Stochastic Processes And Applications to Mathematical Finance: Proceedings of the 5th Ritsumeikan International Symposium, Ritsumeikan University, Japan, 3-6 March 2005

N->oo 2.N + 1

*—I, s=—N

The subindex B is the initial of Bohr, who developed similar ideas in the context of the theory of almost periodic functions. Theorem 2.1. Consider a process p satisfying assumption (H). Then we have: (2)

-!- T(a2) = $ *B <&, where O(fc) = T(dp)(k) 2n

4

for any integer k. Convergence of the convolution product (2) is attained in probability. The proof is based on the ltd energy identity for stochastic integrals (see for instance [39]), after having observed that the drift component b(t) in the semimartingale p(t) does not give contribution to the formula (2). The reconstruction of the function a2(t) from its Fourier coefficients computed by Theorem 2.1, can be obtained as follows. Define
+

°°

VN(k) = ^ - y £

<J>N(s)
S=-oo

Then the Fourier-Fejer summation gives the instantaneous volatility function a2(t) = lim £

(3)

N_>

(1 - j - j ) ^ * ) exp(z'Jfcf) for almost all t e (0,2rc).

° ° \k\
Remark 2.2. We note that the approximating trigonometric functions appearing in the sequence of polynomials (3) are positive (see [43]). It is possible to implement Theorem 2.1 in real terms, by expanding the Fourier transform of the volatility function as a linear combination of sines and cosines (see [41]). This procedure allows us to determine a formula for the volatility a2(t) analogous to (3). The formalism is a little cumbersome and less conceptual than working with complex exponentials; nevertheless the numerical implementation of the real Fourier series is easier. Therefore we recall this procedure. Suppose that the function p(t) satisfies (H). Define »*(p) = n

I J]0,2TI[

cos(kt)p(t)dt,

bk(p) = -

j

sin(kt)p(t)dt,

k>0,

71 J]o,2n[

then a0(dp) = 0,

ak(dp) = kbk(p) + ~(p(2n) - p(0)),

bk(dp) =-kak(p),

k>0.

We define the prolongation to all integers k by parity for ak and by imparity for bk; more precisely let

5

The following Theorem provides the formula of the Fourier coefficients of a2(t) in terms of a series of combination of the Fourier coefficients of dp, previously defined. Theorem 2.3. Consider a process p satisfying hypothesis (H). Define, for 0 < q<2N,

a,(N) =

2NTT I J (^+S^ + b'i+sb*s)' 1

MN) =

v-i

1

s=-N N-q ( +A + * s=-N ~^

2NTT E

h al)

^ -

Denote by aq(o2), bq(a2) the Fourier coefficients of o2(t). Then, for all fixed q > 0, thefollowing convergence in probability holds: (5)

Km aq(N) = - aq(o2), N->+oo

71

lim pq(N) = - bq(a2). N->+oo

71

The above result provides an alternative formula to compute the so called instantaneous volatility. In fact the reconstruction of the function o2(t) from its Fourier coefficients, derived by (5), can be obtained for instance by the classical Fejer inversion formula: N

k

(6) o2(t) = lim Y(l--)(ak(o2)cos(kt)

+ bk

(a2)sin(kt)),

fc=0

for almost all t € [0,2n]. 2.1 Integrated volatility As a byproduct of formula (2), we can also compute the integrated volatility in the time interval [0, 2TZ] through the following identity .277

.2/ o\t)dt.

o Using (2) we have

(7)

^n°2xo) = (ndp) * r(dPw) N

= ^2NTTEW)(SWP)(-S)s=-N

6

This can be expressed using the Fourier coefficients, because (7) is equal to

(8)

$2.2ivTT t l f ^ l 2 = $™2ikit \^2 s=-N

+

^'

s=-N

where a\ and b\ are defined in (4). Finally (9)

f " a\t)dt = n2 lim — ? — f (atf + (b's)2. N Jo ^°° 2N + 1 ^

Therefore (9) provides a measure of the integrated volatility, see [41]. 2.2 Asymptotic analysis The asymptotic properties of the Fourier estimator, i.e., consistency and central limit theorem, have been obtained in [43]. Apart from the statistical importance of these results, they are necessary to understand how the estimator behaves for very small time intervals, due to market frictions. In fact, the presence of market frictions makes this limiting argument not really accurate for very small time intervals. Such difficulties with limiting arguments are present in almost all area of econometrics and the development of a central limit theorem helps in understanding the behavior of the Fourier estimator for relatively small time intervals. We consider the following discrete unevenly spaced sampling of the price process p. We fix a sequence Sn of finite subsets of [0,27i], let S„ := (0 = fo,n < h,n <••• < tk„,n = 2TT( for any n > 1 and denote p(n) :=

max \tj+u - ti„\. 0<j
'

Remark that the Fourier estimation method allows to consider nonsynchronous and random observations. Therefore we could choose the observations randomly in an independent way of W. Nevertheless by splitting the probability space, this kind of sampling reduces to a deterministic sampling. Therefore all our study will be made in the deterministic context. We use the following interpolation formula

Pn(t):=

£PMVW(0,

y=o

where 'i-[tj,„,tH,„i denotes the indicator function of the interval [tj,n,tj+i,n[Then for any integer k, with |fc| < IN compute Ck(dpn) = Y YJ 71

;'=o

ex

P(-iktj.n)&j(p)

wh

e r e 6j(p) := p(fy+i,„) - p(f/,„).

7

Consider now the estimator of the volatility function based on (3) defined by (10)

£(l-^)«t(N,p„)exp(/fcO

^N(£):=

|Jt|
where for any |A:| < N (11)

ak(N,p„) :=

2N + 1

^

ck-s(dpn)cs(dpn).

s=-N

The following theorem shows consistency in probability uniformly in t of the estimator (10). Theorem 2.4. Let p(n) —> 0 as n —> <x>, and consider the estimator of the volatility function defined by (10). Then the following convergence in probability holds: lim sup fflN(t)-o2(t)\=0. We consider now the distribution of the error. The Central Limit Theorem is obtained as p(n) -» 0 and the interval [0,2n] remains fixed. For the existence of a limiting distribution it is necessary that the number of the Fourier coefficients N and the number of observations n increase with a suited order. Therefore let 9(n) be a function such that 8(n) —> oo as n —» oo. Consider the estimator (12)

?2(f):=

£

(l-^-)ak(d(n),Pn)exp(ikt)

of a2(t) as defined in (10), but in (12) we have highlighted dependence between the number of Fourier coefficients 9(n) and the number of observations n. Then: Theorem 2.5. Assume that asn —> oo then p(n) —» 0, &(n) —> oo, p(n)9(n) —> 0 and p(n)(9(n))2 —> oo. Assume furthermore that SUp,(f ;+!,„ - tj,„)

(13)

lim — ^ -

— - = 1.

n->°° inf/(r ; -+i,„ - tj,n)

Then, for any function h e L2(0,2n) satisfying the condition (14)

-}-

Y,

Ic*(ft)l 2 -»0flsn-»oo,

8

the following result holds for'a%(t) defined in (12): (15)

—L= yjp{n) Jo

\\{t)(5l(t)-o2{t))dt

converges in law to a mixture of Gaussian distributions with random variance equalto2J^nh2(t)a4(t)dt. Remark 2.6. Condition (13) is a measure of the regularity of the partition. A different condition called £-balance, with e 6 (0,1), is considered in [9]. Our condition implies e-balance condition for any e e (0,1). In [46] a general study of central limit result is done in the case where prices are recorded at irregular time intervals. 2.3

Cross-volatility computation The Fourier method was proposed in [41] having in mind difficulties arising in the multivariate setting when applying the quadratic variation methodology because of non-synchronicity of prices observed for different assets. Assume that p(t) = ipl{t),... ,pn(t)) are Brownian semi-martingales satisfying the following ltd stochastic differential equations d

(16)

dp>(t) = £ a\{t) dW + b> (t) dt

j=

l,...,n

i=i

where W = (W1,..., Wd) are independent Brownian motions, and a*, and b* are random processes satisfying hypothesis H. There is a large literature in which cross-volatilities are estimated through the quadratic covariation formula. However this formula is not well suited to provide a good estimate of cross-volatilities. Difficulties arise from the absence of synchronous observations. The non-synchronicity trading problem has been studied for quite a long time in empirical finance, e.g. see [52, 37]. The bias caused by non-synchronicity and random sampling for the cross-correlations estimation has been recently highlighted in [29]. The Fourier methodology proposed in [41] is immune from these difficulties due to its own definition, being based on the integration of "all" data. We recall now the Fourier method for computing multivariate volatilities. From the representation (16) we define the volatility matrix, which in our hypothesis depends upon time: d

(17)

E**(f) = £o#)*?(0. 1=1

9

The Fourier method reconstructs £*'*(£) on a fixed time window (which we can reduce always to [0,2n] by change of origin and rescaling) using the Fourier coefficients of dp*(t). First we compute the Fourier coefficients of dpi for;' = 1 , . . . , n defined by (18)

ak(dpi) = -

71

f

cos(kt)dpi(t), bk(dpi) = -

f

n

J%2n{

sin(kt)dp'(t).

J]0,2n[

We then consider the Fourier coefficients of the cross-volatilities (19)

a0(X!'i)=±Z 7 1

f

V'i(t)dt, ak(r-i) = n

J]0,2TI[

(20)

bk(V'i) = -

f

f

cos{kt)l!''(t)dt,

J]0,2TI[

sinWZ'iWdt.

By a polarization argument in [41] it is proved the following Theorem. Theorem 2.7. Fix an integer no > 0, the functions E',;(f) of the volatility matrix have the following Fourier coefficients (21)

ao&'i) = lim

(22)

ak(Z'') = lim

*

V hasWMdpi)

N->oo Jv + 1 — nn *-^ Z s=n0

+ bs(dp%(dp')).

N

"

Y(as{dv>s^¥) i

+

Usidfys^dp1)).

k

AT-»oo T\ + 1 —no s=« —0 N

(23)

ME''>) = lim — 2

V Mdp?ybt+k(dpi) + a,{dpi)bs+k(
N->oo i \ + 1 — HQ ^ - ^ s=n0

Finally using the Fourier-Fejer inversion formula we can reconstruct £' ,; (t) from its Fourier coefficients: (24)

£«(*) = lim I#(f)

where for any t e (0,2n) N

^NW - Z 3.

, ( 1

" ^)(«t(E y )cos(«) +

bk(L^)sm(kt)).

Time Series Analysis This Section has two main goals: to show how the Fourier method can be implemented to compute both volatility and cross volatility; to compare the method with other methods proposed in the literature to compute the integrated volatility.

10

3.1 Volatility computation Efficiency of the Fourier method to compute the integrated volatility of a stochastic process representing the asset price has been analyzed in several papers, see [15,16,33,50,48]. To implement the method, [15, 16] proceed as follows. Given a time series of N observations (f,-,p(f,)), i = 1,... N, data are compressed in the interval [0,2n] and integrals are computed through integration by parts: ~ P ( 0 ) + k hip). n To compute the integrals, we need an assumption on how data are connected. As a matter of fact, high frequency data are not equispaced and therefore there is no constant time length between two consecutive observations. To handle this problem, a time grid with a fixed time interval is chosen (e.g., 30 seconds, 1 minute, 5 minute): h, k = l,2,.... We may not have an observation on a point of the grid. To cope with this problem, two methods have been proposed in the high frequency data literature: interpolation and imputation of data. In the first case p(f,-) and p(f,+i) are connected through a straight line: iff* e [f„ fi+1[, then p(f*) = {tk-U)p{t'^x7tfi). Instead according to the interpolation method p(fjt) = p(f,-) (piecewise constant). In [15,16] the imputation method has been employed, then the integral in (25) in the interval [f„ f,+i] becomes (25)

ak(dp) =

P(27r)

k r f ' +i (26)

-

I

k rtM sin(fcf)p(f)df = p(f,)- I

sin(fcf)df

= p(ti)-[cos{kti) - cos{kti+i)) thus avoiding the multiplication by k which amplifies cancellation errors when k becomes large. The methodology has been applied to compute volatility in a standard GARCH setting. Let p(f) = log S(f), where S(f) is a generic asset price, and rt = p(t) - p(f - 1) is the logarithmic return. It is assumed that the asset price follows the continuous time GARCH(1,1) model proposed in [47]: {

'

rfp(f) = a(f)rfWi(f) do2(t) = d(a> - a2(t))dt + V2A0a2(f)dW2(f),

where Wi, Wz are independent Brownian motions. This model is closed under temporal aggregation in a weak sense, see [22], and its discrete time analogous is given by: (28)

dt£t ~2 = _ 4> ,„ j + .„ 0f +1

. v2a-rf+p-o-f j . a . „2

11

where ef are i.i.d. Normal distributed random variables. The exact relation between (i/», a, ft) and (6, co, A) is derived in [22]. The task is to assess the capability of the Fourier method to reproduce the theoretical volatility of the G ARCH(1,1) model. The analysis is based on Monte Carlo simulations. High frequency unevenly sampled observations have been generated as follows: one day of trading [0,1] has been simulated by discretizing (27) with a time step of one second, for a total of 86.400 observations a day. Then observation times have been extracted in such a way that time differences are drawn from an exponential distribution with mean equal to T = 45 seconds, which corresponds to the average value observed for many financial time series. As a result, we have a dataset [k, pih), k = l,...,N) with tk unevenly sampled. The most used way to compute the integrated volatility is to exploit the quadratic variation and therefore to compute i a2(x)rfr as the sum of squared intra-day returns (realized volatility), see [36] for the relation with the Fourier method. Provided a grid with m points ( - , - , . . . 1), volatility in a day is computed as follows:

(29,

a > ) = £[ p (i)_ p (i^)]

J

.

Theoretically, thanks to the Wiener theorem, by increasing the frequency of observations, an arbitrary precision in the estimate of the integrated volatility can be reached. Note that the Fourier method uses all observations, instead the sum of squared intraday returns uses only a fraction of observations, i.e., for low m some observations are lost. In most of the papers estimating volatility with high frequency data, e.g. see [1], (29) is computed with m = 288 corresponding to five-minute returns. In the simulation setting it is also considered m = 144, corresponding to ten-minute returns, and m = 720 corresponding to two-minutes returns. In the literature the interpolation technique is employed. The performance of the Fourier method is compared to that of (29) with m = 144,288,720 by the statistics: L a2(s)ds - a2

X

°2&ds

/

I

l

„.

..\2"|2i

J 0 V(s)ds-<7 2 ' f

Jo^2(s)ds J

where a2 is the estimate and f o2(s)ds is the integrated volatility generated in a simulation, whose value is known in the simulation setting. We also evaluate the forecasting performance of the model (28), when ex-post volatility is measured by a2. This is done by means of the R2 of the

12

linear regression (30)

d*+l=a + b-ff*+1+ e(.

We recall that without manipulating the data, we should observe smaller ji and std when increasing the frequency. Figure 1 shows the results on simulated time series with a = 0.25,/? = 0.7, i/>(l - a - fi) = 1. First let us consider the realized volatility. The ten-minute estimator provides a downward biased estimate of the integrated volatility with a standard deviation larger than the bias. The five-minute estimator is also downward biased with a standard deviation of the same order of the bias in mean. Increasing further the frequency, the estimator is characterized by a smaller variance but a larger bias is observed. This effect can only be due to the interpolation scheme described above and therefore it can be linked to non-synchronous trading, see also [37]. The Fourier estimator is characterized by the smallest bias in mean and by a variance smaller than that of the 5-10 minutes estimate and slightly larger than that of the 2 minutes estimate. To check the robustness of these results, we repeated the Monte Carlo experiments on a grid of values (a, p, ip = (1 — a — /S)_1) with 2 and 5 minutes returns. Results, reported in Table 1, can be summarized as follows: the estimator (29) turns out to be downward biased (^/ > 0), with a bias increasing in m, while the bias of the Fourier estimator is almost null. If m is chosen in such a way that the bias of (29) is less than its standard deviation, then the Fourier estimate provides a smaller standard deviation. Analyzing the forecasting capability of the discrete time GARCH model (28) we have that the Fourier estimate renders a better performance than the classical estimator computed with 2 and 5 minute returns. The Fourier methods to compute volatility has been applied in several directions. Volatility is not constant over time; empirically it has been shown that days with high (low) volatility are followed by days with high (low) volatility, i.e., there is persistence in the volatility process. In the '90s, a large set of volatility models capturing a persistence component has been proposed in the financial time series literature. GARCH models provided the mile stone. The main problem with these models was that while their forecasting performance in sample was good, the forecasting performance out of sample using rough volatility proxies, e.g., square of the closing price minus the opening price, was very poor. [1] have shown that the problem was given by a poor ex post computation of volatility. Using high frequency data and the cumulative squared intraday returns to compute volatility they show that the forecasting performance of the standard GARCH(1,1) model in predicting the exchange rate volatility is quite good. [16] apply the Fourier method to compute volatility of the Deutsch Mark-Us dollar and of Yien-Us dollar exchange rate considering a one year of high frequency

13 Table 1 p,std,R2 (multiplied by 100) for the three estimators: (29) with m = 720 denoted by 2', (29) with m = 188 denoted by 5' and the Fourier estimator denoted by F, on a grid of values for («,£) in (28), and i/> • (1 - a - f) = 1. All the values are computed with 10000 "daily" replications.

/U a-»

0.5 std R2 0.6 std

R2 0.7 std R2 0.8 std R2 0.9 std R2

2' 23.9 4.4 4.92 23.9 4.4 6.79 23.9 4.4 9.67 23.9 4.4 15.2 23.9 4.4 30.8

0.05 5' 10.0 7.7 4.79 10.0 7.7 6.62 10.0 7.7 9.43 10.0 7.6 14.7 10.0 7.6 29.9

F 0.1 4.9 4.98 0.1 4.9 6.88 0.1 4.9 9.79 0.1 4.9 15.3 0.1 4.8 30.9

2' 23.9 4.5 9.58 23.9 4.5 13.8 23.9 4.4 21.5 23.9 4.4 36.9

0.1 5' 10.0 7.8 9.44 10.0 7.8 13.7 10.0 7.7 21.3 10.0 7.7 36.5

0.15 0.25 0.2 F 2' 5' F 2' 5' F 2' 5' F 0.1 23.910.0 0.1 23.910.0 0.1 23.9 10.0 0.1 5.0 4.5 7.9 5.0 4.6 7.9 5.1 4.6 8.0 5.1 9.70 14.0 13.9 14.2 18.6 18.5 18.8 23.8 23.8 24.0 0.1 23.910.0 0.1 23.910.0 0.1 23.9 10.0 0.1 4.9 4.5 7.8 5.0 4.5 7.8 5.0 4.5 7.8 5.0 14.0 21.9 21.8 22.1 32.5 32.5 33.0 44.6 44.5 45.6 0.1 23.910.0 0.1 23.9 10.0 0.1 23.9 10.0 0.1 4.9 4.4 7.7 4.9 4.5 7.8 4.9 4.5 7.8 4.9 21.7 38.2 37.9 38.7 51.3 51.1 52.2 48.2 48.1 48.8 0.1 23.910.0 0.1 4.9 4.4 7.7 4.9 37.2 54.3 53.8 54.5

Table 2 R2 for the two time series. R2 Estimator DM-$ Y-$ 0.400 0.128 L«NP(258)-PW Fourier 0.470 0.143

observations (from October, 1 st 1992 to September 30th 1993), i.e., the dataset analyzed in [1]. The dataset consists of 1.466.944 quotes for the Deutsch Mark-Us dollar and 567.758 quotes for the Yien-Us dollar exchange rate. Performance of the GARCH(1,1) model has been evaluated when the integrated volatility is computed according to the Fourier method. The parameters of the model are those estimated in [1]. Table 2 provides the corresponding R2. We observe that the GARCH model performs well in forecasting when the Fourier method is employed to compute the integrated volatility. Its performance is better than that associated with the sum of squared intraday returns as an integrated volatility measure. On performance of volatility

14

models and volatility computation see also [31]. As the Fourier method provides an accurate estimate of the volatility, we can handle the volatility as an observable process. In the literature, time varying volatility has been handled as a latent process, e.g., the GARCH process. In [13], this idea has been applied to compute the volatility of the overnight interest rate in the Italian money market and to test for the martingale hypothesis correcting for heteroschedasticity with a good proxy of the volatility and thus avoiding estimation problems connected with the use of a GARCH model. In [17] an autoregressive process for the volatility estimated according to the Fourier method is estimated to forecast oneday return volatility and to define the value at risk threshold. The method performs well in forecasting, better than the GARCH(1,1) model and the exponential smoothing proposed by RiskMetrics. In recent years other volatility models have been proposed to cope with empirical regularities observed for volatility of financial time series. Among the models, we have models with long memory in the volatility process considering a fractional Brownian motion W2 of order d independent of Wi in (27), i.e., W2(t) = J0 nj^)^W(s). To capture sharp increase in volatility, jumps in the volatility process have been introduced adding a Poisson process in (27). In these more general models, the sum of squared intraday returns and the Fourier method do not provide consistent estimators of the volatility. [48] compare through Monte Carlo simulations the bias and the root mean square error of the Fourier method, of the cumulative squared intraday returns and of a wavelet estimator. They show that the Fourier method provides the lowest bias and root mean squared error. They also compare the three methods when a bid-ask bounce effect is inserted, i.e., as a random buy/sell order arrives in the market there is a liquidity effect on the price creating spurious serial correlation in returns and volatility. In this case the realized volatility and the Fourier method are no longer consistent for the integrated volatility. As far as the bias and the root mean square error is concerned, the Fourier method performs better than the wavelet and the cumulative squared intraday returns method. On the comparison between Fourier method and the wavelet method see also [33]. 3.2 Computation of cross volatility In this section we analyze the performance of the method in the bivariate case using Monte Carlo simulations of high frequency asset prices as studied in [44]. As in [51], we simulate two correlated asset price diffusions with the

15

-0.5

-0.4

600

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

0.1

0.2

0.3

0.4

0.5

(b) Mean = 0.10 Std = 0.078

400 200

-0.5

-0.4

-0.3

-0.2

-0.1

0

c Mean = 0.239 Std = 0.045 -0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

-0.5

-0.4

-0.3

-0.2

-0.1

0.1

0.2

0.3

0.4

0.5

Figure 1. Distribution of » _

, where a2 are three different estimators of the integrated

Jo "i2 ( 0 *

volatility: (a) estimator (29) with m = 144; (b) estimator (29) with m = 288; (c) estimator (29) with m = 720; (d) Fourier estimator. The distribution is computed with 10,000 "daily" replications.

bi-variate continuous GARCH(1,1) model: dpi(t) o2(t)dW2(t), ^l7J^e[a\{t)dm(t), da\(t) = X\[w\ - •a\{t)\dt+ da\{t) = A2[w2 - • a\{t)\dt + V2A^<j2(f)rfW4(f), corr(dWi,dW2) = P> dpi(t) •

(31)

and all the other correlations between the Brownian motions set to zero. The choice of this particular model comes from the fact that it is the continuous time limit of the very popular GARCH(1,1) model, and it has been studied extensively in the literature, e.g. see [28]. We will use the parameter values estimated by [1] on foreign exchange rates, i.e. 0i = 0.035,wx = 0.636, Ai = 0.296, d2 = 0.054, w2 = 0.476, A2 = 0.480. We will instead analyze two mirror cases for the correlation coefficient: p = 0.35 and p = -0.35. To get a representation of high-frequency tick-by-tick data, after discretizing (31) by a first-order Euler discretization scheme with a time step

16 Table 3 Average correlation measurement on 10,000 Monte Carlo replications of the model (31). Two generated values of the correlation are considered, p = 0.35 and p = -0.35. We compute the variance-covariance matrix via the Fourier estimator and via the realized volatility estimator (32). L.I. means Linear Interpolation, while P.T. means Previous Tick interpolation. Standard deviations of in-sample measurements are reported in the columns named Std. Estimator Fourier Realized 5', L.I. Realized 5', P.T. Realized 15', L.I. Realized 15', P.T. Realized 30', L.I. Realized 30', P.T.

Generated correlation P = 0.35 Measured Std 0.350 0.039 0.204 0.058 0.060 0.181 0.338 0.090 0.329 0.091 0.345 0.127 0.127 0.342

Generated correlation P = 0.35 Measured Std -0.349 0.039 -0.203 0.055 -0.180 0.058 -0.337 0.090 -0.328 0.092 -0.344 0.126 -0.341 0.126

of one second, we extract observation times drawing the durations from an exponential distribution with mean 30 seconds and 60 seconds respectively. Observation times are drawn independently for the two time series. After simulating the process (31) we compute on it daily (86400 seconds, corresponding to 24 hours of trading, as for currencies) variance-covariance matrix according to the Fourier theory and according to the realized volatility measure of [3], given by:

The choice of m in (32) comes from a tradeoff between increasing precision and cutting out microstructure distortions. A typical choice is m - 288, corresponding to five minute returns. As pointed out above, to obtain an equally spaced time series we can rely upon a linear interpolation (L.I.) or a previous tick interpolation (P.T.). Both methods have been applied for m - 288,96,48 (corresponding respectively to 5,15,30 minute returns), when measuring correlations on Monte Carlo experiments. Table 3 shows the results. First of all we notice that the Fourier estimator performs considerably better than the realized volatility, which is biased toward zero. The bias in the correlation measurement of realized volatility is more and more severe when the sampling frequency is increased. For the fiveminute estimator with the previous-tick interpolation, we get a mean value

17

of 0.181 (-0.180), which is quite far from the true value of 0.35 (-0.35); this bias is completely due to the non synchronicity of quotes. Realized volatility with linearly interpolated returns is closer to the right value, but this is because of the downward bias in the volatility measurement due to the linear interpolation documented in [15, 16]. In these papers, it is shown that the spurious positive serial correlation induced by the linear interpolation technique lowers volatility estimates. Since variances are spuriously measured to be lower, correlations turn out to be spuriously higher, thus compensating in some way the bias due to non-synchronicity. This is also true, but to a much lower extent, for the 15-minute and 30-minute realized volatility estimator. The precision of the Fourier estimator, as measured by the standard deviation of measurement errors across Monte Carlo replications, is always better than the realized volatility estimator. We implemented the Fourier estimator with N = 500 coefficients for the first (30 seconds average spacing) time series, N = 160 coefficients for the second (60 seconds average spacing), and N = 160 coefficients for the computation of covariance. Increasing the number of coefficients would increase the variance measurement precision but not the covariance measurement precision, because of the Epps effect, see [51]. On the other hand, even the gain in precision of the realized covariance measurement obtained when increasing the sampling frequently is cancelled out by the bias. 4.

Applications The potentiality of our method relies in the fact that it reconstructs instantaneous volatility and cross-volatility as functions of time. This feature of the Fourier method is essential when a stochastic derivation of volatility along time evolution is performed as in contingent claim pricinghedging. In this Section we present some developments of the multivariate Fourier estimation procedure which illustrate this point. 4.1 Leverage effect There is a vast empirical literature showing that the asset price and its volatility are related. Among other phenomena, it has been observed the so called leverage effect, i.e., a negative return sequence is associated with a volatility increase. In some recent papers, see [12,10], the integrated volatility estimation methodology based on the so called realized volatility has been extended to allow for a leverage effect. From the mathematical point of view the no leverage hypothesis means that a{t) is independent from the Brownian motion W. The no leverage hypothesis simplifies the study of the properties of the volatility estimator, but is not realistic for the analysis of equity returns. The asset price model H is very general and includes stochastic volatil-

18

ity as well as level dependent volatility models, i.e. volatility is a time independent function of the asset price, it allows for feedback effects of asset price on volatility and in particular the leverage effect. In this section it is proved that these effects can be non-parametrically estimated using the Fourier methodology without knowledge of the exact form of the evolution equation for the volatility process. A first order indicator of the stochastic dependence between the asset price process and the volatility process is obtained and we refer to it as leverage effect, we also consider a second order indicator which is called feedback effect rate. Consider the asset price model which satisfies hypothesis H. Moreover assume that the process a(t) is a random process satisfying (H)

da(t) = a(t)dW(t) + j3(f)dWi (t) + y(t)dt

where a, fS and y are predictable functions (for simplicity suppose they are almost sure bounded) and Wi is another Brownian motion independent of W. Then the leverage effect is defined as B(t) := dp(t) * do2(t), where * denotes the Ito stochastic contraction divided by dt. Therefore B(t) = 2a(t)oz(t). Nevertheless we have no knowledge of the random function a(t). Therefore it is interesting to find a formula for estimating B(t) starting from the price observations. By Ito calculus the following result holds: Theorem 4.1. Let p and a satisfy H and H, then we have B(t) = \(Vol(p + a2)(t) - Vol(p - a2){t)), where Vol(p ± a1) means the volatility function of the stochastic process p ± o2. In order to estimate B(t) from the asset prices we see that the Fourier transform of p can be computed from price observations; the Fourier transform of a2 is computed by Theorem 2.1; therefore the Fourier transform of p + a2 and p - a2 are known; a second application of Theorem 2.1 gives the Fourier transform of the volatility Vol(p ± a2) and finally the usual inversion formula produces the computation of B(t).

19 4.2 Delta hedging (estimation of the gradient) Suppose that the price process p(t) in logarithmic scale satisfies (33)

dp(t) = a(p(t))dW(t) - \a2{p{t))dt

where W is a Brownian motion and a is an unknown C1-function . No hypothesis on the shape of this function are done. In the following it is shown that is possible to estimate the pathwise gradient using Fourier transform methodology, therefore also the Greek Delta can be estimated non-parametrically. In financial applications the Greek Delta is the sensitive which allows to perform the so called Delta-hedging, that is to make the portfolio neutral with respect to small modifications of the initial value of the price. In model (33), the Greek Delta is defined as

A(t)=-fEf(t)[E=0 where pc{t) corresponds to the solution of (36) with pe(to) = p(to) + £• Then (34)

dA(t) = A{t)[a'{p(t)) dW(t) - a(p{t))o'(p(t))dt).

It follows that the computation of A(t) depends on the estimation of the volatility a(p(t)) and the derivative of the volatility a'(p(t)). Denote by B(t) the function obtained in Theorem 4.1, then it holds

(35

>

"'W))-M)Y

This first result expresses the derivative of the volatility function as the ratio of terms which can be estimated using prices data. By substitution of (35) in (34) and using (33), it follows

Note that all terms in the above equation are given or estimated by Fourier method from market asset prices. A more general Delta hedging result has been obtained in [14] by using the Fourier method. 4.3 Feedback volatility rate In [14] we have constructed a second order indicator of the feedback effects of asset price on volatility and we have used the Fourier estimation method in order to compute this effect from asset price data. Working under this general assumption, we produced a time-dependent pricevolatility feedback effect rate function and in the multivariate case a timedependent matrix, which will be called elasticity matrix, implementable in real time from asset price observations.

20

The mathematical theory suggests that eigenfunctions associated with positive eigenvalues of the elasticity matrix correspond to instability directions of the market; eigenfunctions associated with negative eigenvalues correspond to stability directions of the market. The key mathematical tool to construct the elasticity matrix is a new methodology of transferring price perturbations through time, using the inertial frame transport, this transport is designed so that through time the variation is governed by a first order linear ordinary differential equation. The rate of variation through time of the initial perturbation is given by the elasticity matrix. Computation of the elasticity matrix is a three-steps outcome; at each steps it is necessary to compute volatilities of observed quantities or computed in the previous step. Therefore the Fourier methodology works out. We can write explicit expressions in terms of Fourier coefficients of asset prices leading to real-time determination of feedback rates. For simplicity we present the univariate case, but the method can be developed for any finite number of assets (see [14]). Let p(t) be the risky asset price at time t. Suppose it satisfies the stochastic differential equation in logarithmic scale: (36)

dp(t) = o(p{t)) dW(t) -

\a2(p(t))dt

where W is a Brownian motion, o is a fixed but unknown C2(R) function. Let C(f) be the variation process which is solution of the linearized stochastic differential equation dlit) = C(0 k'(p(0) dW(t) - a'(p(t))o(p(t))dt]. We associate to C(0 the rescaled variation defined as z{t) =

a(p(t)Y

In Section 4.2 we have obtained the SDE driving the Delta propagation. It is a remarkable fact that this SDE can be reduced to an ODE at the price of a renormalization of the Greek Delta, as it is showed in the following result. Theorem 4.2. The rescaled variation is a differentiable function with respect to t; denote by A(t) its logarithmic derivative, then


2(0 = exp(

A(T)dx)z(s),

21

Figure 2. Average estimate of daily volatility A(t) on IBM data. On the x axis, the time window [0,2n] corresponds to one trading day (6.5 hours). A(t) displays the typical (i-shape of volatility in stock markets.

where A(0 = -^{a(p{t))a'(p(t)) + a{p(t))a"(p(t))). Definition 4.3. We will call A(t) the price-volatility feedback rate function. Note that in the standard Black-Scholes framework A -0. We derive a nonparametric estimation of the feedback volatility rate by using Fourier methodology. We suppose that we do not know the explicit expression of the function a; we want to obtain a non parametric estimation in real time of A(t) from the observation of a single market evolution. Theorem 4.4. Denoting by • the ltd contraction divided by dt, and define the following cross-volatilities: dp * dp =: A(t), dA * dp =: B(t), dB * dp =: C(t). Then the price-volatility feedback effect rate function A can be expressed as 3B2

IB 4A

2Ar

We stress the point that all terms A, B, C can be obtained from the asset price data through the Fourier cross-volatility estimation. Therefore the feedback rate A(t) can be empirically estimated. The theory suggests that the sign of A is associated with stability of the market: a negative A would witness a period of stability, a positive A would signal instability. Precise estimation of quadratic and higher order variation asks for huge quantities of data; high frequency data are a natural candidate for this

22

purpose. In [14] we have used two data sets: a data set containing quotes of the DEM-USD foreign exchange rates from October 1992 to September 1993 and a data set of IBM quotes from January to December 1999. In the case of the IBM stock price, estimates of A(t), B(t), C(t) have been computed for a one day time window (6.5 hours); the larger number of Fourier coefficients produces a higher resolution of the plots. Figure 2 gives the volatility A(t) averaged over the full year; here we recognize the LZ-shape pattern which is typical of stock market intra-day volatility. Short-horizon estimates of A are the most important for traders. We present in Figure 3 as typical sample of daily (non-averaged) estimates the values for A(t), B(t), C(t) and A(f) for two days in 1999. It is noteworthy that taking the logarithm of the stock price mainly changes the scales of A(t), B(t) and C(t), but lets the shape of the curves more or less invariant. For this reason, in Figures 2,3 the scales have been chosen according to the stock price (without taking logarithms). On January 4th, 1999, the beginning of the trading day reveals positivity of A which detects instability of the market and which is revealed by subsequent large picks of A. Over the whole day the positive values of A dominate and indicate an unstable trading day. On April 9th, 1999, the beginning of the day reveals negativity of A, which indicates stability of the market and which is revealed by a subsequent progressive damping of A. In contrast to January 4th, over the whole trading day, small and mainly negative values of A dominate and indicate stability. As argued above, precise estimation of A could, in principle, result in important consequences for trading strategies. In this context, it can be of some help to analyze the sign of A week-by-week or even day-by-day; our results show that, by using high frequency data, an estimate of this effects can be readily accomplished. 4.4 Dynamic principal component analysis The Fourier method of cross-correlation estimation has been applied in [44] to develop a dynamic principal component analysis. The theory differs from the statistical principal component analysis because we do not assume any linear functional dependence; moreover we describe a dynamic situation, while usually principal component analysis describes a static situation. Classical PCA provides a linear sub-manifold of smaller dimension which carries the essential information coming from the data. Dynamic PCA will produce an abstract curve, which we call the Core, which allows us to determine the eigenvectors of multivariate volatility in continuous time. The analysis is model free. Making mild assumptions on the multivariate prices behavior (Brownian semi-martingale hypothesis), we get by the Fourier method multivariate volatilities as a time dependent quadratic form, let E(f). We suggest a

23 188 187 186 185 184 183 182 181 10

- stock price 1

stock price 1

1

A(t)

_ A(t)

% i

i

i

Figure 3. Daily values of A(t), B(t), C(t), A(t) on IBM data. The time windows [0, In] correspond to two typical trading days in 1999 (6.5 hours each). Jan 4 (left-hand side) displays positivity of A with large picks of A (instable market); April 9 (right-hand side) displays small and mainly negative values of A along with a progressive damping of A (stable market). During the first two hours of trading, A has about the same shape on both days, but A develops dramatically different shapes; computing A in real time could give an indicator forecasting market instability.

