33) when x \￥ O. To do so , it is eno吨h to show that

炖叫才 1' νω 川 ygg

Problem 2: Let

f(8)

6.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

C(K+ 民 T)

- 2C(K , T) x2

+ C(K - x ,T)

where , e.g. , C(K， 1寸) denotes the value of a plain vanilla call option with strike K and maturity T on an underlying asset with spot price 8 following a lognormal distribution. Show that , for a口y continuous function 9 : ~ →~，

~!!J 1: f(S)川8

= g(K)

(6.31)

Let ε> 0 arbitrary. Since 9 is continuous , it follows that there exists 6 > 0 such that Ig(K) - g(7)1 < εfor all 7 such that IK - 71 < 6. Let x ε(0 ， 6) and y ε(0 ， 1). Then jK 一 (K -x 十 xy) I = x(1 - y) < 6 and therefore

Ig(K) - g(K - x 十 xy)1

< 飞

主 (ω(8 一 (K 一川 -2 日lax(8 一 K， O) +max(8 一协叫，0)) 。 … 三 K-x;

旦在纯， ifK-x<8 三 K; qiifk<S三 K+x;

l1' yg νωg川 +忖叫叫♂均 ωUω忡 )d <Jlydu=i

汇 f(8)g(8)d8

In other words , for any

ε>

0 there exists 6 > 0 such that

l1' yg川 + xy) dy

1 1

- g(K)

y dY ! <

- g(K)

泸州)d8 + 儿泸州)d8

主 1" zg(K-x 十忡十刊 ωg(K 十 Z 一 ω)dω7

V0 < x < 6

l' y 叫= 0

r K S 一 (K - x ) r K +x K 十 x-8

儿-X

~，

Then , by definition ,

划 11' yg(K 一叶 xy) dy

Then ,

x < 6, 0 < y < 1.

Therefore , it is easy to see that , for any 0 < x < 6,

Solution: From the definition of f(8) , it is easy to see that f(8) =

\:j 0 <

Problem 3: (i) Show that the approximate formula σ28 2

r

2

e

l 十一一一·一用。

口

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

154

connecting the f and the 8 of plain vanilla European options is exact if the underlying asset pays no dividends ar: d if the risk-free interest rates are zero. In oth~r ';ords , for , e.g. , call options!, σ28

2

f(C)

6.3. SOLUTIONS

f(C)

σ 8V27rT σ8

8(C)

8(C)

2

155

(ii) For q = 0 , we obtain from (6.34) and (6.35) that

AU

1 十一-一·一一­

TO SUPPLEMENTAL EXERCISES

二百万e

_~ 2

- rKe- rT N(d 2 ).

Then , (ii) If q = 0 and r 笋 0 ， show that 2

1 十 σ28 f(C) 一---.一 2 8(C)

σ282

f(C)

2

8(C)

1+ 一一一·一一一­

1 l+~urn:;旦旦~.

σSfii

1 一

2V2irT V

σS ~一手

2V2irT

2rVT N(d 2 )

uS

p.一手

2

8(C)

1

1 + ~ rVT U

if the options are at-the-money, 10% , 20% , 30% , and 50% in-the-money, and 10% , 20% , 30% , and 50% out-of-the-money, respectively. What happens if the asset pays dividends continuously at a 3% rate?

Solution: Recall that the f and the 8 of a plain vanilla European option are

c

e-qT-fi aSV2汗 σ 8e-qT

8(C)

2V27r T

where d1 = (In

(6.34)

7

e

_5. 2

r K e -rT N (d 2 ) ,

十 qSe一叫V(d 1 ) -

(圣)十(叫到T) / (σvr)a叫 =

d1 一 σ ♂

(i) For r = q = 0 , we obtain from (6.34) and (6.35) that

f(C)

σS 刁万 e2;

σS-fi

8(C)

2V27r T

Then ,

1+σ2S2r(C)-1+σ2~2 (_~飞, = O. 2 十一.---一

2θ (C)

十二

2\σ28 )

lNote that~ if r = q = O~ then f( P) = f( C) and

l.- e一手

S . Ke- rT N(d 2 ) ' J2王

1

1 十一一一·一一一

pi

+γKe-rT N(d2 )

2 飞!27rT-

1m

f(C)

+γ Ke- rT N(d 2 )

-

r K e- rT N(d 2 )

(iii) Consider a si议 xmo ∞ 川thsp n 抖la 创i川a ωani 缸 丑川 l让i让llaE盯opea 缸I川 allopμtio ∞ ω1 I丑 on an unde 臼rlyi 创 沁 I口l asset with 印 s po 创t price 50 and volatility 30%. Assume that the interest rates are constant at 4%. If the asset pays no dividends , compute σ28 2

V

-

8(P) = 8(C).

1+ 动T

K:JX:Zii2)

(6.36)

since N'(t) = 在λfor all tεR Recall that the "magic" of Greek computations is due to the following result: 8 N'(d 1 ) = Ke- rT N'(d 2 ); cf. Lemma 3.15 of [2] for q = O. Then , (6.36) becomes σ282

f(C)

1

2

8(C)

l+~ulm 旦旦~.

1 十一一一·一一­

(6.35)

rVT N(d 2 )

(iii) Let 8 = 50 , T = 0.5 ， σ= 0.3 , and r = 0.04. The table below records the values of σ28 2

f(C)

l 十一-一·一一一

2 8(C) (denoted by "Val时') both for q = 0 , and for q = 0.03 , for the following values of the moneyness of the option:

S K

{1 , 1. 1, 1. 2 , 1. 3 , 1. 5, 0.9 , 0.8 , 0.7 , 0.5} ,

corresponding to call options that are at-the-money, 10% , 20% , 30% , and 50% in-the-money, and 10% , 20% , 30% , and 50% out-of-the-money, respectively:

156

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

-SIK Value for q = 0 Value for q = 0.03 1 1. 1 1. 2 1. 3 1. 5 0.9 0.8 0.7 0.5

0.0238 0.0233 0.0144 -0.0187 -0.5503 0.0214 0.0184 0.0155 0.0107

0.1897 0.2582 0.3518 0 .4 711 0.7361 0.1412 0.1068 0.0822 0.0505

6.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

Solution: The spot price S = 3 1. 5 corresponds to a 5% ITM call with K = 30. We find that ~ = 0.692130579727 and r = 0.077379043990. The central 且nite difference approximations ~e and the approximation errors I~e - ~ I are recorded in the table below: dS 0.1 0.01 0.001 0.0001 10- b 10- 0 10-(

We note that the approximation

10-δ

σ2S2

r

2

e

1+ 一一一·一但

10- \:1 O

is better for deep out-of-the-money options (corresponding to small values of S I K) and is worse for deep in-the-money options (corresponding to large values of S I K). Also , for this particular case , the approximation is more accurate if the underlying asset pays dividends. 口 Problem 4: Consider a six months 5% in-the-money plain vanilla European call option with strike 30 on an underlying asset with spot price 20 and volatility 20% , paying dividends continuously at a 2% rate. Assume that the interest rates are constant at 5%. (i) Use central differences to corr职lte the finite difference approximations ~e and f e for ~ and f , respectively, i.e. , ~e

f

e

C(S + dS) … C(S - dS) 2dS ' C(S + dS) - 2C(S) 十 C(S - dS)

(dS)2

for dS = 10-2 with i = 1 : 12 , where , e.g. , C(S 十 dS) = C(S + dS , K , T， σ， r) denotes the Black-Scholes value of the call option corresponding to a spot price S + dS of the underlying asset. (ii) Compute the Delta and Gamma of the call using the Black-Scholes formula , and the approximation errors I~e - ~I and Ife q. Note that these approximation errors stop improving , or even worsen , as dS becomes too small.丑ow do you explain this?

157

10- 1υ

10- 11 10- U

~e

I~e-~I

0.692112731743 0.692131730564 0.692131920566 0.692131922513 0.692131922087 0.692131918000 0.692131916226 0.692131862934 0.692132573477 0.692104151767 0.691890988946 0.687450096848

0.000017847983 0.000001150838 0.000001340839 0.000001342786 0.000001342360 0.000001338274 0.000001336498 0.000001283207 0.000001993750 0.000026427960 0.000239590780 0.004680482879

The approximations became more precise when dS decreased , until dS = 10- 8 ; the best approximation was within about 10- 6 of~. However , for values of dS smaller than 10- 9 , the finite difference approximations deteriorated very quickly. To explain this phenomenon , denote the exact value 2 of Delta by ~exaet. Note that the value of ~ is given by the Black-Scholes formula , i. e. , ~ =

~BS = e- qT N(d 1 ).

This value is computed using a numerical approximation of N(d 1 ) that is accurate within 7.5 . 10- 7; d. [1 ], page 932. In other words , we 0日ly know that (6.37) I~BS - ~exaetl < 10- 6 . When computing the 且nite difference approximation ~e ， we use a numerical estimation of the Black-Scholes formula to compute C(S + dS) and C (S - dS) which once again involves the numerical approximation of the cumulative density of the standard normal variable. In other words , ~e -= CBS(S 十 dS) - CBS(S - dS)

2N ate that

~exact

2dS is a theoretical value , and is 口at the

~

from the table above.

(6.38)

158

CHAPTER 6. FINITE DIFFERENCES. BLACK-SCHOLES PDE.

Deωte by Cexaet(B 十 dS) and C口 mαet(B 一 dB) the exact 刊 va 址 叫lues of the opμt挝io丑 Since the central 缸 fi川 nl让it臼ed 出i宦erence is a sec∞ or丑ld order approx 对im 丑1a 创tion 凡 1， 让 it follows

that , for exαct values of Delta and of the call options ,

~exω= Cexact(B + dB) - C口aet(B - dB) 十 o ((dB)2) , 2dB

(6.39)

•

s dB O. Since CBS(S)=SfqTN(dl)-ke--TTlV(d2)7and since lV(do and lV(d2) are comput~d 'm~merically within 10-6 of their exact value , it follows that

\CBs(B 十 dB) - Cexaet(S 十 dB)1 < α10-j; jCBs(B - dB) - Cexaet(B - dB)j < α10-° ，

(6 .40) (6 .4 1)

where αis a constant proportional to the values of Band K , Usi吨 (6.38) and(6.39) we find that ~e

-

(~e - ~exaet)

~BS

CBs(B +

+ (~exaet -

~BS) dB) - Cexaet(S 十 dB)

2dB CBs(B - dB) - C口创t(B - dB) 2dB 十 ~exaet - ~BS 十 o ((dB?) ,

(6 .4 2)

as dB • O. The only estimate we can find using (6.37) , (6.40) , (6 .41) , and (6 .42) for the approximation of ~BS by ~e as dB • o is \~e

-

~BS\ 三

\CBS(S 十 dB)

- Cex α et(S 十 dB)1 2dS 十 ICBS(B - dB) - Cexaet(B - dB)j 2dB +I~exαet - ~BSj + 0 ((dB)2) α10- 6