24

method to construct an abstract curve which contains the essential information coming from multivariate volatilities. The analysis of the evolution in time of this curve can be used to decipher the stability of the market about an asset, or the degree of market integration of a given asset. We then introduce the concept of reference assets, via a geometric definition: a reference asset is one whose volatility is mainly due to the market volatility instead of its idiosyncratic noise. We omit mathematical details of the construction, while we illustrate the ideas via the analysis of two months, April and May 2001, of highfrequency data for 98 stocks, selected into the S&P 100 index, among the most liquid ones. The results point out the need of a time-dependent analysis versus a static one, since the variance-covariance eigenvalues structure turns out to be deeply time-varying. In particular, there are some days in the market in which correlations are widely distributed, thus less factors are needed to explain the variance-covariance structure. On the whole, we analyze 42 days; for each day, we compute the variance-covariance matrix using the Fourier method. We start by performing PCA for each day, after normalizing the variance-covariance matrix in order to have the variance of every stock equal to 1. The first factor explains, on average, 25.79% of the movements. However in some days the first factor's weight can be as large as 56.09% (April, 18*) and 73.26% (May, 16"1). Moreover, and more interestingly for any financial application, this phenomenon seems to present some degree of persistence. In the second step of our analysis, we define the Core of the market as the vector subspace spanned by the first 30 eigenvectors, and we divide our 42 market days into 6 periods of 7 market days each. In each subperiod, we perform principal component analysis on the aggregate variance-covariance matrix, and we obtain the coordinates of the Core. In each subperiod, we define 15 reference assets as those who have the largest projection onto the Core. We interpret these assets as those who are more correlated with the market itself, or alternatively as the basic constituents of the market. The list of the reference assets in any subperiod is shown in Table 4. Given the low value of the average correlation, we expect that the basket of reference assets is quite variable, given the unalienable noise in the correlation measurements. This is indeed the case. The month of April shows more persistence: 4 reference assets in the first period, out of 15, are in the second period too; and 6 of the second are still reference assets in the third. The month of May shows much more variability, or less "market integration"; only two stocks are reference assets in the third and fourth period, only one in the fourth and fifth period and none out of 15 in the fifth and sixth period. Looking at individual stocks, AES Corporation is a reference asset in the whole sample, with the exception of the sixth period,

25 Table 4 Lists the reference assets by ticker name in the six subperiods considered. They are ranked according to their projection on the core, which is reported in brackets. Also the five assets with the smallest projection on the Core are listed. Each subperiod is composed of 7 market days. In bold face, we indicate those stocks who remain reference assets in the subsequent period.

1 F (74.4%) CCU (72.2%) AOL (71.1%) INTC(70.0%) AES (69.5%) PHA (58.8%) IP (64.6%) UTX (64.1%) CPB (64.0%) GE (63.7%) TYC (63.67o) EK (63.2%) VZ (62.0%) BNI (62 0%) UIS (61.6%) BMY (43.0%) AXP (42.8%) MMM (42.0%) MSFT(391%) MAY (37.4%) BAX (37.0%)

2 AES (93.3%) AOL (86.1%) INTC(84.1%) AA (82.0%) JPM (80.0%) IBM (79.9%) JNJ (78.5%) AEP (77.5%) ATI (74.8%) MRK (70.5%) IP (70.0%) T (69.6%) MCD (67.7%) LU (67.5%) MEDI(66.6%) USB (35.9%) HWP(35.2%) S (35.2%) KO (35.1%) GD(33.1%) VZ (28 5%)

Periods 3 4 AES (84.0%) WY (72.2%) WFC (77.1%) TOY (72.1%) ATI (73.9%) AES (72.0%) LU (73.4%) WMB (68.2%) AEP (72.5%) CSC (67.7%) AVP (72.2%) XOM(672%) TXN (70.3%) MSFT(66.3%) BUD (69.8%) PFE (65.8%) AIG (67.2%) PEP (65.1%) MRK (64.9%) CSCO(65.1%) LTD (64.8%) BAX (64.4%) AMGN(64.7%) BMY (64.3%) SLB (63.1%) MDT (63.5%) JNJ (61.7%) WFC (63.1%) ETR (62.8%) EMC (615%) IBM (40.4%) BNI (41.4%) HON (39.7%) IP (40.3%) DIS (38.1%) LEH (38.3%) AIG (37.6%) IBM (37.9%) DD (37.5%) HD(31.0%) T (29.1%) T(30.3%)

5 AES (88.9%) HIG(78.7%) HAL (76.7%) WMT(71.8%) HWP(71.7%) TYC (70.9%) ONE (68.8%) HNZ (68.3%) F (68.1%) UTX (67.3%) HCA (66.8%) ORCL(66.1%) HET(65.5%) DIS (64.9%) VIAB(64.1%) MER(37.4%) LU (36.3%) NT (34.4%) AA (34.3%) WFC (30.9%) HON (28.0%)

6 HON (86.4%) GM (74.7%) DOW (72.1%) IBM (70.4%) GE (70.4%) LTD (70.4%) BUD (69.7%) FDX (68.4%) JNJ (67.6%) MAY (67.1%) MSFT(66.4%) AIG (65.5%) ETR (65.3%) MMM (64.6%) AXP (63.3%) HCA (37.4%) PFE (36.9%) SLE (36.7%) VIAB(35.4%) CL (34.6%) AES (30.2%)

and in three periods it is the asset with the largest absolute projection on the Core. Table 4 also shows the percentage projection on the Core, and the five assets, in any period, with the lowest projection onto it. For example, in period 2,93.3% of the variance of AES can be considered to be driven by the market, and 6.7% is explained by idiosyncratic fluctuation; thus AES essentially lies on the Core, which is the subspace which explains most of the variance of the whole market; on the other hand, in the same period for Coca-Cola (KO), only 35.1% of its variance is driven by the market, while 64.9% is independent fluctuation. Summarizing, out of the 98 assets, 32 are never reference assets; 46 are only once; 18 are twice, Johnson & Johnson (JNJ) is thrice and AES is five times. Then our analysis identifies nearly 20 assets which had a major role in market integration. In the set of this 20 assets, 8 are among the 20 most capitalized; thus capitalization plays an important role in defining leading assets, but it is not the only factor to be taken in account. For example, in our analysis AES turned out to be the most important stock, but its capitalization (measured as market value) is only about 0.5 of the capitalization of Microsoft, the most capitalized stock in our sample. 5. Laplace Transform Method for Volatility Computation The Fourier theory has been extended in [42] where it is shown that the Laplace transform is an appropriate tool to build estimators of the in-

26

stantaneous volatility based on a long time series of prices by smoothing past data and retaining recent price observations. The Laplace transform estimator has the same advantages of the Fourier estimation procedure in comparison with the quadratic variation methods, being based on the integration procedure of all the data as the Fourier estimation method. Moreover it has further good features: in [41] it was shown that the estimation efficiency is better in the center of the considered time window, while the use of Laplace transform estimator allows to obtain an estimator of instantaneous volatility which becomes less sensitive to boundary effects when approaching to the present time (say going from (t = —oo) to (t = 0)). From a conceptual point of view the introduction of Laplace transform has two advantages: firstly to avoid the artificial modification into periodic functions subjacent to Fourier series; secondly to lead to formula (41) which constitutes a bridge between the two different methods of computation of volatility, the method based on quadratic variation and our approach by Fourier series. On the other side the Laplace method is based on the use of integrals instead of series; the discretization of these integrals will introduce a drawback from the numerical point of view. For simplicity consider (37)

dp(t) = a(t)dW(t)

the log-price process, where W(t) is a Brownian motion on (-oo, 0] and a(t) is a stochastic process adapted to the filtration generated by W. Consider the Laplace transform of dp on (-°o,0] : (38)

<&iv(z):= I

dp{t)exp(zt).

%J — oo

Integrating by parts we have that 5>w(z) can be expressed as (39)

®w(z)=

I

exp(zt)a(t)dW(t).

U—oo

We take z = a + is. In [42] it is proved that the Laplace transform of the volatility process a2(t) (which is expressed by the right hand side of (40)) can be exactly computed through the Bohr convolution product of the Laplace transform of dp(t), which is expressed by the function <&w defined in (39). Then the Laplace inversion formula allows to reconstruct the process a2(t) for any f e (-oo,0]. _ Later on we denote by tp the conjugate of any complex function ip.

27

Theorem 5.1. Let <&w(z) be the Laplace transform of dp as given by (39). Then the following convergence in probability holds (401im — I <$>w(a + is) w(a + is + iv) ds = I

a2(t) exp((2a - iv)t) dt.

Remark 5.2. In formula (40) we have the choice of an arbitrary parameter a; we must consider always a > 0. If we want to damp quickly the effect of the past time we have to take \a\ large. Using the exact formula (40) it is possible to derive an estimator of volatility given a discrete unevenly spaced sampling of the price process p, containing only an averaged sum of the jumps of the prices combined with universal kernels. Denote by £,• the times at which prices are observed, let -oo < ... < tj+i < tj < ... < to <0 and let 6,(p) := - p(fi) + p(f,-_i), for any i>l. Let the integration interval in (40) be fixed equal to [-R, R], then the following approximation formula for the volatility function can be obtained:

(41)

Vol(p){t) * YJ &>(P)¥P)Mt - tjWRiti - ';)) i,)>0

where u?, is defined for 5 > 0 by 1 s sin A u&(s) = g«i(g), "i(s) = l|S|(A) = —^—. 6. HJM Model Looked as an Hypoelliptic Operator Consider the Heath-Jarrow-Morton model [32] for the interest rates in the Musiela parametrization, see [45]. In this parametrization a market state at time fo is described by the instantaneous interest rate curve rto(E.) to a maturity £, defined for E, € R + . The price P(to, T) of a default-free zero coupon bond traded at to with maturity to + T, has the value P(£o,T) = exp( f r t o (£)#)• Jo The interest rate curve takes its value in the infinite dimensional space C of continuous functions on [0, oo[. A remarkable experimental fact is that the rank n of its volatility matrix is very low n - 4, see [18] . Then elliptic models are ruled out and hypoelliptic models are the most regular models still available.

28

By restricting the HJM model to the case of a finite number of scalarvalued driving Brownian motions W\,..., Wn, then the HJM model is expressed by the following SDE: r-.

n

„£

n

drt(Z) = YdBk(t,Z)dWk(t)+\£{S)+Yi k=i

^

^

I k=i

Jo

\

Bk(t,QBk(t,rj)dri\dt, V £ e R + , '

where the drift of the risk-free process is completely determined under the no arbitrage condition by the volatility matrix B,(t, •). In fact it is known from [21] that the risk-free generator associated to the risk free measure has its infinitesimal generator fully determined by its second order terms. For the stock price the drift associated vanishes under the risk free measure. For the interest rate model the drift is determined by the no-arbitrage condition implemented in the HJM model. This key property implies that the infinitesimal generator of the HJM process can be pathwise and model free computed through the computation of the volatility matrix. Under the hypothesis of Markovian completeness of the market, the dependence of the volatility matrix B»(£, •) is factorizable through the final value rt(), therefore it is possible to write n

(42)

drt(E,) =

Yj,Mrt)){l)d\Nk{t) k=l =i

J•{ %& + E X ^'W^faNW ^ } dt> where Ak are "driving vector fields" defined on C. An appropriate notion of "smoothness" of vector fields is a necessary hypothesis in order to prove existence and uniqueness of solutions for (42), see [24]. In [40] a non-parametric estimation of HJM generator is obtained and a method similar to the methodology of the price-volatility feedback rate is developed for the interest rate curve. The question of the efficiency of the mathematical theory proposed in this section to decipher the state of the market has not yet confirmed by numerical computations. The first point is the possibility of measuring, in real time by high frequency market data, the full historical volatility matrix. We remark that, while the correlations between stocks prices at high frequency has no clear economic meaning, on the contrary the high frequency cross-correlations are clearly significant in the HJM model. Assume that the HJM hypothesis for the infinitesimal generator is satisfied; no assumption on the explicit expression of the vector field Ak in (42) are done. It is proved in [40] that an estimation of the maps 11-> Ak(rt) is achieved by the observation of an unique evolution of the market.

29

Fix N different maturities (N can be extremely large and in particular it can ben « N), then define the time-dependent N xN covariance matrix [a2(t)h,n := [drt(E) * drt(»j)]4„

(43)

where £, r/ vary between all maturities and * denotes the ltd contraction divided by dt. For simplicity of notation consider two maturities £,17 and the bidimensional process: t \-> (rt(E,),rt(q)). Then it holds: Theorem 6.1. Given two different maturities £, q we consider the 2-dimensional process: 11-> (rt(£,), rt(r\)); then (44)

[o2(t)hn = \[ Vol(rt(0 + 17(17)) - Vol{rt{E,) - rt{rr))\

The vector Aj(rt) is defined as thef1 column of the matrix yJoHf) Note that the r.h.s. of (44) can be computed using the Fourier methodology for the estimation of volatility. 6.1 Non parametric estimation of Hormander bracket Definition 6.2. Given two vector fields A\, Az denote by A^ the 4-th component of Ak. The Lie bracket of A\ and Az is the vector field having the components:

,

(45)

[

^

dA\

2 ] 4 : =

dA\

^.^

„( =

ndA\

dA\\

j ; ^ . ^ .

The Lie algebra 31 generated by n vector fields A\,..., A„ is defined as the vector space of all fields obtained as linear combinations with constant coefficients of (46)

Ak, [Ak,Ail [[Ak,A,],As],

[[[Ak,A,lAs],Au],...

Given r e R N , let & := {C e R N | C = Z(r) for some Z e &\. Definition 6.3. The vector fields A1,... A„ satisfy the Hormander criterion for hypoellipticity if A\,...,A„ are infinitely often differentiable and if JR(r) = RN for any r e RN. Due to the non-ellipticity of the market, the multivariate price-volatility feedback rate constructed in [14] cannot be applied, so [40] construct a pathwise econometric computation of the bracket of the driving vector of the diffusion.

30

Without any hypothesis on the explicit form of the vector fields Ak{rt), we estimate the bracket [A\, A?\{rt) from a single trajectory. For simplicity we omit writing explicitly the dependence of Ak on rt. In order to get this result, by the definition (45), the main point is the estimation of

= l,2.

P,k drn

This estimation is obtained by the relation

(47)

dA2 *dr? = J^^x

[a2(t)l,lA

n together with the fact that the l.h.s. of (47) can be computed by Fourier methodology. Therefore we can state the result: Theorem 6.4. The brackets of the driving vector fields of the HJM diffusion can be numerically computed from a single time series of market data, without any assumption on the model. 6.2 Global and pathwise compartmentage by maturities Let C the space of continuous functions on [0, oo[. Define the logarithm derivative of a measure ji,\ depending on a parameter A as the function satisfying

h ff(r)

Mdr) = ff(r)

(d, log(^))(r) ^(dr),

for every test function / . Denote nt(r0, dr) the fundamental solution of the heat equation associated to the HJM model; define an Hilbert norm on tangent vectors at r0 by IN? = f [-f

(logfa(r 0 + ez,*)){r)f

nt(r0,dr).

Definition 6.5. We say that the global compartmentage lowing condition holds:

holds if the fol-

the norms ||z||s and ||z||s<, s ± s' are inequivalent. In [40] it is proved that the global compartmentage condition holds generically in the HJM setting. As a consequence, it is showed that it is impossible to hedge an European option at maturity T by trading options on maturities larger than T.

31 The operator Q^ is defined in [40] (5.3) b y solving the linearized HJM model: n

(48)

dCw(f) = Y, Mttt))dWk(t) + A0(at))dt,

w h e r e the function Ajt, taking values in the space of N x N matrices, is defined b y

dA r

l

Choose o n C a n Hilbert metric || * \\c; b y using the canonic Hilbert structure of L 2 ([0,s]; Rn), the adjoint (Q^ )* is defined a n d it is possible to define the Malliavin covariance matrix: °w, = Qws ° (Qw)'Consider a fixed trajectory W of the Brownian motion. Denote the covariance matrix c o m p u t e d on [0,s]. Theorem 6.6. The pathwise compartmentage cr^(s) is a strictly increasing function ofs.

a^(s)

property holds if the range of

Therefore it is possible to d e d u c e the property of pathwise compartm e n t a g e from the behavior of H d r m a n d e r brackets, which w e s h o w e d can b e estimated non-parametrically using Fourier methodology. References 1. Andersen, T. and Bollersiev, T. (1998) Answering the Skeptics: Yes, Standard Volatility Models do Provide Accurate Forecasts. International Economic Review, 39, 885-905. 2. Andersen, T., Bollersiev, T. and Diebold, F. (2002) Parametric and Nonparametric Volatility Measurement. Handbook of Financial Econometrics. 3. Andersen, T., Bollersiev, T., Diebold, F. and Labys, P. (2003) Modeling and forecasting realized volatility. Econometrica, 71,579-625. 4. Andersen, T., Bollersiev, T., Diebold, F. and Labys, P. (2001) The distribution of exchange rate volatility. Journal of the American Statistical Association, 96, 42-55. 5. Andersen, T, Bollersiev, T. and Lange, S. (1999) Forecasting Financial Market Volatility: Sample Frequancy vis-a-vis Forecast Horizon. Journal of Empirical Finance, 6,457-477. 6. Bandi, F. M. and Phillips, P. C. B. (2003) Fully nonparametric estimation of scalar diffusion models. Ecometrica, 71(1), 241-283. 7. Bandi, F. M. and Russel, J. R. (2003) Microstructure noise, realized volatility and optimal sampling. Preprint.

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8. Barndorff-Nielsen, O. E. and Shephard, N. (2002) Econometric analysis of realised volatility and its use in estimating stochastic volatility models. Journal of the Royal Society, Series B, 64, 253-280. 9. Barndorff-Nielsen, O. E. and Shephard, N. (2003) Realised power variation and stochastic volatility. Bernoulli, 9, 243-265. 10. Barndorff-Nielsen, O. E. and Shephard, N. (2005) Power Variation and Time Change. Probability Theory and its Applications (forthcoming). 11. Barndorff-Nielsen, O. E. and Shephard, N. (2005) Variation, jumps, market frictions and high frequency data in financial econometrics. Preprint. 12. Barndorff-Nielsen, O. E., Graversen, S. E., Jacod, J. and Shephard, N. (2005) Limit theorems for bipower variation in financial econometrics. Preprint. 13. Barucci, E., Impenna, C. and Reno, R. (2003). The Italian overnight market: microstructure effects, the martingale hyphotesis and the payment system. Temi di Discussione, Bank of Italy. N. 475. 14. Barucci, E., Malliavin, P., Mancino, M. E., Reno, R. and Thalmaier A. (2003) The price-volatility feedback rate: an implementable mathematical indicator of market stability. Mathematical Finance, 13,17-35. 15. Barucci, E. and Reno, R. (2001) On measuring volatility of diffusion processes with high frequency data. Economics Letters, 74,371-378. 16. Barucci, E. and Reno, R. (2002a) On measuring volatility and the GARCH forecasting performance. Journal of International Financial Markets, Institutions and Money, 12,183-200. 17. Barucci, E. and Reno, R. (2002b). Value at risk with high frequency data. In C. Zopounidis (Ed.), New Trends in Banking Management, 223-232. Physica-Verlag. 18. Bouchaud, J. P., Cont, R., El-Karoui, N., Potters, M. and Sagna, N. (1999). Phenomenology of the interest rate curve. Appl. Math. Finance, 6(3), 209-232. 19. Campbell, J., Lo, A. and MacKinlay, C. (1997) The Econometrics of Financial Markets. Princeton Univesrity Press. 20. Comte, F. and Renault, E. (1998) Long memory in continuous time stochastic volatility models. Math. Finance, 8,291-323. 21. Delbaen, F. and Schachermayer, W.: The No-Arbitrage Property under a Change of Numeraire, Stochastics Stochastics Rep. 53 (1995), no. 3-4, 213-226. 22. Drost, F. and Werker, B. (1996). Closing the GARCH Gap: Continuous time GARCH Models. Journal of Econometrics, 74,31-57. 23. Engle, R. F. and Russel, J. R. (1998) Analysis of high frequency data. In Y. Ait-Sahalia and L. P. Hansen Handbook of Financial Econometrics, North-Holland. 24. Filipovic, D. and Teichmann, J. (2003) Existence of invariant manifolds for stochastic equations in infinite dimension. /. ofFunct. Anal, no. 2, 398-432. 25. Foster, D. P. and Nelson, D. B. (1996) Continuous record asymptotics for rolling sample variance estimators. Econometrica, 64,139-174. 26. Genon-Catalot, V, Laredo, C. and Picard, D. (1992) Nonparametric estimation of the diffusion coefficient by wavelet methods. Scandinavian Journal of Statistics, 19, 317-335. 27. Ghysels, E., Harvey, A., and Renault, E. (1996) Stochastic volatility. Handbook of Statistics, vol.14 Statistical methods in finance, 119-191. North-Holland. 28. Kroner, K. and Ng, V. (1998) Modelling asymmetric comovements of asset

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returns. Review of Financial Studies, 11, 817-844. 29. Hayashi, T. and Yoshida, N. (2004) On covariance estimation of nonsynchronously observed diffusion processes. Bernoulli, 11, 359-379. 30. Hansen, P. R. and Lunde, A. (2003) An optimal and unbiased measure of realized variance based on intermittent high-frequency data. Preprint. 31. Hansen, P. R., Lunde, A. and Nason, J. (2003) Choosing the best volatility models: the model confidence set approach. Preprint. 32. Heath, D. Jarrow, M. and Morton A. (1992) Bond pricing and the term structure of interest rates: a new methodology for contingent claims valuation. Econometrica, 60(1), 77-105. 33. Hog, E. and Lunde, A. (2003) Wavelet estimation of integrated volatility. Preprint. 34. Hsieh, D. (1991) Chaos and nonlinear dynamics: application to financial markets. Journal of Finance, 46,1839-1877. 35. Jacod, J. and Protter, P. (1998) Asymptotic error distributions for the Euler method for stochastic differential equations. Annals of Probability, 26, 267-307. 36. Kanatani, T. (2004) Integrated volatility measuring from unevenly sampled observations. Economics Bullettin, 3,1-8. 37. Lo, A. and MacKinlay, C. (1990) The econometric analysis of nonsynchronous trading. Journal of Econometrics, 45,181-211. 38. Malliavin, P. (1995) Integration and Probability. Springer-Verlag. 39. Malliavin, P. (1997) Stochastic analysis. A series of comprehensive studies in mathematics, vol. 313. Springer-Verlag, 1997. 40. Malliavin, P. and Thalmaier A. (2005) Stochastic Calculus of Variations in Mathematical Finance. Springer Finance, 2005. 41. Malliavin, P. and Mancino, M. E. (2002). Fourier series method for measurement of multivariate volatilities. Finance and Stochastics, 4,49-61. 42. Malliavin, P. and Mancino, M. E. (2004). Nonparametric Volatility Estimation using Laplace Transform. Preprint Dept. Math, for Decisions, University ofFirenze. 43. Malliavin, P. and Mancino, M. E. (2005). A Fourier transform method for nonparametric estimation of volatility. Preprint Dept. Math, for Decisions, University of Firenze. 44. Mancino, M. E. and Reno, R. (2005). Dynamic Principal Component Analysis of multivariate volatilities via Fourier analysis. Applied Mathematical Finance, 12(2), 187-199. 45. Musiela, M. (1993). Stochastic PDEs and term structure models. Journees Internationales de Finance, IGR-AFFI, La Baule, 1993. 46. Mykland, P. and Zhang, L. (2005). Anova for diffusions. Annals of Statistics, 33. 47. Nelson, D. (1990). Arch Models as Diffusion Approximations. Journal of Econometrics, 45, 7-38. 48. Nielsen, M. O. and Frederiksen, P. H. (2005). Finite sample accuracy of integrated volatility estimators. Preprint Department of Economics, Cornell University. 49. Ocone, D. and Karatzas, I. (1991). A generalized Clark representation formula, with application to optimal portfolios. Stochastics and Stochastic Reports, 34,187220. 50. Precup, O. V. and Iori, G. (2003) A comparison of high-frequency cross-

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correlation measures. Preprint. 51. Reno, R. (2003) A closer look at the Epps effect. International Journal ofTheoretical and Applied Finance, 6,87-102. 52. Scholes, M. and Williams, J. (1977) Estimating betas from nonsinchronous trading. Journal of Financial Economics, 5, 309-327. 53. Schwert, G. (1990) Stock market volatility. Financial Analyst Journal, 46, 23-34. 54. Zamansky, M. (1950) Classes de saturation des precedes de sommation des series de Fourier et applications aux series trigonometriques. Annales scientifiques de 1'E.N.S. y serie, t.67,161-198. 55. Zhang, L., Mykland, P. and Ait-Sahalia, Y. (2003) A tale of two time scales: determining integrated volatility with noisy high frequency data. Forthcoming Journal of the American Statistical Association.

Hedging of Credit Derivatives in Models with Totally Unexpected Default Tomasz R. BIELECKI1, Monique JEANBLANC2, and Marek RUTKOWSKI3 1

Department of Applied Mathematics, Illinois Institute of Technology, Chicago, IL 60616, USA 2 Departement de Mathematiques, Universite d'Evry Val d'Essonne, 91025 Evry Cedex, France 3 School of Mathematics, University of New South Wales, Sydney, NSW 2052, Australia & Faculty of Mathematics and Information Science, Warsaw University of Technology, 00-661 Warszawa, Poland The paper analyzes alternative mathematical techniques, which can be used to derive hedging strategies for credit derivatives in models with totally unexpected default. The stochastic calculus approach is used to establish abstract characterization results for hedgeable contingent claims in a fairly general set-up. In the Markovian framework, we use the PDE approach to show that the arbitrage price and the hedging strategy for an attainable contingent claim can be described in terms of solutions of a pair of coupled PDEs. Key words: arbitrage pricing, hedging, credit risk, default intensity.

Introduction This paper presents some methods to hedge defaultable derivatives under the assumption that there exist tradeable assets with dynamics allowing for elimination of default risk of derivative securities. We investigate hedging strategies in alternative frameworks with different degrees of generality, an abstract sernirnartingale framework and a more specific Markovian set-up, and we use two alternative approaches. On the one hand, we use the stochastic calculus approach in order to establish rather abstract characterization results for hedgeable contingent claims in a fairly general set-up. We subsequently apply these results to derive closed-form solutions for prices and replicating strategies in particular models. 35

36

On the other hand, we examine the PDE approach in a Markovian setting. In this method, the arbitrage price and the hedging strategy for an attainable contingent claim are described in terms of solutions of a pair of coupled PDEs. Again, for some standard examples of defaultable claims, we provide explicit formulae for prices and hedging strategies (for further examples of trading strategies involving tradeable credit derivatives, we refer to Laurent [31] or Bielecki et al. [8]). As expected, both methods yield identical results for some special cases considered in this work. For the sake of simplicity, we only deal with financial models with no more than three primary assets (models with an arbitrary number of primary assets were studied in Bielecki et al. [6]). Also, it is postulated throughout that the default time is the same for all defaultable securities. An extension of our results to the case of several (possibly dependent) default times is crucial if someone wishes to cover the so-called basket credit derivatives (in this regard, see Section 6 in Bielecki et al. [7]). Let us comment briefly on the terminology used in this work. Traditionally, credit risk models are classified either as structural models (also known as value-of-the-firm models) or as reduced-form models (also termed intensity-based models). In their original forms, the two approaches, structural and reduced-form, are extreme cases, in the sense that the default time is modelled either as a predictable stopping time (the first moment when the firm's value hits some barrier, as in Black and Cox [9]), or by a totally inaccessible stopping time (defined via its intensity, as in Jarrow and Turnbull [25]). However, as argued by several authors (see, for instance, Duffie and Lando [17], Giesecke [21]-[22], Jarrow and Protter [24], Jeanblanc and Valchev [28], or Guo et al. [23]), probabilistic properties of default time are directly related to the publicly available information (it is important, for instance, whether the value of the firm and/or the default triggering barrier are observed by the market with absolute accuracy). In fact, in several structural models the default time is no longer predictable, as it was the case in classic models with deterministic default triggering barrier and full observation of the firm value process (see, Merton [33] or Black and Cox [9]). For this reason, we decided to refer to credit risk models considered in this work as models with totally unexpected default (the strict mathematical term, totally inaccessible stopping time, seems to be rather cumbersome for a frequent use). For a more exhaustive presentation of mathematical theory of credit risk, we refer to Arvanitis and Gregory [1], Bielecki and Rutkowski [3], Bielecki et al. [4], Cossin and Pirotte [15], Duffie and Singleton [18], Lando [30], or Schonbucher [36].

37

1. Totally Unexpected Default In this section, we describe briefly the fundamental features of the credit risk models with unexpected default. Also, we collect here few technical results that are used in subsequent sections. 1.1 General set-up We assume that we are given a probability space (Q, Q, P) and a nonnegative random variable T on this space. We always postulate that x is strictly positive with probability 1. Note that the probability measure P represents the historical probability reflecting the real-life dynamics of prices of primary traded assets, rather than some martingale measure for our financial model. We first focus on different definitions of default intensity encountered in the literature. 1.1.1 Intensity of a stopping time Suppose that (£2, Q, P) is endowed with some filtration G such that x is a G-stopping time. Let H be the default process, defined as Ht = l(t>T| (note that H is a bounded G-submartingale). We say that x admits a G-intensity if there exists a G-adapted, nonnegative process A such that the process WAT _^

pt

(1)

Mt = Ht-

Audu = HtJo

Kdu Jo

is a G-martingale (the second equality in (1) follows from the fact that the process H is stopped at x). Then M is called the compensated Gmartingale of the default process H. In order for a G-stopping time x to admit a G-intensity A, it has to be totally inaccessible with respect to G, so that P(x = 9) = 0 for any G-predictable stopping time 9. The simplest example is the moment of the first jump a Poisson process. Note that the intensity A necessarily vanishes after default. Remark 1.1. Some authors define the intensity as the process A such that Ht - jC A„ du is a G-martingale. In that case, the process A is not uniquely defined after time x. 1.1.2 F-intensity of a random time We change the perspective, and we no longer assume that the filtration G is given a priori. We assume instead that x is a positive random variable on some probability space (Q,£,P). Let H = {'Ht, t > 0) be the natural filtration generated by the default process (Ht, t > 0), and let F = (ft, f > 0) be some reference filtration in {£l,@, P). We assume throughout that the information available to an investor is modeled by the filtration G = F V H . Consequently, we can reduce our

38

study to the case where the default intensity (if it exists) is G-adapted, meaning that the process M given by (1) is a G-martingale for some Gadapted process A. In this setting, there exists a process A = (At, t > 0), called the W-intensity of T, which is F-adapted and equal to A before default, so that Afli(
Cil|«T) = W W

Vf€R+.

//, in addition, the inequality Ft := IP(T < t \ ft) < 1 holds for every ( e E + then the process C satisfying (2) is unique. Of course, we have that WAT

Mt = Ht -

I

Jo

/->(AT

Au du = Ht -

I

A„ dw.