〈一一 + 10- 6

dB

+

0 ((dB)2) ,

(6 .4 3)

as dB • O. While the approximation error j~e - ~j may be better in practice , the bound (6.43) provides the intuition behind the fact that , for dS too small , t_h~ numerical approximation error I~e - ~I = I~e - ~Bsl deteriorates as 斗主­ becomes large. The central finite difference approximations errors Ire - r j are recorded in the table below:

re

and the approximation

6.3. SOLUTIONS TO SUPPLE1VIENTAL EXERCISES

159

dB re We - f/ 0.1 0.077370009586 0.000009034404 0.01 0.077371495486 0.000007548504 0.001 0.077371502982 0.000007541008 0.0001 0.077371709040 0.000007334951 10-b 0.077271522514 0.000107521476 10-0 0.074606987255 0.002772056735 10-( -0.355271367880 0 .4 32650411870 10-1:> 71. 054273576010 70.976894532020

For dB 三 10- 9 ， the values of r e increased dramatically, reaching 109 for dB = 10-12 , and were no longer recorded. The finite difference approximations of r became more precise while dB decreased to 10-4 , but were much worse after that; the best approximation was within 10-5 of r. The reason for this is similar to the one explained above for the finite difference approximations of ~.口

ap 4Ler

阿t

冒 n

c

Multivariable calculus: chain rule , integration by sUbstitution , extremum points. Barrier options. Optimality of early exercise. 7.1

Solutions to Chapter 7 Exercises

Problem 1: For q = 0 , the formula for the Gamma of a plain vanilla European call option reduces to

r

=s中叫一叮?〉

(7.1)

where

d1 (S)

=

In (丢)十 (γ 十号 )T 识 飞IT

(7.2)

I

Show that , as a function of S > 0 , the Gamma of the call option is first increasing until it reaches a maximum point and then decreases. Also , show that (7.3) 31lr(S)=O and liIII Y( 句=

°

S→∞

Sol仰on:

From (7.1)

we 自时 that

r can be written as

/ (d 1 (S)? r(s) = 一古二 exp (一 飞

2

\

ln(S) ---\'- / J) ,

(7.4)

where d1 (S) is given by (7.2). Since r(S) > 0, it follows that the functions r(S) and ln(r(S)) have the same monotonicity intervals. Let 1 : (0 ，∞)→1R given by 1(S) = ln(r(S))

=一叩2 -ln(S) -1巾而)

CHAPTER 7. MULTIVARIABLE CALCULUS.

162

7.1. SOLUTIONS TO CHAPTER 7 EXERCISES

163

from (7 .4), we conclude that

Then ,

1' (S)

d1 (S)

1

Sσn

S

1 ( (d 1 (S))2 \ li~ 一万亏 exp (一 门 -ln(S))

li~ r(S) -

3 、v

-~ (1+ 扭)

a 、uσV

L:7r 1

\

中/

= O.

口

(7.5) Problem 2: Let D be the domain bounded by the x-axis , the y-axis , and the line x 十 ν= 1. Compute

Recall from (7.2) that

(7.7)

fLEjdzdu

d1 (S)

Solution: Note that

D = {(x , y)

It is easy to see that d1 (S) is an increasing function of S and that gsdl(S)= 一∞;

lim 州)=∞

(7.6)

S→∞

From (7.5) we find that I(S) has one critical poi风 denoted by S* , with d1 (S*) = 一 σn. From (7.5) and (7.6) it follows that 1'(S) > 0 if 0 < S < S* and 1' (S) < 0 if S* < S. In other words , the function I(S) = ln(r(S)) is increasi 吨 when 0 < S < S* and is decreasing when S* < S. We conclude that r(侈 S)川is 址 aIsωo 沁1 丑肌 C盯reaωasl 且i恒 I丑1 when 0 < S < S* and decreasing when S* < S. We now compute lir町、o r(S) and lims→∞ r(s). Note that lims→∞ d 1 (S) = ∞. Therefore , Fm r(S) =

S→。G

"

Fro皿 (7.2) ，

(

(d1 (S))2

s

n-WM/

\

I

=一∞，

\

2σ2T(1日 (S))2

2σ 2T

/一 (d 1 (S)?

( 门 一 ln(S)

\

I

乙/

It is easy to see that (x , y)

n

ε D

= {(8 , t)

if and only if (8 , t)

I0 三 S 三

εn ，

where

1 ， -8 三 t 三 8 }.

The Jacobian of the change of variable (x , y) εD → (8 ， t) E dxdy

=

lθzθuδzθyl I 一一一一 ~V I lθsθtθtθ81

j 儿可叫

= O.

d8dt -

n is

~d8dt ，

2I

'

ij1:(l:tdt)ds=0

1 ) 1口 (S) J

f 儿 :jM=f(/:二战)

fL7fjdzdu

l' U'-Y~号刊 dy

L((x - 2yln(川)) I~二tu 句)

l'

Since Ii町、。 (exp (一 (ln(S)?)) = 0, we obtain that

S\O\

We use the change of variables 8 = X 十 y and t = x - y , which is equivalent to S 十 t 8 - t x - -2-; y - -2-

it follows that

1: … {(ln(S) - 1日(K) +(γ+ ~2) T) s~o

1}

and therefore -ln(S) \ I)

and using the fact that lims \o ln(S)

2-n 一 -cυ

1ιs

mwa

…-

s - 2 7飞

Ju

1

lim 一?古 exp (一

S→∞ σV 'l.作 l\2

I x 主 O， y 三 O， X 十 y 三

= 0

1- y

+ 勾 ln(y)

ds

CHAPTER 7. MULTIVARIABLE CALCULUS.

164 si丑ce limy\o y2 In (y) = O.

7. 1. SOLUTIONS TO CHAPTER 7 EXERCISES

,

口

,

,

(i) For a call option V(S T) = max(S - K O). From

,

u(x O) = Problem 3: Use the change of variables to polar coordinates to show that the area of a circle of radius R is 1f R2 , i.e. , prove that

,

= exp(αx) max(S - K O)

max(Ke X- K , O) -

,

(7.8) , we find that

,

Kexp(αx) max(e X 一 1 ， 0).

,

(7.8) , we 且时 that

(ii) For a put option V(S T) = max(K - S O). From u(x ， O)

fL)ldzdu=d2

exp(αx)V(S， T) exp(αx)

-

165

工 exp(αx)V(S， T)

-

exp(αx)max(K

,

= exp(αx) max(K - S O) Kexp(αx) max(1 - eX , 0).

- KeX , O) -

口

Solution: We use the polar coordinates change of variables

,

,

(x y) = (r cos e r sin e)

with

(r

,e) εit = [0,R] x [0 ， 2作) .

Problem 5: Solve for

(

Recall that dxdy = rdedr. Then ,

川旷 dxdy 工 flo川价:叫R 巾=作R27 which is equal to the area of a circle of radius

,

R.

Solution: From

1

一~， 2'

Usi吨 (7.9) ，

1~

b=

(γ -q

I 一 n 飞 σ2

\2

十一 i '2 J

十一τ 俨

This is the change ofvariables that reduces the Black-Scholes PDE for V(S , t) to the heat equation for u(x ,T). (i) Show that the boundary condition V(S , T) = max(S - K , O) for the European call option becomes the following boundary condition for u(x , T) at time T = 0: u(x , O) = K exp(αx) max(eX - 1, 0). (ii) Show that the boundary condition V(S , T) = max(K - S , O) for the European put option becomes

u(x , O) = K

exp(αx)

川一半- 0; b+ ια(1-~) ~去 z 。

the 且rst

equation , it is easy to see that

(7.9)

, ,

,

b the following system of equations:

口

Problem 4: Let V(S t) = exp( 一 αx - bT)U(X T) where L (S \ (T - t) σ2γ -q x = In I -:;:-; I T =' ~ ， α= … n -飞 K J" 2σ2

αand

max(l - eX , 0).

b

=

we note that the second equation can be

written 创

~a2 - a (1 一勺牛二 2

一(子 -D 一(子 -D'2G 一子)+Z (子 ~r 十 2(子 -D 十二 (子 -32+ 主 (子 +;)2-2(γ) 十三 (子寸 )24 口 2

Solution: Note that t = T if and only if T = O. Then ,

V(S , T) =

exp( 一 αx)u(x ， O).

Here , x = In (丢)， which can also be written as S = K eX

(7.8)

Problem 6: Assume that the function V(S , I , t)

satis 自 es

the following PDE:

θV θVθV 1 2S2 θ 2V rS 一 - rV = O. σ 一十 S 一十一 一一 ' 2θ S2 + θS θtδI

(7.10)

CHAPTER 7. MULTIVARIABLE CALCULUS.

166

7.1. SOLUTIONS TO CHAPTER 7 EXERCISES By substituting

Consider the following change of variables:

V(川)=5H川 7where

=i

R

(7.11)

inω(7.10)，

o θ V θV

167

it follows that

1 <)_<)θ2V

θV

=一一十 S一一十一 σ，c， 5，c， 一一一十 r5一一 θ t θI ' 2θ52θS

rV

θHθH 1 '> _，> R θ 2H _ (θH\ S一-十 S一一十… σ于一一一十 γ 5IH-R 一~ 1 2

Show that H (R , t) satisfies the followi吨 PDE: θH

1 '>_'>θ2H 万J+2σ".
+

θ t θR

θH

θH

(1- rR) 否五口。

(7.12)

Note: An Asian call option pays the maximum between the spot price 5(T) of the underlying asset at maturity T and the average price of the underlying asset over the entire life of the option , i.e. , /III-\

T

--T

T PI''''nu

Cυ

cυ

max

\liI/

T

T

Thus , the value V (5, I , t) of an Asian option depends not only on the spot price 5 of the underlying asset and on the time t , but also on the following random variable: I(t) =

I

5(7) d7.

JO

It can be shown that V(5 , I , t) satis丑es the PDE (7.10). Similarity sol沁 叫 u 1 of the type (7.11) are good candidates for solving the PDE (7.10). The PDE (7.12) satisfied by H(R ,T) can be solved numerically, e.g. , by using finite differences. Sol仰 on: Let V(5 , I , t)

W一仿W-MW-mω

that

θ2V δ52

= 5 H(R , t) , with R

=~' Using Chain Rule , we 白缸 fin 丑l

S 豆豆 . θt '

S 纽纽一 θH 一

θRθIθR' θHθRδH ( I \IθH H+5 一一…一 =H+5 一一!一一 l=H 一一一一 θRθSθR 飞

H-R

52

J

~~

δH 一一: δR J

2H θR _ I θ2H θHθRθRθHθ 一一一一一一一一一 - R 一一一一… - R 一一一一一 θRθSθSθRθR2

R 2 θ 2H

S

θR2'

δ5

-

.L"

1 叮叮 θ2H

52 θ R2

5

θR2

飞

θ RJ

- r5H

θH

S一十 ::'-(7".
By dividi吨 by 5 , we conclude that H(R , t) satis且es the PDE θH

1 叮叮 θ2H

θH

37+rzRZ否E 十 (1 - r R) 百五 which is the same as

JU

' 2-- 5

(7.12).

0,

口

Problem 7: The price of a non-dividend-paying asset is lognormally distributed. Assume that the spot price is 40 , the volatility is 30% , and the interest rates are constant at 5%. Find the Black-Scholes values of the IT JvI put options 0丑 the asset with strikes 45 , 48 and 51 , and maturities 3 months and 6 months. For which of these options is the intrinsic value max(K - 5 , 0) larger than the Black-Scholes value of the option (in which case the ∞ c orrespo丑时 剑i丑 d Arne 臼flca 创 缸I卫1 put is guaranteed to be worth more than the European p 叫 ? Solution: The values of the out options are summarized in the table below: Option Type Strike Jvlaturity Value K-5 6 months 5.8196 45 5 Put 5 3 months 5.3403 Put 45 8.0325 8 6 months 48 Put 3 months 7.8234 48 8 Put 6 months 10 .4 862 11 51 Put 3 months 10.5476 11 51 Put

θR

In general , the values of deep-in-the money European put options are lower than the premium K - 5. This feature was observed for the options priced here: the values of the 51-puts and of the three months 48-put are below their intrinsic value K - 5. Also , note th创 for the 51-puts (i.e. , for deep in the money puts) , the values of the short dated options are higher than the values of the long dated

CHAPTER 7. l\1ULTIVARIABLE CALCULUS.

168

options. This is to be expected; for example , if the spot price is 0 , the value of a European put is K e- r1\ in which case longer dated puts are worth less than short dated ones. 口 Problem 8: Show that the premium1 of the Black-Scholes value of a European call option over its intrinsic value max( 8 - K , 0) is largest at the money. In other words , show that the maximum value of

7.1. SOLUTIONS TO CHAPTER 7 EXERCISES

plain vanilla call with strike K when B \、 O. For simplicity, assume that the underlying asset does not pay dividends and that interest rates are zero. Sol毗 on: Let t = 0 in formula (7.13). For r = q = 0 , we find that α=-i Therefore , the value of the down-and-out call is

V(8)

8 _. I β2\

B

81B

B

if 8 三 K; f(S)=i cm(S)-S+K7if S >K OBs(8) ,

d1 =

Note that j (8) is a continωus function , but it is not differentiable at 8 = K. For 8 < K , the function j(8) is the value of a call with strike K , 缸 a丑 n1 tr扭lerefore is ir丑lcreasir卫19. For 8 > K , we find that 1 < N(d 1 )

…

0(8) -

1 =-N(-d1 )<0 ,

and therefore the function j(8) is decreasing. We conclude that j(8) has an absolute maximum point at 8 = K.

口

γ 一/一

= 0(8 , K , t) 一(计

o

IB2

\

(7 叫

γ 一/一-

…

σ 飞IT

Note that 0 < N(d 1 ) < 1. Then , 且也 BN(d 1 ) = O.

(7.15)

Using I'H6pital's rule , we obtain that 8K

E~ ~;.，~ B\() B

N(d 2 ) θ出 8K 问丁-zSKB可LNf(d2) 一

N(d 2 )

θB

1

li囚一二eXD

B\、0 飞12 1f

I

I-

品\

d~ \2

I

-~.4

~J -

.一一τ B σ 飞IT

\ I口 (B) ) " }

(7.16)

As B \、 0 ， the term 一手- In(B) is on the 。阳. of 一(ln( B))2 , and 阳efore I

-3 2

』史坷吃~~←一 7?一-In 峭 n

\

(7.17)

From (σ7.16 叫 创) a叫 (σ7.17η) we 直nd that 6 8K

口

Problem 10: Show that the value of a down-and-out call with barrier B less than the strike K of the call , i.e. , B < K , converges to the value of a lThis premium is also called the time value of the option.

(7.14)

I川剧 _ ~2T =

28K.. I d~ τF王三 Ulr~ exp ( - ~.4

\ ~ , K ,t ) ,

$3.398883.

d?

-and

σ 〉乙 1f T B\J\~

whereα= 手- ~' to find the value of a six months down-and-out call on a non-dividend-paying asset with price following a lognormal distribution with 30% volatility and spot price 40. The barrier is B = 35 and the strike for the call is K = 40. The risk-free interest rate is constant at 5%.

Solution: The value of the down-and-out call is

I

8K ~~~ N(d 2 ) , B

+

BN(d 1 )

σ 飞IT

Problem 9: Use the formula V(8 , K , t)

\

8

l叫到十卓T

8K

IB \2α

飞

where 0(8) is the value of the plai口 vanilla call wi出 strike K and

Solution: Let j(8) = OBs(8) - max(8 - K , O). It is easy to see that

-

飞 8 2

0(8) 一一( ~n -N(d ) ,--.J.1 I) - -KN(d - _ . ' - "2 'I J

is obtained for 8 = K , where OBs(8) is the Black-Scholes value of the plain vanilla European call option with strike K and spot price 8.

1' (8) = ~(CBS) - 1 = e- qT N(d1 )

I: J)

0(8) 一;， 0

=

OBs(8) - max(8 - K , O)

J

169

问~~~ N(d 2 )

= 0,

(7.18)

and , fro皿 (7.14) ， (7.15) , and (7.18) , we conclude that (

_. _.

U~ V(8) = h~ (0(8) \O\.0 飞

_

_ _, _ ,

8K

\

BN(d 1 ) 十一:;N (d 2 )) B _. '--"'I J

= 0 (8) .口

CHAPTER 7. MULTIVARIABLE CALCULUS.

170

Problem 11: Compute the Delta and Gamma of a down-and-out call with

7.2. SUPPLEMENTAL EXERCISES

7.2

171

Supplemental Exercises

B
V(8) =

B

2α

CB s(8) 一 ~')n 2α 8

(B \ 2

CBS (

~n

飞 8

(7.19)

).

}

1. Compute

D

(7.20)

e doma n D ·唱 BE-

\一/

QU A?U

larger ，俨 or 7r e ?

2. Which number is

l丑(是) + (γ-q 十号 )T

u-zm any

、i

=

TLU

咽''BA

d1 (x)

×

,

一一

where ~(CBS)(X) = e- qT N(d 1 (x)) , with

c

BU

τG

j

QU

吨，

飞

-

a -naf- 5] z UU n ''SA

υj

hI MDTK euu C41 DE-si an n-g pe

臼 2

(B2 \

Lυ1i

•

V 忻州J

,_.

P 叹口

飞 8 )

+1

.

{(川

= 唱 hund

8

B2α+2

十一丁 ~(CBS) (一)

牛E U

(B2 \

E凶

2αB2α

~(CBs)(8) + 一一-CBsl-i 2α

xdX

where

By differentiati吨 (7.19) with respect to 8 , we obtain that ~(V) =

J!n

3. Let 叩) : [0 ，∞)→ [0 ，∞) be two continuous functions with positive values. Assume that there exists a constant M > 0 such that

u(x) 三 M+fa 叫 t)v( 忖 \f x~O 2α(2α 十 1)B 2 α

r(V) = r(CBs)(8) 一例。 8'2α十万 (4α +

Show that

u(x) 三 M 叫俨(t) dt) ,

2)B 2a + 2 ( B 2\ B h + 4 / \ ~(CBS) r(CBs) (τ) , \ ~ )一百 a+4

where

/,

11\

,TLU

fu日 ction /III-\ PI'120 z

u

ex D&

\It's//

,?b

,d

ι，ν

A'LU

If x ?: 0

、飞Bt/

口

U

7α

J

、、.-、BES''''

V

/

Z υ

，e，

/III-\

、、 l'/

x σ 飞/三7r 1\

Hint: Investigate the monotonicity of the

v'

ι'U

J \ _ d11 (x)2 ~exp ( ': I

(

十

(7.20).

e- qT

flf 。

r(cBS)(x) = with d1 (x) given by

(B 2 \ CBS ( 一) 飞 8 J

,

Note: This is a version of Gronwall's inequality, and it is needed , e.g. , to prove the uniqueness of the solution of an initial value problem for ordinary differential equations. 4. What does the boundary condition V(B , t) = R for a down-and-out call with barrier B and rebate R > 0 correspond to for the function 也 (x ， T) defined as follows: V(8 , t) 二 exp( 一 αx - bT)U(X , T) , where

(8\

x = In I =-_ I. 飞 K J I

T

二

(T - t) σ2γ -q

1

1-._

(γ - q

,

1\2

,

2q

α= 一一「一丁、 b=l 一"十丁 1+ 一τ· σ4 ~'\σ :L 2σ :L ~J 龟

CHAPTER 7. IvIULTIVARIABLE CALCULUS.

172

7.3

5. Assume that the function V(S , I , t) satisfies the PDE θ V θVI') ~')θ2V δV 一+ InS一十一σ乎一一 + rS一 - rV δtθ I

2θS2

θS

7.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

= O.

(7.21)

Solutions to Supplemental Exercises

Problem 1: Compute

Consider the following change of variables: V(S , I , t)

Show that F(y , t)

I

= F(y , t) , where y = satis且es

the

+ (T - t) InS ,

2T2

8 y2

D

followi吨 PDE:

主主十 σ2(T - t)2 ~2: + (r 一到丘 t 8F δt

where

,- T

\2)

f儿z 伽

= {何?们}R2 I x ~ 0 , 1 三时 2 ， 1 才三 2}

Sol仰on: The change of variables 8 = xy and t

rF

=

0

7. For the same maturity, options with different strikes are traded simultaneously. The goal of this problem is to compute the rate of change of the implied volatility as a function of the strike of the options. In other words , assume that S , T , q and r are given , and let C (K) be the (known) value of a call option with maturity T and strike K. Assume that options with all strikes K exist. Define the implied volatility σimp(K) as the unique solution to

~ is equivalent to

when x 主 o and y ~三 O. This change of variables maps the domain D into the recta吨Ie n = [1 , 2] x [1 , 2]. It is easy to see that

6. One way to see that American calls on non-dividend-paying assets are

never optimal to exercise is to note that the Black-Scholes value of the European call is always greater than the intrinsic premium S - K , for S>K. Show that this argument does not work for dividend-paying assets. In other words , prove that the Black-Scholes value of the European call is smaller than S - K for S large enough , if the underlying asset pays dividends continuously at the rate q > 0 (and regardless of how small q is).

=

x-~andY=而?

Tθu

Note: The values of Asian options with continuously sampled geometric average satisfy the PDE (7.21).

173

dxdv _ IJ

-

|θzβ钊

θzβ111

I:~ ~~i1 一一一立 I Iθsθtθtθ81

d8dt

|土豆(-均丘[出dt 2VSt 20 \ 2t0) 2ft

i2t dsdt Then ,

J!n

XdX

[[~去 ω H[ VS dS ) ([忐 dt)

i(;二1:)(剖)

C(K) = CBs(K， σ问 (K)) ，

where CBs(K， σimp(K)) = CBs(S , K , T， σimp(K) ， r， q) represents the Black-Scholes value of a call option with strike K on an underlying asset following a lognormal model with volatility 向mp(K). Find an implicit differential equation satis自ed byσ问 (K) ， i.e. , fi日d δσimp(K) θK

aωsaf 旬 un 肌ct杜10 ∞丑 of σ 町irrη1以 K).