Jo

Suppose that the reference filtration is chosen in such a way that the default events {T < t} are not in F. Then the F-intensity A is uniquely defined after x and, typically, does not vanish after r. 1.2 Hypothesis (H) In this section, we focus on the invariance property of the so-called hypothesis (H) under an equivalent change of a probability measure. Definition 1.1. We say that filtrations F and G, with F c G, satisfy the hypothesis (H) under P whenever any F-local martingale L follows also a G-local martingale. Remark 1.2. We emphasize that, in general, an F-martingale may fail to follow a G-martingale. As a trivial example, consider a fixed date T > 0 and take Qt = TT for every t e [0, T]. Then any F-martingale L satisfies Ep(L( | @s) = Lt for s < t, and thus L is not a G-martingale, in general. It is even possible, but more difficult, to produce an example of an Fmartingale, which is not a semi-martingale with respect to G. For other counter-examples, in particular those involving progressive enlargement of filtrations, we refer interested reader to Protter [35], or Mansuy and Yor [32]. The original formulations of the hypothesis (H) refer to martingales (or even square-integrable martingales), rather than local martingales. We

39

shall show that in our set-up the definition given above is equivalent to the original definition. In fact, the hypothesis (H) postulates a certain form of conditional independence of a-fields associated with F and G, rather than a specific property of F-(local) martingales. In particular the following well known result is valid. Lemma 1.2. Assume that G = F v H, where F is an arbitrary filtration and M is generated by the process Ht - t\T
F(T
nT
(i') For any t e R+, we have (4)

F(T
=

F(r
(ii) For any t € R+, the a-fields Too and Qt are conditionally independent given Tt under F, that is, F,P(Hn\Tt) =

F.A£,\Tt)Fw{n\Tt)

for any bounded, Too-measurable random variable E, and bounded, Qi-measurable random variable r\. (Hi) For any t e R+, and any u > t the a-fields Tu and Qt are conditionally independent given Tt(iv) For any t e R+ and any bounded, Too-measurable random variable E,:

Ep(ei^) = E P (||r ( ). (v) For any t e R+, and any bounded, Qrmeasurable random variable r\: Fv(r]\Tt) = FF(rj\Too). Proof. The proof of equivalence of conditions (i')-(v) can be found, for instance, in Section 6.1.1 of Bielecki and Rutkowski [3] (for related results, see Elliott et al. [20]). Using monotone class theorem it can be shown that conditions (i) and (i') are equivalent. Hence, we shall only show that condition (iv) and the hypothesis (H) are equivalent. Assume first that the hypothesis (H) holds. Consider any bounded, T^o-measurable random variable £. Let Lt = E P (£ | Tt) be the martingale associated with E,. Then, (H) implies that L is also a local martingale with respect to G, and thus a G-martingale, since L is bounded (recall that any bounded local martingale is a martingale). We conclude that L( = Ep(£ | Qt) and thus (iv) holds. Suppose now that (iv) holds. First, we note that the standard truncation argument shows that the boundedness of E, in (iv) can be replaced by the

40

assumption that £, is P-integrable. Hence, any F-martingale L is an Gmartingale, since L is clearly G-adapted and we have, for every t < s, Lt = Er(Ls\rt)

= Ep(Ls\St).

Now, suppose that L is an F-local martingale so that there exists an increasing sequence of F-stopping times T„ such that lim„-,M T„ = oo, for any n the stopped process U" follows a uniformly integrable F-martingale. Hence, LT" is also a uniformly integrable G-martingale, and this means that L follows a G-local martingale. a 1.2.1 Hazard process Let T be a random time on a space (Q, Q, P) such that the filtrations F and G = F v H satisfy the hypothesis (H). Then, from (4), the process Ft = P(T < 11 ft) is increasing. We make the standing assumption that Ft < 1 for every t e R+, and we define the ¥-hazardprocess Tby setting Tt = - In (1 -Ft). Let, in addition, the process F be absolutely continuous with respect to the Lebesgue measure, so that -i = I fu du, Vf e R + , Jo for some F-progressively measurable (or F-predictable) process / . Then the F-hazard process satisfies Tt==

I yyudu, udu,

Jo

VfeR+,

where the F-intensity y is given by

(5)

y

=

t^F'

vte

K+.

i — tt

From now on, we take (5) as the definition of the F-intensity y, and we make the standing assumption that the hypothesis (H) holds under P. The following auxiliary result is standard (see, for instance, Elliott et al. [20] or Blanchet-Scalliet and Jeanblanc [10]). Lemma 1.3. For any V-integrable, fj-measurable random variable X we have, forte[0,T], E P ( X l | T < r , \Qt) = l, / < r , e r 'E P (Xe-^ | Tt). 1.2.2 Canonical construction We now describe the canonical construction of a random time with a given F-hazard process. Let W be an F-adapted, increasing, nonnegative

41

process with WQ = 0 and lim^oo ^ t = oo. We define a nonnegative random variable x by setting T = inf(f 6M + : ^ , > 0 ) , where 0 is a random variable independent of F, with the exponential distribution of parameter 1. Of course, the existence of 0 on the original probability space (Q, Q, P) is not guaranteed, so that we allow for an extension of the underlying probability space. We shall now find the process Ft = P{x < t \ Tt\. Since clearly {T > t} {& > Vt), we get P{x >t\Ta,)

= P{0 > W, \r„} = e~w>.

Consequently, i - Ft = P{T > 11 Tt) = E F ( P { T >t\rea}\ri)

= e~w>,

and so F is an F-adapted, continuous, increasing process. We conclude that for every t e Pv+ (6)

Ft = l-e~%

= P{x < t\Too) = P{T < t\Tt),

and thus W coincides with the F-hazard process T of x and the hypothesis (H) is valid. It is also not difficult to show that the process Mt = Ht- TJAT = fit - ^JAT follows a G-martingale. The following result shows that under the hypothesis (H), for any random time x with continuous hazard process T, the auxiliary random variable 0 can be constructed on the original probability space, using x and T (see El Karoui [19] or Blanchet-Scalliet and Jeanblanc [10]). Lemma 1.4. Let x be a random time on a probability space (CI, Q, P) such that the F-hazard process Tofr under P is continuous and the hypothesis (H) holds. Then there exists a random variable 0 on (CI, Q, P), independent of F and with the exponential distribution of parameter 1, such that (7)

x = inf{£ e R + : Tt > 0 | .

Proof. It suffices to check that the random variable 0 = TT has the desired properties. Indeed, we have, for every t € R+, P ( 0 > 11 Too) = P(r T >t\Tlx,) = P(x > At I Too) = exp(-r A ,) = e~l, where A is the left inverse of T, so that r^, = t for every t e R+.

•

42

1.3 Change of a probability measure Kusuoka [29] shows, by means of a counter-example, that the hypothesis (H) is not invariant with respect to an equivalent change of the underlying probability measure, in general. It is worth noting that his counterexample is based on two filtrations, H 1 and H 2 , generated by the two random times T 1 and T 2 , and he chooses H 1 to play the role of the reference filtration F. We shall argue that in the case where F is generated by a Brownian motion (or, more generally, by some martingale orthogonal to M under P), the above-mentioned invariance property is valid under mild technical assumptions. 1.3.1 Preliminary lemma Let us first examine a general set-up in which G = F V H , where F is an arbitrary filtration and H is generated by the default process H. We say that Q is locally equivalent to P if Q is equivalent to P on (Q, Qt) for every t e R + . Then there exists the Radon-Nikodym density process r\ such that (8)

dQla =

tydP|ft,

VfeR+.

Part (i) in the next lemma is well known (see Jamshidian [27]). We assume that the hypothesis (H) holds under P. Lemma 1.5. (i) Let Q be a probability measure equivalent to P on (Q, Qt) for every t e R+, with the associated Radon-Nikodym density process n. If the density process r\ is F-adapted then we have Q(T
Q(r
F f : = Q ( x < f | ^ ) = Q(T
WeR+.

43

Note that since the random variables 77tl(T
n /
EpP/ti|T
Ep^ooiir^iiyco) Ep(rjoolfoo)

QimtlTo,).

We conclude that the hypothesis (H) holds under Q if and only if (9) is valid. D Unfortunately, straightforward verification of condition (9) is rather cumbersome. For this reason, we shall provide alternative sufficient conditions for the preservation of the hypothesis (H) under a locally equivalent probability measure. 1.3.2 Case of the Brownian filtration Let W be a Brownian motion under P with respect to its natural filtration F. Since we work under the hypothesis (H), the process W is also a Gmartingale, where G = F V H. Hence, W i s a Brownian motion with respect to G under P. Our goal is to show that the hypothesis (H) is still valid under Q e Q for a large class Q of (locally) equivalent probability measures on (Q, Q). Let Q be an arbitrary probability measure locally equivalent to P on (Cl,§). Kusuoka [29] (see also Section 5.2.2 in Bielecki and Rutkowski [3]) proved that, under the hypothesis (H), any G-martingale under P can be represented as the sum of stochastic integrals with respect to the Brownian motion W and the jump martingale M. In our set-up, Kusuoka's representation theorem implies that there exist G-predictable processes 6 and C > - 1 , such that the Radon-Nikodym density 1] of Q with respect to P satisfies the following SDE (11)

dilt = ilt-(etdWt

+ i:tdMt)

with the initial value rjo = 1- More explicitly, the process 77 equals

(12)

Tit = St If 6U dWu\ &t If C« dMu\ = TJS?>

where we write

(13)

TJ!1' = 8tIf

9UdWu\ = expIf BudWu-^f

d\du),

44

and (14)

qf^&tl

f QudMu\ = expl f in(l + Qu) dHu - f

Z„yudu).

Moreover, by virtue of a suitable version of Girsanov's theorem, the following processes W and M are G-martingales under Q (15)

Wt = Wt-

f 9U du,

Mt = Mt-

Jo

f l{u
Proposition 1.1. Assume that the hypothesis (H) holds under P. Let Qbe a probability measure locally equivalent to F with the associated Radon-Nikodym density process IJ given by formula (12). If the process 6 is F-adapted then the hypothesis (H) is valid under Q and the F-intensity of r under Q equals y~t = (1 + Qt)yt, where C is the unique ^-predictable process such that the equality Cfl|t
(16)

dF | Sl = St If

L dMu\ dP\0l = r,f] dF\0r

We claim that the hypothesis (H) holds under P. From Girsanov's theorem, the process W follows a Brownian motion under P with respect to both F and G. Moreover, from the predictable representation property of W under P, we deduce that any F-local martingale L under P can be written as a stochastic integral with respect to W. Specifically, there exists an F-predictable process £ such that U = U+

\ Jo

ludWu.

This shows that L is also a G-local martingale, and thus the hypothesis (H) holds under P. Since dQ | g ^StijOudWuV = &t 9udWu\dF \§i> by virtue of part (i) in Lemma 1.5, the hypothesis (H) is valid under Q as well. The last claim in the statement of the lemma can be deduced from

45

the fact that the hypothesis (H) holds under Q and, by Girsanov's theorem, the process Mt=Mt-

I llu
I 1{U
is a Q-martingale.

a

We claim that the equality P = P holds on the filtration F. Indeed, we have dP | r, = T]tdF\ri, where we write rff = Ep(r/f * \ Tt), and (17)

E P (fjf |F») = Epl£».IftudMu

r„

=1,

VteK+,

where the first equality follows from part (v) in Lemma 1.2. To establish the second equality in (17), we first note that since the process M is stopped at x, we may assume, without loss of generality, that C = C where the process C is F-predictable (see Lemma 1.1). Moreover, in view of (7) the conditional cumulative distribution function of x given To has the form 1 - exp(-r ( (c))- Hence, for arbitrarily selected sample paths of processes C, and t, the claimed equality can be seen as a consequence of the martingale property of the Doleans exponential. Formally, it can be proved by following elementary calculations, where the first equality is a consequence of (14)), E P 1st If

lu dMu\ | Too) = E P f(l + l)fet,C"i) exp ( (l + l { f e B | C a ) e x p ( - J

f

Zuyu du) | ToA

Zvyvdv)yue-&'r*dvdu\TJ

= Ep I I (l + C«)y« exp ( - I (1 + Cv)Yv dv)du | Too J + exp ( - j ZvYv dv) E P I J

yue~ X" ^ *d« | Tco)

= I ( l + C«)y« exp ( - I (1 + Ci>)yi> dv)du QoYvdv) J = l-exp(-

yue-£r°dvdu

I (1 + Cv)yv dv) + exp ( - I C ^ d») exp ( - I yvdv) = l,

where the second last equality follows by an application of the chain rule.

46

1.3.3 Extension to orthogonal martingales Equality (17) suggests that Proposition 1.1 can be extended to the case of arbitrary orthogonal local martingales. Such a generalization is convenient, if we wish to cover the situation considered in Kusuoka's counterexample. Let N be a local martingale under P with respect to the filtration F. It is also a G-local martingale, since we maintain the assumption that the hypothesis (H) holds under P. Let Q be an arbitrary probability measure locally equivalent to P on (C1,Q). We assume that the Radon-Nikodym density process q of Q with respect to P equals (18)

drlt = tlt-(eldNt + QdMt)

for some G-predictable processes 9 and C > —1 (the properties of the process 6 depend, of course, on the choice of the local martingale N). The next result covers the case where N and M are orthogonal G-local martingales under P, so that the product MN follows a G-local martingale. Proposition 1.2. Assume that the following conditions hold: (a) N and M are orthogonal G-local martingales under P, (b) N has the predictable representation property under P with respect to F, in the sense that any W-local martingale L under P can be written as Lt = L0+ \ £udNUl Jo

V£eR+/

for some W-predictable process £,, (c) P is a probability measure on (Q, Q) such that (16) holds. Then we have: _ (i) the hypothesis (H) is valid under P, (ii) if the process 6 is W-adapted then the hypothesis (H) is valid under Q. The proof of the proposition hinges on the following simple lemma. Lemma 1.6. Under the assumptions of Proposition 1.2, we have: (i) N is a G-local martingale under P, (ii) N has the predictable representation property for W-local martingales under P. Proof. In view of (c), we have d¥\Sl = if^ dV\Sl, where the density process r/2) is given by (14), so that dnf* = rjf}C,tdMt. From the assumed orthogonality of N and M, it follows that N and ;/ 2) are orthogonal G-local martingales under P, and thus NT/ 2 ) is a G-local martingale under P as well. This means that N is a G-local martingale under P, so that (i) holds. To establish part (ii) in the lemma, we first define the auxiliary process rf by setting r\t = Ep(j;|2) | ft). Then manifestly dP| r > = r\tdV\Tl, and thus

47

in order to show that any F-local martingale tinder IP follows an F-local martingale under P, it suffices to check that /ft = 1 for every t e R+, so that P = P on F. To this end, we note that ipv(if\Tt)

-Me'{l

= EP\6t\

| QudMu

Tco) = l,

VfeR+,

where the first equality follows from part (v) in Lemma 1.2, and the second one can established similarly as the second equality in (17). We are in a position to prove (ii). Let L be an F-local martingale under P. Then it follows also an F-local martingale under P and thus, by virtue of (b), it admits an integral representation with respect to N under P and P. This shows that N has the predictable representation property with respect to F under P. • We now proceed to the proof of Proposition 1.2. Proof of Proposition 1.2. We shall argue along the similar lines as in the proof of Proposition 1.1. To prove (i), note that by part (ii) in Lemma 1.6 we know that any F-local martingale under P admits the integral representation with respect to N. But, by part (i) in Lemma 1.6, N is a G-local martingale under P. We conclude that L is a G-local martingale under P, and thus the hypothesis (H) is valid under P. Assertion (ii) now follows from part (i) in Lemma 1.5. • Remark 1.3. It should be stressed that Proposition 1.2 is not directly employed in what follows. We decided to present it here, since it sheds some light on specific technical problems arising in the context of modeling dependent default times through an equivalent change of a probability measure (see Kusuoka [29]). Example 1.1. Kusuoka [29] presents a counter-example based on the two independent random times xi and X2 given on some probability space (Q, Q, P). We write M\=H\- £Ml yi(u) du, where H\ = 1((>T,| and y{ is the deterministic intensity function of T,- under P. Let us set dQ \gt = r\t dV \ gt, where r\t = r/| 77; and, for i = 1,2 and every t e R+,

1{P = 1 + j T rfiLtf dK = St (JT Ci? dM<) for some G-predictable processes C(,)/ i - 1/2, where G = H 1 V H 2 . We set F = H 1 and H = H 2 . Manifestly, the hypothesis (H) holds under P.

48

Moreover, in view of Proposition 1.2, it is still valid under the equivalent probability measure P given by

It is clear that P = P on F, since

EP(r/f> I Tt) = E P U If £® dMj J | W,1 J = 1, V r e R + . However, the hypothesis (H) is not necessarily valid under Q if the process C'1' fails to be F-adapted. In Kusuoka's counter-example, the process C(1> was chosen to be explicitly dependent on both random times, and it was shown that the hypothesis (H) does not hold under Q. For an alternative approach to Kusuoka's example, through an absolutely continuous change of a probability measure, the interested reader may consult Collin-Dufresne era/. [13]. 2.

Semimartingale Model with a Common Default In what follows, we fix a finite horizon date T > 0. For the purpose of this work, it is enough to formally define a generic defaultable claim through the following definition.

Definition 2.1. A defaultable claim with maturity date T is represented by a triplet (X, Z, T), where: (i) the default time T specifies the random time of default, and thus also the default events (T < /} for every / e [0, T], (ii) the promised payoff X e TT represents the random payoff received by the owner of the claim at time T, provided that there was no default prior to or at time T; the actual payoff at time T associated with X thus equals Xl(r)1(7-)! {7>T| = ZT1)T>T|.

49

When dealing with replicating strategies, in the sense of Definition 2.2, we will always assume, without loss of generality, that the components of the process are F-predictable processes. 2.1 Dynamics of asset prices We assume that we are given a probability space (Q, Q, P) endowed with a (possibly multi-dimensional) standard Brownian motion W and a random time T admitting an F-intensity y under P, where F is the filtration generated by W. In addition, we assume that T satisfies (4), so that the hypothesis (H) is valid under P for filtrations F and G = F v E Since the default time admits an F-intensity, it is not an F-stopping time. Indeed, any stopping time with respect to a Brownian filtration is known to be predictable. We interpret T as the common default time for all defaultable assets in our model. For simplicity, we assume that only three primary assets are traded in the market, and the dynamics under the historical probability P of their prices are, for i = 1,2,3 and t e [0, T], (19)

dY\ = Y\_(in,t dt + ox dWt + Ki4 dMt),

or equivalently, (20)

dY\ = Y\_({^tt - K;,fy(l,f
The processes (jij, a,-, K,) = (/i,-^, cr,-,t, K,/(, t > 0), ;' = 1,2,3, are assumed to be G-adapted, where G = F V H In addition, we assume that K; > - 1 for any i = 1,2,3, so that V are nonnegarive processes, and they are strictly positive prior to T. Note that, according to Definition 2.2, replication refers to the behavior of the wealth process V((p) on the random interval [0, T A T] only. Hence, for the purpose of replication of defaultable claims of the form (X, Z, x), it is sufficient to consider prices of primary assets stopped at T A T. This implies that instead of dealing with G-adapted coefficients in (19), it suffices to focus on F-adapted coefficients of stopped price processes. However, for the sake of completeness, we shall also deal with T-maturity claims of the form Y = G(Y\, Y\, Y\,HT) (see Section 5 below). 2.1.1 Pre-default values As will become clear in what follows, when dealing with defaultable claims of the form (X, Z, T), we will be mainly concerned with the so-called pre-default prices. The pre-default price Y' of the fth asset is an F-adapted, continuous process, given by the equation, for i = 1,2,3 and t e [0, T], (21)

dY\ = Yt((n» - Ki,tyt) dt + aiit dWt)

50

with Yj, = Y[,. Put another way, Y1 is the unique F-predictable process such that (see Lemma 1.1) YJ1(,\ dY\ = #Y t (m A dt + oiJt dWt) - |YjKyy, dt. The last term may be here separated, and formally treated as an effect of continuously paid dividends at the dividend rate K,,t}'[. However, this interpretation may be misleading, since this quantity is not directly observed. In fact, the mere estimation of the drift coefficient in dynamics (21) is not practical. Still, if this formal interpretation is adopted, it is sometimes possible make use of the standard results concerning the valuation of derivatives of dividend-paying assets. It is, of course, a delicate issue how to separate in practice both components of the drift coefficient. We shall argue below that although the dividend-based approach is formally correct, a more pertinent and simpler way of dealing with hedging relies on the assumption that only the effective drift jU,/t - K^yt is observable. In practical approach to hedging, the values of drift coefficients in dynamics of asset prices play no essential role, so that they are considered as market observables. 2.1.2 Market observables To summarize, we assume throughout that the market observables are: the pre-default market prices of primary assets, their volatilities and correlations, as well as the jump coefficients Kj/t (the financial interpretation of jump coefficients is examined in the next subsection). To summarize we postulate that under the statistical probability P we have (22)

dY\ = Y\_ (Jliit dt + ait dWt + K,-, dHt)

where the drift terms fT,/( are not observable, but we can observe the volatilities a,,f (and thus the assets correlations), and we have an a priori assessment of jump coefficients K,> In this general set-up, the most natural assumption is that the dimension of a driving Brownian motion W equals

51

the number of tradable assets. However, for the sake of simplicity of presentation, we shall frequently assume that W is one-dimensional. One of our goals will be to derive closed-form solutions for replicating strategies for derivative securities in terms of market observables only (whenever replication of a given claim is actually feasible). To achieve this goal, we shall combine a general theory of hedging defaultable claims within a continuous semimartingale set-up, with a judicious specification of particular models with deterministic volatilities and correlations. 2.1.3 Recovery schemes It is clear that the sample paths of price processes Y' are continuous, except for a possible discontinuity at time T. Specifically, we have that A Y ; := r T - r T _ = K,-TY*T_, so that YT = Y;_(l + K,-T) = Y\_{\ + K,>). A primary asset Y' is termed a default-free asset (defaultable asset, respectively) if %i = 0 (K,- £ 0, respectively). In the special case when K,- = —1, we say that a defaultable asset Y' is subject to a total default, since its price drops to zero at time T and stays there forever. Such an asset ceases to exist after default, in the sense that it is no longer traded after default. This feature makes the case of a total default quite different from other cases, as we shall see in our study below. In market practice, it is common for a credit derivative to deliver a positive recovery (for instance, a protection payment) in case of default. Formally, the value of this recovery at default is determined as the value of some underlying process, that is, it is equal to the value at time x of some F-adapted recovery process Z. For example, the process Z can be equal to 5, where 5 is a constant, or to g(t, 5Yt) where g is a deterministic function and (Yf, t > 0) is the price process of some default-free asset. Typically, the recovery is paid at default time, but it may also happen that it is postponed to the maturity date. Let us observe that the case where a defaultable asset Y' pays a predetermined recovery at default is covered by our set-up defined in (19). For instance, the case of a constant recovery payoff 6/ > 0 at default time T corresponds to the process K,/t = 6;(YJ_)_1 - 1. Under this convention, the price Y' is governed under P by the SDE (23)

dY\ = Y;_(^,,t dt + at,t dWt + (SKYlJ-1 - 1) dMt).

If the recovery is proportional to the pre-default value Y\_, and is paid at default time T (this scheme is known as the fractional recovery of market value), we have K,/f = 6 , - 1 and (24)

dY\ = Yit_(^ii)t dt + on dWt + (6,- - 1 ) dMt).

52

2.2

Risk-neutral valuation To provide a partial justification for the postulated dynamics of the price of a defaultable asset delivering a recovery, let us study a toy example with two assets: a savings account with constant interest rate r and a def aultable asset Y represented by a defaultable claim (X, Z, T). In this toy model, the only source of noise is the default time, hence, the only relevant filtration is H (in other words, the reference filtration F is trivial). We assume that by choosing today's prices of a large family liquidly traded defaultable assets, the market implicitly specifies a martingale measure R such that the random variable h(j) is

53

Q-integrable. Then Ml = Mh0+ f(h(u)-g(u))dMu=M» + f (h(u) - Mhu_) dMUr Jo Jo where we write g(t) = er^EQ(l[t
Remark 2.1. Using the above proposition, it can be easily shown that on (Q, QT) we have

dF = 8T(-fzudMu\dQ, for some H-predictable process C 2.2.1 Price dynamics of a survival claim (X, 0, T). In what follows, we shall refer to a defaultable claim of the form (X, 0, T) as a survival claim. By virtue of the risk-neutral valuation formula, the price of the payoff 1{T<TIX that settles at time T equals, for every t e [0, T], Yt =

ertEQ(llT
Note that X is fr-measurable, and thus constant since the rr-field TT is trivial. To find the dynamics of the price process, it suffices to apply Proposition 2.1 to the function h(u) = l|M>7|e~rTX. For the Q-martingale M'l = e-rtYh we thus get, for every t e [0, T], rt „-rfv _ Y v .eYt = 0

I „-™ e~ruY

Jo )o

u^dMu.

Suppose that Y(t) — J y(u)du. Then an application of Ito's formula yields (26)

dY, = rYt dt - Yt- dMt = (r + l{t<^y(tj)Yt

dt - Y(_ dHt.

We deal here with an example of a defaultable asset that is subject to the total default, meaning that its price vanishes at and after default. 2.2.2 Price dynamics of a recovery claim (0, Z, T). Recall that our standard convention stipulates that the recovery Z is paid at the time of default. Hence, the price process Y of (0, Z, T) is given by the expression yf = e rt E Q (l, T > T| e-"Z(T)|'H ( )We now have h(u) = \\M<j\e~ruZ.(u). Consequently, -rtYf = Y0 + j

(e-ruZ(u)-e-ruYu-) dMu.

54

By applying Ito's formula, we conclude that the dynamics under Q of an asset that delivers Z(x) at time x are dYt = rYt- dt + (Z(t) - Y,_) dMt = (r + l{l
- l{t
+ (Z(t) - Yt-)dHt.

2.2.3 Price dynamics of a defaultable claim (X, Z, x). By combining the formula above with (26), and using Remark 2.1 together with Girsanov's theorem, we arrive at the following result. Proposition 2.2. The price process Y of a defaultable claim (X, Z, x) satisfies under Q dYt = rYt- dt + (Z(f) - Yt-)dMt with the initial condition Y0 = EQ(l{T
+ l (T > T , e —Z(x)) = e-(rT+F(T))X + f Z(u)y{u)e-fw

du.

Under the statistical probability F, the price process Y satisfies dYt = (rYt- + 1,« T| (Z(0 - Yt-)y{t)C(t))dt + (Z(f) - Yt-)dMt, where the G-martingale M under F equals M( = M f + I Jo

t{u
Remark 2.2. Proposition 2.2 can be extended to the case when the recovery is random, and is given in the feedback form as Z(t) = g(t,Yt~) for some function g(t, y), which is Lipschitz continuous with respect to y. Assume, for instance, that the claim is subject to the fractional recovery of market value, so that Z(t) - 6Y(_ for some constant 5. If, in addition, C and y~are constant, then we obtain (cf. (24)) dYt = Yt-((r + l ( t < t | (6 - l)yX) dt + (5- 1) dMt). Note that here the drift coefficient pt = r + l[f
55

3. Trading Strategies in a Semimartingale Set-up We consider trading within the time interval [0, T] for some finite horizon date T > 0. For the sake of expositional clarity, we restrict our attention to the case where only three primary assets are traded. The general case of k traded assets was examined by Bielecki et al. [5]. We first recall some general properties, which do not depend on the choice of specific dynamics of asset prices. In this section, we consider a fairly general set-up. In particular, processes Y', i = 1,2,3, are assumed to be nonnegative semi-martingales on a probability space (Q, Q, P) endowed with some filtration G. We assume that they represent spot prices of traded assets in our model of the financial market. Neither the existence of a savings account, nor the market completeness are assumed, in general. Our goal is to characterize contingent claims which are hedgeable, in the sense that they can be replicated by continuously rebalanced portfolios consisting of primary assets. Here, by a contingent claim we mean an arbitrary ^-measurable random variable. We work under the standard assumptions of a frictionless market. 3.1 Unconstrained strategies Let (j) - (0 1 , ' is predictable with respect to the filtration G. The wealth of equals

V,(4>) = £ # Y ( , Vfe[0,n, 1=1

and a trading strategy (p is said to be self-financing if Vt(4>) = V0(

eft dY'u, V t e [0, T].

Let <& stand for the class of all self-financing trading strategies. We shall first prove that a self-financing strategy is determined by its initial wealth and the two components <j)2,(p3. To this end, we postulate that the price of Y1 follows a strictly positive process, and we choose Y1 as a numeraire asset. We shall now analyze the relative values: Vj(cp) := Vt(cp)(YJy\

Y? := Y^Y,1)"1.

Lemma 3.1. (i) For any cp e O, we have 3

„(

V)($) = V10(cj>) + Y J

h2 0


V t e [0,T].

56

(ii) Conversely, let Xbea Qj-measurable random variable, and let us assume that there exists x e Rand G-predictable processes (ft', i = 2,3 such that (

(27)

3

X = Y\ x + 7=2

Then there exists a G-predictable process (ft1 such that the strategy (ft = ((ft1, (ft2, (ft3) is self-financing and replicates X. Moreover, the wealth process of (ft (i.e. the time-t price ofX) satisfies Vt{(ft) = V\Y\, where 3

(28)

V}=x + Y

M

(ftl dY*,

V t e [0, T].

Proof. The proof of part (i) is given, for instance, in Protter [34]. In the case of continuous semimartingales, this is a well-known result; for discontinuous processes, the proof is not much different. We reproduce it here for the reader's convenience. Let us first introduce some notation. As usual, [X, Y] stands for the quadratic covariation of the two semi-martingales X and Y, as defined by the integration by parts formula: XtYt = X0Y0 + f Xu.dYu+ Jo

[ Yu-dXu + [X,Y]t. Jo

For any cadlag (i.e., RCLL) process Y, we denote by AYf = Yt- Yt- the size of the jump at time t. Let V = V((ft) be the value of a self-financing strategy, and let V1 = V 1 ^ ) = V((ft)(Yl)~l be its value relative to the numeraire Y1. The integration by parts formula yields dV} = V,-d(Y}rl + (Yl)~ldVt +

d[(Y1)-\V]t.

From the self-financing condition, we have dVt = 2Z,?=i
+(ft3tYld(Y]rl

+ (YJLrVI dY) + (Yjj-V? dY\ + (YlT1^ dY) + (ft] d[(Yl)-\ Yx]t + (ft2 d[{Yxr\

Y2]t + (ft3 dWT1)-1, Y1],

= (ft}(Y}_ d(Y})~1 + (Yl)-1 dY} + d[(YlT\ Y1],) + (ft2(Yf_ d(YJrl + (YlT1 dYl + d[(Yl)~\ Y2]t) + (ft3t(Y3_ d(Y}yl + (Y}_yl dYl + d[{Yly\

Y3]t).

57

We now observe that Yld{Y)yl

+ {Y}_)-xdY} +d[(Y1)-\Y1]l = d^Y, 1 )" 1 ) = 0

and

Y\_ d{Y]Tl + (YJL)-1 dY\ + d[{YlY\ Y% = d^Y,1)-1 YJ). Consequently,

dVJ = 3, and we define the process V1 by setting (cf. (28)) v

}=x

+

f\

f

tidY*,

Vfe[0,T].

Next, we define the process (p1 as follows: 3

3

cp] = Vj - £ &Yf = (Y})-\vt - £ $*), i=2

i'=2

where Vt = V/ Y). Since dV,1 = £f=2 <# dYJ'1, we obtain dV, = ^(V/yj1) = VJLdY? + YldV} + d[Y\ Vl\t 3

= V}_dYJ + £ Vfil AYf + d[Y\ Y-%). From the equality dY\ = d{Y>;lYJ) = Y\\dYJ + YJ_dY\x + d[Y\ Y a ] f , it follows that 3

3

3

dvt = vldYj + £ #(dy; - yj^dyj) = (v^ - £ # Y^JdY,1 + £ $ dY|, i'=2

1=2

i'=2

and our aim is to prove that dVf = £f=1 } dy|. The last equality holds if 3

(29)

3

# = V,1 - £ cftYJ'1 = y/_ - £ <#Y*, I'=2

1=2

58

i.e., if AVJ = Lf=2^fAV;'1, which is the case from the definition (28) of V1. Note also that from the second equality in (29) it follows that the process (p1 is indeed G-predictable. Finally, the wealth process of

X = Y4
X = V0(cp) + t

T

f
Suppose that there exists an e.m.m. for some choice of a numeraire asset, and let us restrict our attention to the class of all admissible trading strategies, so that our model is arbitrage-free. Assume that a claim X can be replicated by some admissible trading strategy, so that it is attainable (or hedgeable). Then, by definition, the arbitrage price at time t of X, denoted as n f (X), equals Vt(cp) for any admissible trading strategy (ft that replicates X. In the context of Lemma 3.1, it is natural to choose as an e.m.m. a probability measure Q 1 equivalent to P on (CI, QT) and such that the prices Y1,1, i = 2,3, are G-martingales under Q 1 . If a contingent claim X is hedgeable, then its arbitrage price satisfies 7t,(X) = YjE Q l (X(Yj.)- 1 |^). We emphasize that even if an e.m.m. Q 1 is not unique, the price of any hedgeable claim X is given by this conditional expectation. That is to say, in case of a hedgeable claim these conditional expectations under various equivalent martingale measures coincide. In the special case where Y\ = B(t,T) is the price of a default-free zero-coupon bond with maturity T (abbreviated as ZC-bond in what follows), Q 1 is called T-forward martingale measure, and it is denoted by QT. Since B(T, T) = 1, the price of any hedgeable claim X now equals 7i((X) = B(f,T)E QT (X|£ f )3.2 Constrained strategies In this section, we make an additional assumption that the price process Y3 is strictly positive. Let tp = ((p1, 3) be a self-financing trading strategy satisfying the following constraint: 2

(30)

YJ&YI

= Zt, Vie[0,T],

59

for a predetermined, G-predictable process Z. In the financial interpretation, equality (30) means that a portfolio

3

AVt(<j>) = £ 4>[AYt = VtW - £ #Y{_. i=i

i=i

By combining this equality with (30), we deduce that

Vt_(0) = £<#YJ_ = Z(+4>3Y<_. i=i 3

1

Let us write YJ' = YJ(Yf)- , Z 3 = Zt{Y3t)-1. The following result extends Lemma 1.7 in Bielecki et ah [4] from the case of continuous semimartingales to the general case (see also [5]). It is apparent from Proposition 3.1 that the wealth process V((p) of a strategy

(31)

V f a ) = V3(c/>) + f & frfY2'3 - ^ dY\A + f 1L dYY. Jo

\

Yu'_

)

Jo Y„'_

Proof. Let us consider discounted values of price processes Y1, Y2, Y3, with Y3 taken as a numeraire asset. By virtue of part (i) in Lemma 3.1, we thus have 2

(32)

rt

V3(4>) = V3((/>) + y TT i=i Jo

The balance condition (30) implies that

M? - Zf,

cgdY^

60

and thus (33)

^) = {Y^r\z]-^Y^).

By inserting (33) into (32), we arrive at the desired formula (31).

D

The next result will prove particularly useful for deriving replicating strategies for defaultable claims. Proposition 3.2. Let a Qj-measurable random variable X represent a contingent claim that settles at time T. We set •2,3

(34)

Y; dY\ = dYf - - f 3 dY]'3 = dY2'3 - Yf:1 dY]'3,

where, by convention, Y*0 = 0. Assume that there exists a G-predictable process c()z, such that

(35)

X = Y\ L + JfrtfdY] + £ ^ dY]A.

Then there exist G-predictable processes cp1 and cp3 such that the strategy

(36)

Vt() = Y3(x+ JT*<j>ldYl + ^ ^ d Y l A .

Proof. As expected, we first set (note that the process
(37)


n.

and

Jo

Jo Y]f_

Arguing along the same lines as in the proof of Proposition 3.1, we obtain 2

Vf = V3+Y,

M

^dy«3-

61

Now, we define 2

2

3

_1

tf = f? - £ <^' - (V?) (Vt - £ # YJ), /=i

i=i

where Vt = V3 Y3. As in the proof of Lemma 3.1, we check that

i=i

and thus the process t^3 is G-predictable. It is clear that the strategy

) = X, so that (p replicates X. Finally, equality (37) implies (30), and thus (p belongs to the class <1>(Z). D Note that equality (35) is a necessary (by Lemma 3.1) and sufficient (by Proposition 3.2) condition for the existence of a constrained strategy that replicates a given contingent claim X. 3.2.1 Synthetic asset Let us take Z = 0, so that

(38)

dV*() =

tildYV-$LdY1A

The process Y2 = Y3Y', where Y* is defined in (34) is called a synthetic asset. It corresponds to a particular self-financing portfolio, with the long position in V2 and the short position of Yt'_ number of shares of Y1, and suitably re-balanced positions in the third asset so that the portfolio is self-financing, as in Lemma 3.1. It can be shown (see Bielecki et al. [5]) that trading in primary assets Yl,Y2,Y3 is formally equivalent to trading in assets Yl,Y2,Y3. This observation supports the name synthetic asset attributed to the process Y2. Note, however, that the synthetic asset process may take negative values. 3.2.2 Case of continuous asset prices In the case of continuous asset prices, the relative price Y* = Y2(Y3)~l of the synthetic asset can be given an alternative representation, as the following result shows. Recall that the predictable bracket of the two continuous semi-martingales X and Y, denoted as (X, Y), coincides with their quadratic covariation [X, Y]. Proposition 3.3. Assume that the price processes Y1 and Y2 are continuous.

62

Then the relative price of the synthetic asset satisfies

r;= fVK'W-

d?u,

Jo where Yt := Yf'V'" and (39)

at :=
In terms of the auxiliary process Y, formula (31) becomes (40)

Vf(
where
$(YfT1ea<.

f

^dY1/,

Proof. It suffices to give the proof for Z = 0. The proof relies on the integration by parts formula stating that for any two continuous semimartingales, say X and Y, we have Yj\dXt - Y;xd{X,Y)t)

= d(XtYjl) -

XtdYj\

provided that Y is strictly positive. An application of this formula to processes X = Y2,1 and Y = Y3,1 leads to ( Y 3 ' 1 ) - 1 ^ 2 ' 1 - {Y]'l)-xd(Y2\

Y3*1),) = d(Yfl(Yflrx)

- Yf

d{Y^)~\

The relative wealth V3(
Jo 3

= V 0(4>)+ f ^ ( Y 3 / 1 ) " 1 ^ dYu, Jo = V30(
f^udYu

where we denote (pt = (/>2(Y3'1)_1e'''. Remark 3.1. The financial interpretation of the auxiliary process Y will be studied in Sections 4.1.6 and 4.1.8 below. Let us only observe here that if Y* is a local martingale under some probability Q* then Y is a

63

Q*-local martingale (and vice versa, if Y is a Q-local martingale under some probability Q then Y* is a Q-local martingale). Nevertheless, for the reader's convenience, we shall use two symbols Q* and Q, since this equivalence holds for continuous processes only. It is thus worth stressing that we will apply Proposition 3.3 to predefault values of assets, rather than directly to asset prices, within the set-up of a semimartingale model with a common default, as described in Section 2.1. In this model, the asset prices may have discontinuities, but their pre-default values follow continuous processes. 4. Martingale Approach to Valuation and Hedging Our goal is to derive quasi-explicit conditions for replicating strategies for a defaultable claim in a fairly general set-up introduced in Section 2.1. In this section, we only deal with trading strategies based on the reference filtration F, and the underlying price processes (that is, prices of defaultfree assets and pre-default values of defaultable assets) are assumed to be continuous. Hence, our arguments will hinge on Proposition 3.3, rather than on a more general Proposition 3.1. We shall also adapt Proposition 3.2 to our current purposes. To simplify the presentation, we make a standing assumption that all coefficient processes are such that the SDEs appearing below admit unique strong solutions, and all stochastic exponentials (used as Radon-Nikodym derivatives) are true martingales under respective probabilities. 4.1 Defaultable asset with total default In this section, we shall examine in some detail a particular model where the two assets, Y1 and Y2, are default-free and satisfy dY\ = Yj(ju,v dt + aiit dWt),

i = 1,2,

where W is a one-dimensional Brownian motion. The third asset is a defaultable asset with total default, so that dY] = Yf_(ju3/( dt + o3,t dWt - dMt). Since we will be interested in replicating strategies in the sense of Definition 2.2, we may and do assume, without loss of generality, that the coefficients \L\,t, Oi,t, i = 1,2, are F-predictable, rather than G-predictable. Recall that, in general, there exist F-predictable processes ^3 and <J3 such that (41)

?3,tl{t
ff3,(l|f
= ff3,fl(t
We assume throughout that Y0 > 0 for every i, so that the price processes Y1, Y2 are strictly positive, and the process Y3 is nonnegative, and has strictly positive pre-default value.

64

4.1.1 Default-free market It is natural to postulate that the default-free market with the two traded assets, Y1 and Y2, is arbitrage-free. More precisely, we choose Y1 as a numeraire, and we require that there exists a probability measure P 1 , equivalent to P on (Q, TT), and such that the process Y2,1 is a P 1 -martingale. The dynamics of processes (Y 1 ) -1 and Y2,1 are d(Y})-1 = {Y})-%a\t

(42)

- pu) dt - au dWt),

and rfY2'1 = Y?1^

- (iU + ai,t(oi,f - a2,t)) dt + (az,t - oi,t) dWt),

respectively. Hence, the necessary condition for the existence of an e.m.m. P 1 is the inclusion A c B, where A = {(t, w) e [0, T] x Q : a\,t{ai) = a2,tMl and B = \(t, a>) e [0, T] x Q : p.\,t{w) = ^{oS)). The necessary and sufficient condition for the existence and uniqueness of an e.m.m. P 1 reads (43)

EPi&rlfdudWu)\

=l

where the process 8 is given by the formula (by convention, 0/0 = 0) (44)

el=a

VlLlBL,

We[0,T].

0\,t ~ 02,t

Note that in the case of constant coefficients, if o\ = a2 then the model is arbitrage-free only in the trivial case when ^ 2 = fiiRemark 4.1. Since the martingale measure P 1 is unique, the default-free model (Y1, Y2) is complete. However, this is not a necessary assumption and thus it can be relaxed. As we shall see in what follows, it is typically more natural to assume that the driving Brownian motion W is multidimensional. 4.1.2 Arbitrage-free property Let us now consider also a defaultable asset Y3. Our goal is now to find a martingale measure Q 1 (if it exists) for relative prices Y2,1 and Y3,1. Recall that we postulate that the hypothesis (H) holds under P for filtrations F and G = F V H, The dynamics of Y3,1 under P are rfY3,i = y 3,i j

^3 (_^

+ ait{aii

_ Q3t)yt

+ (ff3 (

_ (ji () dWt _ dMf j

Let Q 1 be any probability measure equivalent to P on (Q, Qj), and let q be the associated Radon-Nikodym density process, so that (45)

dQ1\gl=TjtdP\e„

65 where the process t] satisfies (46)

djjt = Tjt-(0tdWt + C(iM t )

for some G-predictable processes 8 and C, and JJ is a G-martingale under P. _ From Girsanov's theorem, the processes W and M, given by (47)

Wt = Wt-

f 6U du,

Mt=Mt-

f l,„< r »U« du,

Jo

Jo

are G-martingales under Q 1 . To ensure that Y2,1 is a Q 1 -martingale, we postulate that (43) and (44) are valid. Consequently, for the process Y3-1 to be a Q 1 -martingale, it is necessary and sufficient that C satisfies ytQt - i*3.t - Lii,t

(ff3,t - o\it). 0~\,t -

Ol,t

To ensure that Q 1 is a probability measure equivalent to IP, we require that Cf > - 1 . The unique martingale measure Q 1 is then given by the formula (45) where r\ solves (46), so that r\t = £ / (

f 0udWu\stl

f QudMuY

We are in a position to formulate the following result. Proposition 4.1. Assume that the process Q given by (44) satisfies (43), and (48)

Q = - L < ~ iiu ~ ^——(°v ~ °u)\ > - I yt \ o\,t - o~i,t i 1 2 3 Then the model M = (Y , Y , Y ; <E>) is arbitrage-free and complete. The dynamics of relative prices under the unique martingale measure Q 1 are dYtl = dYf

Yf1(o2,t-0i.t)dWt,

= Y^({fj3jt ~ aif)dWt - dMt).

Since the coefficients ^,,t, a^t, i - 1,2, are F-adapted, the process W is an F-martingale (hence, a Brownian motion) under Q 1 . Hence, by virtue of Proposition 1.1, the hypothesis (H) holds under Q 1 , and the F-intensity of default under Q 1 equals ~

/1

r s

(

yt = yt(l + Qt) = yt + \ ii3,t - pu \

W

~ l"2'(

I

\

(03,1 - o\,t) • o-\,t - <Ji,t I

66

Example 4.1. We present an example where the condition (48) does not hold, and thus arbitrage opportunities arise. Assume the coefficients are constant and satisfy: jii = ^2 - ai = 0, ^3 < —y for a constant default intensity y > 0. Then Y? = 1„), where V() represents the wealth of a self-financing strategy (^>1,^)2,0) with (p2 = f-. Hence, the arbitrage strategy would be to sell the asset Y3, and to follow the strategy
dY\ = Y3((/T3,f + yt) dt + a3,t dWt)

with Y^ = Y^. This strictly positive, continuous, F-adapted process enjoys the property that Y3 = 1{«T)Y3. Let us denote the pre-default values in the numeraire Y3 by Y'f = V](Y3)-1, i = 1,2, and let us introduce the pre-default relative price Y* of the synthetic asset Y2 by setting _ _ Y2'3 dTt :=dY^-^dY^

=

-

Y?((^-iii,+(J3Acyi,t-°2,t))dt+(02,t-ou)dWt),

67 and let us assume that a\it - (J2,t =£ 0. It is also useful to note that the process Y, defined in Proposition 3.3, satisfies dYt = Yt(Jji2,t - \i\,t + 03,tiau ~ C2,t))dt + (a2,t -

0irt)dWt).

In Sections 4.1.6 and 4.1.8, we shall show that in the case, where a given by (39) is deterministic, the process Y has a nice financial interpretation as a credit-risk adjusted forward price of Y2 relative to Y1. Therefore, it is more convenient to work with the process Y* when dealing with the general case, but to use the process Y when analyzing a model with deterministic volatilities. Consider an F-predictable self-financing strategy cp satisfying the balance condition
V^):=X>;Y; = ^Y?. Let Vt{<§) := $Y\. Since the process V(<$) is F-adapted, we see that this is the pre-default price process of the portfolio (f>, that is, we have l(T>t|V"(() = l(T>f|V*((^>); we shall call this process the pre-default wealth of (p. Consequently, the process Vf((p) := Vt((j))(Y^-i - ^,3 j s t e r m e c i m e relative pre-default wealth. Using Proposition 3.1, with suitably modified notation, we find that the F-adapted process V3(c/>) satisfies, for every t e [0, T],

Vf(
Define a new probability on (Q, Tr) by setting dQ* = rfTdW, where dvft = rftd\ dWt, and ,5Q,

0

,

=

W

- Vi,t + g3,?(gy ~ Q2,t)

The process Y*, t e [0, T], is a (local) martingale under Q*. We shall require that this process is in fact a true martingale; a sufficient condition for this is that

X

T

_

2

3

EQ. (Yf (o2,t - au))

dt < oo.

68

From the predictable representation theorem, it follows that for any X e Tj, such that X(Y 3 ) -1 is square-integrable under Q*, there exists a constant x and an F-predictable process (p2 such that

(51)

X = Y^(* + J^ (tfdnj.

We now deduce from Proposition 3.2 that there exists a self-financing strategy

(52)

cgdYu.

Jo

Moreover, it satisfies the balance condition
= l{T) = l(T<x}X,

and thus this strategy replicates the survival claim (X, 0, T). In fact, we have that Vt(
n f (X,0,x) = YfE Q .(X(Y 3 r 1 |!F0,

V£s[0,T].

The process n(X, 0, x)(Y 3 ) -1 is an F-martingale under Q*. Proof. Since X(Y 3 ) -1 is square-integrable under Q*, we know from the predictable representation theorem that cp2 in (51) is such that EQ. I jT ((p2)2d(Y*)t\ < oo, so that the process V3 given by (52) is a true martingale under Q*. We conclude that (X, 0, x) is attainable. Now, let us denote by nt(X, 0, x) the time-t price of the claim (X, 0, x). Since

n^(X,0,x) = l{T>tlVt((p) = =

l{T>t]Y3Ev(X(Y3Tr1\Tt)

l{,>nY3EQ,(V3T\rt)

69

for each t e [0, T]. This proves equality (53).

•

In view of the last result, it is justified to refer to Q* as the pricing measure relative to Y3 for attainable survival claims. Remark 4.3. It can be proved that there exists a unique absolutely continuous probability measure Q on (G, QT) such that we have 1(T>T|X

Gi

=llT>f>Y?EQ.

However, this probability measure is not equivalent to Q", since its RadonNikodym density vanishes after x (for a related result, see Collin-Dufresne et al. [13]). Example 4.2. We provide here an explicit calculation of the pre-default price of a survival claim. For simplicity, we assume that X = 1, so that the claim represents a defaultable zero-coupon bond. Also, we set yt = y = const, Hiit = 0, and oltt = a,, i = 1,2,3. Straightforward calculations yield the following pricing formula n0(l,0,T) = Y30e-ly+i°Vr. We see that here the pre-default price 7To(l, 0, T) depends explicitly on the intensity y, or rather, on the drift term in dynamics of pre-default value of defaultable asset. Indeed, from the practical viewpoint, the interpretation of the drift coefficient in dynamics of Y2 as the real-world default intensity is questionable, since within our set-up the default intensity never appears as an independent variable, but is merely a component of the drift term in dynamics of pre-default value of Y3. Note also that we deal here with a model with three tradeable assets driven by a one-dimensional Brownian motion. No wonder that the model enjoys completeness, but as a downside, it has an undesirable property that the pre-default values of all three assets are perfectly correlated. Consequently, the drift terms in dynamics of traded assets are closely linked to each other, in the sense, that their behavior under an equivalent change of a probability measure is quite specific. As we shall see later, if traded primary assets are judiciously chosen then, typically, the pre-default price (and hence the price) of a survival claim will not explicitly depend on the intensity process. Remark 4.4. Generally speaking, we believe that one can classify a financial model as 'realistic' if its implementation does not require estimation of drift parameters in (pre-default) prices, at least for the purpose of hedging

70

and valuation of a sufficiently large class of (defaultable) contingent claims of interest. It is worth recalling that the drift coefficients are not assumed to be market observables. Since the default intensity can formally interpreted as a component of the drift term in dynamics of pre-default prices, in a realistic model there is no need to estimate this quantity. From this perspective, the model considered in Example 4.2 may serve as an example of an 'unrealistic' model, since its implementation requires the knowledge of the drift parameter in the dynamics of Y3. We do not pretend here that it is always possible to hedge derivative assets without using the drift coefficients in dynamics of tradeable assets, but it seems to us that a good idea is to develop models in which this knowledge is not essential. Of course, a generic semimartingale model considered until now provides only a framework for a construction of realistic models for hedging of default risk. A choice of tradeable assets and specification of their dynamics should be examined on a case-by-case basis, rather than in a general semimartingale set-up. We shall address this important issue in the foregoing sections, in which we shall deal with particular examples of practically interesting defaultable claims. 4.1.4 Hedging a recovery process Let us now briefly study the situation where the promised payoff equals zero, and the recovery payoff is paid at time T and equals ZT for some Fadapted process Z. Put another way, we consider a defaultable claim of the form (0, Z, T). Once again, we make use of Propositions 3.1 and 3.2. In view of (35), we need to find a constant x and an F-predictable process cj)2 such that

(54)

4>T := - J Yi **\* = * + f tit*!.

Similarly as in Section 4.1.3 we conclude that, under suitable integrability conditions on i/>r, there exists (p2 such that dipt = (p2dY't, where \pt = EQ.(I/>T I "Ft). We now

set

v? = *+ f
f§-3dY¥, Jo Y/

so that, in particular, V3, = 0. Then it is possible to find processes (p1 and (p3 such that the strategy (p is self-financing and it satisfies: Vt((p) - V3Y3 and Vt( T).

71

4.1.5 Bond market For the sake of concreteness, we assume that Y\ - B(t, T) is the price of a default-free ZC-bond with maturity T, and Yf = D(t, T) is the price of a defaultable ZC-bond with zero recovery, that is, an asset with the terminal payoff Y| = 1|T< T |. We postulate that the dynamics under P of the default-free ZC-bond are (55)

dB(t,T) = B(t,T)(n(t,T)dt

+ b(t,T)dWt)

for some F-predictable processes \i{t, T) and b(t, T). We choose the process Y) = B(t, T) as a numeraire. Since the prices of the other two assets are not given a priori, we may choose any probability measure Q equivalent to P on (Q, QT) to play the role of Q 1 . In such a case, an e.m.m. Q 1 is referred to as the forward martingale measure for the date T, and is denoted by QT. Hence, the Radon-Nikodym density of Qr with respect to P is given by (46) for some F-predictable processes 8 and C and the process Wf = W,-

f 6ttdu, Jo

Vte[0,T],

is a Brownian motion under Qj. Under Q j the default-free ZC-bond is governed by dB(t, T) = B(t,1) (ji(tt T)dt + b{t, T) dWj) where Ji(t, T) = j.i(t, T) + 9tb(t, T). Let T stand for the F-hazard process of T under Q T , so that? f = - ln(l - f t ) , where ft = QT(T < 11 Tt). Assume that the hypothesis (H) holds under Qr so that, in particular, the process T is increasing. We define the price process of a defaultable ZC-bond with zero recovery by the formula D(t,T):= Ba,T)E Qr (l,T< T | \Qt) = \{t
\Tt),

where the second equality follows from Lemma 1.3. It is then clear that Yf'1 = D(t, T)(B(t, T))- 1 is a Q T -martingale, and the pre-default price D(t, T) equals D(t,T, =

B(t,T)EQT(e?>-?T\rt).

The next result examines the basic properties of the auxiliary process T(t, T) given as, for every t e [0, T], T(t,T) = Y f = Dit.TKBitrT))-1

=

EQ^-^lTt).

72

The quantity T(t, T) can be interpreted as the conditional probability (under Qr) that default will not occur prior to the maturity date T, given that we observe ft and we know that the default has not yet happened. We will be more interested, however, in its volatility process ji(t, T) as defined in the following result. Lemma 4.1. Assume that the W-hazard process F of T under Q j is continuous. Then the process T(t, T), t e [0, T], is a continuous W-submartingale and (56)

d?(t, T) = T(t, T)(dTt + p(t, T) dWf)

for some F-predictable process fi(t,T). The process T(t, T) is offinitevariation ifand only if the hazard process T is deterministic. In this case, we have T(t, T) = er'~Tr. Proof. We have T(t, T) = EQr(eF'-?'-1 Tt) = eT>Lt, where we set Lt = TEQT(e~Tr 17rX Hence, T(t, T) is equal to the product of a strictly positive, increasing, right-continuous, F-adapted process er', and a strictly positive, continuous F-martingale L. Furthermore, there exists an F-predictable process f>(l, T) such that L satisfies

dLt = LtJ(t,T)dWj with the initial condition

LQ

= E Q J ^ 7 ] . Formula (56) now follows by an

application of Ito's formula, by setting /3(t, T) = e_r,/3(f, T). To complete the proof, it suffices to recall that a continuous martingale is never of finite variation, unless it is a constant process. D Remark 4.5. It can be checked that |3(f, T) is also the volatility of the process T(t,T) =

Er(er'-rT\rt).

Assume that Tt - J^ y~u du for some F-predictable, nonnegative process y~. Then we have the following auxiliary result, which gives, in particular, the volatility of the defaultable ZC-bond. Corollary 4.2. The dynamics under QT of the pre-default price D(t, T) equals db(t, T) = D(t, T)((f?(^, T)+b{t, T)p(t, T)+ yt)dt+(b(t, T)+p(t, T))d(t, T)dWj).

73

Equivalently, the price D(t, T) of the defaultable ZC-bond satisfies under QT dD(t, T) = D{t,T)((fT(t, T) + b(t, T)p(t,T))dt + d(t,T)dWj - dMt). where we set d(t, T) = b(t, T) + p(t, T). Note that the process p(t, T) can be expressed in terms of market observables, since it is simply the difference of volatilities d(t, T) and b(t, T) of pre-default prices of tradeable assets. 4.1.6 Credit-risk-adjusted forward price Assume that the price Y2 satisfies under the statistical probability P (57)

dY2t = Y2(^2/, dt + at dWt)

with F-predictable coefficients y. and a. Let F^itJ) = Y2(B(f,T))_1 be the forward price of Y2. For an appropriate choice of d (see 50), we shall have that dFyiit, T) = Fy2(f, T)(a, - b(t, TJ) dWj. Therefore, the dynamics of the pre-default synthetic asset Y't under QT are dY] = Yf3(ot - b(t, T)) (dWj - fi(t, T) dt), and the process Yt = -Y^e - "' satisfies dY, = Y,(a, - b(t, T)) (dWj - p(t, T) dt). Let Q be an equivalent probability measure on (Q, Qr) such that Y (or, equivalently, Y") is a Q-martingale. By virtue of Girsanov's theorem, the process W given by the formula Wt = Wj-

f p(u,T)du, Jo

Vte[Q,T],

is a Brownian motion under
dFyiit, T) = Fyi(t, T)(oi - b(t, T))(dWt + p(t, T) dt).

It appears that the valuation results are easier to interpret when they are expressed in terms of forward prices associated with vulnerable forward contracts, rather than in terms of spot prices of primary assets. For

74

this reason, we shall now examine credit-risk-adjusted forward prices of default-free and defaultable assets. Definition 4.2. Let Y be a ^r-measurable claim. An ^-measurable random variable K is called the credit-risk-adjustedforward price of Y if the pre-default value at time t of the vulnerable forward contract represented by the claim TL(T
Lemma 4.2. The credit-risk-adjusted forward price Fy(t, T) of an attainable survival claim (X,0, T), represented by a Qj-measurable claim Y = Xl( T
is worthless at time t on the set {t < T). It is clear that the pre-default value at time t of this claim equals nt(X, 0, T) - KD(t, T). Consequently, we obtain FY(t,T) = nt(X,0,T)(D(t,T))-K • Let us now focus on default-free assets. Manifestly, the credit-riskadjusted forward price of the bond B(t,T) equals 1. To find the credit-riskadjusted forward price of Y2, let us write fy2(t,T)

:= FY2(t,T)eaT-a'

= Y2'1^-'",

where a is given by (see (39)) (59) at=

f (ou-b(u,T))p(u,T)du=

f

(au-b(u,T))(d(u,T)-b(u,T))du.

Lemma 4.3. Assume that a given by (59) is a deterministic function. Then the credit-risk-adjusted forward price ofY2 equals F^it, T)for every t £ [0, T]. Proof. According to Definition 4.2, the price Fyi{t, T) is an Tf-measurable random variable K, which makes the forward contract represented by the claim D(T,T)(Yj - K) worthless on the set {t < T}. Assume that the claim Y2, - K is attainable. 1 Since D(T,T) - 1, from equation (53) it follows that the pre-default value of this claim is given by the conditional expectation D(t,T)E^(Y2T-K\rt). 1 Attainability of this claim can be shown in a similar way as the attainability of a vulnerable call option considered in Section 4.1.7.

75

Consequently, Tyi(t, T) = E^(Y2T I T>) = E ^ F ^ T , T) \ Tt) = F^t, T) e^a>, as was claimed.

•

It is worth noting that the process Fy^f, T) is a (local) martingale under the pricing measure Q, since it satisfies (60)

dFyi(t, T) = f W , T)(ot - b(t, T))dWt.

Under the present assumptions, the auxiliary process Y introduced in Proposition 3.3 and the credit-risk-adjusted forward price Fy2(f, T) are closely related to each other. Indeed, we have Fy2(t,T) = Yte"T, so that the two processes are proportional. 4.1.7 Vulnerable option on a default-free asset We shall now analyze a vulnerable call option with the payoff CT

=

fl{T<Ti(*Y

_

^)+-

Our goal is to find a replicating strategy for this claim, interpreted as a survival claim (X, 0, T) with the promised payoff X = Cj = (Y2 — K)+, where Cr is the payoff of an equivalent non-vulnerable option. The method presented below is quite general, however, so that it can be applied to any survival claim with the promised payoff X = G(Y^.) for some function G : R —» R satisfying the usual integrability assumptions. We assume that Y) - B(t, T), Yf = D(t, T) and the price of a default-free asset Y2 is governed by (57). Then CT = 1{T
=

!{T
.

We are going to apply Proposition 3.3. In the present set-up, we have Y2,1 = Fy2(t, T) and Yt = Fy2(f, T)e~°". Since a vulnerable option is an example of a survival claim, in view of Lemma 4.2, its credit-risk-adjusted forward price satisfies Tc<,(t, T) = Cdt (D(f, T)) -1 . Proposition 4.2. Suppose that the volatilities a, b and f> are deterministic functions. Then the credit-risk-adjusted forward price of a vulnerable call option written on a default-free asset Y2 equals (61)

?c,(t, T) = Fy2(f,T)N(d+(Fy2(f,T), t, T)) - KN(d-(fy2(t,T), t, T))

where

d±(f,t,T) =

\nf-]nK±\v1{t,T)

v(t,T)

76

and v\t,T)=

f

(au-b(u,T))2du.

The replicating strategy cj) in the spot market satisfies for every t e [0, T], on the set [t < T), 4>\B{t,T) = -tfY*,

2 = D{t,T)(B{t,T))-lN{d+(t,T))ea^',

<$D(t,T) = Cdt,

where d+ (t, T) = d+ (F^ (t, T),t, T). Proof. In the first step, we establish the valuation formula. Assume for the moment that the option is attainable. Then the pre-default value of the option equals, for every t e [0, T], (62) Cj = D(t,T)E6((Fy2(T,T)-K)+

\Tt) = D(t,T)E^F^T,T)-K)+

\Tt).

In view of (60), the conditional expectation above can be computed explicitly, yielding the valuation formula (61). To find the replicating strategy, and establish attainability of the option, we consider the Ito differential dFcj(t, T) and we identify terms in (52). It appears that (63)

d?c*(t, T) = N(d+(t, T)) dFyi(t, T) = N(d+(t, T))e"T dYt = Af(d+(t,T))Y*'1eai~a' dTt,

so that the process (p2 in (51) equals (pf

=

Yf1N(d+(t,T))e"T-a'.

Moreover, cj)1 is such that c$B{t,T) + (pjYf = 0 and $ = Cdt{D(t,T))-x. It is easily seen that this proves also the attainability of the option. • Let us examine the financial interpretation of the last result. First, equality (63) shows that it is easy to replicate the option using vulnerable forward contracts. Indeed, we have Cd rT FCJ(T,T) = X = ^ - ^ + D(0,T) Jo

N(d+(t,T))dFyi(t,T)

and thus it is enough to invest the premium CdQ - CQ in defaultable ZCbonds of maturity T, and take at any instant t prior to default N(d+(t, T)) positions in vulnerable forward contracts. It is understood that if default

77

occurs prior to T, all outstanding vulnerable forward contracts become void. Second, it is worth stressing that neither the arbitrage price, nor the replicating strategy for a vulnerable option, depend explicitly on the default intensity. This remarkable feature is due to the fact that the default risk of the writer of the option can be completely eliminated by trading in defaultable zero-coupon bond with the same exposure to credit risk as a vulnerable option. In fact, since the volatility j3 is invariant with respect to an equivalent change of a probability measure, and so are the volatilities a and b(t, T), the formulae of Proposition 4.2 are valid for any choice of a forward measure QT equivalent to P (and, of course, they are valid under IP as well). The only way in which the choice of a forward measure Qr makes an impact on these results is through the pre-default value of a defaultable ZC-bond. We conclude that we deal here with the volatility based relative pricing a defaultable claim. This should be contrasted with the more popular intensity-based risk-neutral pricing, which is commonly used to produce an arbitrage-free model of tradeable defaultable assets. Recall, however, that if tradeable assets are not chosen carefully for a given class of survival claims, then both hedging strategy and pre-default price may depend explicitly on values of drift parameters, which can be linked in our setup to the default intensity (see Example 4.2). Remark 4.6. Assume that X = G(Y2) for some function G : R —» R. Then the credit-risk-adjusted forward price of a survival claim satisfies Fx{t, T) = v(t, Fyi (t, T)), where the pricing function v solves the PDE

dtv(t,f)+ \{at - b(t, T))2?d~v{tJ) = 0 with the terminal condition v(T,f) = G(/). The PDE approach is studied in Section 5 below. Remark 4.7. Proposition 4.2 is still valid if the driving Brownian motion is two-dimensional, rather than one-dimensional. In an extended model, the volatilities at, b(t, T) and p(t, T) take values in R 2 and the respective products are interpreted as inner products in R 2 . Equivalently, one may prefer to deal with real-valued volatilities, but with correlated one-dimensional Brownian motions. 4.1.8 Vulnerable swaption In this section, we relax the assumption that Y1 is the price of a defaultfree bond. We now let Y1 and Y2 to be arbitrary default-free assets, with

78

dynamics dY\ = Y\[jiii,t dt + out dWt),

i = 1,2.

We still take D(t, T) to be the third asset, and we maintain the assumption that the model is arbitrage-free, but we no longer postulate its completeness. In other words, we postulate the existence an e.m.m. Q 1 , as defined in Section 4.1.2, but not the uniqueness of Q 1 . We take the first asset as a numeraire, so that all prices are expressed in units of Y1. In particular, Y)'1 = 1 for every t € R + , and the relative prices Y2'1 and y 3 ' 1 satisfy under Q 1 (cf. Proposition 4.1) dY? = dYf

Y\\a2jt-alJt)dVitl

= Y**({o3jt - au)dWt - dMt).

It is natural to postulate that the driving Brownian noise is twodimensional. Under this assumption, we may represent the joint dynamics of y 2,1 and y 3 ' 1 under Q 1 as follows

dYf dY3,i =

=Y$\a2Jt-au)dW\. Yf({0Xl

- ou)dW2 - dMt),

where W1, W2 are one-dimensional Brownian motions under Q 1 , such that d( W1, W2)t = pt dt for a deterministic instantaneous correlation coefficient p taking values in [-1,1]. We assume from now on that the volatilities cr„ i = 1,2,3 are deterministic. Let us set (64)

at =
pu(a2,u -ffi,«)(ff3,«- ffi,«) du, Jo

and let (Q be an equivalent probability measure on (Q, Qj) such that the process Yt = Y2,1e~a' is a Q-martingale. To clarify the financial interpretation of the auxiliary process Y in the present context, we introduce the concept of credit-risk-adjusted forward price relative to the numeraire y 1 . Definition 4.3. Letybea^r-measurableclaim. An ^-measurable random variable K is called the time-f credit-risk-adjusted Y1-forward price of Y if the pre-default value at time t of a vulnerable forward contract, represented by the claim

ipvriO^r^Y-KYj.) = i^nOT^r 1 -K), equals 0.