Problem 2: Which number is larger ，♂ or 7r e ? Solution: We show that 7r e <♂. By taking the natural logarithm , it is easy to see that l叫7r) / 1 _ ln(e) 作e< ♂仁中 eln(作) <作仁学一一<一一 -7· 7r e e

(722)

174

CHAPTER 7. MULTIVARIABLE CALCULUS.

Let f(x) = ln~x) with f : (0 , 00) • JR. Then f'(x) = 弓凶 The function f (x) has one crit iC al point correspondi吨 tox = 飞 IS 1肌reasi吨 on the interval (0 , e) and is decreasing on the interval (e ， ∞). We conclude that f (x) has a global maximum point at x = e, i.e. , f (x) < f (e) = ~ for all x > with x 护民 and therefore

In other words， ω( x) is a decreasi吨 function on the 凶 i nt怡 er凹 va 址I [归 0 ，∞) an 丑l therefore ω(0) 三四 (x) for all x 主 0. Since ω(0) = M , and using (7.23) , it follows that

M:: : (M + 1" u(t)v(叫叫- 1" v(t) dt)

°

ln(作)

忡忡 -r<;7 which is

eq山Talent to 作 e

< e1r ; cf.

(7.22).

:::: u(x)exp(-

Show that

1 x

u(t)v

(7.23)

\I

(7.24)

x :::: 0

L

( 8 \ ( T - t) σ2γ -q ~_ I 、

K

J1

r=

、

ι'U

，α

U

Z flo

也 (x ，

(7.25)

,TL

-

u

Z

where the derivative is computed with respect to x. Using the Product 丑ule， we find that

州=叫x)υ(以P ( -

I

1\2

. 2

J

2q

σ4

v(t) dt)

the boundary condition corresponding to V (B , t) = R for all °三 t 三 Tis

7)

叫咱) +b 7 ) 川) (;)α 川，

V

°三 γ 三字口

θ V θV 1 ()_()δ2V θV 一一十 In8~ 十一σ'l， 8'l， 一一十 r8 一一 θtθ I ' 2θ8 2θS

F(y , t) , where

satis且es

y

2

I

2T2

,Tb

a-cu-

1Ei-

…

the following PDE:

但十 σ2(T - t)2a F 十 (r _ ~2) 丘1 笠 θt

(7.27)

0.

、、 11'/­

Show that F(y , t)

=

/lt\-

二T

V(8 , I , t)

V(t)dt) (7.26)

rV T 一

Consider the following change of variables: ri 一

1" u(t)v(叫叫- 1"

exp(αz 十 br)V(8， t) ,

Problem 5: Assume that the function V(8 , I ,t) satis丑es the PDE

(M + 1" u(t)υ(叫(→(叫(- 1" V(t)dt))

巾) (u仲 M -

=

十一

+

1"

r)

、飞，，/

,?-b

(r-q

L

b=l 一"一十一 1+ 一τ.

2 1\σ ~

Solω 毗4ω 仰 i仰阳 on 7ηZ

\iI/ ，α

们U

/1 、

/ I I I -飞 \

Z U Z

、1 2 /

、ls/

?-U

们 U

JU,

J''t 飞

、 、l /

υ

、飞，，/

J F E E飞

,Tb

一一

ρ'''。 'U ''

4·L

1

α: 一"一一、

2σ~

u (In ( ; ) , \Ill-/ WU

x::::O ,

°

\111/

that /III-\ Z

\l

corresponds to °三 7 三字 Since 丛' U

ρIIt'nu

ex p

,d

，于心

U

、、 B，/

ιι

ι'u

u

、、3，/

+

ρI''。 ''

一­

/，tt、、

ω

Z

M

V(t)dt)

Problem 4: What does the boundary condition V(B , t) = R for a downand-out call with barrier B and rebate R > correspond to for the fu日ction u(x , r) defined as follows: V(8 , t) = exp( 一 αx - br)u(x , r) , where

x = In I飞

functionω: [0 ，∞)→ [0 ，∞)as \1111/ /III-\ Z z

/III-\

丑ecall

°

也(x) 三 M 叫1" v(t) dt) ,

Solution: Define the

1"

口

Problem 3: Let 叩) : [0 ，∞)→ [0 ，∞) be two continuous functions with positive values. Assume that there exists a constant M > such that

u(x) 三 M +

175

7.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

θν2\

2)

T

θu

rF =

°

CHAPTER 7. MULTIVARIABLE CALCULUS.

176

Solution: Using chain rule , it is easy to see that

7.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

which is what we wanted to

show.

177

己

θVθFInSθF δtθt

T

θν7

δV

1θF

θI

T

δV

1 T-t θF

θu?

δS S T

θ2V

θν?

1 ( (T - t)2 θ2F

一一_.

θS2

Problem 7: For the same maturity, options with different strikes are traded simultaneously. The goal of this problem is to compute the rate of change of the implied volatility ωa function of the strike of the options. In other words , assume that S , T , q and r are given , and let C(K) be the (know叫 value of a call option with maturity T and strike K. Assume that options with all strikes K exist. Defi 缸 fine 也 t he implied volatility σ 叽inη1以 K) 创 as the unique solution to C(K) = CBs(K， σimp(K)) ，

S2 飞

T-t θF\

一一一一.

T2

θy2

T

θy)

Then , the PDE (7.27) for V(S , I , t) becomes the followi鸣 PDE for F(y , t):

o=

θVθ V

一一

+

1 ')~')护 VθV

1口 S 一一 +一 σL. SL. 一一τ +γS一一 ' 2- .- 8S 2 θS

rV

θ t θI

θ FInS θF 一一一一一一一十 θt

T

θy

__1δF

lnS~~

-T

θσimp(K)

θu

十 1σ2 i( (T - t)Z 一一一一一一一一一…一一一 θ2F T-t θFV+T-tθF I + r 一一一一一一一一 2\

T2

δy2

T

θF σ2(T - t)2 θ2F , 一一+ n ~ ~十 θt'

2T2

where CBs(K， σ 叽仰nη以 1 阮cl挝 S hoωoles 臼s value of a call 叩 0 pt挝io ∞ n with strike K on an unde 臼rl勾i坊抖 y 甘切 切 rin 吨 1鸣 ga 部ss附 创t ，f'01 e 扣如 low 叭in 吨 ga log 伊 nω10ωr m 口 口 r丑na 1恼al 叫 model 呐 w it由 h volatility σ叫 (K). Find an implicit differential equ 任 tion satisfied byσ叫 (K) ， i.e. ，缸d

θy)

,

•

T

θu

-

~

(σ2\ T-t θF Ir 一 -=::-1 一一一一一 - rF.

8γ\

2)

T

θu

θK

邸saf缸 a un 旧ct钊io ∞丑 of σ 叽imη1以 K). 口

Solution: We first 自nd the partial derivative of the Black-Scholes value CBs(K) of a call option with respect to its strike K. Recall that CBs(S , K) = Se- qT N(d 1 )

Problem 6: One way to see that American calls on non-dividend-paying assets are never optimal to exercise is to note that the Black-Scholes value of the European call is always greater than the intrinsic premium S - K , for S>K. Show that this argument does not work for dividend-paying assets. In other words , prove that the Black-Scholes value of the European call is smaller than S - K for S large enough , if the underlying asset pays dividends continuously at the rate q > 0 (and regardless of how small q is). Solution: We want to show that , if the dividend rate of the underlying asset is q > 0, then CBS(S , K) < S - K for S large enough. Note that CBS(S , K) = Se- qT N( d1 )

-

K e- rT N( d2)

< Se- qT ,

since N(d 1 ) < 1 and N(d 2) > O. If Se- qT < S - K , which is equiv，由lt to S > 亡ιT > 0 since q > 0, it follows that CBs(氏 K) < S - K. We conclude that CBS(S , K) < S - K , V S

> 4K 川

Ke- rT N(d 2).

-

Then , θd 1

θCBS

一币

俨

EJd

Se一句'(d 1 ) ;~一严 N(d 2 ) - Ke- r :J.'N'(d 2 ) 坛

θK

Also , recall that

Se- qT N'(d 1 )

Ke- rT N'(d 2);

-

(7 到 (7.29)

d. Lemma 3.15 of [2]. From (7.28) and (7.29) , we find that 到C 'R.c:

#二 Since d 1

~'T' _ _,

• ,

~~

rr ~ TI I

(8d 1

8ι\

- e- rT N(d2 ) 十 Ke一叩'(也) \拔一括)

工 d 2 十 σ叮，

_",

"

(7.30)

it follows that θd 1

θd 2

θK

θK

We co肌lude from (7.30) tl川 θCBS θK

- e- rT N(d 2 )

(7.31 )

CHAPTER 7. MULTIVARIABLE CALCULUS.

178

We now differentiate with respect to K the formula

C(K)

CBs(K， σ问 (K))

=

θC θK

δCBS

IθCBS

一一一·

θK

θσθK

(7.32)

where θCBS

_-aT

Lagrange multipliers. N- dimensional Newton's method. Implied volatility. Bootstrapping.

I T_ '!J.

万 =Se-qVV=dse-PNf(do C1

V'T Ke- rT N'(d 2 )

-

K产品生

d. (η9). We conclude that the implied differential equation (7.32) can be written as

IT VK with

e

+iu r

θσimp(K)

-e一句(d2 ) + 鸣a(CBs) 叫ffk)?

vega(CBs)

ch ap

民U

which is the de直缸缸 旧 nl让i扰 山tior丑1 of σ 町仰mη1以 K). Note that C (K) is assumed to be known for all K , as it the market prices. Using Chain Rule and (7.31) we find that

一兰 θσimp(K) 2

θ-y- = 在 + e一叩 (d 2 ) ，

dry = In (去)十 (r - q)T 一切 (K)VT h .. (j imp(K)VT 2' ~

8.1

Solutions to Chapter 8 Exercises

Problem 1: Find the maximum and mi山num ofthe function f(Xl' X2 ， 句)= 4X2 - 2X3 subject to the constraints 2X1 - X2 - X3 = 0 and xI + x~ = 13.

Solution: We reformulate the problem as a constrained optimization problem. Let f : JR3 • JR and 9 : JR3 • JR be defined as follows:

削 =

4X2 -

2叫仰)

=

(专JZ二3)

where x = (X l, X2 , X3). We want to 五nd the maximum and minimum of f (x) on JRδsubject to the constraint g(叫 =0. We 自rst check that ra出(飞7 g(x)) = 1 for any x such that g(x) = O. Note that \1 g(x) =

(2;，二 en

It is easy to see that rank( \7 g( x)) = 2, unless Xl = X2 = 0, in which case g(x) 并 O.