79

The credit-risk-adjusted Y1-forward price of Y is denoted by Fy|Yi (t, T), and it is also interpreted as an abstract defaultable swap rate. The following auxiliary results are easy to establish, along the same lines as Lemmas 4.2 and 4.3. Lemma 4.4. The credit-risk-adjusted Y1-forward price of a survival claim Y = (X, 0, T) equals F^ y l (t, T) = nt(X\ 0, x)(D(f, I))" 1 where X1 = X(Y^)' 1 is the price ofX in the numeraire Y1, and nt{Xx,Q, x) is the pre-default value of a survival claim with the promised payoff X1. Proof. It suffices to note that for Y = 1(T
•

Lemma 4.5. The credit-risk-adjusted Y1 -forward price of the asset Y2 equals (65)

FY2|y, (t, T) = Y2'1 eaT~a' = YteaT,

where a is given by (64). Proof. It suffices to find an ^-measurable random variable K for which

Consequently, K = Fy2|Yi (t, T), where

fyy. (t, T) = E^(Y2/ I Tt) = Yf1 e"T~a' = Yt e"\ where we have used the facts that Yt = Y2,le~a' is a Q-martingale, and a is deterministic. • We are in a position to examine a vulnerable option to exchange defaultfree assets with the payoff (66)

CdT = 1| T < T ) (Y^)- 1 (Y2 - KY\)+ = 1,T
The last expression shows that the option can be interpreted as a vulnerable swaption associated with the assets Y1 and Y2. It is useful to observe that

ci

i |T
Y2T Y1

-K

80

so that, when expressed in the numeraire Y1, the payoff becomes C1T4=D1(T,T)(y%l-K)+, where C\A = Cd{Y}yx and D\t,T) = D{t,T){Y\)~x stand for the prices relative to Y1. It is clear that we deal here with a model analogous to the model examined in Sections 4.1.5 and 4.1.7 in which, however, all prices are now relative to the numeraire Y1. This observation allows us to directly derive the valuation formula below from Proposition 4.2. Proposition 4.3. The credit-risk-adjusted Yx-forward price of a vulnerable call option written with the payoff given by (66) equals Ftfiyi (t, T) = Fy^iy,(t, T)N(d+(FY2iy, (f, T), I, T)) - KN(d_(Fy2|Y1 (t, T), t, T)) where

__

\nf-\nK±\v\t,T)

r

and

T

v\t,T) = j{oi,u-oiM)2du. The replicating strategy

T)Y*-a>,


where d+(t, T) = d+(f^(t, T), t, T). Proof. The proof is analogous to that of Proposition 4.2, and thus it is omitted. D It is worth noting that the payoff (66) was judiciously chosen. Suppose instead that the option payoff is not specified by (66), but it is given by an apparently simpler expression (67)

CdT =

l{T
Since the payoff CdT can be represented as follows (-7- — CJ(I J, YJ, I J) — YjiYj — KYj) , where G(yi, 1/2/J/3) ~ 1/3(1/2 - Ky\)+, the option can be seen an option to exchange the second asset for K units of the first asset, but with the payoff

81

expressed in units of the defaultable asset. When expressed in relative prices, the payoff becomes Cj

= 1(T < T |(Y T ' -K)

,

where 1 |T
dY\ = YJ_(fii/( dt + aiit dWt - dMt),

where W is a one-dimensional Brownian motion, so that Yt

~ l{t
Yt = l ( f < T ) Y t ,

with the pre-default prices governed by the SDEs (69)

dY\ = YJ((^,f + Yt) dt + aif dVi,).

The wealth process V associated with the self-financing trading strategy ((p1,
where Y)x = Yf/Y]. Since both primary traded assets are subject to total default, it is clear that the present model is incomplete, in the sense, that not all defaultable claims can be replicated. We shall check in Section 4.2.1 that, under the assumption that the driving Brownian motion W is onedimensional, all survival claims satisfying natural technical conditions are hedgeable, however. In the more realistic case of a two-dimensional noise, we will still be able to hedge a large class of survival claims, including options on a defaultable asset (see Section 4.2.2) and options to exchange defaultable assets (see Section 4.2.3). 4.2.1 Hedging a survival claim For the sake of expositional simplicity, we assume in this section that the driving Brownian motion W is one-dimensional. This is definitely not

82

the right choice, since we deal here with two risky assets, and thus their prices will be perfectly correlated. However, this assumption is convenient for the expositional purposes, since it ensures the model completeness with respect to survival claims, and it will be later relaxed anyway. We shall argue that in a model with two defaultable assets governed by (68), replication of a survival claim ( X , 0 , T ) is in fact equivalent to replication of the promised payoff X using the pre-default processes. Lemma 4.6. If a strategy ', i = 1,1, based on pre-default values Y', i = 1,1, is a replicating strategy for an Tr-measurable claim X, that is, if

}Yj +

VT(cp) = X, then for the process Vt((p) - §)Y) + (p2Y2 we have, for every t e [0,T], dVt{2dY2, VT() = X l | T < t | . This means that a strategy ) = 1(KT)Vt(cp). From 4>] dY) + $ dYf = -{] dY) + tf dYJ), it follows that (j)] dY} + tfdYf = -Vt((p)dHt + (1 - Ht-)dVt(
_ ] dY) + <$dY] = d(l{t).

It is also obvious that VT(
• 1

Combining the last result with Lemma 3.1, we see that a strategy (t/) ,
Y\(x+ f $dY?)

=X

for some constant x and some F-predictable process cp2, where, in view of (69), dY2'1 = Y2t'\(p.2rt - nu + au(au

- av))dt

+ (ov -

au)dWt).

83

We introduce a probability measure Q, equivalent to F on {D.,QT), and such that Y 21 is an F-martingale under Q. It is easily seen that the RadonNikodym density rj satisfies, for t e [0, T],

(70)

dQ\e,= r\t dP | a = St IJ 0S dWs\ dV \ g,

with f*2,t - i"l,t + Ol,t(aU

~

ff

2,f)

ff -

,

provided, of course, that the process 6 is well defined and satisfies suitable integrability conditions. We shall show that a survival claim is attainable if the random variable X(Y*,)-1 is Q-integrable. Indeed, the pre-default value Vt at time t of a survival claim equals

Vt =

Y}^{x{Y\Tl\Tt),

and from the predictable representation theorem, we deduce that there exists a process (j>2 such that EQ(X(Y1 } - I

!^j

= E_(x(?i)-i)

+ f fi &Yf. •JO

1

The component (p of the self-financing trading strategy cf> = (cp1, (p2) is then chosen in such a way that }Y}+tf?i = Vt,

Vt6[0,T].

To conclude, by focusing on pre-default values, we have shown that the replication of survival claims can be reduced here to classic results on replication of (non-defaultable) contingent claims in a default-free market model. 4.2.2 Option on a defaultable asset In order to get a complete model with respect to survival claims, we postulated in the previous section that the driving Brownian motion in dynamics (68) is one-dimensional. This assumption is questionable, since it implies the perfect correlation of risky assets. However, we may relax this restriction, and work instead with the two correlated one-dimensional Brownian motions. The model will no longer be complete, but options on a defaultable assets will be still attainable. The payoff of a (non-vulnerable) call option written on the defaultable asset Y2 equals CT = (Y 2 --K) + =l,r< T |(Y 2 -IC) + ,

84

so that it is natural to interpret this contract as a survival claim with the promised payoff X = (Y2 - K)+. To deal with this option in an efficient way, we consider a model in which (71)

dY\ = YtJjiij dt + a if dW[ - dMt),

where W1 and W2 are two one-dimensional correlated Brownian motions with the instantaneous correlation coefficient pt. More specifically, we assume that Y] = D(t,T) — l(« T |D(f, T) represents a defaultable ZC-bond with zero recovery, and Y2 = 1{«T}Y* is a generic defaultable asset with total default. Within the present set-up, the payoff can also be represented as follows CT = G{Y\,Y2) = {Y\-KY\Y, where #(1/1,1/2) = (1/2 - KyiY', and thus it can also be seen as an option to exchange the second asset for K units of the first asset. The requirement that the process Y2'1 = Y^Y,1)-1 follows an Fmartingale under Q implies that

(72)

dYf1 = YY((avpt ~ au) dWJ + a2,t y/l-pfdWf),

where W = (W1, W2) follows a two-dimensional Brownian morion under Q. Since Y^, = 1, replication of the option reduces to finding a constant x and an F-predictable process cj)2 satisfying

rT

(p2 dY2'1 = (Y2 - K)+. Jo To obtain closed-form expressions for the option price and replicating strategy, we postulate that the volatilities a\,t, 02,t a n d the correlation coefficient pt are deterministic. Let Tyi(t,T) = Y2t{D{t,T)Yl (?c(t,T) = Q(D(f,T)) -1 , respectively) stand for the credit-risk-adjusted forward price of the second asset (the option, respectively). The proof of the following valuation result is fairly standard, and thus it is omitted. x+

Proposition 4.4. The credit-risk-adjusted forward price of the option written on Y2 equals ?c(t,T) = ?Yi(t,T)N(d+(Fyi(t,T),t,T))

-

KN(d-(?y2(tfT),t,T)).

Equivalently, the pre-default price of the option equals C, = Y2N(d+(F^(t, T), t, T)) - KD(t, T)N(d4fy2(t, T), t, T)),

85 where In / - In

ijJ,t,T)

K±\v2{t,T)

v(t,T)

and v2(t,T)=

I

(o\M+o\u-2puoiM02,u)du.

Moreover the replicating strategyfyin the spot market satisfies for every t e [0, T], on the set \t < T), co) = -KN(d-(fyi(t,

T), t, T)),

$ = N(d+(FY2(t, T), t, T)).

4.2.3 Option to exchange defaultable assets We work here with the two correlated one-dimensional Brownian motions, so that (73)

dY\ = Y;_((U,/( dt + aiA dW\ - dMt),

i =

1,2,

where d{ W1, W2)t = pi dt for some function p with values in [—1,1]. The model is no longer complete, but it is still not difficult to establish a direct counterpart of Proposition 4.4 for the exchange option with the payoff (Y2, KYj)+. In fact, the next result shows that the pricing formula expressed in terms of pre-default prices has the same shape as the standard formula for the option to exchange non-defaultable assets with dynamics (68). It is notable that we do not need to make any assumption about the behavior of the default intensity. We only assume that the coefficients in (73) are such that there exist an e.m.m. for the process Y2,1, where (74)

dYt = Y\{{pi,t + yt)dt + GltdW^

i = 1,2,

so that we implicitly impose mild technical conditions on drift coefficients. Proposition 4.5. Assume that the volatilities o~\,02 and the instantaneous correlation coefficient p are deterministic. Then the pre-default price of the exchange option equals Q = Y2N(d+(Y2'\ t, T)) - KY}N(d-(Y2-\ t, T)), where d t T)

^' =

\ny-\nK±\v2(t,T) v(t,T)

86

and v2(t, T) = I (o2lu + a\u - 2puo\,ualM) du. Moreover the replicating strategy cf> in the spot market satisfies for every t e [0, T], on the set {t < T},
t, T)),

$ = N(d+(Y2'\ t, T)).

The pricing formula for the option on a defaultable asset (see Proposition 4.4) can be seen as a special case of the formula established in Proposition 4.5. Similarly as in Sections 4.1.7 and 4.1.8, we conclude that the pricing and hedging of any attainable survival claim with the promised payoff X = g(Yj, Y2,) depends on the choice of a default intensity only through the pre-default prices Y\ and Y2. This property shows that we have correctly specified the hedging instruments for a claim at hand. Of course, the model considered in this section is not complete, even if the concept of completeness is reduced to survival claims. Basically, a survival claim can be hedged if its promised payoff can be represents as X = Y^hiY2:1). 5. PDE Approach to Valuation and Hedging In the remaining part of the paper, we take a different perspective, and we assume that trading occurs on the time interval [0, T] and our goal is to replicate a contingent claim of the form Y = l ( T> T |gi(^, Y2, Y3T) + l,r
Y\,Hr),

which settles at time T. We do not need to assume here that the coefficients in dynamics of primary assets are F-predictable. Since our goal is to develop the PDE approach, it will be essential, however, to postulate a Markovian character of a model. For the sake of simplicity, we assume that the coefficients are constant, so that dY\ = Y\_(in dt + at dWt + K,- dMt),

i = 1,2,3.

The assumption of constancy of coefficients is rarely, if ever, satisfied in practically relevant models of credit risk. It is thus important to note that it was postulated here mainly for the sake of notational convenience, and the general results established in this section can be easily extended to a non-homogeneous Markov case in which /i, t - jii{t, Y)_, Y2_, Y^_, Hf_), <J, t = Oi{t,Y\_,Y2_,Yl,Ht.),eic.

87

5.1 Defaultable asset with total default We first assume that Y1 and Y2 are default-free, so that KI = K2 = 0, and the third asset is subject to total default, i.e. K3 = - 1 , dY3 = Y3_(^3 dt +ff3dWt - dMt). We work throughout under the assumptions of Proposition 4.1. This means that any QMntegrable contingent claim Y = G(Y\, Y2, Y\; HT) is attainable, and its arbitrage price equals nt(Y)=:Y}EQi(Y(YlTr1\St),

(75)

Vfe[0,T].

The following auxiliary result is thus rather obvious. Lemma 5.1. The process (Y1^^3,!!) has the Markov property with respect to the filtration G under the martingale measure Q 1 . For any attainable claim Y = G(Y\, Y2, Y\;HT) there exists a function v : [0,T] x R 3 x {0,1} -> IR such thatnt{Y) = v{t,Y),Y],Y3;Ht). We find it convenient to introduce the pre-default pricing function v(;0) = v{t,yi,y2,y3',0) and the post-default pricing function u ( ; l ) = v(t,y\,y2,y3;l). In fact, since Y3 = 0 if Ht = 1, it suffices to study the post-default function v(t, 1/1,1/2; 1) = v(t, y\, y2,0; 1). Also, we write ai = fa - ai

^1 - P-2 , b = (ju3 - jUiXffi -
Let y > 0 be the constant default intensity under P, and let C > - 1 be given by formula (48). Proposition 5.1. Assume that the functions v{-; 0) and v{-; 1) belong to the class C 1 ' 2 ^ , T] x R 3 , R). Then v(t, yx, y2,1/3; 0) satisfies the PDE 2

dtv(-; 0) + ^

x

3

aiyidiVi-; 0) + {a3 + Qy3d3v(-; 0) + - ^

1=1

a/CT;y,i//<9,7t;(-;0)

i,j=i

- aiv(-; 0) + (y -

g

I [v(t, t/i, y2; 1) - v(t, ylf y2, y3; 0)] = 0

subject to the terminal condition v(T,yi,y2,y3;0) v(t, 1/1,1/2; 1) satisfies the PDE

= G(yi,y2,y3;0),

2

dtv(-; 1) + YJ oayidiv(-; 1) + 2 H o~i°jyiyjdijv{-; 1) - axv(-; 1) = 0 1=1

!,/=l

subject to the terminal condition v{T, yi, y2; 1) = G(y\, y2,0; 1).

and

Proof. For simplicity, we write Q = nt(Y). Let us define Au(f,yi, y 2 ,ys) = v(t,yi,y2; 1) -

v(t,yx,yz,y3;0).

Then the jump AC( = Ct - Cf_ can be represented as follows: AQ = l | T = / ) (v(t, Yj, Y); 1) - v(t, Y), Y], Y)_; 0)) = l | T = f ) Av(t, Y), YJ, Y3t J. We write
f Jo

g{u,Yl,Yl)du

since Y„ and Y\_ differ only for at most one value of u (for each co). Let E,t = l((
dCt = dtvdt + ]T divdY] + - Y i=i

OiOjY^Yldijvdt

i,;=l

+ (Az; + Yf_^3f)rfH( 3

1

3

= dtvdt + Y d{odYit + 2 Xi '=i

a a Yi Y

i i t- [-diivdt

',/=i

+ (Au + Y^uJfdM, + E,tdt), and this in turn implies that 3

3

dC( = dtvdt + Y Yit_div(^idt + oidWt) + ~Y '=i

ajaf^^Y'^dijVdt

i',/=i

+ Ai;dMf + (A» + Yf_<93i;)^f dt = \d,v + Y mYt_diV + i Y OiOjYiYldijV + (Av + Yld3v)tt \ dt { '=1 y=i ] 3

( J ] cr/Y^-p) rfW, + Au dM,.

89 We now use the integration by parts formula together with (42) to derive dynamics of the relative price Q = C^Y,1)-1. We find that dQ = Ct-[(-m + of) dt - en dWt) + (YlT1

Utv + J^^Yldiv

+l~Y^oiOjY^Yldijo

+ (Av + Y?_<93z>)6 dt

3 1

3 l

+ (Yjj-

-1

J^ OiY\_divdWt + {Y)_Y &vdMt - (YJL) CTI £

0iY\_djvdt.

Hence, using (47), we obtain dCt = Q_( - (Lii + of)dt + Q_( - (Ji dWt + (Yl)-1

Utv + f^ inY'_diV + 1 £

OiOjYLYiBijV + (Av + Y?_«93t>)6 1A

3 Y 1

oiddt)

3

1

1

1

aiYit_ddivdt

+ ( t -)" £ ffiYJ_d,i>dW, + (Y, .)" £ i=i

/=i 3

+ (Yj_)_1AodMt + (Y3L)_1C6Apdt - (YJL)_1ffi £ oiYit_d{Ddt. This means that the process C admits the following decomposition under Q1 dQ = C(_f — |Ui + fff - oi9)dt + (YJL)-11 ^

+£

[

<=i

/x,-YJ_^,-i» + ^

OiofaYldijD

+ (Av + Yld3v)tt

\ dt

i,j=i

3

+ C^1-)-1 J ] , OiY\_ddiVdt + {Y}_TX&tAvdt f=i 3

- (Y,1.)"1^! 2 2 <JiY\_diVdt + a ((^-martingale. i=i

From (75), it follows that the process C is a martingale under Q 1 . Therefore, the continuous finite variation part in the above decomposition necessarily

90

vanishes, and thus we get Ct-(Yl)-1(-ii1+(%-o1d)

0=

+ (YJL)"1 Utv + J^ fiiYldiV + ^ £

aiOjYlYidijv

3

+ (Av + Y=Lfci>) i, 3

+ (Yj-)"1 X ff/YUe^ + (VjL)_1KtAp - (Yji)- 1 ^ £ a.-Y*.^. 1=1

!=1

Consequently, we have that

Q=

Ct-(-\ii+a\-axd) 3

3

+ dtv + £ mYldfo + - £ o{OjYit_Yidijo + (Av + Yld3v) 6 !=1

!,/=l

3

3

1=1

1=1

Finally, we conclude that 2

dtv + ^

3

«/Y|_d,o + (a 3 + £0 Vf-^3f + ^ £

i=l

- aiQ_

OiO-jYlY'-^jV

i',y=l

+ (1 + Q & A P = 0.

Recall that £t = l(« T |y. It is thus clear that the pricing functions v(-, 0) and v(-; 1) satisfy the PDEs given in the statement of the proposition. D The next result deals with a replicating strategy for Y. Proposition 5.2. The replicating strategy
= ~Av(t, Y), Y2t, Y?_) = v(t, Y\, Yf, Y?_; 0) - v(t, Y\,Y\; 1), 3

$Yj{02 - o\) = -(<TI - o?,)Av - o\v + J^ Y(_cr,(9,t>, i=i

<#Y? = i;-
-1

1

rfQ = -(Yj) o-iodWt + (Yj)- ]T aiYil_d(odYtt + (Yj)_1ApdM,. ;_1

91

The self-financing strategy that replicates Y is determined by two components <^>2, (/>3 and the following relationship: dC, =

tfdY^+tfdYf1

= $Y2t'\02-ox)dWt+<$Y^

((a 3 - ai)dWt - dM>).

By identification, we obtain ^ Y ^ 1 = (Y])-1Av and 3

(f)fYf(o2 - cri) - (a3 - oi)Av = -aiQ + ]T Yit_aidiv. 1=1

This yields the claimed formulae.

•

Corollary 5.1. In the case of a total default claim, the hedging strategy satisfies the balance condition. Proof. A total default corresponds to the assumption that G(t/i, y2,1/3,1) = 0. We now have v{t,yi,y2;Y) = 0, and thus $Y> = v(f,Y),Y],Y]_;0) for every I € [0,T]. Hence, the equality cp)Y) + (£,Y2 = 0 holds for every t € [0, T]. The last equality is the balance condition for Z = 0. Recall that it ensures that the wealth of a replicating portfolio jumps to zero at default time. • 5.1.1 Hedging with the savings account Let us now study the particular case where Y1 is the savings account, i.e., dY] = rY\ dt, Y\ = 1, which corresponds to jUi = r and o\ = 0. Let us writeT= r + f, where f= y{\ + Q = y + fi3 - r + — (r - ^>) 02

stands for the intensity of default under Q 1 . The quantity Thas a natural interpretation as the risk-neutral credit-risk adjusted short-term interest rate. Straightforward calculations yield the following corollary to Proposition 5.1. Corollary 5.2. Assume that o2±0 and dY] = rYj dt, dY^ = Yj(^i2dt + a2dWt), dY] = Y?_(fi3 dt + 03 dWt - dMt).

92

Then the function v(-;0) satisfies dtv(t, y2,1/3;0) + ry2d2v(t,y2, y3;0) + 7y3d3v(t, y2,y3;0) -n;(f, y 2 ,y 3 ;0) 1 3 _ + 2 X , CiCjyiyjdijvit' V2, ys; 0) + yi7(t, y2; 1) = 0 with v(T, y2/ y3; 0) = G(y2, y3; 0), and the function v(-; 1) satisfies dtv(t, y2; 1) + ry2d2v(t, y2; 1) + -ajyld^vit,

y2; 1) - rv(t, y2; 1) = 0

with v(T, y 2 ; 1) = G(y2,0; 1). In the special case of a survival claim, the function v(-;l) vanishes identically, and thus the following result can be easily established. Corollary 5.3. The pre-default pricing function v(-;0)ofa survival claim Y = 1{T
ai(J

iViVidiMt'

y*> y*> °) - w^> y^ y^ °) = °

with the terminal condition v(T, y2, y3; 0) = G(y2r y3). The components
4>2a2Y2 = £ OiY\_diV{t, Y], yf_;0) + a3v{t, Y2, Y?_;0), i=2

tfYl

= v(t,Yl,Yl;0).

Example 5.1. Consider a survival claim Y = 1 {T<x\g{Y\), that is, a vulnerable claim with default-free underlying asset. Its pre-default pricing function v(-; 0) does not depend on y3, and satisfies the PDE (y stands here for y2 and a for a2) (76)

dtv(t, y; 0) + ryd2v(t, y; 0) + ^tfdzrft,

with the terminal condition v(T, y;0) =

y; 0) - Tv(t, y; 0) = 0

1(«T|^(J/)-

v(t, y) = eP-'W-D vr*2(t, y) = e^^

The solution to (76) is t>r*2(f, y),

where the function o r * 2 is the Black-Scholes price of g{YT) in a Black-Scholes model for Y> with interest rate r and volatility a2.

93 5.2 Defaultable asset with non-zero recovery We now assume that dYf = Y3_(^3 dt + 03 dWt + K3 dMt) with K3 > - 1 and K 3 # 0. We assume that Y3 > 0, so that Y3t > 0 for every t e R+. We shall briefly describe the same steps as in the case of a defaultable asset with total default. 5.2.1 Arbitrage-free property As usual, we need first to impose specific constraints on model coefficients, so that the model is arbitrage-free. Indeed, an e.m.m. Q 1 exists if there exists a pair (9, Q such that dt(Oi

- CTi) + C f & ^ - j

= |Ul - jU; + (7l(<J; - CJi) + £ , ( K ; - Ki)——,

Z= 2,3.

1 + K\ 1 + Ki To ensure the existence of a solution (6, Q on the set T < t, we impose the condition \l\ -\l2

Ml ~ j " 3

01

= CTl CTi - 02

, 0\~

03

that is, Mi((j3 - a2) + [iz{o\ - 03) + 1*3(02 ~ 01) = 0. Now, on the set T > t, we have to solve the two equations dt(02 - 01) = (ii - ^2 + ai(a2 - cri), fft(cT3 - 0i) + QtyK3 = Hi - 1*3 + 0l{03 - 0l)If, in addition, (02 - <JI)K3 + 0, we obtain the unique solution Hi-Hi

0-01

= 01

0\ -02

[ii-li3 01 -

,

03

C = 0>-1, so that the martingale measure Q 1 exists and is unique. 5.2.2 Pricing PDE and replicating strategy We are in a position to derive the pricing PDEs. For the sake of simplicity, we assume that Y1 is the savings account, so that Proposition 5.3 is a counterpart of Corollary 5.2. For the proof of Proposition 5.3, the interested reader is referred to Bielecki et al. [7]. Proposition 5.3. Let CT2 * 0 and let Y1, Y2, Y3 satisfy dY\ = rY) dt, dY^ = Y^2dt 3

+ 02dWt),

dY* = Y _(^3 dt + 03 dWt + K3 dMt).

94 Assume, in addition, that a2{r - ^3) = 03(r - p2) and K3 4- 0, K3 > - 1 . Then the price of a contingent claim Y = G(Y*, Y*., HT) can be represented as nt(Y) = v(t, Y2t, Y3, Ht), where the pricing functions v(-; 0) and v(-;l) satisfy thefollowing PDEs dtv{t, y2,1/3; 0) + ryzdzv(t, y2, y3; 0) + t/3 (r - K3y) d3v{t, y2,1/3; 0) - rv{t, y2,1/3; 0) +

1 3 2^

aia

)yiVidi)v(t> 3/2' ys; 0) + y(v(t, y2, y3(l + K 3 ); 1) - i>(£, 1/2,1/3; 0)) = 0

'./=2

and dtv(t, y2,1/3; 1) + ry2d2v(t, y2, y3; 1) + ry3d3v(t, yz, j / 3 ; 1) - rv(t, y2, y3; 1) 1 3 + 5 X , aiajyiyjdijv(t>y2,y3;

i) = o

W=2

subject to the terminal conditions v(T, y2, y3; 0) = G(y2, i/3; 0),

y(T, y2,1/3; 1) = G(y2, J/3; !)•

The replicating strategy

1

& = ~vi

YjO-iyidivit^lYlMt-)

°2Yt

/=2

- - ^ U t . Yl Yf_(l + K3); 1) - v(t, Y2, Y3_; 0)), 02K3l(


Yl Yl{\ + K3); 1) - v(t, Y2, Yf_;0)),

and $ is given by
l{T
Then the post-default pricing function v$(-; 1) vanishes identically, and the pre-default pricing function v%(-; 0) solves the PDE div*(-; 0) + ry2d2v*{-; 0) + y 3 (r - K3y) <93^(-; 0) 1

3

+ 2 TJ WiMiWi-; ;,/=2

0) - (r + y W ; 0) = 0

95

with the terminal condition ^(T,yz, 1/3;0) = g(y3). Denote a = r- x3y and P = 7(1 + K 3 ). It is not difficult to check that v&(t, yz, y3; 0) = e^T-t)va'^(t,y3) is a solution of the above equation, where the function w(t, y) = v"-g,i(t, y) is the solution of the standard Black-Scholes PDE equation \ dtw + yadyW + -aly2dyyw -aw = 0 with the terminal condition w(T, y) = g(y), that is, the price of the contingent claim g(Yr) in the Black-Scholes framework with the interest rate a and the volatility parameter equal to a3. Let C( be the current value of the contingent claim Y, so that

The hedging strategy of the survival claim is, on the event (f < T}, ^3y3

=

_le^(T-f)t;«,S,3(f/y3)

=

K3


_ !

C

K3

Y?) - 0?Y?).

5.2.4 Hedging of a recovery payoff As another illustration of Proposition 5.3, we shall now consider the contingent claim G(Y2,Y^,HT) = l[T>i)g(Y2), that is, we assume that recovery is paid at maturity and equals g{Y2). Let » s be the pricing function of this claim. The post-default pricing function v%{-; 1) does not depend on y3. Indeed, the equation (we write here t/2 = y) dtvS{-; 1) + rydyv*(-; 1) + \o\fdyyv*{-;

1) - ro*(-; 1) = 0,

with v%(T, y; 1) = g(y), admits a unique solution vr%-2, which is the price of g(Yj) in the Black-Scholes model with interest rate r and volatility ozPrior to default, the price of the claim can be found by solving the following PDE dt&{.-, 0) + ry2d2vS(-, 0) + y3 (r - K3y) d3v*(-, 0) 1 3 + 2 E OiOjyiyjdijvH-,0) - (r + y)v«(-,0) =

-yv^t.yrA)

with vs(T, y2, yy, 0) = 0. It is not difficult to check that vS{t,yz,y3;Q) = (1 - e * ' - 7 V * 2 & ¥2)The reader can compare this result with the one of Example 5.1.

96

5.3 Two defaultable assets with total default We shall now assume that we have only two assets, and both are defaultable assets with total default. We shall briefly outline the analysis of this case, leaving the details and the study of other relevant cases to the reader. We postulate that (77)

AY\ = YtJjn At + at AWt - AMt), i = 1,2,

so that

_ Yt

=

^-{t
_ Yt =l(t< T |Y f ,

with the pre-default prices governed by the SDEs AYt = r ; ( ( ^ + y) At + oi AWt), i = 1,2. In the case where the promised payoff X is path-independent, so that X1|T
for some function G, it is possible to use the PDE approach in order to value and replicate survival claims prior to default (needless to say that the valuation and hedging after default are trivial here). We know already from the martingale approach that hedging of a survival claim XI |T
i = 1,2.

We need not to worry here about the balance condition, since in case of default the wealth of the portfolio will drop to zero, as it should in view of the equality Z = 0. We shall find the pre-default pricing function v(t,y\,y2), which is required to satisfy the terminal condition v(T, 1/1,1/2) = G(y\,y2), as well as the hedging strategy (\Y) +
(78)

Proposition 5.4. Assume that a\ 4- 02- Then the pre-default pricing function v satisfies the PDE dtv + yi Uii + y \ + ^{yl^nv

ffi 1 d\v + y2 [ui + y - az ——— I d2v 02 — 0\ I \ C>2 — 0\l

+ y\a\dnv + 2yiyzoio2di2v)

= ( ^ +y-

with the terminal conAition v(T, t/i, t/2) = G(i/i, 1/2)-

01

_^]v

97

Proof. We shall merely sketch the proof. By applying It6's formula to v(t, Y], Y2), and comparing the diffusion terms in (78) and in the ltd differential dv(t, Y), Yj), we find that y\a\d\v + y2a2dzv =
(79)

^yiait

where cj>1 = cj>'(t,yi,y2). Since
a2-a\

On the other hand, by identification of drift terms in (79), we obtain dtv + yi((Ui + y)d\v + yzi^z + y)d2v

+ 2(yi°iduV = ^yiii-'i

+

y\aldv-v + 2yiy2
+ r) + 2y2(i"2 + y)-

Upon elimination of (p1 and (j)2, we arrive at the stated PDE.

•

Recall that the historically observed drift terms are /T; = \n + y, rather than |L(,. The pricing PDE can thus be simplified as follows: dtv + yiljii -ax— + 2(y\a\dnv

-\div + y2\jiz-0z—

-\d2v

+ y\a\dvLv + 2yiy2ffiff2^i2u) = vI/?i - e n — - — j .

The pre-default pricing function v depends on the market observables (drift coefficients, volatilities, and pre-default prices), but not on the (deterministic) default intensity. To make one more simplifying step, we make an additional assumption about the payoff function. Suppose, in addition, that the payoff function is such that G(yi,y2) = y\g(yzly\) for some function g : R+ —> R (or equivalently, G(yi,y2) = yihiyilyi) for some function h : R+ -» R). Then we may focus on relative pre-default prices Q = Ct(Y])~l and Y2,1 = Y2{Y])~l. The corresponding pre-default pricing function 1u{t,z), such that Q = zF(t, Y2,1) will satisfy the PDE dtv+-{o2-ai)2zzdzzv

=0

98 with terminal condition IftTrZ) = g(z). If the price processes Y 1 a n d Y 2 in (68) are driven b y the correlated Brownian motions W a n d W w i t h the constant instantaneous correlation coefficient p, then the PDE becomes dtv+ x(°\ + al~

2poi02)z2dzzv'=

0.

Consequently, the pre-default price Q = Y ^ r , Y2'1) will not d e p e n d directly on the drift coefficients /?i a n d J12, and thus, in principle, w e s h o u l d b e able to derive an expression the price of the claim in terms of m a r ket observables: the prices of the u n d e r l y i n g assets, their volatilities a n d the correlation coefficient. Put another way, neither the default intensity nor the drift coefficients of the u n d e r l y i n g assets appear as i n d e p e n d e n t parameters in the pre-default pricing function. Before w e conclude this w o r k , let u s stress once again that the martingale approach can be used in a fairly general set-up. By contrast, the PDE methodology is only suitable w h e n dealing with a Markovian framework. In a forthcoming p a p e r [8], w e analyze a more general situation w h e r e a traded defaultable asset is a credit default s w a p , so that its dynamics involve also a continuous d i v i d e n d stream. Acknowledgments. Some results of this work were presented by Monique Jeanblanc at the "International Workshop on Stochastic Processes and Applications to Mathematical Finance" held at Ritsumeikan University on March 3-6, 2005. She deeply thanks the participants for questions and comments. The first version of this paper was written during her stay at Nagoya City University on the invitation by Professor Yoshio Miyahara, whose the warm hospitality is gratefully acknowledged. The work was completed during our visit to the Isaac Newton Institute for Mathematical Sciences in Cambridge. We thank the organizers of the programme Developments in Quantitative Finance for the kind invitation. References 1. Arvanitis A. and J. Gregory, Credit: The Complete Guide to Pricing, Hedging and Risk Management, Risk Publications, 2001. 2. Ayache, E., P. Henrotte, S. Nassar and X. Wang, Can anyone solve the smile problem? Wilmott (2004), 78-96. 3. Bielecki, T. R. and M. Rutkowski, Credit Risk: Modelling, Valuation and Hedging. Springer-Verlag, Berlin Heidelberg New York, 2002. 4. Bielecki, T. R., M. Jeanblanc and M. Rutkowski, Hedging of defaultable claims, Paris-Princeton Lectures on Mathematical Finance 2003, R. A. Carmona, E. Cinlar, I. Ekeland, E. Jouini, J. E. Scheinkman, N. Touzi, eds., Springer-Verlag, Berlin Heidelberg New York, pp. 1-132, 2004. 5. Bielecki, T. R., M. Jeanblanc and M. Rutkowski, Completeness of a general semimartingale market under constrained trading, to appear, Proceedings of International Lisbonn Conference, Springer,2005.

99 6. Bielecki, T. R., M. Jeanblanc and M. Rutkowski, Completeness of a reducedform credit risk model with discontinuous asset prices, to appear, 2005. 7. Bielecki, T. R., M. Jeanblanc and M. Rutkowski, PDE approach to valuation and hedging of credit derivatives, Quantitative Finance 5, (2005), 257-270. 8. Bielecki, T. R., M. Jeanblanc and M. Rutkowski, Pricing and trading credit default swaps, working paper, 2005. 9. Black, F. and J. C. Cox, Valuing corporate securities: some effects of bond indenture provisions, Journal of Finance 31 (1976), 351-367. 10. Blanchet-Scalliet, C. and M. Jeanblanc, Hazard rate for credit risk and hedging defaultable contingent claims, Finance and Stochastics 8 (2004), 145-159. 11. Bremaud, P., Point Processes and Queues. Martingale Dynamics, Springer-Verlag, Berlin Heidelberg New York, 1981. 12. Carr, P., Dynamic replication of a digital default claim, working paper, 2005. 13. Collin-Dufresne, P., R. S. Goldstein and J.-N. Hugonnier, A general formula for valuing defaultable securities, Econometrica 72 (2004), 1377-1407. 14. Collin-Dufresne, P. and J.-N. Hugonnier, On the pricing and hedging of contingent claims in the presence of extraneous risks, working paper, 1999. 15. Cossin, D. and H. Pirotte, Advanced Credit Risk Analysis, J. Wiley, Chichester, 2000. 16. Dellacherie, C , B. Maisonneuve and P.-A. Meyer, Probability et potentiel, chapitres XVII-XXIV, Hermann, Paris, 1992. 17. Duffie, D. and D. Lando, The term structure of credit spreads with incomplete accounting information, Econometrica 69 (2001), 633-664. 18. Duffie, D. and K. Singleton, Credit Risk: Pricing, Measurement and Management, Princeton University Press, Princeton, 2003. 19. El Karoui, N., Modelisation de l'information, CEA-EDF-INRIA, Ecole d'ete, unpublished manuscript, 1999. 20. Elliott, R. J., M. Jeanblanc and M. Yor, On models of default risk, Mathematical Finance 10 (2000), 179-195. 21. Giesecke, K., Default and information, working paper, Cornell University, 2001. 22. Giesecke, K., Correlated default with incomplete information, Journal ofBanking and Finance 28 (2004), 1521-1545. 23. Guo, X., R. A. Jarrow and Y. Zheng, Information reduction in credit risk models, working paper, Cornell University, 2005. 24. Jarrow, R. A. and P. Protter, Structural versus reduced form models: A new information based perspective, Journal of Investment Management 2/2 (2004), 1-10. 25. Jarrow, R. A. and S. M. Tumbull, Pricing derivatives on financial securities subject to credit risk, Journal of Finance 50 (1995), 53-85. 26. Jacod, J. and A. N. Shiryaev, Limit Theorems for Stochastic Processes, SpringerVerlag, Berlin Heidelberg New York, 1987. 27. Jamshidian, R, Valuation of credit default swap and swaptions, Finance and Stochastics 8 (2004), 343-371. 28. Jeanblanc, M. and S. Valchev, Partial information, default hazard process, and default-risky bonds, IJTAF 8 (2005), 807-838. 29. Kusuoka, S., A remark on default risk models, Advances in Mathematical Eco-

100

nomics 1 (1999), 69-82. 30. Lando, D., Credit Risk Modeling, Princeton University Press, Princeton, 2004. 31. Laurent, J.-P, Applying hedging techniques to credit derivatives, Credit Risk Conference, London, 2001. 32. Mansuy, R. and Yor, M., Random Times and Enlargement ofFiltrations in a Brownian Setting, forthcoming, Springer-Verlag, Berlin Heidelberg New York, 2005. 33. Merton, R. C , On the pricing of corporate debt: The risk structure of interest rates, Journal of Finance 29 (1974), 449-470. 34. Protter, P., A partial introduction to financial asset pricing theory, Stochastic Processes and Their Applications 91 (2001), 169-203. 35. Protter, P., Stochastic Integration and Differential Equations, 2nd edition, version 2.1, Springer-Verlag, Berlin Heidelberg New York, 2005. 36. Schonbucher, P. J., Credit Derivatives Pricing Models, J. Wiley, Chichester, 2003. 37. Vaillant, N., A beginner's guide to credit derivatives, working paper, Nomura International, 2001.

A Large Trader-Insider Model Arturo Kohatsu-Higa and Agnes Sulem INRIA-Rocquencourt Domaine de Voluceau, Rocquencourt, B.R 105, F-78153 Le Chesnay Cedex, France We give some remarks on the anticipating approach to insider modelling introduced by the authors recently. In particular, we define forward integrals by using limits of Riemmann sums. This definition is well adapted to financial applications. As an application, we consider a portfolio maximization problem of a large trader with insider information. We show that the forward integral is a natural tool to handle such problems and we compute the optimal portfolios for an insider and a small trader. Key words: Anticipating Calculus, Information asymmetry, large traders.

1.

Introduction In this article, we would like to explain the anticipating approach to insider information. The section on the forward integrals properties relies on Chapter 3 of Nualart (1995). Nevertheless, as we have not found a standard reference for this material in the form of the forward integral we will do it here in detail. For this we need to introduce the basic tools of differentiation on the Wiener space. Consider the interval [0, T] and a complete probability space (Q, T, P) on which a standard one dimensional Brownian motion W is denned; {Ft \teio,T] denotes the filtration generated by W, augmented with the P-null sets and made right continuous. Since all the results in the paper rely heavily on Malliavin calculus, we introduce some of its terminology briefly. We denote by C~(R") the set of C°° bounded functions / from R" to R, with bounded derivatives of all orders. If S is the class of real random variables F that can be represented as /(W f ,,...,W t n ) for some n e N, t\,...,t„ e [0,T] and / e C£°(R"), we can complete this space under the 101

102

Sobolev norm || • ||i,p given by 2

||F||^=E(|F|") + E ( ( J \DsF\

A df

where D is defined as DSF = > — (Wtu...,

ds)*),

Wt„)l[o,ff](s), obtaining a Ba-

nach space, usually indicated with D 1,p . Analogously, we can construct the space Dkp by completing S under the Sobolev norm

||F||£p=E(|Fn + £E[(J" ...J where D^

S] F

\Dl,SlF\2dSl...ds^]r

= DSi... DSiF. Finally, we denote D°° = p | Q D*'P. P>i

fei

We denote the adjoint of the closable unbounded operator D : D 1 ' 2 c L2(Q) —> L 2 ([0,T]xQ) by 5g. This operator is called the Skorohod integral. The domain of 6^ is the set of all processes u in L2([0, T] x Q) such that

;( f DtFutdt)


VFeS,

for some constant C possibly depending on u and ||F||2= E(|F|2)1/2. If u e Dom(<5j), then 5j(w) is the square integrable random variable determined by the duality relation E(<5j(u)F) = E( f D(Fufdf)

Jo

VF 6 Du.

Note that the above construction can be carried through for any fixed time interval [s, S], in the space L2([s, S] x Q). We will also use the notation 6 0 T ( u)=

\ u<(s)6W(s). Jo

For a stochastic process (p, we say that (f) e L1,2 if the following norm is finite: 11,2

=E

J |«/>(s)|2ds + E[JT JT |D^(s)|

dsrfw

103

2.

The Forward Integral Consider an insider, that is an agent that has sensible information about the future values of a stock, who may also have an influence on the evolution of the stock price. This is called a large trader-insider. In general one would like to study models of the type S(t) = S(0) + f fi(s,7i(s))S(s)rfs + f ff(s,7i(s))S(s)rf-W(s). Jo Jo Here n represents the insider's strategy which is adapted to a filtration Q, which may be bigger (or just different) than the filtration generated by the Wiener process W with natural filtration f. Therefore S is also adapted to Q (if T c Q) and the above stochastic integral will be an anticipating integral commonly known as the forward integral of Russo-Vallois. Next, we define the forward integral. For this, given any partition 0 = to < ... < tn = T such that max{f,+i - (,•; i = 0,..., n - 1 | —> 0 as n —* <x>, let rj(s) := maxjf,-; f,- < s}.

Then we can define the forward integral as follows: Definition 2.1. Let (/> : [0, T] X Q -» be a measurable continuous process. The forward integral of

/

0

(t)
y^ti)(w(tM)-mti)),

'="

if the limit exists in probability and is independent of the partition sequence taken. This definition does not coincide exactly with the original definition of Russo-Vallois, unless we put some additional assumptions. Note that the above definition is local. That is, let be forward integrable such that for a measurable set A c Q we have that (plA — 0, Then JT cj)(t)lAd-W(t) = 0. In that sense, as in Nualart, (1995), page 45 we will use the local defintion of all the spaces to appear below. First let us start proving that the expectation of this integral is not zero and therefore the usual rules of calculus do not apply. In particular, usual martingale properties are not true. For parallel martingale properties of anticipating integrals, see the interesting articles of Tudor (2004) and Pecatti-Theieullen-Tudor (2005). Definition 2.2. Let : [0, T] X D —> R be a measurable process such that

104

(f>(t) e L 1 ' 2 . We say that § e L*'2 if the following stability property is satisfied: for any sequence of partitions 0 = to < ... < t„ = T such that its norm tends to zero as n —» <x>, there exists the trace process Ds+
-*0

In such a case we say that (j> € L+2 and we define 2 1,2

1,2,+

+E

ds Jo

This norm will serve to control the variance of the forward integral as it is shown in the next Theorem. Theorem 2.1. Suppose that cp e L+ 2 . Then the forward integral ofcp exists, the limit in the definition 2.1 being satisfied in LJ(Q) and furthermore

\
If

(p(t)d-W{t)

IL-

<2

Proof. In order to prove that the integral exists we use the following formula (see formula (1.12) in page 130 in Nualart (1995a)) (j)(ti)(W(ti+1) - W(ti)) = f '+1 (p(ti)5W(s) + f '+1 Ds(P(ti)ds. Then the existence of the forward integral follows from Definition 2.2. Furthermore w e h a v e that each element in this expression belongs to L 2 (Q) a n d therefore w e have that YdE[p(tiW{tM)-Vf(ti))] i=o

=E

Ds<J>(ij(s))di l-Jo I Ds+4>di Jo

105

The last estimate is obtained similarly. We have Y


f cj>(1](s))5W(s)+ f Ds
TJT 1=0

Therefore, ^2^

'n-\

j](/)(f,)(w(f,+1)-wa,)) ,1=0

<2lE

U

cj>(r,(s))5W(s)\ +EU

Ds4,(n(s))ds)

Then the Riemmann sum sequence is bounded in L2(Q) and therefore converges in L2(Q) as it converges in L^Q). Then taking limits in the above inequality we obtained the desired result. • Next we prove that the integral process is a continuous process. \P/2

Theorem 2.2. Suppose that § e L* , such that E

X T (X T |D s ^( M )| 2 ds) P du

oo for some p > 2 then the process {jC cj)(s)d W(s);t e [0,T]} has a continuous version. Proof. Use Proposition 5.1.1 in Nualart (1995a).

•

Now we give the formula for the quadratic variation. Theorem 2.3. Given any sequence of partitions of the interval [0,t], nn : 0 to < ... < t„ = t such that max{£,+i - f,-; i = 0,..., n - 1 } —> 0 as n —> oo, we have that

5(1

4>(s)d-W(s)\ -» I \(j)(s)\ dsa.s.

for (p e L1,2 Proof. First suppose the simple case that there exists a fixed partition 0 = so < ... < sm = t such that m-l

cf)(s) = ^ 1=0

Fjl(si <s< s,+i),

106

where F, 6 D1-2. In such a case we obviously have that cp is forward integrable and furthermore (/.(s)d-W(s) = V F,(W(s,+i) - W(Si)). to

J

o

We then also have that for the sequence of partitions n'n = {£,•; i = 0,..., n) U {s/;/ = 0, ...,m\ then ( P + I c/)(s)rW(s)) - V

V

( P' + ' c/)(s)d-W(s)

•0,

as «—> oo because the partition {sy;/ = 0,..., m) is fixed and the forward integrals are L2- continuous in the time variable. Therefore without loss of generality we will suppose that [sf, j = 0,...,m\ c nn. Then we have that \2 \*

m-1 m-1

« - l i/ n-1

ntu „t M

£

I (s)d-W(s)\ = £ F

1=0 \J'>

'

;=0

2

E ( w (^0 - W(W)2

Sj
As the partition (s ; ;; = 0, ...,m} is fixed we have that (W(t/ + i)-W(f,-)) 2 -»S; + i-Sy

£ S;<(;<S/+i

as n —> oa. Therefore n-l h] " - 1 /i ~t, rli>-[

E

\2

(^-W(s)

m-1

-+£F*(S,-+1-S/). ;=0

Finally the result follows from the following density argument: E

£(J(

< £

4>(s)d-W(s)j - £ ( J (

"_1 / r !,+1

*2lk-4AJ*-

\ 2 1) 1 / 2

«s)fW( S )j ( T"-1 / r' /+i

\2

1/2

llu,+ •

n Now we give the ltd formula that is necessary for our calculations. Before we need a preliminary Lemma.

107

Lemma 2.1. Suppose that (p e L*' 2 n L 2 , 4 with Ds+d> e L 1 / 2 and b is a stochastic process with b e L 1 , 2 . Define the process X(t) = x + \ b(s)ds + \ Jo Jo

(p(s)d~W(s).

Then /(•, X)0 e l};2 for any f e C**2([0, T] x R). Proof. First, note that j£ b(s)ds e L+' 2 . In fact, Du £ b(s)ds = £ Furthermore, one clearly has that \DU Jo

b(s)dsJo

Dub(s)ds du

Dub(s)ds.

•0.

Jo

The other properties being clear, the assertion J^ b(s)ds e L+ 2 follows. Next, consider Du I f 4>(s)d-W(s)\ = Dulf


= (j)(u)l(u < rj(t)) + I Dud)(s)5W(s) + / DuDs+$ds. Jo Jo Therefore we have that Du+ I f 4>(s)d-W(s)\ = f Duds. Finally by the chain rule a n d product rule, w e h a v e that Ds (f{t, X{t))d>(tj) = it(t, X(t))DsX(t)cp(t) + f{t, Xit))Ds) = -£i-, X)DS+X4> + /(s, X(s))Ds+c/>.

D

Theorem 2.4. Suppose that cp € L j , 2 n L 2 ' 4 with Ds+(j> € L 1 , 2 and b is a stochastic process with b e L 1 ' 2 then for any f e C*'2([0, T] x R) we have that fit,Xit))

= /(0,x) + j

^ ( s , X ( s ) ) + ^-is,Xis))bis)

+ J^(s,X(s))4,( S )rf-W(s),

+

l

-^is,Xis))4>is)2ds

108

for X(t) = x + f b(s)ds + [ cp(s)d-W(s). Jo Jo Proof. In order to prove that the integral exists we find first a smooth approximation of the process

<ns) = £Fp(s,-<s<s,- +1 ) 1=0

where F" = cp(Sj) e D1-2 and 0 = So < ... < s„ = T is a fixed partition and

as n —> oo. Note that in this case one has that f
F?(W(sM) - W(Si)).

Now define rji(s) = inf(s,;s,- > s) and m(s) - sup {s,-;s,- < s). We define similarly the approximation process X"(t) = x+ f - T T ^ — r x

f

b(u)duds+ f d>n(s)d-W(s).

Jo m(s)-m(s)Jm(s)

Jo

Consider any partition 0 = to < ... < tm = t such that it contains all the points Sj, j = 0,..., n. Using the Taylor expansion we have m-l

f(t,X»(t)) = f(0,x) + £

(dxf(ti/X"(ti))(X"(tM)

- X"(f;)) + dt/faritiMtM

1=0 m-l

1

+ r £d xx f(ti,X n (ti))(X n (t M )

-

Xn(td)2.

2

i-o

Here X ((,) denotes a value between X"(f,) and X"(t!+i) and F, a value between t\ and t{+\. Obviously, f(t,Xn(t)) converges a.s. to f(t,X(t)) as n —> oo. The last term above, as in the previous Theorem 2.3 converges to ^ f

dxxf(s,X(s))cp(s)2ds.

- td)

109

In fact, one can easily reduce the problem to the calculation of the limit of £ dxxf(th X (ti)) I J

cj)n(s)d- W(s)\

.

= £ (d»f^' x"{ti)) ~ d**f(Tn(ti)>xn(ri2(tj))j) 1=0

4>"(s)d- w(s) ^

f

<

'

«-l

i=0

Sj
The first term converges to zero as n —» oo and the second converges first as m —> oo to f < W M * ) , X»(7j2(s)))0"(j72(s))ds, Jo and to j[ dxxf(s, X(s))(p(s)2ds asn -> oo. The other terms converge clearly to j r | ^ ( S / X ( s ) ) + ^(s,X(s))b(s))ds. So we only have to consider the last d term which is YZo xf{U, X"(f/)) J~* "(s)d-W(s). First, as m -» oo this term converges a.s. as all the other terms converge. Therefore this limit is the forward integral JQ dxf(s,Xn(s))" such that sup (s . T |X(f) - X"(f)| < n~e for e > 1/2. We will then have that for the same sequence f; = Ti/m J

(dxf(s,X(s))ct>(s) - dxf(s,Xn(s))"(s))d-W{s) m-\

= Um £ (dx/(fc, X(.ti))<Wd - ^/(fc, X"(f,-))"(f;)) (W(f,-+i) - Wfr)). i'=0

Now we consider the subsequence for which n = m to obtain that the above limit converges to zero. Then the result follows. D Remark 2.1. l.The previous proof also gives a sense to the integral J 0 f ^/(s,X(s))d-X(s).

110

2. In fact the original definition of the forward integral by Russo-Vallois is somewhat different to the one given here. In general, their definition is more general. Nevertheless, once one wants that this integral becomes the limit of Riemman sums then one is forced to the above framework. Still, we remark that the above conditions can be somewhat relaxed but the general idea remains. 3. For example, the above proof is also satisfied in local form. That is, the result is also satisfied if cj) e L ^ n L ^ with Ds+cp e L ^ and b is a stochastic process with b e L^ 2 and / € C1'2([0, T] x R). For the definition of these spaces see Nualart [25]. 4. The fact that the above Ito formula demands an extra condition (Ds+(j) e L1,2) in comparison with its counterpart in Skorohod integral form is well documented in the literature. In particular, in the case of the Stratonovich-Skorohod integral. Nevertheless as our restriction comes from the financial interpretation of the models to be used we accept them as natural. 3.

A First Toy Example Rather than following the general theory exposed in Kohatsu-Sulem (2006), we will give some examples in order to illustrate the theory. In this section, we consider a first toy model where the dynamics of the prices are given by (2)

dS{t) = S(f)(/i + bW(T))dt + oS(t)d-W(t)

where fj. and b are real numbers, a > 0. We suppose moreover that p(t) = p = constant. The interpretation of this model when b > 0 is that the insider introduces a higher appreciation rate in the stock price if W(T) > 0. Given the linearity of the equation of S this indicates that the higher the final stock price the bigger the value of the drift in the equation driving S. Some cases of negative values for b can also be studied but the practical interpretation of such a study is dubious. Furthermore we remark that usually in this model we assume that the trades of the insider are not revealed to the public. This is also an interesting modelling issue which is also assumed by Kyle and Back. They assume that the cumulative trades of the insider plus a Wiener process in the insider's filtration are public information. The Wiener process is interpreted as the effect of the so-called noise traders. This interpretation can also be applied in any of the cases studied with the enlargement of filtration approach and as we will see it can also be applied here. The difference here is that we will introduce large trader-insider models with finite utility where there can also be small traders that act rationally.

Ill

In order to compare with the theory given in our previous article, we decide to first give an approach which is easier to introduce at this stage but that later will not be possible to apply This is the set-up of enlargement of filtration. For this, consider the filtration Qt = %V a(W(T)). In this filtration it is well known that W is a semimartingale and its semimartingale decomposition is given by

ww-ww+if

m^>ds,

Jo

where W is a Wiener process in Q. Therefore in this case, as the forward integral becomes a semimartingale integral we have that the model for S is

dS(t) = S(t)• (1,11 + bW(T) + g W ( 7 2 ~ W ( f ) ) dt + aS(t)dW(t). S(t) = S(0) exp 11 JJ - ^ 11 + bW(T)t + aW(t) Therefore the optimization of the logarithmic utility for this model is done through classical methods. Briefly, one has that the wealth process associated with this price process is given by V(t) = V(0) + f ^^-dS(s) s s Jo ( )

+ f (1 - n(s))V(s)rersds. Jo

Then the discounted wealth, V(t) = e~r'V(t) can be written as

•n(s)V(s)ds + f on(s)V(s)dW(s). Jo The solution to the above equation is

V(t) = V(0) exp ( ^ \p - r + bW(T) +

a ^ ^ ^ )

•"(s)-yrr(s) 2 rfs+ |f an(s)dW(s)\ an(s)dW{s)\. Therefore if we consider the optimization of the logarithmic utility we have the following problem max J(n)

112

where J(n) = Mt,n)

=E

IV

W(T)-W(s)) r + bW(T) + a T-s J

•TT(S)-—rc(s) z ds + | ci7i(s)dVV(s) Jo

and for any filtration "H QQ satisfying the usual conditions we define &H(t) = 17i is rH adapted; I \n{sf ds < oo 1. We then have the following theorem Theorem 3.1. Assume that rH is any filtration included in Q. Then the optimal portfolio for the above problem is given by 7t(s)= ^ + E

-^W(T) + a

_t W(T) - W(s) T-s

n

and the optimal value is given by (H-rft 2a2

. 1

+ ^

[

E\E\bW(T)

W(T)-W{s) """'IKS T-s

+ a--yZ

ds.

In particular, lim/&-(f,ft) = oo,

(3) while

lim/^(t,Tt) < oo for "Ht = a(S(s);s < t). Furthermore the functions Jg(t,n) and J^(t,n) are increasing in b. A far more general theorem was given in Kohatsu-Sulem (2006). Proof. In order to obtain the result first note that given that n € Ji
0.

Jo

Next the function fs(n) = \p.-r + E

I —s

n-—T?

113

is a strictly convex function adapted to the filtration ti. Therefore the maximal value is obtained for the value n given in the statement of the theorem. The limit wealth for the full insider is infinite because

mT),a^izmlg,

= b2T + +2ba +

T-s

The last result follows by noting that n = o(bW(T)s + aW(s);s
E[W(T) - W(t)\
Therefore the result follows because 21

IE[W(T)-W(t)\n] \ T-t

b2t (b2T + 2ba)t + a2'

To finish one only needs to note that W(T) - W(s) K T-s

(V-rft 2o2

b2 2a2 Jo

l2l

ds

((bT + a) + a(T - s)f ds. (b2T + 2bo)s + a2

Finally differentiating with respect to b it follows that ]
• There are various other interesting remarks that are made in KohatsuSulem (2006) with respect to the interpretation of this result. This result says that in various situations the insider which acts as a large trader may have effects in the market and the small trader only uses a projection of this market in order to optimize its utility. This projection does not transfer the information from the insider to the small investor. This example also reflects the fact that there is not only one insider but various insiders that may act depending on the nitration that

114

one takes between *H( = a{S(s);s < t)and(?t = Tt Va(W(T)). Finding examples where the calculations can be done explicitely will be an interesting subject of future research. This toy example, which can be solved using the simple technique showed here was solved in Kohatsu-Sulem (2006) using a powerful technique consisting on optimization in an anticipating framework. We will show in the next section an example which can be considered as a nontrivial application which cannot be solved using the previous technique. Before that we will discuss another issue related with (3). In fact with a small modification we can obtain that the optimal logarithmic utility of the insider is finite. Theorem 3.2. Consider the filtration Q't = TtVa (W(T) + W((T - s) 9 ); s < t) where W is another Wiener process independent of W and d e (0,1). Then we have that limfoit,ft) < oo. Proof. First note that the first part of Theorem 3.1 can be applied to the filtration Qt = TtV a(W(T)) V £f(W'(s);s < Te). Therefore we only need to compute / W(T) - W(s) I \ \ T-s /&7

=

W(T) - W(s) + W'((T (T-S)B T-S +

sf)

From here it follows that the logarithmic utility is finite if 6 < 1

•

To finish we prove a theorem that can be interpreted as the non-existence of arbitrage or the issue of non-conspicuous insider trader. Theorem 3.3. For any filtration 'H included in Q such that S is fi-adapted, suppose that there exists an "H-optimal portfolio ft e L^.'2 which leads to a finite logarithmic utility. Then there exists an 'N-Wiener process W^ such that log(S(0/S(0)) = f (r + a2fi{s))ds +

aYf^t).

Proof. Just to avoid explicit notation let fis(a>) = p+bW(T). If there exists an optimal portfolio ft then it minimizes the logarithmic utility of this trader which is EI f (ps - r)n(s) - °— n(sfds + f an(s)dW(s) J.

115

Applying variational calculus to the above expression we obtain that E ( f (/is - r) - a2n{s)ds + a(W(t) - W(u))
f <72fz(s)ds

Therefore by Levy's characterization of the Wiener process we have the result. • Note that in the classical Merton model fl(s) - £ ? . Therefore the previous theorem states that the small trader will not find any anomaly in his trading of the stock even if this is influenced by an insider. This result also says that if we interpret W
Continuous Stream of Information In this section, we consider for 5 > T fixed

(fi + bW(s + 5))S(s)ds+ f ffS(s)
Proposition 4.1. W is not a semimartingale on the filtration CFt+6)te[o,T] • Proof. Consider the definition of semimartingale as given in Protter 's book page 52. If W is a (^r(+6)-semimartingale, then for any partition whose norm tends to zero and always smaller than 5, consider the process n-l

E

H(t) = £(W(f, + 1 ) - W(t/))l(ww](0i=0

116

This process is then (9^+6)-adapted and converges uniformly to zero but its stochastic integral converges to the quadratic variation of W leading to a contradiction. D This shows that the insider filtration does not even correspond to (Tt+d)teio,T\- Th e definition for the insider's filtration in the particular case t h a t 5 > T is Qt = TtV e(W(T)) V a(W(s + 6) - W(T);s < 0Then the calculations can be carried out as in the previous section. Nevertheless, we need to be more precise here in the general case. We do this here. In such a situation, we have to clearly use the anticipative set-up given in the first section. Therefore we have to find the solution for the equation of the prices. Proposition 4.2. S(t) = S(0)exp(L - iff 2 )t + b J

W(s)ds + aW(t)\

2 is the unique solution of equation (4) in the space L*'loc' The proof of this result follows directly from the Ito formula given in theorem 2.4. We are interested in computing the optimal policy of the small investor with filtration % = a(Ss; s < t). From the previous proposition, we have that
where Y(s) — b L W(r)dr + CTW(S). N O W we study the wealth process associated with this price process. The wealth process is defined as the solution of V(t) = V(0) + f ^p.dS(s) s s Jo ( )

+ f (1 - n(s))V(s)rersds Jo

where the interpretation of d~S(t) is as in Definition 2.1. Note that in order that this equation among others has a sensible financial interpretation we introduced in Section 2 the forward integral as a limit of Riemmann sums. Then the discounted wealth, ty(t) - e~rtV(t) can be written as 9(t) = V(0) + f (li-r Jo

+ bW{t + 6))n(s)V(s)ds + f Jo

on(s)V(s)d-W(s).

117

As before the solution to the above equation is 9{t) = V(0)exp

(Jf<-

r + bW(t + 5))n(s)

n(sfds + | cm(s)rW(s)|. Jo

We will later show that the optimal portfolios proposed satisfy the conditions stated in Section 3. With these assumptions, we have that the limit of the logarithmic wealth process can be written as J(n) =

Mt,n):=Elog(V(t))-log(Vo)

= E J (n(s)(ti - r + bW(s + 5)) - ^a2n{s)2)ds + a f

n(s)itW(s)

The class of admissible portfolios is given by ft = \n is 'H adapted; n e

V/}.

Theorem 4.1. Define the following portfolio n{s)

\i — r

+ o-2E(bW(s + 5)fH s) + <j_1a(s)

where a(s) = L 1 ( Q ) - l i m £

W(s + h)- W(s)

h->0

n,

//ft e L+2 then ft is the optimal portfolio for the above problem for any filtration 'H and the optimal value is given by J{t,n)=-E

f fc(s)2ds Jo

A more general theorem was proved in Kohatsu-Sulem (2006). Proof. In order to obtain the result we have to prove first that the functional / is strictly convex. For this, let UQ and n\ e ft. Then we have that for any a 6 (0,1)

J(an0 + (1 - a)7ii) < aj(n0) + (1 - a)J(n{). This property clearly comes from the factor - y n{s)2 in the expression for /. Next, we find the first directional derivative of /. Consider for n,v eJl, then DnJ(n): = l i m M e->0

Jo

'

n

'

r + bW(s + S))v(s) - o2n(s)v(s)ds +

Jo

av(s)d'W(s)

118

If we set the above equation equal to zero for all v e J{ and in particular for v = Xl [s0,t0] for X e D1-2 we have by a density argument that z

J ^ r + bW{s + 5))ds - a n{s)ds + a (W(t ) 0

W(s0))
Now note that ft satisfies the above equation. In fact, replacing ft in the above equation, we have

J ~°

limE

•J So

— -a lim £ ft-»0

W(s + h)- W(s)

•H, ds + a(W(t0)-W(so))

r ° w(s + h) - W(s) ds + (W(t0)-W(s0)) h Js0

H

Ho = 0,

by continuity of the paths of the Wiener process. Therefore ft has to be optimal. In fact, for all f> e J{ and E 6 (0,1), we have J(ft + e/0 - /(ft) = /((l " £) ^

+ tf) - /(")

>(l-^)/(T^) + ^)-/(ft)

Now, with

= 1 + n we have 1 - £

'

lim - ( / ( - ^ - ) - /(ft)) = lim ^ ( / ( f t + jjft) - /(ft)) = Dft/(ft). t-»o t

1 —t

i;—>o

;]

Then we get D„/(ft) = lim -(/(ft + e/3) - /(ft)) > Dft/(ft) + /(/J) - /(ft). We conclude that /0?) - /(ft) < D„/(ft) - Dft/(ft) ; ft,/J e # . In particular, using that DpJ(n*) = 0, we get

/(/?)-/(O<0, which proves that n* is optimal.

119

To find the optimal expression for the utility it is enough to note that E I (n - r + bW(s + 5))ft(s) - a2n(s)ft(s)ds + ( oft(s)d-W(s)

= 0,

therefore the optimal utility is E

f (n(s)(p-r

-

i (fi-r Jo

+ bW(s +

5))-^a2ft(s)Ads

+ bW(s + 5))ft{s) -

a2n{s)fi(s)ds)

From here the result follows.

•

A very useful property is that the optimal portfolios in a smaller filtration is just a projection. Proposition 4.3. Let 'Hl c fi2 c Q be two filiations satifying the usual conditions such that there is an optimal portfolio ft2 in *H2 within a class of protfolios tft-Hi. If&H< c &

then there is an optimal portfolio fti in
ft1(s) = E[n2(s)/'H}]f Jw(t,ni)

<

Jw(t,n2).

Therefore in order to prove the existence of the optimal portfolio it is essential to compute a or at least obtain its existence and some regularity properties. We do this, first in the case that 5>T. This is done in the next proposition. Proposition 4.4. Suppose that 5 >T. The optimal logarithmic utility portfolio in the filtration K c Q is given by ft(s) = £ - = - + a~2E

bW(s + 5) +

,W(Tl-W^lH

T-s

The optimal value is given by 2o2 In particular,

mi ^^..ftai/ T-s

lim Js(t, ft) = <x>,

ds.

120

while Vaa.J.H(t,ft)<°° t—>T

for
Proof. Define Y(t) = bj6 limE sit

W(r)dr + oW(t). Then for 6 > T

W(s)-W(Q/ r t s-t

= bM f g(t,u)dY(u). Jo

E[W(t + 5)/H] = (b(t + 6) + a)M \

g(t,u)dY(u

Jo \-i

where M = Mt = a~l {{bd + 2a)(e^ - l) + a(e2« + l))

and g(t,u) =

e±{2t-u)+e±u_

In fact, note that Y is a Gaussian process. Therefore E[W(s)/'W(] = Jij h(s, t, u)dY(u) for a deterministic function h. To compute h we compute the covariances between W(s) and the stochastic integral and Y(v) for some v
=

n

b2

h(s,t,9i)(eiAe2 + 5)de2de1

+2bav> fh(s, t, 6)d0 + a2[ h(s, t, 8)d8. Jo Jo

(5)

Therefore the above two expressions have to be equal. After differentiation of the equality with respect to v < t three times, we obtain -,d2h -b2h{s, t, u) + cr 2 ^-r(s, t, u) = 0. A aU

Solving this differential equation gives h(s,t,u) = Ci(s,f)e"»" + C2(s,f)e°". Next one verifies that for the following constants, the covariances coincide. C2(s, t) = a~\bs + a) ((b5 + 2a) (e? - l) + a (e^ + l)) _ 1 Ci(s,t) = e^C2(s,t).

121

Therefore, we have that E (W( S )-W(Q^j =

|

kjs^-hit^u)dy{u)

Then the result follows. Next, using Theorem 4.1, we have that the possible optimal portfolio 7i* defined by

is in L+'2. In fact, all the properties are obtained through the process Y. We do not give the details of this verification. Then the optimal utility is finite as it is given by /(f,7T*) = log(Vo) + y E

(sYds

Jo

o

Remark 4.1. When s < T, we have that

E[JV(s)/7/ r ]= f h(s,T,u)dY(u)+

f

h(s,T,u)dY{u),

where h(s, t, u) = dis, t)e-«u + C2(s, t)e«u C2(s, t) = a'1 (1 + ah(s, T, s)) (e^

+ev \

M

Ci(s,t)=e-C2(s,t). This shows that even the information on all the prices of the interval [0, T] does not reveal the information held by the insider to the small trader. As before we can also show that the insider's utility is finite if we use the filtration Q't = TtVa (W(s + 6) + W'((T - t)e); s < t) for 6 < 1. Similarly we can also obtain a representation theorem such as Theorem 3.3. Instead we will take a look at the case & < T. We use a different shortcut through the anticipating Girsanov's theorem. For details and notation we refer to Chapter 4 in [25]. Theorem 4.2. Consider the case 5
122

Proof. We apply Theorem 4.1.2 in [25] in the interval [0,T + 6] with the transformation T(co) = CJ + b\{- < T) f co(s + 5)ds, Jo defined in C[0, T + 5]. Then we have that if T(a>) = 0 then co(t) = 0 for all t e [T,T + 6]. Therefore T(a>) = a* + bl(-
X Jo

a>(s + S)ds.

That is, by finite induction we have that T is an injection. To prove that it is surjective one follows a similar pattern. Next we have that det 2 (/ + Du) > 0 for us(a>) = b\{s < T)co(s + 5) and that under the change of measure j p = det 2 (/ + Du)exp(-

f

bW(s + 5)dW(s)-h-

f

W(s + dfds)

then W = T(W) has the law of a Wiener process under Q. Therefore there exists an equivalent martingale measure for this problem. In order to compute the optimal portfolio one uses the dual method. That is, denote m = o~2(fi - r) and define dQ'

j

,

„

,

- ^ - = det 2 (Ir + DM)exp • ( - f (bW(s + 6)-m)dW(s)-l

f

(bW(s + 6) + m)2ds\.