The Lagrangian associated to this problem is F(x ， λ)

4X2 - 2♂3 十 λ1(2x1 - X2 - X3) 十七 (xr + x~ - 13) ,

whereλ=(λ1 ， λ2)t εJR2

(8.1)

is the Lagra口ge multiplier. We now find the critical points of F ( x ， λ). Let 均 x 0=(阳 均0 ， 1 ， X协 Zωa Z 缸 a丑 n1 沁0=(λ λ 沁0，1 ， λ 沁O叩 ω ，2公). From (8.1) it follows that \7 (x，)..)F(xo， λ0) = 0 is equivalent 179

180 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S JVIETHOD. eη

飞八 ZZ

缸， 2 队，川

-3

"ZZ

在-·龟，

0 ，

hm

12132

ZZ

mm 一一十=

λλJzll

十十

22J4

句B A - -

矶，

认入川，

;「

(ii) Find the asset allocation for a maximum expected return portfolio with standard deviation of the rate of return equal to 24%.

0; 0; 0; 13.

Solution: For i 二 1 : 4 , denote by Wi the weight of asset i in the portfolio. Recall that the expected value and the variance of the rate of return of a portfolio made of the four assets given above are , respectively, 、 、E E， ， ，

f'51 飞

。，血

。δ

才t牛

nuO&

一­

飞八

门4

一一

巧t

人

吼叫

一

一一

nu 4t4

and

XO ,l = -2; XO ,2 = 3; XO ,3 = -7;λ。， 1 = -2;λ0， 2 = -1. (8.3) For the 在rst solution (8.2) , we compute the Hessian D 2 月 (x) of 凡 (x) = f(x) + λ切 (x) ， i.e. , of 凡 (x) -

4x2-2x3-2(2x1-x2-x3)+xi+x~-13

xi+x~-4x1 +6x2-13

and obtain I 2 0 0 飞 I 0 2 0 \ 0 0 0 I

which is (semi)positive definite for a町 Z εJR3. This allows us to conclude directly that the point (2 ，一 3 ， 7) is a minimum point for f(x). Note that f(2 , -3 , 7) = -26. Similarly, for the second solution (8.3) , we 自nd that 月 (x)

- xi - x~ - 4X1

=

I -2 0 0 飞 。 一-2 0 I 飞 o 0 0 I

I

which is (semi)negative de主nite for any x 巴 JR3. We conclude that the point (-2 ， 3 ，一 7) is a maximum point for f (x ). Note that f (… 2 , 3, -7) = 26. 口 Problem 2: Assume that you can trade four assets (and that it is also possible to short the assets). The expected values , standard deviatio风 and correlations of the rates of return of the assets are: μ1 μ2 μ3 = μ4 =

0.08;σ1 0.12;σ2 0.16;σ3 0.05;σ4

- 0.25; P1 ,2 = - 0.25; - 0.25; P2 ,3 - - 0.25; - 0.30;ρ1. 3 工 0.25; - 0.20; PiA - 0, V i = 1 : 3.

(8.4) (8.5)

十2 (ω1ω伊1σ2P1 ，2 十 ω2ω3σ2{J3P2 ，3 十 ωlω3σlσ3P1 ，3) ，

= 0 for i = 1 : 3. We do not require the weightsωi to be positive , i. e. , we allow taking short positions on each one of the assets. However , the following relationship between the weights must hold true: si口ce Pi ,4

ω1 十 ω2 十 ω3+ω4

= 1.

(8.6)

(i) We are looki吨 for a portfolio with given expected rate of return E[R] = 0.12 and minimal variance of the rate of return. Using (8 .4-8.6) , we obtain that this problem can be written as the following constrained optimization problem: 且ndω° such that

gErof(ω) = f(ω0) ，

+ 6X2 + 13

and 2 D 几位 )

E[R] = ω1μl 十 ω2μ2+ω3μ3 十 ω4μ4; var(R) = ω?σ? 十 ω2σ~+ω;σ; 十 ωiσ~

I

D 2 凡 (x) 工

181

(i) Find the asset allocation for a mi山nal variance portfolio with 12% expected rate of return;

000020·7 。;

hL

，，"

，

一­

4EA

diU2'

们向

Cο.

O

EflJll

σba

WUW

1μe

4ιu·τ···4

ρUQU

ωt

f w u H M 4L m h z-7 'b s QU m m E S

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

呐咄 w t扭ler陀e ω

(8.7)

=( ωi) 山 )i= 吐= f(ω 叫) = 0.0625ωi + 0.0625ω;+0Oω9ω:+O 04tωρi 一 0.03125w1W2 - 0.0375ω2ω3 十 0.0375ω1ω3;

(8.8) 飞

/ωl 十 ω2 十 ω3 十 ω4- 1

g(ω) = \ 0 伽1 十 O 山2 + 0.16ω3+ 0 伽4 一- 0.12 )

(8.9)

It is easy to see that rank( \7 g(ω)) = 2 for 缸lY ωεJR 4 ， since

The Lagrange multipliers method can therefore be used variance portfolio.

to 丑nd

the minimum

182 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

Denote by λ1 and λ2 the Lagra且ge multipliers. From (8.8) and (8.9) , we obtain that the Lagrangian associated to this problem is

Note that the D 2 Fo(w) is equal to twice the covariance matrix of the rates of return of the four assets , i.e. ,

F(ω7 人) = 0.0625ωi + 0.0625ω2+O 09吟 +0.04ω2

-

0.03125ωlω2 一 0.0375ω2ω3

+λ1 (ω1 十 ω2 十 ω3+ω4 +λ2 (0.08ω1

The gradient of the

Lagra吨ian

\7(叫) F(ω?λ)=

+

(8.10)

D2

0.0375ω1 W3

- 1)

0.08ω4+λ1 十 0.05λ2 ω1 十 ω2 十 ω3 十 ω4- 1 0.08ω1 十 0.12ω2 + 0.16ω3 十 0.05ω4

- 0.12

To 自nd the critical points of F(叽 λ) ， we solve \7 (w ，λ) F(ω?λ) = 0 , which can

be written as a linear system as follows: 0.125

一 0.03125

0.0375

0.125 -0.0375

。

。

l

1 0.12

0.08

0.0375 O -0.0375 。 0.18 O O 0.08 l

0.16

1 1 1 1

0.08 0.12 0.16 0.05

。

。

0.05 0

。

也)1

000σ~

J

We co叫ude that D 2Fo (ω) is a positive definite matrix Therefore , the associated quadratic form q( v) = v t D 2 Fo(wo)v is positive definite , and so will be the reduced quadratic form corresponding to the linear constraints 飞7g(ω。 )υ=0.

We conclude that the point ωo = (0.1586 0 .4143 0.3295 0.0976) is a constrained minimum for f (ω) given the constraints g(ω) = O. The portfolio with 12% expected rate of return and minimal variance is invested 15.86% in the first asset , 4 1. 43% in the second asset , 32.95% in the third asset , and 9.76% in the fourth asset. The minimal variance portfolio has a standard deviation of the expected rate of return equal to 13.13%.

包)3

W4

/σ?σ1 0"2P1 ，2 σ1σ3P1 ，3

0\

|σlσ3P1 ，3 σ2σ3P2，3

O~

M=Iσ向P1，2

λ1 λ2

一

(8.11 ) / 0.1586\ I 0.4 143 I 0:3295 I;λω= 0.0112;λ0.2 =一 0.3810 \0.0976 /

,

σ§

\000σ1

μtω;

E[R] var(R)

ω tMω7

J

J

(8.12) (8.13)

-iqAnOA

\飞 -litz-1/

也

μ'μ'μ'μ'

μ'

//IIll--1 飞

where 一一

凡 (ω) = 0.0625叫 + 0.0625w~ + 0.09咛 + 0.04wl - 0.03125w1 ω2 - 0.0375w2W3 + 0.0375w1W3 + 0.0112 (ωl 十 ω2+ω3 十 ω4 - 1) 一 0.3810 (0.08ω1 + 0.12ω2 + 0.16ω3 + 0.05ω4 - 0.12) ,

O"~σ2σ向3 ~ I

the covariance matrix of the rates of return of the four assets. Let σp = 0.24 be the required standard deviation of the rate of return of the portfolio. If Wi denotes the weight of the asset i in the portfolio , i = 1 : 4 , it follows from (8 .4) and (8.5) that

Let 月 (ω) = F(ω7λ0，1， λ0.2) ， i.e. ,

and compute its Hessian

σ:3

0\ ~ I U I

(ii) Denote by

W2

The solution of the linear system (8.11) is ωo =

iσlσ3ρ1 ， 3σ2σ3P2 ， 3

+ 0.12ω2 + 0.16ω3 + 0.05ω4 - 0.12).

0.125w1 - 0.03125ω2 + 0.0375ω3+λ1 十 0.08λ2 、 t 0.125ω2 - 0.03125ω1 - 0.0375ω3+λ1 + 0.12λ2 0.18ω3 十 0.0375ω1 - 0.0375ω2 十沁十 0.16λ2

一 0.03125

/ σ? σ1σ2ρ口内 σ3P1 ，3 月 (ω ) = 21σ1σ2P1， 2 作 σ2匀2，3 飞

is the following (row) vector:

183

is the vector of the expected values of the rates of return of the four assets. The problem of 五日ding a portfolio with m以imum expected rate of return and standard deviation of the rate of return equal to σp can be formulated as a constrained optimization problem as follows: find Wo such that ~~n_ f(ω) = f(ω。)，

g(ω)=0

(8.14)

184 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.1.

where ω = (Wi)仨吐

However , it is easy to see that f(ω)

(ωtkLz)7

σp

(8.16) We can now proceed with finding the portfolio with maximum expected return using the Lagrange multipliers method. Denote by λ1 and λ2 the Lagrange ml削pliers. From (8.15) and (8.16) , we obtain that the Lagrangian associated to this problem is

where

0)

1

F(ω7λ) =μtω+λ1 (Itω-1) 十 λ2(ωtMω - (J'~)

G(ω?λ1 ， λ2)

1

C

σ2

P

.

)

It (川

飞7(叫) G(ω?λ1， λ2)

2(Mω )t

ω

2

wt !VIω - (J'~

\飞 III-f/

l t M- 1 1

(μ+λ11 + 2川) Itω-1 ,.

This is done using a six dimensional Newton's method; note that the gradient of G(叽 λ1 ， λ2) is the followi吨 6 x 6 matrix: M00 -inunu

We find that the Lagrangian (8.21) has exactly one critical point given by

(8.20)

From (8.19) and (8.20) , we find that , ifthere exists ωεJR4 such that g(ω) = 0 and ra出 (\7g(ω)) = 1, then

11111/

C

\飞

C 孚

= ~1~ω= 2 2

JR6 is given by

。白

乎

~~

ω

2

守J

C

nL

。

r

飞八 1if

σ各=一 ωι1

ω

中=丰〉

?b

wtMω=σ主

(8.19)

，

1=;山一11;

、‘ A气 ，

{=中

μ'

=1

一一

(8.18)

•

From (8.16) it follows that , if g(ω) = 0 , then l tw = 1 and wtMw 工作­ Usi吨 (8.18)， we find that l tw

飞八

where G : JR6

;M九

2-UM" M2p 111

\7 (ω，，\) F(ω7λ) = 0 ,

G(ω?λ1 ， λ2)

In order to use the Lagrange multipliers method for solvi吨 problem (8.14) , we first show that the matrix \7 g(ω) has rank 2 for any ωsuch that g(ω) = O. Note that ra此 (\7g(ω)) = 1 if and only if there exists a constant C εR such that 2!VIw = C 1. Using the fact that the covariance matrix !VI of the assets considered here is nonsingular , we obtain that w

/1 、、

飞， I/

/'41 、、

。"

'?b

\l/

直缸缸时 r1

\飞 III/

!ω

rM

F

ω

To /III-\

ω

(8.17)

飞A

2(Ax)t.