Then the optimal portfolio value is

Q

-vdQ'

The optimal portfolio value is finite because £ I log (-^p- J < oo. Off course an interesting problem is to compute explicitely the optimal portfolio for the case 5 < T. Although one may consider that the large trader effect is somewhat hidden in this paper through the process appearing in the drift. We remark that this may be considered as a first learning step towards more complex models. Some of these models were presented in Kohatsu-Sulem (2006) or Kohatsu (2005).

n

123

References 1. Amendinger, J., Imkeller, P., and Schweizer, M., 1998. Additional logaritmic utility of an insider. Stochastic Proc. Appl. 75, 263-286. 2. Amendinger, J., 2000. Martingale representations theorems for initially enlarged filtrations. Stochastic Proc. Appl. 89,101-116. 3. Amendinger, J., Becherer, D., and Schweizer, M., 2003. A monetary value for initial information in portfolio optimization. Finance and Stochastics 7, 29-46. 4. Baudoin, E, 2003. Modelling anticipations in financial markets. In Paris-Princeton Lectures on Mathematical Finance 2002. Lect. Notes in Maths. 1814, SpringerVerlag. Berlin. 5. Baudoin, E, 2002. Conditioned Stochastic Differential Equations and Application to Finance, Stochastic Processes and their Applications, Vol. 100,109-145. 6. Back, K., 1992. Insider Trading in Continuous Time. Review of Financial Studies 5, 387-409. 7. Biagini, E and 0ksendal, B.: A general stochastic calculus approach to insider trading. Preprint Series, Dept. of Mathematics, Univ. of Oslo, 17/2002. 8. Corcuera, J. M., Imkeller, P., Kohatsu-Higa, A., and Nualart, D., 2004. Additional utility of insiders with imperfect dynamical information. Finance and Stochastics 8,437-450. 9. Chaumont, L. and Yor, M., 2004. Exercises in Probability, Cambridge University Press, 2004. 10. Elliot, R. J., Geman, H., and Korkie, B. M., 1997. Portfolio optimization and contingent claim pricing with differential information. Stochastics and Stochastics Reports 60,185-203. 11. Grorud, A., 2000. Asymmetric information in a financial market with jumps. International Journal of Theoretical and Applied Finance 3,641-659. 12. Grorud, A. and Pointier, M., 1998. Insider Trading in a continuous Time Market Model. International Journal of Theoretical and Applied Finance 1, 331-347. 13. Imkeller, P., 1996. Enlargement of the Wiener filtration by an absolutely continuous random variable via Malliavin's calculus. Probab. Th. Rel. Fields 106, 105-135. 14. Imkeller, P., 1997. Enlargement of the Wiener filtration by a manifold valued random element via Malliavin's calculus. In Statistics and Control of Stochastic Processes. The hipster Festschrift, Y. M. Kabanov, B. L. Rosovskii, and A. N. Shiryaev (eds.) World Scientific, Singapore. 15. Imkeller, P., 2002. Random times at which insiders can have free lunches. Stochastics and Stochastics Reports 74, 465-487. 16. Imkeller, P., Pontier, M., Weisz, E, 2001. Free lunch and arbitrage possibilities in a financial market with an insider. Stochastic Proc. Appl. 92,103-130. 17. Jacod, J., 1985. Grossissement initial, hypothese (H'), et theoreme de Girsanov. In Grossissements de Filtrations: Exemples et Applications, T. Jeulin, and M. Yor (eds.) Lect. Notes in Maths. 1118. Springer-Verlag. Berlin. 18. Jeulin, T, 1980. Semi-Martingales et Groissessement de Filtration. Lect. Notes in Maths. 833. Springer-Verlag, Berlin. 19. Karatzas, I. and Pikovsky, I., 1996. Anticipative portfolio optimization. Adv. Appl. Prob. 28,1095-1122.

124 20. Kohatsu-Higa, A., 2005. Insider models with finite utility. Lecture Notes. 21. Kohatsu-Higa, A. and Sulem, A., 2006. Utility maximization in an insider influenced market, Mathematical Finance 16,153-179. 22. Kyle, A., 1985. Continuous Auctions and Insider Trading. Econometrica 53, 1315-1335. 23. Liptser, R. S. and Shiryaev, A. N., 1997. Statistics of Random Processes I. General Theory. Springer-Verlag. New York. 24. Mansuy, R. and Yor, M., 2004. Harnesses, Levy processes and Monsieur Jourdain. to appear in Stochastic Process. Appl. 25. Nualart, D., 1995. The Malliavin Calculus and Related Topics. Springer-Verlag. Berlin. 26. Nualart, D., 1995a. Analysis on Wiener space and anticipating calculus. In Lectures on Probability Theory and Statistics. Ecole d'ete de Probability de Saint-Flour XXV. Lect. Notes in Maths. 1690. Springer-Verlag. 27. 0ksendal, B. and Sulem, A.: Partial observation in an anticipative environment. Preprint University Oslo 31/2003. 28. Peccati, G., Thieullen, M., and Tudor, C , 2005. Martingale structure for Skorohod integral processes, to appear in The Annals of Probability. 29. Protter, P., 2004. Stochastic Integration and Differential Equatwns. A New Approach. Springer-Verlag. New York. 30. Russo, F. and Vallois, P., 1993. Forward, backward and symmetric stochastic integration. Probab. Th. Rel. Fields 97, 403-421. 31. Russo, F. and Vallois, P., 2000. Stochastic calculus with respect to continuous finite quadratic variation processes. Stochastics and Stochastics Reports 70,1-40. 32. Russo, F. and Vallois, P., 1995. The generalized covariation process and Ito formula. Stochastic Process. Appl. 59, 81-104. 33. Seminaire de Calcul Stochastique 1982/83, Universite Paris VI, 1985. Grossisements de Fitrations: Exemples et Applications. T. Jeulin and M. Yor (eds.) Lect. Notes in Maths. 1118. Springer-Velag. Berlin. 34. Tudor, C , 2004. Martingale type stochastic calculus for anticipating integrals, Bernoulli 10(2), 313-325. 35. Yor, M., 1985. Grossissement de nitrations et absolue continuite de noyaux. In Grossissements de Filtrations: Exemples et Applications. T. Jeulin and M. Yor (eds.) Lect. Notes in Maths. 1118. Springer-Verlag. Berlin.

[GLP & MEMM] Pricing Models and Related Problems Yoshio Miyahara Graduate School of Economics, Nagoya City University Mizuhochou Mizuhoku, Nagoya, 467-8501, Japan The [GLP & MEMM] pricing model (= [Geometric Levy Process & Minimal Entropy Martingale Measure] pricing model) has been introduced as a pricing model for the incomplete financial market. This model has many good properties and is applicable to very wide classes of underlying asset price processes including the geometric stable processes. We explain those good properties and see several examples of this model. After that we investigate the calibration problems of [GLP & MEMM] model. Key words: Geometric Levy Process, Relative entropy, Minimal entropy martingale measure, Stable process, Calibration 1.

Introduction The [Geometric Levy Process & MEMM] pricing model was first introduced in [36]. This model is one of the incomplete markets, and is based on the geometric Levy process and the minimal entropy martingale measure (= MEMM). We assume that the value process of bond is given by Bt = exp{rf(, where r is a positive constant. The price process of the underlying asset is denoted by Sf. 1.1 Black-Scholes model The explicit form of Black-Scholes model (Geometric Brownian motion model) is given by (1)

S, = S0e(''-^2)f+l7W',

and the stochastic differential equation (SDE) form is given by (2)

dSt = St{iidt + 0dWt),

where Wt is a standard Wiener process. 125

126

The risk neutral measure Q is uniquely determined by the Girsanov's lemma. Under the Q the process W, = W, + (^ - r)a~xt is a Wiener process and the price process S, is expressed in the form of (3)

St = S0e('-*ff2)f+'TlV'

or dSt = S, (rdt + adWt).

The price of an option X is given by e~rTEQ[X\. The theoretical B-S price of the European call option, C(So, K, T), with the strike price K and the fixed maturity T is given by the following formula (4)

CK = C(S0, K, T) = e-rTEQ[(ST - K)+] = S 0 *(di) - e-rTK$(d2),

where <5(d) is the normal distribution function and (5)

log | +{r+£)T di= BK .,

rf2=

log|+(r-4)T ,K V=dl-ayff.

1.2 Properties of B-S models 1.2.1 Distribution of log returns The log return is the increment of the logarithm of St, 1

(6)

•>

AlogS f = logS t+At - logS, = (ju - -a1)At +

CTAW,,

and the log return process is {ji - \a2)t + aWt. The distribution of the log return (or the log return process) of the B-S model is normal. This is convenient for the calculation of the option prices. For example we have obtained the explicit formula of the price of European call options. But it is said that the distributions of the log returns in the real market usually have the fat tail and the asymmetry. These facts suggest us the necessity to consider another models. 1.2.2 Historical volatility and implied volatility Under the setting of the B-S model, the historical volatility of the process is defined as the estimated value of a based on the sequential data of the price process S,. We denote it by~5. On the other hand the implied volatility is defined as what follows. Suppose that the market price of the European call option with the strike K, say C£, were given. Then the value of a which satisfies the following equation (7)

S0(d1)-e-rTK(d2) = cf\

is the implied volatility, and this value is denoted by a^\ We remark here that the implied volatility a^ depends on the strike value K, and that on the contrary the historical volatility "a does not depend on K.

127

We first consider the case where the market value of options obey to the Black-Scholes model, and so the market price Cj? is equal to the theoretical B-S price Q . In this case the solution of the equation for the implied volatility is equal to the original a and it holds true that ff^m) = a = constant. This means that if the market obeys exactly to the Black-Scholes model, then the implied volatility o^ should be equal to the historical volatility ~5. But in the real world this is not true. It is well-known that the implied volatility is not equal to the historical volatility, and the implied volatility o™ is sometimes a convex function of K, and sometimes the combination of convex part and concave part. These properties are so-called volatility smile or smirk properties. 1.3 Generalization of B-S model 1.3.1 Geometric Levy Process models We start from the explicit form of Geometric Brownian motion: St = Soe'' ,-? " V+aW'. It may be a natural idea to replace the Wiener process with the more general Levy processes Z ( and set (8)

St = S0ez'.

This type processes called the Geometric Levy Processes (GLP). The [GLP & MEMM] pricing model is one of this type of generalisation of B-S model. The class of Levy processes are very wide and the distributions of St may have the fat tail property and may be asymmetric. 1.3.2 Stochastic volatility models We start from the SDE form dSt = St {pdt + adWt) • When we replace the Brownian motion with a Levy process, we obtain the equation described in the previous subsection (see §2). When we randomize the volatility a as follows (9)

dS, = S,^dt

+ a,dWt),

where at is a stochastic process, then we obtain the so-called stochastic volatility models. 1.4 Our Goal The purposes of this lecture are, 1) we introduce the [GLP & MEMM] pricing model and see that this model has many good properties, and next 2) we review some relating problems of this model, in theoretical sense and (or) in practical fence (for example, the fitness analysis and calibration analysis).

128

2.

Geometric Levy Process Pricing Models We assume that the value process of bond is given by

(10)

Bt = explrt],

where r is a positive constant. A pricing model consists of the following two parts: (A) The price process St of the underlying asset. (B) The rule to compute the prices of options. For the part (A) we adopt the geometric LeVy processes, so the part (A) is reduced to the selecting problem of a suitable class of the geometric Levy processes. For the part (B) we adopt the martingale measure method, so the part (B) is reduced to the selecting problem of a suitable martingale measure Q, and then the price of an option X is given by e_rTEQ[X]. 2.1 Geometric LeVy processes The price process St of a stock is assumed to be defined as what follows. We suppose that a probability space (Q, f, P) and a filtration \ft, 0
St = SQez>,

0
Throughout this paper we assume that f — cr(Ss, 0 < s < t) = a(Zs, 0 < s < t) and f = fT. We give here the definition of Levy process and the characterization of it (see [45]). Definition 2.1. A stochastic process [Zt] on Rd is a Levy process if the following conditions are satisfied. 1) For any choice of n > 1 and 0 < to < ti • • • < t„, random variables Z(o, Zfl - Z(0, Z(2 — Z(,,..., Ztn - Zfo„_i, are independent (independent increments property). 2) Z 0 = 0 as. 3) The distribution of Z s+( - Z s does not depend on s (temporal homogeneity or stationary increments property). 4) It is stochastically continuous. 5) There is Qo 6 f with P(Qo) = 1 such that for every a> e Q 0 , Zf(a>) is right-continuous in t > 0 and has left limits in t > 0. In this lecture we discuss the case of d = 1. The Levy process Zt is characterized by the generating triplet (a2, v(dx), b), where a2 is a non-

129

negative constant, v(dx) is a measure such that

X

oo

{\x\2 A l)v{dx) < oo

oo

and b is a constant. By the use of this generating triplet, the characteristic function of Zt is (13)

] = exp{txj>(u)}

(14)

xp(u) = --a2u2

+ ibu+ I

(e™* - 1 - iuxl{M
Using Ito formula, we know that St satisfies the following stochastic differential equation (15)

dSt = St-dZt,

where Z t is another Levy process given by (16)

Zt=Zt

+ \{Z% + £

(eAZs - 1 - AZS).

0<s
And the price process St has the following expression (17)

St = S0£iZ)t

where £(Z) ( is the Doleans-Dade exponential (or stochastic exponential) of £(Z) f = g2.-2H<*.*>. Y[(l + AZs)e-AZ»

(18)

The generating triplet of Ztl say (a1, v(dx), b), is (19) (20) (21)

a2 = a2 v(A) =

flA(e*-l)v{dx),

b = b + -a2 + I ((e* - l)l,|e»_n
Remark 2.1. (i) It holds that supp v c ( - 1 , oo). (ii) If v(dx) has the density n(x), then v(dx) has the density n(x) and n(x) is given by (22)

n(x) =

^n(log(l+x)).

130

(iii) The relations between Zf and Zt are more precisely discussed in [30], where the stochastic logarithm of Xt, £(X)t, is defined and the following relations are obtained. (23)

Zf = log£(Z) f ,

Zt = £{ez)t

Many candidates for the suitable Levy process have been proposed. We give some examples below. 1) Stable process (Mandelbrot and Fama (1963)) 2) Jump diffusion process (Merton (1973)) 3) Variance Gamma process (Madan (1990)) 4) Generalized Hyperbolic process (Eberlein (1995)) 5) CGMY process (Carr-Geman-Madam-Yor (2002)) 6) Normal inverse Gaussian process (Barndorff-Nielsen (1995,1977)) 2.2 Equivalent martingale measures A probability measure Q on (Q, T) is called an equivalent martingale measure of St if Q ~ P and e~rtSt is (Ti, Q)-martingale. Since the geometric Levy process model is incomplete in general, there are many equivalent martingale measures. For the part of (B) of the pricing model we have to select a special martingale measure. Many candidates for the equivalent martingale measure have been proposed as follows. 1) Minimal Martingale Measure (MMM) (Follmer-Schweizer (1991)) 2) Variance Optimal Martingale Measure (VOMM) (Schweizer (1995)) 3) Esscher Martingale Measure (ESMM) (Gerber-Shiu (1994), B-D-E-S (1996)) 4) Minimal Entropy Martingale Measure (MEMM) (Miyahara (1996), Frittelli (2000)) 5) Utility Martingale Measure (Utility-MM) 6) Mean Correcting Martingale Measure (MCMM) 3. Esscher Transformed Martingale Measures 3.1 Esscher transforms The Esscher transform is very popular and thought to be very important method in the actuary theory (see [24] ). Esscher has introduced the risk function and the transformed risk function for the calculation of collective risk. His idea has been developed in his work [18] and by many authors, and played very important roles in the option pricing theory. 3.1.1 Esscher transforms and risk processes We give several definitions. Definition 3.1. Let R be a random variable and h be a constant. Then the

131

probability measure Pjf?S) defined by

dP(Rfr !lS)

(24)

dP

^ •\r ,r

-

E[ehR]

is called the Esscher transformed measure of P by the risk variable R and the index h, and this measure transformation is called the Esscher transform by the risk variable R and the index h. Definition 3.2. Let Rt, 0 < t < T, be a stochastic process. Then the Esscher transformed measure of P by the risk process Rt and the index process hs is the probability measure P^SS\ defined by dP<£SS), .ChM. — 1 5 — \ r = —~f

25

( )

dP

E[eX

'^]

This measure transformation is called the Esscher transform by the risk process Rt and the index process hs. Definition 3.3. In the above definitions, if the index index process is chosen so that the P(„ ' is a martingale measure of St, then P„£SS). is called K[0,T1,"[0,T]

°

K[0,T|,H[0,T1

the Esscher transformed martingale measure of St by the risk process Rt, and it is denoted by PJfs^ or simply P<£ss). 3.1.2 Simple return process and compound return process When we give a certain risk process Rt, we obtain a corresponding Esscher transformed martingale measure if it exists. As we have seen in the previous section, the GLP has two kinds of representation such that St = Soe2' = So£(Z)t. So the processes Zt and Zt are both candidates for the risk process. We shall see the economical meaning of them. For this purpose, we will review the discrete time approximation of geometric Levy processes. Set (26)

s[ n) = S t / 2 .,

k=l,2,....

According to the above two kinds of expression of St, we obtain two kinds of approximation formula. First one is

(27)

Sf = Sa
k=l,2,....

132

Second approximation is k

(28)

S f = S0£(Y("))Jt = [ ] (l + (Yf > - Y p ) ,

it = 1,2,...,

where £(Y(n))jt is the discrete time Doleans-Dade exponential of Yj. , and Y* is defined from the following relations (29)

A '-IN' +

(Y< n )

-*Pl>),

A: = 1 , 2 , .

So we obtain (30)

7(») 7<»)

e<]

= (1

+

(y(«)_y

From this we obtain c(")

(31)

AY<"> =

e< - l

AS<"> - 1 = —— c(") An) *k-l

n)

and we know that A Y[ is the simple return process of Sj[ . On the other hand, we obtain from the definition of AZ™ (32)

' AS<"^ * A Z f = logS<">-logS<"_)1 = log 1 + V

s?> °jt-i

y

and we know that AZ["' is the increment of log-return and it is called the compound return process of S[ . Remark 3.1. The terms "simple return" and "compound return" were introduced in [1], and well known in economics (See also [47]). For t e ( ^ i , |r) we define z| n ) = zf,

Y((n) = Y*0. Then it is easy to

see that the process Z, converges to the process Zt when n goes to oo. On the other hand the process if' converges to the process Z ( as follows. The process St satisfies the stochastic differential equation dSt = St-dZt. From this it follows that dZt = j ^ . (For the justification of this formula, see [30] .) Comparing this formula with (31), we know that the process Y, is the approximation process in the procedure of solving the stochastic

133

differential equation for Z f . This fact means that the process Y" converges to the process Z ( . Based on the above observation, it is natural for us to give the following definition. Definition 3.4. The process Zf is called the "simple return process" of St, and the process Zt is called the "compound return process" of St. 3.2 Esscher transformed martingale measures for geometric Levy processes We next study the existence and uniqueness of Esscher transformed martingale measures. 3.2.1 Esscher Martingale measure (ESMM) Suppose that Zf is adopted as the risk process. Then, if the corresponding Esscher transformed martingale measure Pz is well defined, then it should be called the 'compound return Esscher transformed martingale measure'. This is the Gerber-Shiu's Esscher martingale measure introduced in [24], and the term 'Esscher martingale measure' is usually suggesting this compound return Esscher transformed martingale measure Pz '. We suppose that the expectations which appear in what follows exist. Then the martingale condition for an Esscher transformed probability measure Q = P(7£SS), is ^

(33)

Z[0,T|,/l

EQ[fTrSi] = e-rS0EQ[ez^] =
= So-

This condition is equal to the following condition Ep[e(',+1)Zl] = e r E P [^ z '],

(34)

and this is also equivalent to the following expression,
(35)

where (p(u) is the characteristic function of 2,\ (
fVSMM>(h) =

b + (L+

h)a2 +

J

((£x _ 1)ehX _ xl{m^

Then we obtain Theorem 3.1. If the equation (37)

fESMM)(h)

= Xi

v(dx)

134

has a solution h\ then the ESMM ofSt, P<£SMM>, exists and (3g\ *

p(ESMM) _ p(ESS)

_ p(£SS)

Z[O,T]^I-

'

Zr,h'

The process Zt is also a Levy process under p( £SMM > and the generating triplet of Z, Under p(ESMM)f

say

(39)

a(ESMM)2 =

(40) (41)

{O(ESMM)Z/

v{ESMM\dx) b(ESMM)

v(ESMM)(dx)/

b(ESMM));

{s

a2f

= eh'xv{dx), = b + h'a2 + f

x(^'x -

l)v{dx).

(Proof) The equation (37) is equivalent to the condition (3.16). Therefore P , ,, is a martingale m e a s u r e of St. Z|o,r|,«

°

The characteristic function of Z, u n d e r p( ESMM > = P{ESS)., b y definition (42)

(ESMM)/iA _

r >z,, = J

F

™'(u) = EP(ESMM,


is

EP[e,uZleh'zT] EP[eh'zr]

And this is equal to 1

'

Ep[e('"+h')z'] EP[e»-z<]

<M»-' f t ')
By simple calculation w e obtain (44)


x ^ * 1 - l)v(dx))u

+ £>»" - 1 - Hurl^^M***))} • Using this formula, w e c a n s e e that t h e martingale condition (3.16) is reduced to the equation (3.18). (Q.E.D.) 3.2.2 Minimal Entropy Martingale Measure (MEMM) Next w e consider t h e case w h e r e Zt is a d o p t e d as the risk process. If the corresponding Esscher transformed martingale m e a s u r e p ' £ S S ) exists, Z[0,T|

then it should be called the "simple return Esscher transformed martingale measure". We will see here the relation b e t w e e n the Esscher transform a n d the minimal entropy m a r t i n g a l e m e a s u r e (MEMM). We first give the definition of the M E M M .

135

Definition 3.5. If an equivalent martingale measure P* satisfies (45)

H(P*|P) < H(Q\P)

VQ:

equivalent martingale measure,

then P* is called the "minimal entropy martingale measure (MEMM)" of St. Where H(Q\P) is the relative entropy of Q with respect to P

(46)

H(Q\P)=lIo^dS^if

Q«P>\.

[ oo,

otherwise, J

From the proof of [23] it follows that Proposition 3.1. The simple return Esscher transformed martingale measure Py of St is the minimal entropy martingale measure (MEMM) ofSt. Based on the above results, we give the following definition. Definition 3.6. (i) The compound return Esscher transformed martingale measure Pz is called the "Esscher martingale measure (ESMM)" and £SA1M) denoted by P< . (ii) The simple return Esscher transformed martingale measure Pis Z

I0,T|

called the "minimal entropy martingale measure (MEMM)" and denoted by P (or p(MEMM)). As we have mentioned above, the MEMM, P*, is the simple return Esscher transformed martingale measure. (P* = P?SS)). The existence Z

|0.T1

theorem of the MEMM is obtained in [23]. Set i

(47) / (M£MM »(0) = & + ( - + d)a2 + J

r°°

((e* - l)ee^^

Then the following result is obtained ([23]). Theorem 3.2. If the equation (48)

/ ( M £ M M ) (0) = r

has a solution 8', then the MEMM ofSt, P\ exists and /4m

p * _ p(MEMM) _ p(ESS) Z[O.T\,0'

_ p(ESS) ZT,d'

- xl, Ws i,(x)) v(dx)

136

The process Zt is also a Levy process under P* and the generating triplet of Zf under P*, say (a*2, v*(dx), b'), is (50) (51)

a'2 = a2, v(dx) = e^-Vvidx),

(52)

b' = b + 9'a2 + f

x(eB'{e"-l) - l)v(dx).

(Proof) The results of this theorem follows directly from the proof of [23]. (Q.E.D.) Remark 3.2. (i) The above result is improved to the multi dimensional cases (see [22] or [17]). (ii)The function / ( M £ M M ) (6) is also expressed as follows (xeex - xl l w
CT*2

= a2 = a2,

(55)

v(dx) = e°'xv(dx),

(56)

b' = b + 6'52+

x(ed"x - l)v(dx).

| IW<1|

(iii)The function f(MEMM)(Q) is a non-decreasing function of 9. (iv) If S( is integrable, then it holds that (57)

E(St) = S0 exp(f/ (MEMM) (0)).

(vi) If the condition that /<MEMM»(0) > r is satisfied, then the solution 6' of (48) is negative (9* < 0) if it exists. Such cases occur very often. 3.2.3 Mean Correcting Martingale Measure (MCMM) For the jump-diffusion type models, the Brownian motion can be adopted as the risk process. In that case the corresponding Esscher transformed martingale measure is the mean correcting martingale measure (MCMM) (see [46] or [8]). We suppose that the expectations which appear in what follows exist. Then the martingale condition for an Esscher transformed probability

137

measure Q = P\^S), (58)

is

.zn-^c^e2'^] EQle-'SJ = e-'S0EQ[^ ] = ^ " % E p [ g h W l ] = So-

This condition is equal to the following condition Ep[ez^m']

(59)

= erEP[ehw'l

and this is also equivalent to the following expression, !//(-/) +hi2 + oh = r+ hi2.

(60)

So the martingale condition is r°°

1

(61)

rff2 + b +

(ex-l-xl{M<1)(x))v(dx)

+ oh^r.

To formulate the existence theorem, we set (62)

f(MCMM){h) = b +

1 a2 +

h(J +

j

(ex

_ J _ xl(Wsl((je))

v(dx)

Then we obtain Theorem 3.3. If the equation (63)

f(MCMM)(h) = ^

/ws a solution h', then the MCMM ofSt, p(M™M), exists and (54)

p(MCMM) _ p(£SS)

_ p(£SS)

Tfte process Zt is also a Levy process under p(MCMM) and the generating triplet of Z, under P ^ C M A ^

(65) (66) (67)

say

{(J(MCMM?I v{MCMM)(dx)/b(MCMM))r js

aWCMM)2 =

ait

v{MCMM\dx) = v(dx), b(MCMM) = h + h*a 1

= r--a2*~

f°°

I \)—QO

(e*-l-xl l w
138

3.3 Comparison of ESMM and MEMM The ESMM, MEMM and MCMM are all obtained by Esscher transforms, but they have slightly different properties. The MCMM is supported in only the case of a > 0. So we restrict our attention on the ESMM and MEMM, and we will survey the properties and the differences of them. 3.3.1 Corresponding risk process The risk process corresponding to the ESMM is the compound return process, and the risk process corresponding to the MEMM is the simple return process. The simple return process seems to be more essential in the relation to the original process rather than the compound return process. In this sense we can say that the MEMM is more reasonable martingale measure than the ESMM. 3.3.2 Existence condition As we have seen in the previous section, for the existence of ESMM, P (£SMM) , the following condition f \(e* - l)eh'x\v(dx) < oo J|W>i| is necessary. On the other hand, for the existence of MEMM, P*, the corresponding condition is (68)

(69)

f |(e* - l)eB"(eX-l)\ v(dx) < oo. J(M>1)

This condition is satisfied for wide class of Levy measures, if 6* < 0. Namely, the former condition is strictly stronger than the latter condition. This means that the MEMM may be applied to the wider class of models than the ESMM. This difference does work in the stable process cases. In fact the MEMM method can be applied to the geometric stable model but the ESMM method can not be applied to this same model. 3.3.3 Corresponding utility function The ESMM is corresponding to power utility function or logarithm utility function (see [24] or [25] ). On the other hand the MEMM is corresponding to the exponential utility function (see [21] or [25]). We remark here that, in the case of ESMM, the power parameter of the utility function depends on the parameter value h* of the Esscher transform. 3.3.4 Properties special to MEMM a) Minimal distance to the original probability: The relative entropy is very popular in the field of information theory, and it is called Kullback-Leibler Information Number(see [27] ) or Kullback-Leibler distance (see [12] ). Therefore we can state that the MEMM is the nearest equivalent martingale measure to the original probability P in the sense of Kullback-Leibler distance. Recently the idea of

139

minimal distance martingale measure is studied. In [25] it is mentioned that the relative entropy is the typical example of the distance in their theory. b) Large deviation property: The large deviation theory is closely related to the minimum relative entropy analysis, and the Sanov's theorem or Sanov property is well-known (see, e.g. [14]). Applying this theorem on the paths spaces of the price process, we can conclude that that the MEMM is the most possible empirical probability measure of paths of price process in the class of the all equivalent martingale measures. In this sense the MEMM should be considered to be the exceptional measure in the class of all equivalent martingale measures. The Sanov's theorem is Theorem (Sanov) [14] Let fi be a probability measure on the Polish space £ and let /2„ e Mi(Mi(L)) be the distribution under u" of the \ £" =1 baj. Then H(\ji) is a good, convex rate function on Mi(L) and [fi„ : n > 1) satisfies the full large deviation principle with rate function H(-|ju). c) Indifference Price: The MEMM is related to the indifference prices with the exponential utility in the following sense. Let pr(X) be the indifference price of X, where y is the parameter for the absolute risk aversion. Then it holds that limp y (X) = e-rrEp.[X] y—>0

We also remark that, in the case of MEMM, the relation of the MEMM to the utility indifference price is known (see [44], [23] and [48]). As the result of the above discussions, we can say that the MEMM has many better properties than the ESMM in the theoretical sense. 4.

[GLP & MEMM] Pricing Model In this section we explain the [GLP & MEMM] pricing model and see examples of the model. 4.1 Model Now we give the definition of the [GLP & MEMM] Pricing Model. The [GLP & MEMM] Pricing Model is such a model (see [36]):

(A) The price process St is a geometric Levy process (GLP). (B) The price of an option X is defined to be e~rTEp-[X], where P* is the MEMM. Of course this model can be considered for the cases where the MEMM exists. The existence condition is given in Theorem 2 of Section 3.

140

4.2 Examples of [GLP & MEMM] Pricing Model In this section we see several examples of [GLP & MEMM] Pricing Models. To do this, we have to check the existence condition of the MEMM, i.e. we have to examine that the given geometric Levy process St = So exp Z ( satisfies the conditions of Theorem 2. We denote the function /(MEMM)(Q) by f(9) for simplicity, namely (70)

f(6) = b + d + 6)o2+

({
Then the condition is that the following equation has a solution ff. (71)

f{9) = r.

4.2.1 Geometric Variance Gamma Model The Levy measure of Variance Gamma process is of the following form (see [31]). (72)

v{dx) = C {l[x0| exp(-c 2 |x|)) \x\~ldx,

where C, c\, d are positive constants. The following results are obtained (see [23]). Proposition 4.1. 1) If ci < 1, then the equation f(6) = rhasa unique solution 9", and the solution is negative. 2) lfc2 > 1 and /(0) > r, then the equation f(9) = r has a unique solution 9*, and the solution is non-positive. 3) If ci > 1 and /(0) < r, then the equation f(9) -rhasno solution. 4.2.2 Geometric CGMY Model The L6vy measure of the CGMY process is (73)

v{dx) = C (I(x0] exp(-M|x|)) |x|-<1+YW,

where C > 0,G > 0,M > 0, Y < 2 (see [3]). If Y < 0, then G > 0 a n d M > 0 are assumed. We mention here that the case Y - 0 is the VG process case, and the case G - M = 0 and 0 < Y < 2 is the symmetric stable process case. In the sequel we assume that G, M > 0. For this model the following results are obtained (see [41]). Proposition 4.2. 1) IfM < 1, then the equation f(9) -rhasa unique solution 9', and the solution is negative. 2) IfM > 1 and /(0) > r, then the equation f{9) = rhasa unique solution 6*, and the solution is non-positive. 3) If M > 1 and /(0) < r, then the equation f(8) = rhasno solution.

141

4.2.3 Geometric Stable Model We consider the stable model. Suppose that Zt is a stable process and let (0, v(dx), b) be its generating triplet. The Levy measure is (74)

v(dx) = Cll[x<0]\x\-^dx

+ c2I{x>0]\x\-("+»dx,

where 0 < a < 2 and we assume that (75)

ci > 0,

c2 > 0,

c = cl+cz>

0.

The following results are obtained (see [23]). Proposition 4.3. Under the assumption c\,c2 > 0, the equation f(9) = rhas a unique solution 6% and the solution d* is negative. Remark 4.1. Consider the case where c\,c2 > 0. Under the original measure P, St, t > 0 is not integrable. But under the MEMM P", any moments Ep* [\St\k], k = 1,2,..., of St are finite. This fact follows easily from the result that 6' is negative, and this property is very useful for the study of option pricing of this model. 4.3

Option Pricing and Volatility Smile/Smirk Properties In order to apply the [GLP & MEMM] Pricing Models to the financial problems, we have to establish the methods to compute the option prices. Namely we have to compute the expectations £p.[F(o))], where F(a)) is a functional of Levy process. 4.3.1 European Type Options If a contingent claim C is depending only on the terminal value of the stock price ST - S^e7-1', then we can compute the price of C as what follows. Let C = f(ST) = / ( S o ^ ) = F(ZT), (F(z) = f{SQ
(77)

dC(t,z) dz

•£

(c(f,z + z) - C(t,z) - z ^ % ^ l , | 2 | < 1 ( ( z ) | v(dz) dz -rC(t,z), 0
Solving this equation, we obtain the option price C(0, So) = C(0,0).