=

it is easy to see that \7g(ω)

/III-11\

节V

(去无)

(8.21)

The gradient of the Lagrangian is 十

Recall that , if the function h : JRn • JR is given by h(x) = xtAx , where A is an n xηsymmetric square matrix , then the gradient of h( x) is

Usi吨 (8.17) ，

185

山飞= 80.01 并 17.36 二 jf

(8.15)

g(ω)

Dh(x)

SOLUTIONS TO CHAPTER 8 EXERCISES

Wo

(om)

0.6450 0.6946 J' -0.3503

λ0 ， 1

=

-0.0738;λ0.2

= -0.8510.

Let 月 (ω) = F(ω7λ0 ， 1 ， λ0.2) ， i.e. , 几 (ω)

μtω - 0.0738(l t ω - 1) -

0.8510(ωtMω - (J'~).

186 CHAPTER 8. LAGRANGE 1VIULTIPLIERS. NEWTON'S METHOD.

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

The Hessian of Fa (ω) is

where B = 100 +去 and y = 0.03340 1. We obtain that the duration of the bond is 4.642735 and the convexity of the bond is 22.573118. 口

D 2 凡 (ω )

= -

0.8510· 2JvI

187

= - 1. 7019M.

Since the covariance matrix M of the rates of return of the four assets is a positive definite matrix , it follows that D 2 Fo (ω) is a negative defi时te matrix for any ω. Therefore , the associated quadratic form q( v) = v t D 2 Fo (ωo)v is negative definite , and so will be the reduced quadratic form corresponding to the linear constraints \7 g(ω。 ) v = O. We conclude that the point ωa = (0.0107 0.6450 0.6946 - 0.3503) is a constrained maximum for f (ω) given the constraints g(ω) = O. The portfolio with 24% standard deviation of the rate of return and maximal expected return 1.07% in the first asset , 64.50% in the second asset , 69 .46% in the third asset , while shorting an amount of asset four equal to 35.03% of the value of the portfolio. For example , if the value of the portfolio is $1 ,000 ,000 , then $350 ,285 of asset 4 is shorted (borrowed and sold for cash) , $10 ,715 is invested in asset 1 , $644 ,965 is invested in asset 2 , and $694 ,604 is invested in asset 3. This portfolio has an expected rate of return equal to 17.19%. 口

Problem 4: Recall that 缸ding the implied volatility from the given price of a call option is equivalent to solving the nonlinear problem f (x) = 0 , where

f(x) = Se- qT N(d 1 (x)) - Ke- rT N(d 2 (x)) - C and d1 (x)

一叫圣) (叫+手)T - ---\1'../ +xJr"':l r ,d

2

(x)

=

叫到十(叫一手)T xJT \1'>./

(i) Show that limx~∞ d 1 (x) = ∞ and li皿z→∞ d 2 (x) = 一∞， and conclude that lim f(x) = Se 一qT _ C. (ii) Show that {一∞，

ItIr! d 1 (x) = ItIr! d 2 (x) = { 、。

叭。

I

0,

∞，

if Se(r-q)T < K; if Se(r-q厅 K; if Se(r-q)T > K.

(Recall that F = Se(r-q)T is the forward price.) Conclude that Problem 3: Use Newton's method to find the yield of a five year semiannual coupon bond with 3.375% coupon rate and price 100 古 What are the duration and convexity of the bond? Solution: Nine $1. 6875 coupo丑 payments are made every six months , and a final payment of $101.6875 is made after 5 years. By writing the value of the bond in terms of its yield , we obtain that /IIiI\

,

时i

nu

42·A

吃$ 4

"U 1-9"

exp

十

寸t4

noQO

\飞 III/'

9ZM

vhυ 巧t

1-m 一-

寸t4

+

nunu

nhuQO vhu

exp

Uu

vhu

(8.22)

We solve the nonlinear equation (8.22) for y using Newton's method. With initial guess Xo = 0.1 , Newton's method converges in four iterations to the solution y = 0.03340 1. We conclude that the yield of the bond is 3.3401 %. The duration and convexity of the bond are given by

D

~ (飞J、 Z16吨 9 2 exp

B

·2

(iii) Show that

if Se(r-q)T

-C.

三 K;

> K.

f (x) is a strictly increasi吨 function and

-C < f(x) < Se- qT - C , if Se(r-q)T 三 K; Se- qT - Ke- rT - C < f(x) < Se- qT - C, if Se(叫)T > K. (扣) For what range of call option values does the problem f(x) = 0 have a positive solution? Compare your result to the range

Se- qT - Ke- rT

< C <

Se- qT

required for obtaining a positive implied volatility for a value C of the call option. Solution: (i) Note that

= ;(主 1ω巾

C =

(

1Se- qT - K产- C, if Se(r-q)T

lV(叫-

d1 ‘

ltt/\

飞Z \/飞)、 + 10 1. 6875· 25 exp(一切) )、IJ‘ ,

(x) Z

出 (x)

= In (丢) + (γ - q)T , x vIT ZZ飞IT 十 2

In (圣)

+ (r - q)T x vIT

x vIT 2

(8.23) (8.24)

188 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

It is easy to see that

(iii) Differentiati 吨 f (x) with respect to x is the same as computing the derivative of the Black-Scholes value of a European call option with respect to the 飞Tolatility 矶 which is equal to the vega of the call. In other words ,

l豆豆 d1(x) = ∞

J豆豆 d 2 (x) = 一∞?

and

and therefore

f' (x)

We conclude that lim f(x) -

In (去)

xVT

= 一节言十一一;

xvT

I

Se- qT - C.

'

2

也 (x)

In (去)

L/Tr

-ln( 是) + (r+手)T where d l = d d z ) - d Thus , f'(x) > 0 , \:j x > 队 and f (x) is strictly increasi吨­ Recall that limx---t∞ f(x) = Se- q'1' - C and

(ii) Let F = Se(r-q)T be the forward price. From (8.23) and (8.24) it follows that d1 (x) and d2 (x) can be written as d1(x)

=仇一听「Le一手 7 飞/

J马 N(d 2 (x)) = 0

出 N(ddz))=laad

= vega(C)

189

xVT

xVT

• If F < K , then In (去) < 0 and

r

炖忡忡 1 Since

f (x) is strictly increasi吗，

-

we co叫ude

C < f(x) <

< K: F 二 K7

if F

if

C,

that

Se- qT -

C , if F 三 K; C, if F > K.

(iv) If F 三 K ， the problem f(x) = 0 has a solution x > 0 if and only if

Ther陀 e£扣 ore 且 I im♂\oN(何 出l(归 d x)) 工 1且im 八 也2(μ d x)) 工 Oa丑 x \ oN(伺

:炖咆 f 仲(归x)尸巳一 C

-C.

K;~rT

-C < f(x) < Se- qT

Se- qT - Ke- rT -

:坦白(z)zESd2(z)= 一∞

Se- qT -

。< C

<

Se一ι(8.25)

If F > K , the problem f(x) = 0 has a solution x > 0 if and only if Se- qT - Ke- rT

• If F = K , 出nd1(x)= 孚 and 也 (x) 口 一千， 缸aI丑侃圳 l叫d时 山阳re伽 缸 b f扣 ωr O臼

<

C

<

Se-qT.

(8.26)

Note that

:炖巧 d出1叫巾(归 仰 Z) = :炖坷 d 也2纣 州(x)忏 =0

Se- qT - Ke- rT

Thus ,1im x\ a N(d1(x)) = lim八a N(d 2 (x)) = ~ and 一俨r

w(Z)=;(SfqT-ke-TT)-czET(F-K)-c

=

e-rT(Se(叫)T _ K)

From (8.25) and (8.26) , we conclude that the problem f(x) = 0 has a positive solution if and only if C belongs to the following range of values: r max (Se一 qT 一 Ke-气 O吗)

= -C.

= e-rT(F - K).

< C <

Se- qT

• If F > K , then In (去) > 0 and ~t~ d1 (x) = l~~ d2 (x) = ∞-

z \v

X \.U

Therefore li叫\\\、、、oN \V(付 出l(仪 d x)川) = 且 I im♂八\oN(何 也2(归 d x)) = 1 a丑 :炖巧 f 仲 (μ x)忏 =S 仇 仇 s f旨e一-qT 一 Ke巳f ♂一刊 T叮 T 一 C

Problem 5: A three months at-the-money call on an underlying asset with spot price 30 paying dividends continuously at a 2% rate is worth $2.5. Assume that the risk free interest rate is constant at 6%. (i) Compute the implied volatility with six decimal digits accuracy, using the bisection method on the i口terval [0.0001 , 1]' the secant method with initial guess 0.5 , and Newton's method with initial guess 0.5.

190 CHAPTER 8. LAGRANGE

(ii) Let

MULTIPLIER丘 NEWTON'S METHOD.

σ问 be

the implied volatility previously computed method. Use the formula

σ

usi吨 Newtor内

r-.JV2在 C 一与庄s r-.J百万

imp ,approx

to compute an approximate value compute the relative error

σimp， apprωfor

the implied volatility, and

|σzmp， αpprox 一 σ vzmp σimp

Solution: (i) Both the secant method with ♂ -1 = 0.6 and Xo = 0.5 and Newton's method with initial guess Xo = 0.5 converge in three iterations to an implied volatility of 39.7048%. The approximate values obtained at each iteration are given below:

k Secant :Nlethod Newton's Method 。

1 2 3

0.5 0.3969005134 0.3970483533 0.3970481868

0.5 0.3969152615 0.3970481867 0.3970481868

[0.375063 , 0.5];

[0.375063 , 0 .406309];

zmp，α，pprox

一 σ| v

zmp I

巳

0.000948

=

0.0948%.

口

σzmp

(

xf + 2XlX2 十 zi-2223 十 9\

( 2XI + 2XlX~ 十 zizi-ziz3 一 2 \

Z问仇 +zi-23-zd-4

'--J/

f 1 \

and x'

=

for

Xo

= I

2

I

飞 3

J

(-n 扣r Xo 工 0)

The iteration counts are given in the table below: Iteration Count Iteration Count Iteration Cou日t Approximate 时ewton Approximate Newton Newton's Method Central Differences Forward Differences

( 1\

{~ }

9

9

9

40

65

43

\ 2

Problem 6: Let F : :lR3 • :lR3 given by

F(x) -

,

I 1. 9831072429 I 飞 -0.8845580785 J

旷工

Xo

39.6672%.

If σimp = 0.3970481868 is the implied volatility obtained using Newton's method , then |σ

Fi(x + hej) 2h - -Fi(x - '--J/ hej) , j = 1 :风 £\--

~£\~

Solution: We use Newton's method and the approximate Newton's method both with forward difference approximations and with cen~ral difference approximations with toLconsec = 10- 6 and toLapp r
[0.390686 , 0 .406309];

0.3966718145 -

-

where ej is a vector with all entries equal to 0 with the exception of the j-th entry, which is equal to 1. (i) Solve F(x) = 0 using the approximate Newton:s algorithm obtai~ed by substituting ~cF(Xold) for ~F(Xold)' Use h = 10- b , toLconsec 干 10-° ， and toLapprox = 10- 9 , and two different initial guesses: Xo = (1 2 3)t and Xo = (2 2 2)t. (ii) Compare these results to those corresponding to the approxim~t~ Newtor白 method with forward finite difference approximations for ~F(x).

[0.375063 , 0 .4 37556];

(ii) The approximate value for the implied volatility given by (8.27) is σimp，αpprox =

~c，jFi(X)

/ -1. 6905507599\

The bisection method on the interval [0.0001 , 1] co盯erges in 30 iterations to the same implied volatility of 39.7048%. The 五rst five iterations generate the following intervals: [0.250075 , 0.5];

The approximate gradi怡e时 ~cF(x 叫) = (~c，jFi(仪 X) )i队ω i 庐产 ，j= j户= i.e. ,

put臼 ed us剖ir卫19 cent忧ra 址1 也 d ifference approx 对im 丑lations ，

(8.27)

1 一呼E

191

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

I

/

We note that usi鸣 central 丑时te differences approximates the gradient

DF(x) of F(x) more accurately than if forward finite differences are used ,

192 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

+ 102 .4375 e- 2r (O ,2) 2.4375 e-O. 5r (O ,O.5) + 2.4 375 exp ( - r(旧的十 0 叭

十 2 .4 375

resulting in algorithms with iteration counts closer to the iteration counts for Newton's method. 口

e- 1. 5r(O ,1. 5)

A

十2 .4 375

Problem 7: (i) Use bootstrappi 吨 to obtain a zero rate curve from the following prices of Treasury instruments with semiannual coupo口 payments: 3623510 -

IVIonth T-bill IVIonth T-bill Year T-bond Year T-bond Year T-bond Year T-bond

°

°

If we let x = r(O , 2) , we

自nd

r(O , 0.5)

+ 0.5x

t ε[0.5 ， 2].

0.5r(0 , 0.5) 1. 5

(8.28)

+x

Recall that the price of the two year bond is the discounted present value of all the future cash flows of the bond. Then , 100 +立= 2 .4 375 32

e-O. 5r (O，O.5) 十 2.4375 e- r(O ,l)

I

LV

)

e-2x

= 4.7289%.

We proceed by assuming that the zero rate curve is linear between two years a时 three years. We note that r(O , 0.5) ， γ(0 ， 1) ， γ(0 ，1. 5) ， and γ(0， 2) are known. If we let x = γ(0 ， 3) ， the price of the three year bond can be written as 100 +主 32

2.4 375

e-O. 5r (O，O.5) 十 2 .4 375 e- r (O， l) 十 2 .4 375

+ 2.4375

e- 2r (O，2) 十 2 侃侃p

/ r(O , 2) \ -2.5 ' \V'~d

e- 1. 5r(O ,1. 5)

+x\ I

LV

)

+ 102 .4375 exp (-2x)

r(0 , 3) = 4.7582%.

from (8.28) that

T(071)=;γ(0， 5) 1. 5 . ,-, 1. _._/

\

Z 二二 0.047582.

Bootstrapping is needed to obtain the 2-year , 3-year , 5-year and 10-year zero rates. For example , for the two year bond , if the zero rate curve is assumed to be linear between six months and two years , then

+ (t - 0.5)r(0 , 2) 1. 5

V.VI

Using Newton's method to solve the nonlinear problem above , we obtain that Therefore

( 100 \ = 4ln (一~: I = 0.052341 = 5.2341 %; 飞 98.7 ) ( 100 \ r(0 , 0.5) = 2ln (一一 I = 0.050636 工 5.0636%. 飞 97.5 J

γ(0 ， 0.25)

(2 - t)r(O , 0.5)

.....5 0.5γ(0 ， 0.5) 十 z\ exp (( -1. V~V J + 102.4375 r- \ _.1. 5 J

γ(0 ， 2)

(i) For the Treasury bills , the zero rates can be computed directly:

俨 (07027V

v:

J

飞1. 5

Using Newton's method to solve the nonlinear equation above for x , we obtain that x = 0.047289 , and therefore

Coupon Rate Price 98.7 97.5 100~ 4.875 4.875 10。在 4.625 99 玄 4.875 101 主

Assume that interest is continuously compounded. (ii) How would the zero rate curves obtained by bootstrappi吨 from the bond prices above , one corresponding to semi-annually compounded interest , and the other one corresponding to continuously computed interest , compare? In other words , will one of the two curves be higher or lower than the other one , and why? Sol仰on:

193

Using bootstrapping , we obtain similarly that

r(0 , 5) = 4.6303%

and

r(0 , 10) = 4.6772%.

Thus , we found the following zero rates corresponding to the maturities of the four given bonds , i.e. , 3 months , 6 months , 2 years , 3 years , 5 years , and 10 years: γ(0 ， 0.25) = 5.2341%;γ(0 ， 0.5) = 5.0636%;γ(0 ， 2) = 4.7289%; γ(0 ， 3) = 4.7582%;γ(0 ， 5) = 4.6303%;γ(0 ， 10) = 4.6772%.

Since we assumed that the zero rate curve is linear between any two consecutive bond maturities , the zero rate r(O ,t) is known for any time between the shortest and longest bond maturities , i. e. , for any t ε[0.25 ， 10]. (ii) Denote by rc(O , t) and r2(0 , t) the zero ra~e curves co.rrespo~di吨 to identicai discount factors , with rc(O , t) correspondi卫g to continuously compounded

195

194 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.1. SOLUTIONS TO CHAPTER 8 EXERCISES

interest , and with r2(0 , t) corresponding to semi-annually compounded interest. Then ,

Assume that the overnight rate is 5% and that the zero rate curve is linear on the following time intervals:

产t衍 f 川川 T川 叫州 呻 (Oρ 川

[0 , 0.5];

By solving for rc(O , t) in (8.29) , we 在nd that

几(0， t) =咐 +23二~r) 巧(0， t)

In (

(1 + r2(~' t) )"(~"))

巧(0， t) In ( (1 十 2/句:OJ))2/T2仰

r 51 11, ~

I 5 10 I I~' -3-1;

I;

[0.5 , 1];

110 I I 言， 51

Solution: We know that r(O , O) = 0.05. The 览e six mo ∞ I丑1 叫 computed from the price of the 6ι 一I囚工丑1∞ 0I丑1此 时ths 曰s zero ∞ c oupon bOI口1d as ( 100\ 飞 97.5 J

r(0 , 0.5) = 2ln (一一 ) = 0.050636 = 5.0636%

(8.30)

Using (8.30) , we can solve for the zero rate r(O , 1) from the formula given the price of the one year bond , i.e. , 100 _ 2.5 e- O.5r (O ,O.5) + 102.5 e- r (O ,l) and obtain that γ(0 ， 1) = 0.049370 = 4.9370%.

for the last inequality we used the fact that

(1+ ~r <民 'v'x > 0 , for x = 2/r2(0 , t). In other words the semi-annually compounded zero rate curve is higher than the continuously compounded zero rate curve if both curves have the same discount factors. While a rigorous proof is much more technical , the same happens if the two curves are obtained by bootstrapping from the same set of bonds , i.e. , the zero rates corresponding to each bond maturity are higher if interest is compounded semi-annually than if interest is compounded continuously. This is done sequentially, beginning with the zero rates corresponding to the shortest bond maturity and moving to the zero rates corresponding to the longest bond maturity one bond maturity at a time. 口

The third bond pays coupons in 2, 8 , 14 , and 20 months , when it also pays the face value of the bond. The凡 103 =

3 叫去(喝)

3 叫去(叫)

P(马(山( 20 ( \ ) (一))叫-ZT(072))(8 叫 12

飞，

12 ) } \ \

}J

Since we assumed that the zero rate curve is linear on the intervals [0 , 0.5] and [0.5 , 1], the zero rates r(O， 是) and r(O ， 击) are known and can be obtained by linear interpolation as follows:

γ(咐

命 (070)7(0705)= 0.0502 叫

(8.32)

r(叫

4r(0 , 0.5) + 2γ(0 ， 1) = 0.050214.

(8.33)

Let x = γ(0 ，

f§).

Since r(O , t) is linear 0川he interval [1 , f§ J, we find that

(D , 12}

Problem 8: Use bootstrapping to obtain a continuously compounded zero rate curve given the prices of the following semiannual coupon bonds: NIaturity Coupon Rate Price 6 months 0 97.5 1 year 5 100 20 months 6 103 40 months 5 102 4 5 years 103

+

衍 (0 ， 1) + 2x

(8.34)

8

From (8.34) , it follows that the formula (8.31) can be written as (

103 =

1

(一\\(

2

(~

8\\

3 仪p ( -~r ( 0，专门+ 3 叫了( 0, 1~2) )

-i .

/6γ(0 ， 1) 3 叫 \ VI \ V ~

,

+ 2x \ ( 5 飞 ~W) + 103 叫 \ -~x ), I

(8 蚓

196 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.2. SUPPLEMENTAL EXERCISES

where r (0 ， 元) and r (0 ， 击) are given by (8.32) and (8.33) , respecti叫y. Using Newtor内 method to solve for x in (8.35) , we 自nd t1时 x = 0.052983 , and therefore ( 20\ (0、~= I = 5.2983%.

8.2

飞

'12

J

Bootstrapping for the fourth and fifth bonds proceed similarly. For exampIe , the fourth bond makes coupon payments in 4 , 10 , 16 , 22 , 28 , 34 , and 40 months. The zero rates corresponding to coupon dates less than 20 months , i.e. , to the coupon dates 4 , 10 and 16 months , can be obtained from the part of the zero curve that was already determined. By setting x = γ(0 古) and assuming that the zero rate curve is linear between 20 months and 40 months , the zero rates corresponding to 22 , 28 , 34 , and 40 months can be written in terms of x. Thus , the pricing formula for the fourth bond becomes a nonlinear equation in x which can be solved using Newton's method. The zero rate r (0 , ~) is then determined Using bootstrapping and Newton's method we obtain that

。一)

(40 , 12

=

4.5帆 r(0， 5)

=

3叫

Summarizing , the zero rate curve obtained by bootstrapping is given by

r(O , O) = 0.05;

(

riO , 飞，

2\/ 1-("\

12

I

riO , 1~("\ I = 0.050214; 飞， 12 }

= 0.050212;

}

民\

72)=om3; ~D = 0.052983; r叫叶= (叫 0.0铅326;γ(0， 5)

(0 ,

and is linear on the intervals

vhu

Supplemental Exercises

1. (i) If the

c盯rent

η (0， t)

zero rate curve is =

0.025 由叫古)十 Jv

find the yield of a four year semiannual coupon bond with coupon rate 6%. Assume that interest is compounded continuously and that the face value of the bond is 100.

(ii) If the zero rates have a parallel shift up by 10 , 20 , 50 , 1~0 ， and ?O~ ,

bs.:sis points , respectively, i.e. , ifthe zero rate curve changes from rl(O , t)

to r2(0 ,t) = rl (0 , t) +dr , with dγ= {0.001 , 0.002 , 0.005 , 0.01 , 0.02} , find out by how much does the yield of the bond increase in each case. Note: In general , a small parallel shift in the zero rate curves results in a shift of similar size and direction for the yield of most bonds (possibly with the exception of bonds with 10吨 maturity). This assumption will be tested for the bond considered here for parallel shifts ranging from small shifts (ten basis points) to large s1曲 2. Consider a six months at-the-money call on an underlying asset following a lognormal distribution with volatility 30% and paying dividends continuously at rate q. Assume that the interest rates are constant at 4%. Show that there is a unique positive value of q such that .6.( C) = 0.5 , and find that val~e using Newton's method. How does this value of q compare to r 十亏? 3. The following prices of the Treasury instruments are given:

口

m-3 『 gaii-」

俨ittEtlttgL

5-3

叮 3111111334

ω-3

俨s t t 5 3 3 3 3 1 L

气 333111'iid

「1 1 3 1 3 3 3 3 1 h

434

vhu-qο

nu vhu

咱t 4

nu nu vhu

= 0.032119 ,

197

612 23510 -

lVlonth T-bill Month T-bill Year T-bond Year T-bond Year T-bond Year T-bond

Coupon Rate 0 0 2 4.5 3.125 4

Price 99.4565 98.6196 1011号 107~I 102 立

g?t

103~';

The Treasury bonds pay semiannual coupons. Assume that interest is continuously compounded. (i) Use bootstrappi 吨 to obtain a zero rate curve from the prices of the 6-months and 12-monthsτ'reasury bills , and of the 2-year , 5-year and 10-year Treasury bonds;

198 CHAPTER 8. LAGRANGE lVIULTIPLIERS. NEWTON'S METHOD. (ii) Find the relative prici吨 error correspondi吨 to the 3-year Treasury bond if the zero rate curve obtained at part (i) is used. In other words , price a 3-year semiannual coupon bond with 4.5 coupon rate and find its relative error to the price 107去 of the 3-year Treasury bond

8.3

8.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

(ii) If the zero rates increa毗 the value of the bond decreases , and therefore the yield of the bond will increase. Our goal here is to investigate whether a parallel shift of the zero curve up by dr results in an increωe of the yield of the bond also equal to dr\ When the zero rates increase from r1 (0 , t) to 俨2(0 ， t) = r1(0 ,t) + dr , the value of the bond decreases from B 1 given by (8.36) to B2

Solutions to Supplemental Exercises

= 0.025

- 了 3 叫 (飞飞， -r2 ( 0 , ~ )斗+ 103 吨(-4州， 4)) 仨7 2) 2)

The new yield of the bond , denoted by Y2 , will be larger than the initial yield and is obtained by solving

Problem 1: (i) If the current zero rate curve is

η (0， t)

到，

+忐叫古)+τ:71

B2 = 且nd the yield of a four year semiannual coupon bond with coupon rate 6%.

Assume that interest is compounded continuously and that the face value of the bond is 100. (ii) If the zero rates have a parallel shift up by 10 , 20 , 50 , 100 , and 200 basis points , respecti叫y， i. e. , if the zero rate curve cha吨es from 'r1 (0 , t) to r2(0 , t) = η(OJ)+d71with dT={0.00170.00270.00570.0170.02}7and out by how much does the yield of the bond increase in each case.

B乌1 工 兰 3 倪吨叫叶 寸(-一仆俨 p( η1 = 106.1995.

(8.36)

The yield of the bOIId is found by solving the formuia for the price of the bond in terms of its yield , i.e , by solving

,

=

~ 3 exp ( -y

D

+ 103

exp(一句)

(8.37)

for U7WIleEe BIis given by (836)7i.e-7BI=1061995.Using NewtOI内 method , we obtain that the yield of the bond is Y = 0.042511 = 4.2511%.

￡ 3 仪叫叫叶 寸(-一叶 p( Uω2

for Y2 , where B 2 is given by (8.38). For parallel shifts equal to dr = {0.001 , 0.002 , 0.005 , 0.01 , 0.02} , we obtain the following bond prices and yields: Zero rate shift New bond price New yield Yield increase dr B2 的 -Y Y2 0.043511 10bp = 0.001 105.8150 0.00099979 0.044510 0.00199957 20bp = 0.002 105 .4 319 0.047510 50bp = 0.005 104.2915 0.00499893 0.052509 100bp = 0.01 102 .4 199 0.00999784 0.062506 200bp = 0.02 98.7829 0.01999562

SoJUUonJ(i)TEle bond provides coupOBpaymmts equal to3every sixmo毗1S until 3.5 years from now , and a final cash flow of 103 i且 four years. By disc，?u~t，i?g t~is ca~h flo;V~ t~ the. I? resent using the zero rate cur~e r1 (0 , t) , we 直nd that the value of the bond is

B

199

As expected , the increase of the yield of the bond is slightly smalle飞 very close to , the parallel shift of the zero rate curve , i.e. ，的 -Y 自 dr.

but 口

Problem 2: Consider a six months at-the-money call on an underlying asset following a lognormal distribution with volatility 30% and paying dividends continuously at rate q. Assume that the interest rates are constant at 4%. Show that there is a unique positive value of q such that .6.( C) = 0.5 , an 白II口1d that value us岳i口g Newtωon f 旷 l'S method. How does this value of q compare to

r+ 号?

Solution: Recall that the Delta of a plai口 vanilla call option on an underlying asset paying dividends continuously at rate q is

.6.( C) = e- qT N(d 1 ) ,

(8.39)

200 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S METHOD.

8.3. SOLUTIONS TO SUPPLEMENTAL EXERCISES

where

In other words , if the interest rates are flat at 4% , the Delta of a six months at-th e-money call option on a口 u日derlying asset with volatility 30% is equal to 0.5 if the underlying asset pays 6.69% dividends continuously.

d1 =

In (去) + (r-q 十号 )T \(/

‘

σ 飞IT

If q= γ + 兰 we 直缸nd from (何8.ι40创) and (8 .4钊钊 1) tha 创td 出l=Oa卫

For an at-the-money option , i.e. , for S = K , we find that

d1

~(C)

= 川♂一-σf

(8 .40)

From (8.39) and (8.40) , we find that ~(C)

/ (r - q) 叮

σd 飞

)

= e- qT N(d1 ) = e- qT N Iσ; ~

It is easy to see that

~(C)

201

(8 .4 1)

=

巳f 一叫\V(仰0) = 0.5e一q矿 T

< 0.5.

Since ~(C) is a decreasing function of q , we obtain that the value of q such that ~ (C) = 0.5 must be lower than r 十号 Indeed， the value previously obtained for q satisfies this condition , i.e. , q = 0.066906

<

0 附 - r 十二口

is a decreasing function of the dividend rate

q , smce

Problem 3: The following prices of the Treasury instruments are given: θA θq

-Te 句 (d 1 ) 十 f 俨卢 q叩咐川 T 川/气(刷 N' d ω1) 出

8d1 dr

I

‘~ jr中 F 飞

卡 J 一 口e一庐句川 T 川(刷 N 仇1创) 十+ 俨e一吁q飞 d TτE 乒 =e一亏叮{ 一工立土

V 'l:7r

l

\σ/

< O.

…

When q = 0 , we find that

6. (0) = N

(手+手)

> N(

5

Also , since 0 < N(d 1 ) < 1, it follows that

J!!炉 (C) = J豆豆 (e-句 (d 1 )) = 0 We conclude that ~(C) is a decreasing function of q and that , for q 主 0 ， the values of ~(C) decrease from N(O) > 0.5 to O. Therefore , there exists a unique value q > 0 such that ~(C) = 0.5. From (8 .4 1) , it follows that this value can be obtained by solving for x the nonlinear equation f (x) = 0, where / (r - x) 叮 σn\ f( 叫 = e- X1 ' N I \. ~/v ~ + ~.L I - 0.5. v

1σ

二~

612 23510 -

Month T-bill Month T-bill Year T-bond Year T-bond Year T-bond Year T-bond

Price 99 .4 565 98.6196 1011旦 107里 102 立

~?b

103~'~

The Treasury bonds pay semiannual coupons. Assume that interest is continuously compounded. (i) Use bootstrapping to obtain a zero rate curve from the prices of the 6一 months and 12-叮τ￥easur)γy bonds; (ii) Find the relative pricing error corresponding to the 3-year Tr easury bond if the zero rate curve obtained at part (i) is used. In other words , price a 3-year semiannual coupon bond with 4.5 coupon rate and 五日d its relative error to the price 107~ of the 3-year Treas町 bond

Solution: (i) The 6-mo叫1S and 12-months zero rates can be obtained directly from the prices of the τ'reasury bills , i.e. , r(0 , 0.5)

I 、、， 1''''

nu

才t4

俨'

/IE 飞

Using Newton's method , we obtain that q = x = 0.066906.

Coupon Rate 0 0 2 4.5 3.125 4

21n

(品5)

叫品)

= 1.09%; =

1 究

202 CHAPTER 8. LAGRANGE MULTIPLIERS. NEWTON'S IvIETHOD.

Using bootstrapping , we obtain the following 2-year , 5-year and 10-year zero rates: γ(0 ， 2) 1' 1'

(0 , 5)

(0 , 10)

1. 2099%; 2.6824%; 3.7371%.

Bibliography

(ii) The zero rates computed above correspond to the followi吨 zero rate curve which is piecewise linear between consecutive bond maturities:

1'

(0 , t)

=

r (到一 1)γ( 0, 1) + 2(1 - t) l' (0 , 0.5) , if 0.5 三 t 三 1; J (t -1) 1' (0 ， 2) 十 (2 - t) 1' (0 , 1) , if 1 三 t 三 2; )平 γ(0， 5) +于 γ(0， 2) ， if 2:::;t:::;5; t孚 γ(0 ， 10) +半响的， if 5 < t < 10

l

[1] Milton Abramowitz a时 Irene Stegun. Handbook of Mathematical Functions. National Bureau of Standards , Gaithersburg ，扣1aryland ， 10th corrected printing edition , 1970. [2] Dan Stefa旧ca. A lVlathematical Primer with Numerical lVlethods for Financial Engineering. FE Press , New York , 2007.

t-2-z-2

/III-\ /It--\ \1111/ \liI/

nu qd

飞

exDA

/1

vhu

币' 门。

。，"

nu

。中

才i

nu

十

ex ny

γ

vhu

门，由

2

nu

、毛 J

τi

川队

一一一­

B

5γμHm

With this zero rate curve , the value of the 3-year semiannual coupon bond with 4.5 coupon rate is

The price of the 3-year Treasury bond was given to be 107.5625. Therefore , the relative pricing error given by the bootstrapped zero rate curve which does not include the 3-year bond is 1107.5625 - 108.19301 i = 0.005862 107.5625

0.59%.

口

203