142

4.3.2 FFT Method for European Call Options The fast Fourier transform method (FFT method) is very useful for the computation of option prices. We need to compute the such an expectation Ep-[F(a>)], and in the case of European type options such type of expectations Ep-[G(Sr)]- If we know the distribution function p*T(z) of ZT under P\ then E/* [G(ST)] = J_x G(z)p*T(z)dz. Levy process is characterized by the generating triplet, and the generating triplet is given explicitly in the characteristic function. So we can assume that the characteristic function
ct>;(u) = ] = exp(i},'t(u)) = exp(ty'(u)),

i = V=T,

where ip'(u) = ty\{u). Let [i]{dz) be the distribution of Zt under the MEMM P", and assume that juj(dz) = p't{z)dz. Then eiuYt(z)dz, CO

The price of European call option is (Soe2 - K)+p'T(z)dz. CO

Set K/So = e*, and define c(k; S0, T) = C(S0, S 0 e\ T). Then {
We introduce the so-called time value of option (82)

c(k; So, T) = c(k; S0, T) - (S0 - e~rTK)+ = c(k; S0, T) - S 0 (l - ek-rT)+

and let C(v; S0, T) be the Fourier transform of c(k; S0, T) eivkc(k;S0,T)dk. oo

143

Using (4.12)

03

xJz

and (85)


The characteristic function (p'T(u) is computed directly from the generating triplet {a2, b", v*{dx)), so Q(v; So, T) is obtained from the above formula. Next, by (4.14), c(k; So, T) is obtained by the inverse Fourier transform i

(86)

c(k; S0, T) = ^ J

r°°

e-llcvC(v; S0, T)dv

and (87)

c(k; So, T) = c(k; So, T) + (S0 - e- rT K) + ,

K = Soc*.

Finally we obtain the price of the European call option C(So, K, T) as (88) C(S0, K, T) = c(log(K/S0); So, T) = c(log(X/S0); S0, T) + (S0 - £-rTK)+. 4.3.3 Volatility Smile/Smirk Properties The volatility smile/smirk properties are reported for many market prices of options. This fact tells us that the Black-Scholes model is not necessarily best model, and that we should study other models which may have the volatility simile/smirk properties. It is known that the [GLP & MEMM] models have those properties (see [39]). 5. Physical World and MEMM World The behavior of the price process Sf is governed by the original probability P, and the movement of St is observable. This is the real world (=Physical world). On the other hand the price of an option X is computed as the expectation e~rTEp»[X], namely the process St is supposed to obey the MEMM P*. This world is differ from the real world, and this world should be called the imaginary world (=MEMM world). 5.1 From Physical World to MEMM World Suppose that the price process St = Soe2' is given and the generating triplet of Z ( is (a2, v, b). Let d* is the solution of f(0) - r, where the function f(d) is defined by (4.6). Then, by Theorem 3 in §4.2, the generating triplet (a*2,v*,fc*)ofZ, u n d e r P i s

144

(89) (90)

a'2 = a2, v{dx) = ee'(e"-l)v{dx),

(91)

V = b + 6*o2 + \

xd(v* - v)

= b + 9*o2 + f x(e e " ( '"- 1 ) -l)v(dx). JM
1

r°°

^

%J—oo

b' + -a'2 +

(ex-l-xl{M<1)(x))v*(dx)

= r.

We should notice that the 9* does not appear explicitly in this formula, and that this formula is just the same condition that P* is a martingale measure of the price process St. Concerning to the martingale condition for more general cases of semimartingales, see [47]. 5.2 From MEMM World to Physical World We study the inverse problem of the previous subsection. Suppose that the generating triplet (a , v*, b*) of Zt under P* is given. Since we assume that P" is martingale measure, the condition (M*) is satisfied. We try to construct a probability P such that under P the price process St = Soe2' is geometric Levy process and the MEMM of St = Soe2' with P is P*. Let 9* be any real number (it is usually supposed that 9" < 0) and set (93) (94) (95)

a2. = a*2 ve.(x) = e - ^ - V f d x ) be- =V-

0*ff2 + f

x (e~e'^-V - l) V'(dx),

where we assume that all integrals are converge. Then suppose that we could construct the probability measure Pe- such that under Pe- the process Zt is a Levy process with the generating triplet (o2e.,ve-,be-). It is easy to see that P* is the MEMM of St = Soe2' with Pe-. We remark here that there are many geometric Levy processes whose MEMM is just the same P*.

145

5.3 Example: Geometric Stable Process Case • Parameters in the physical world: (a,C\,C2,b),

0 < a < 2, C\,Ci > 0,

C\ + Ci > 0, - o o < b < oo.

The triplet is (0, v, b), where (96)

v(dx) =

—

dx.

• Parameters in the MEMM world: (0*, a*, c\,c\, V), c\, c* > 0, c*x+c*2> 0, -oo < b* < oo, where v (dx) = e 9 <^ «

(97)

^ ^

9* < 0, 0 < a* < 2,

dx.

and the following martingale condition (e*-l-xl ( W s l l (*))v'(dx) = r CO

must be satisfied. So, if we have given the values of (0", a*, c*, c*,), then the value of b* is determined by the above condition (M*). 5.4 Diagram of Physical World and MEMM World Physical World MEMM World P

P* 2

S, = Soe ' = S0S(Z)t 2

(o ,v,b) under P

Zt

(cr*2, v*,b*) under P*

(a2, v, 5) under P

Z(

(a*2, v", 5*) under P*

Z t : log-return process, Zf: simple return process 6. Estimation of Levy Processes in the Physical World Usually this procedure is carried on under the restriction of the class of Levy processes, for example the stable process class, VG process class, etc. Therefore the estimation problem of the process is reduced to the parameter estimation problems. There are many papers on this subject, (see [37] for example). If the MLE is possible and easy, then this method may be good. But this method is not easy to apply our cases.

146

6.1 Characteristic Function Method of Moments 1) Characteristic Function and Moment Generating Function: The characteristic function (in the sense of distribution) cp(u) of X is defined by (98)

cf>(u) = cp(u; X) = E[eiuX] = exp \i\>(u)\, i = V^l.

The sample characteristic function $„{u) is given by (99)

§n{u) =

- Y ie'u^i,

- o o < u < oo.

n *r-' ;=i

Note that $„(«) is a consistent estimator of <$>(u): (100)

lim &„{u) =
The moment generating function M(u) of X is defined, if it exists, by (101)

M(u) = M(u; X) = E[euX],

-co < u < oo,

and the sample moment generating function JVI(u) is given by (102)

1 " M„(u) = - ^ eui>,

-co < u < oo.

The sample moment generating function lsA(u) is a consistent estimator ofM(u). 2) Moment Equations for Characteristic Function: When we take the function e'uX as the function /„(X) for the generalized method of moments, then the generalized moment equations are (103)

(p{u) = (\>n{u),

- o o < u < oo.

k

If the moment E[X ] exists, then it is well-known that (104)

mk = E[Xk] = jEWXf]

= ^ ( 0 ) ,

and the classical moment equations are ldk
(io5)

T*~dSi0) = 7hk'

k=

12

' >----

3) Moment Equations for Moment Generating Function: Suppose that the moment generating function M(u) exists. In such cases we can take the function euX as the function fu(X) for the generalized method of moments, and then the generalized moment equations are (106)

M(u) = Mn(u),

-co < u < oo.

147

6.2

Estimation of Levy Processes Set Z = Z\. The corresponding characteristic function
(107)

(u) = E[eiuZ] = exp {^(u)} = exp < ——u2 + ibu+ I

(e'"x - 1 - mxl{|X|Slf(x)) v{dx) \

What we have to do is to estimate the generating triplet (a1, v(dx), b) of the distribution of Z. These parameters explicitly contained in the characteristic functions. So it is natural for us to apply the characteristic function method to those estimation problems. Set (108)

Ij = Zj-Zhuj

= l,2

then [Ij,j = 1,2,...} is i.i.d. with the same distribution as Z\, since Levy process has temporally homogeneous independent increment. So, if we are given a sequential data of a Levy process Z ( , then we can apply the method described above to estimate the distribution of Z = Z\, namely we can apply the generalized moment equation or the classical moment equation when it exists (see [37] or [2]). 7. Fitness Analysis of the Models Suppose that the sequential data of the price process S< of underlying asset and the data of market prices of options. From these data, we have to select a model which is most fitting to the given data. This is the fitness analysis of the models. 7.1 Procedure of fitness analysis • Collecting data: the sequential data of the price process St, and the data of market prices of options. • Selection of the most fitting model to the obtained data. To solve this problem, we first fix a type of model (for example, [Gstable-P & MEMM]), and we take the following steps. 1) Estimation the price process of the underlying asset in the physical world from the sequential data of it. 2) Calculation of the MEMM from the estimated parameter, and computation of the theoretical prices of options in the estimated MEMM world. 3) Analyzing the fitness of the theoretical prices to the market prices. We carry on the above procedures for several types of models, and the final step is 4) Determination of the most reasonable model.

148

7.2 Fitting error of the model Denote the estimated probability by P, (or equivalently, the estimated generating triplet by (a2,"?",ft)),and let P* be the corresponding MEMM. Then the theoretical price of option C is Ep.[Ce_rT]. We denote this value byC*. Suppose that the data, //;, / = 1,2,..., L, of market prices of options Q are given. Then we define the fitting error of the model by r

* = \Yu

e

(109)

\Q-rji\

Procedure to obtain the fitting error: Physical World

MEMM World

Data: {E,j\ (time series data) Estimated: P _

Transformed: P*

(European Call Options) Theoretical prices: Q Data: {ly} (European Call Options, in the Market)

* Fitting

error:

l

v icT - nil

e =—>

Remark 7.1. the second candidate for the fitting error is 1

(no)

AT }

7.3

—

Y \ci - n,? 1=1

Example: Geometric Stable Process Case Parameter in the physical world: (a,c\,C2, b)

/,,-n

,, s

(111)

v(dx) =

Cil, x < o|(x) + C 2 1 ( I >0|W

—

dx.

Estimators: (a,Ci,C2,&) (112)

v(dx) = — • — • — _ K

v

\xf+V

dx.

149

6* is determined by (113)

?+J

{{?-l)l?V-»-xlltei)(x))v{dx) = r.

The process which determines the theoretical option prices is (114)

V(dx) =

g W i W ' ^ W * ^

X

CO

xl, Ms i,(ac)d(7-v). CO

7.4 Fitness analysis of the estimated model As the results of the above procedure, if the value e* nlQ*-jj/|

' =

&

is small, then the fitness of the model to the data is good. The value e* depends on the model, namely on the selected class of the process (for example, class of stable processes, class of CGMY processes, etc.) We can conclude that the class whose fitness error is the smallest is the best class (see [40].) 8.

Calibration The calibration is similar to the fitness analysis, but usually the calibration is done based on only the data of option prices in the market (see [11] ). Namely the calibration solves the following minimization problem L

(116)

mjne^^ij^

IQ^-ipl

where y* is the parameter of a model in the risk neutral world. For example, in the geometric stable process case, the calibration means the estimation of y* = (8', a*, c\, c*2, b') in the MEMM world. Suppose that the above minimization were attained at y w = y(cfl,), then the calibration error is (117)

ARPE« = 1 £ - | C ' 1=1

'"'

150

The second candidate for the calibration error is the root mean-square error (RMSE) given by

RMSE M ) = min

(118)

*

iLfT/=i

m

12

9. Calculation of the Option Prices: Expectation of Functionals of Levy Processes In order to apply the [GLP & MEMM] Pricing Models to the financial problems, we have to establish the methods to compute the option prices. Namely we have to compute the expectations Ep-[F(o;)], where F(CL>) is a functional of Levy process. 9.1 Asian Option Let A(S, K) be the Asian option, namely (119)

A(S,K)=max(i f T Jo

Stdt-K,Q).

Then the price of A(S., K) is e~rTEP.[A(S., K)]. For the computation of the above value, in [49] the following interest results have been obtained. Set qt = Jf (l - e~r(T_f)J, and let Xt be a stochastic process determined by the following equation (120)

dXt = cjtdSt + r(Xt- - qtSt-)dt,

(121)

0
X0 = <7oSo-e-rTK

Then it holds that (122)

XT=A(S,K)

and so the calculation problem of the price A'(S0,K) = e~rTEP-[A(S(a>),K)] is reduced to the calculation of e~rTEp>[Xr]. Next we set (123)

Yt = %•

and define a new probability measure Q* by

151

then the process Yt is a Markov process under Q* and solves the following equation (125) odWt + {V - -a2)dt + \

xW*p(dudx) - I

-^—N*p(dudx) ,

Next we define V(t, y) as the solution of the following equation (126)

(127)

^

+

La2iZM{qt-y)2

= 0, 0 < t < T, V(T,y) = So max{y, 0},

where v*(dx) is the Levy measure of Zf under the probability P*, and Zt is the Levy process which appears when St is expresses in the form of S - S0G(Z)t (Doleans-Dade exponential). Then V(Q, ^ ) = V(0,q0 - e~rT^) is the price of the Asian option A(S, K). The points of this result is that the problem to obtain the price of Asian option is reduced to the problem to obtain the price of European type option of Markov process. 10. Utility Indifference Prices and Risk Measure We consider the exponential utility function (128) and set (129)

Ua(x) =

l-e-"x,

}a(c, B) := sup EP[Ua{c + G(0) r - B)] = sup £ P [1 - exp{-a(c + G(d) - B\]

where ® is a suitable set of strategies, G(0) is the gain of a strategy 6, B is a contingent claim. We give the definition of the utility indifference price p„(c, B) (see [13] §4.2, or [26] ). Definition 10.1. The value p„{c, B) which satisfies the following equation (130)

Jn(c + p«(c,B),B) = Ja(c,0)

is called the utility indifference price of B.

152

It is easy to see that the value pa(c, B) does not depend on c, so we use the notation pn(B). The quantity Ja(c, B) is related to the relative entropy by the following duality relation (131)

;„(c, B) = 1 - exp{- inf^ (H(Q\P) + ac- EQ[aB])} = 1 -e-"cexp{a

sup

(EQ{B)

-

QeMV

-H(Q\P))}, a

'

where At is a convex subset of local martingale measures corresponding to 0 (see [22]). And the utility indifference price pa(B) has the following property (132)

limp a (B) = EP.[B]. aj,0

(See [44], [23] and [48].) This result suggests us that -Ep. [B] may be an example of the reasonable coherent risk measure. 11. Generalization of the [GLP & MEMM] Pricing Model 11.1 Multi dimensional cases The multidimensional cases are very important in the practical sense, because of the fact that many options are based on the index, for example Nikkei 225, and the index is the combination of the multi dimensional price processes of underlying assets. Suppose that the price processes are given by (133)

S'^S'^,

j = l,...,d

where Zt = ( Z j , . . . , Z\)T is a d-dimensional Levy process. This process is equivalent to (134)

ds[ = s[_dZ\,

j = l,...,d

where Zf = (Z*,..., Zdt ) T is the corresponding d-dimensional Levy process, and the price processes S[ have the following expression (135)

S{ = sfaZ'),,

j=

\,...,d

where &{Z>)t is the Doleans-Dade exponential of Z\. The results described in the previous sections for 1-dimensional case are generalized to the multi dimensional cases (See [22] and [17]).

153

The points are the following two. 1) The MEMM P* is obtained by the Esscher transform by Z. 2) The processes Zt and Zf are also Levy processes under P*. Remark 11.1. The Esscher transform by Zf is unique if it exists, but the Esscher transform by Zt is not necessary unique in the multi dimensional cases (See [30]). 11.2 [GLP & MEMM] models with defaultable risk We have started from the following type models St = Soez', 0 < t < T, or the following SDE dSt = St-dZtr where Zt is a Levy process such that supp v c (-1, oo). In this case the model is without defaultable risk. If we permit the case supp v c [-1, oo), then the defaultable risk is in mind. To do this we introduce a new Levy process Z\ (A > 0), defined on a new probability space (Q*A)!F(A), P'A)), whose Levy measure v(A)(dx) is vw{dx) = v{dx) + A5t-i}(dx).

(136)

And let SJA) be the solution of dSJA) = SJA)dZ
T(A) = inf\t > 0; S 0; 2| A) = -1).

It is easy to see that T(A) is independent of S(, Zt and Zt, and that (138)

P(T ( A )
= A f

e-^ds

= 1-

e~M.

Jo A)

It is obvious that S| = Sfl|T<.i>>f|. Suppose that the MEMM of S|A), P ( A ) \ exists. Then 2[A) is Levy process under P*A'* and the LeVy measure of Z\A) under P* is (139)

vw'(dx) =

edW'xvm(dx),

where 0(A)" is the solution of the following equation for 6 (xeex-xlM
(140)

= r.

oo

We can see that Zf is also a Levy process under P(A)* and the Levy measure of it is eeW xv{dx). Therefore Z ( is also a Levy process under P(A)*.

154 Remark 11.2. The Levy measure of Z( under P(A>* (= eeW'xv(dx)) is different from the Levy measure of Zt under P*, which is ee'xv(dx). From these results we know that T(A) is independent of St and Zt under P >*, and (A

f e-Ae-"W'sds = 1 e^"^1. Jo The theoretical prices of options can be computed as the expectation with respect to P(A)*. In particular, The prices of European type options are easily computed, using the above properties of T (A) . (141)

pW*(T
Remark 11.3. The arguments of this subsection are possible in the MEMM setting, but not possible in the ESMM setting because the process Z)' such that S), = Soe2'1' is not well-defined. 11.3 semimartingale process model The martingale theory is established in the framework of semimartingale process. So, in the theoretical or mathematical sense, it is natural to study the semimartingale process models. In fact many articles are studied under the semimartingale setting. Among them the generalization of the [GLP & MEMM] Pricing Model is discussed in [9] , where the entropy-Hellinger martingale measure is introduced. Acknowledgments. The author likes to thank professor Albert Shiryaev for valuable comments. References 1. Buhlmann, H., Delbaen, R, Embrechts, P., and Shiryaev, A. N. (1996), Noarbitrage, Change of Measure and Conditional Esscher Transforms, ONI Quarterly 9(4), 291-317. 2. Carrasco, M. and Florens, J.-P. (2002), Efficient GMM Estimation Using the Empirical Characteristic Function (preprint). 3. Carr, P., Geman, H., Madan, D. B., and Yor, M. (2002), The Fine Structure of Asset Returns: An Empirical Investigation, /. of Business, 2, 305-332. 4. Carr, P. and Madan, D. (1999), Option valuation using the fast Fourier transform. Journal of Computational Finance 2,61-73. 5. Carr, P. and Wu, L. (2003), "The Finite Moment Log Stable Processes and Option Pricing," The Journal of Finance 58(2), 753-777. 6. Carr, P. and Wu, L. (2004), "Time-changed Levy Processes and Option Pricing," journal of Financial Economics 71,113-141. 7. Chan, T. (1999), Pricing Contingent Claims on Stocks Derived by Levy Processes. The Annals of Applied Probability 9(2), 504-528.

155 8. Cheang, Gerald H. L. (2003), "A simple approach to pricing options with jumps" (preprint) 9. Choulli, T. and Strieker, C. (2004), Minimal entropy-Hellinger martingale measure in incomplete markets, to appear in Mathematical Finance. 10. Cont, R. and Tankov, P. (2004a), Non-Parametric Calibration of Jump-Diffusion Option-Pricing Models, J. of Computational Finance 7(3), 1-49. 11. Cont, R. and Tankov, P. (2004b), Financial Modelling with Jump Processes, CHAPMAN & Hall/CRC. 12. Cover, T. and Thomas, J. (1991), Elements of Information Theory, Wiley. 13. Delbaen, F., Grandits, P., Rheinlander, T., Samperi, D., Schweizer, M., and Strieker, C. (2002), "Exponential Hedging and Entropic Penalties," Mathematical Finance, 12(2), 99-124. 14. Deuschel, J.-D. and Stroock, D. W. (1989), Large Deviations, Academic Press. 15. Eberlein, E. and Keller, U. (1995), Hyperbolic distributions in finance. Bernoulli 1, 281-299. 16. Edelman, D. (1995), A Note: Natural Generalization of Black-Scholes in the Presence of Skewness, Using Stable Processes, ABACUS 31(1), 113-119. 17. Esche, F. and Schweizer, M. (2005), Minimal Entropy Preserves the Levy Property: How and Why, Stochastic Processes and their Applications 115,299-327. 18. Esscher, F. (1932), On the Probability Function in the Collective Theory of Risk, Skandinavisk Aktuarietidskrift 15,175-195. 19. Fama, E. F. (1963), Mandelbrot and the Stable Paretian Hypothesis. /. of Business, 36, 420-429. 20. Follmer, H. and Schweizer, M. (1991), Hedging of Contingent Claims under Incomplete Information. In M. H. A. Davis and R. J. Elliot (ed.): Applied Stochastic Analysis, Gordon and Breach, 389-414. 21. Frittelli, M. (2000), The Minimal Entropy Martingale Measures and the Valuation Problem in Incomplete Markets, Mathematical Finance 10, 39-52. 22. Fujiwara, T. (2002), The Minimal Entropy Martingale Measures for multidimensional Geometric Levy Processes and the optimal strategies for the exponential utility maximization (preprint) 23. Fujiwara, T. and Miyahara, Y. (2003), The Minimal Entropy Martingale Measures for Geometric Levy Processes, Finance and Stochastics 7, 509-531. 24. Gerber, H. U. and Shiu, E. S. W. (1994), Option Pricing by Esscher Transforms, Transactions of the Society of Actuaries XLVI, 99-191. 25. Goll, T. and Riischendorf, L. (2001), Minimax and minimal distance martingale measures and their relationship to portfolio optimization, Finance and Stochastics, 5(4), 557-581. 26. Hodges, S. D. and Neuberger, A. (1989), "Optimal Replication of Contingent Claims under Transaction Costs," Review of Futures Markets 8, 222-239. 27. Ihara, S. (1993), Information Theory for Continuous System, World Scientific. 28. Jacod, J. and Shiryaev, A. (1987), Limit Theoremsfor Stochastic Processes, SpringerVerlag. 29. Kallsen, J. (2002), Utility-Based Derivative Pricing in Incomplete Markets, in Mathematical Finance - Bachelier Congress 2000, Springer, 313-338. 30. Kallsen, J. and Shiryaev, A. N. (2002), The Cumulant Process and Esscher's

156 Change of Measure, Finance and Stochastics 6(4), 397-428. 31. Madan, D. and Seneta, E. (1990), The variance gamma (vg) model for share market returns. Journal of Business 63(4), 511-524. 32. Madan, D., Carr, P., and Chang, E. (1998), The variance gamma process and option pricing. European Finance Review 2, 79-105. 33. Mandelbrot, B. (1963), The variation of certain speculative prices. /. of Business 36, 394^19. 34. Miyahara, Y. (1996), Canonical Martingale Measures of Incomplete Assets Markets, in Probability Theory and Mathematical Statistics: Proceedings of the Seventh Japan-Russia Symposium, Tokyo 1995 (eds. S. Watanabe et ah), pp. 343-352. 35. Miyahara, Y. (1999), Minimal Entropy Martingale Measures of Jump Type Price Processes in Incomplete Assets Markets. Asian-Pacific Financial Markets 6(2), 97113. 36. Miyahara, Y (2001), [Geometric Levy Process & MEMM] Pricing Model and Related Estimation Problems, Asia-Pacific Financial Markets 8(1), 45-60. 37. Miyahara, Y (2002), Estimation of Levy Processes, Discussion Papers in Economics, Nagoya City University No. 318, pp. 1-36. 38. Miyahara, Y (2004), A Note on Esscher Transformed Martingale Measures for Geometric Levy Processes, Discussion Papers in Economics, Nagoya City University No. 379, pp. 1-14. 39. Miyahara, Y. and Moriwaki, N. (2004), Volatility Smile/Smirk Properties of [GLP & MEMM] Pricing Models, Discussion Papers in Economics, Nagoya City University No. 405, pp. 1-16. 40. Miyahara, Y and Moriwaki, N. (2005), Application of [GLP & MEMM] Model to Nikkei 225 Option (preprint). 41. Miyahara, Y. and Novikov, A. (2002), Geometric Levy Process Pricing Model, Proceedings ofSteklov Mathematical Institute, 237, pp. 176-191. 42. Overhaus, M., Ferraris, A., Knudsen, T., Milward, R., Nguyen-Ngoc, L., and Schindlmayr, G. (2002), Equity derivatives: Theory and Applications, Wiley. 43. Rachev, S. and Mittnik, S. (2000), "Stable Paretian Models in Finance", Wiley. 44. Rouge, R. and El Karoui, N. (2000), Pricing via Utility Maximization and Entropy, Mathematical Finance 10(2), 259-276. 45. Sato, K. (1999), "Levy Processes and Infinitely Divisible Distributions", Cambridge University Press. 46. Schoutens, W, (2003), Levy Processes in Finance: Pricing Financial Derivatives, Wiley. 47. Shiryaev, A. N. (1999), Essentials of Stochastic Finance: Facts, Models, Theory, World Scientific. 48. Strieker, C. (2004), Indifference Pricing with Exponential Utility, Progress in Probability, 58,323-328. Birkhauser 2004. 49. Vecef, J. and Xu, M. (2001), "Pricing Asian Options in a Semimartingale Model" (preprint) 50. Xiao, K., Misawa, T., and Miyahara, Y. (2000), Computer Simulation of [Geometric Levy Process & MEMM] Pricing Model, Discussion Papers in Economics, Nagoya City University No. 266, pp. 1-16.

Topics Related to Gamma Processes Makoto Yamazato Department of Mathematical Sciences University of the Ryukyus, Senbaru 1, Nishihara-cho, Okinawa, 903-0213 Japan The aim of this paper is to explain important but not popular properties related to gamma processes and show the applicability of these properties. We define subclasses (CME and its subclasses) of the class of infinitely divisible distributions, which are generated by mixtures and convolutions from gamma distributions, and study their properties. Then we apply the obtained results to the unimodality of the distributions in the above classes, the boundedness in space-time parameters of transition densities of subordinators generated by CME distributions and the determination of the class of hitting time distributions of 1-dimensional generalized diffusion processes. Finally, we remark that some subclasses of the class CME and the class of selfdecomposable distributions are often used in mathematical finance. Key words: gamma process, convolution, mixture

1.

Introduction Gamma process appears in various fields both theoretical and practical. Its 1-dimensional distribution - gamma distribution has various remarkable properties (refer [23]). The aim of this paper is to introduce and explain important but not popular (in Japan) properties related to gamma processes and show the applicability of these properties. We restrict ourselves to 1-dimensional processes. We define subclasses ME, CE, CG, CEME and CME of the class of infinitely divisible distributions, which are generated by mixtures and convolutions from gamma distributions, and study their properties. For this purpose, we mainly use the fact that the distributions in these classes can be decomposed into exponential distributions and gamma distributions (Sections 3-6). Then we apply the results obtained in Sections 3-6 to unimodality of the distributions in the above classes (Section 7), boundedness in space-time 157

158

parameters of transition densities of CME+ processes (Levy processes generated by CME distributions on [0, oo)) in Section 8 and determination of the class of hitting time distributions of 1-dimensional generalized diffusion processes (Section 9). In the final section, we remark that distributions of some subclasses of the class CME and, class L (class of selfdecomposable distributions) and related stochastic processes are often used in mathematical finance. Almost all facts described in this paper are already known. However, some results are new (Theorems 3.3, 3.4,6.3,8.2, 8.3, Example 8.1) and the proof of Theorem 3.5 is simplified. 2. Infinitely divisible distributions and Levy processes Definition 2.1. A probability measure p on R is said to be infinitely divisible if for every n > 1, there is a probability measure pn on R such that p = (/in)"* where (pn)n* is the n-fold convolution of pn. Theorem 2.1. A probability measure p on R is said to be infinitely divisible if and only if its characteristic function is represented as follows: Tp(z) = I eizxp{dx) = exp[zyz - ^-z 2 + ( z

J

(eizu - 1

-izulw
JK\(0|

where, y e R, a > 0 and v is a measure, called the Levy measure of p, satisfying < oo.

We denote by I the class of infinitely divisible distributions on R. Definition 2.2. A probablity measure p on R is said to be selfdecomposable if for any 0 < c < 1, there is a probablity measure pc on R such that Tp(z) =

Tp(cz)Tpc(z).

We denote by L the class of selfdecomposable distributions. The following characterization of the class L is well known. Theorem 2.2. A probability measure ponM. is selfcecomposable if and only if it is infinitely divisible and its characteristic function is represented as a1 C Tp(z) = exp[fyz - —z 2 + I 1

(e'zu - 1 -

izul^^u^k^du]

JE\(OI

where, y e R, a > Qandk(u) is a nonnegative measurable function nondecreasing on (-oo,0) and nonincreasing on (0, oo) satisfying / ( M _ 1 A |x|)fc(x)dx < oo.

159

Characteristic function of an exponential distribution /.i with density Ae Ax is represented as ru{z) = —2-r- = expt (eizu A - iz Jo

l)u-le-Audu\.

Hence the exponential distribution is selfdecomposable with Levy measure M-ig-Aw jfae class of exponential distributions is regarded as a subclass of the class of gamma distributions. A probability measure p is said to be a gamma distribution with parameter (a, |3) if it is absolutely continuous with density m

=

\ 0

lmxa~le^Xf°rx>0

otherwise.

The parameter a is called the shape parameter and j3 is called the scale parameter. The gamma distribution is selfdecomposable with Levy measure a u -l e -fu_

Definition 2.3. A stochastic process [Xt : t > 0| on R is a Levy process if the following conditions are satisfied: (1) For any choice of n > 1 and 0 < to < h < ••• < t„, random variables Xf„, Xt] - X ( o ,..., Xf„ - Xf„_! are independent (2) X0 = 0 a.s. (3) The distribution of Xs+t - Xs does not depend on s. (4) lim u o PflXd > e) = 0 for e > 0. (5) Xt is right-continuous in t > 0 and has left limits in t > 0. Theorem 2.3. Let [Xt\ be a Levy process on R. Then the distribution ofXt is infinitely divisiblefor every t > 0. If the characteristicfunction of X\ is represented as (1)

E(e''2Xl) 2 = exp[iyz - °-z + f z

(eizu - 1 - izul, Msl ,)v(du)] == ^ ( z ) <

JR\{0|

then the characteristic function ofXt is represented as E(eizX>)=e^z). Let {Xt} be a Levy process. If its distribution at t = to > 0 is an exponential distribution with density jSe - ^, (jS > 0), then the distribution at t > 0 is a gamma distribution with density nJ7^x
160

3.

Mixtures of exponential distributions We say that a probability measure ji is an ME+ distribution if there is a probability measure G on (0, oo] such that (2)

H([0,x])= f

(1 - e'xu)G{du)

forx>0.

J(0roc]

The measure y may have a point mass G({oo}) at the origin and the distribution function of y is infinitely differentiable on (0, oo). Similarly, we say that a probability measure JJ. is an ME- distribution if there is a probability measure G on [-oo, 0) such that (1 -e~xu)G{du)

P<[x,0])= f

forx<0.

J[-°o,0)

Theorem 3.1. ([22]) A probability measure p. on [0, oo) is an ME+ distribution if and only if there is a nonnegative and absolutely continuous measure Q on (0, oo) with density bounded by 1 a.e. satisfying JT u~lQ{du) < oo such that, for s e R , (3)

£p(s) = exp[ f

(e'sx - 1)( f

J(0,°o)

=exp

f

e~xuQ{du))dx\

J(0,°o)

{

^ks-\)Q{du)'

J(0,oo) U + S

U

where £p stands for the Laplace transform of y. We have G ( M ) = lim £fi(s) = exp(- f

-Q(du)).

We call measures G appearing in (2) and Q appearing in (3) G-measure and Q-measure of p € ME+, respectively. We denote ME = ME+ * ME_. Theorem 3.2. ([29]) Let p+ e ME+ and p.- e ME_ and let p - p+ * /i_ e ME. Let G+ and G_ be mixing distributions of' p+ and p-, respectively. Denote d- = sup{y < 0 : G_([i>,0)) > 0} and d+ = ini{v > 0 : G+((0,i>]) > 0}.

161

lfd-< d+, then the Laplace transform £.\i{s) of \i exists for -d+ < s < -d_ and is represented as S

J(-oo,0) ;(-oo,o)

+

J(0,oc)

+

V

S+U

Theorem 3.3. Let p. 6 ME+ and let q(u)du be its Q-measure. Assume that C ^q(u)du = oo, equivalently, p is absolutely continuous w.r.t. Lebesgue measure. Let f be the density of p(dx). Then f is bounded if and only if £° ±(1 - q(u))du < oo. Proof. Let G be the G-measure of \i. Since f(x) is bounded iff J^ uG(du) < oo and since G(du) —» I uG(du) S+U Jo as s —» oo, we have that f(x) is bounded if and only if lim,,-^ s£.p(s) < oo. The quantity sXfvfc) is written as r

U

°°

i

i

(

r i )q{u)du +

i -du

u + s uny Jj u J =- I — zq(u)du + 1 -(1 - q(u))du J0 u(u + syy Ji uy r i r°° s 0

+ I

q(u)du - I

—;

-q(u)du.

The first and the second terms tend to T1 1 f °" 1 - I -q(u)du > -oo and I - ( 1 - q(u))du < oo, Jo u Ji u respectively, by monotone convergence theorem. The third term minus fourth term is equal to r1

i

r°° -q{su)du -

which is bounded in s. If J oo exists, liml I

I

—.

—q{su)du,

£(1 - q{u))du < oo, then since lim^oo s£.pi{s) < -q(su)du-

I

—

— q(su)du)

162

exists and finite. Hence, lim,,-^ SXJU(S) < oo. Converse is obvious.

•

Theorem 3.4. Let p. e ME+ and let q(u)du be its Q-measure. Assume that C° ^q(u)du < oo, equivalently, \i has a point mass exp(- j ^ \q{u)du) at the origin and is absolutely continuous w.r.t. Lebesgue measure on (0, oo). Let f be the density of the absolutely continuous part of [i(dx). Then lim^o f(x) = C° q(u)du exp(- L° ^q(u)du) and hence f is bounded if and only if J q(u)du <

Proof. Let G be the G-measure of p. j(Qm)ue-^G{du)dx, s{Lii(s)-G{[oo)))=

f

Since ji(dx) - G({oo|)60(dx) +

J^G(du)^

J(0,°o)

U + S

f J(0,oo)

uG(du) <)

as s —» oo. Hence we have lims(£/i(s)-G(M))=/(0+). The left hand side of the above equality is written as exp(- I J0

-q(u)du)[exp( I u J0

q(u)du) - l}s. u+s

By l'Hospital's rule,

r°° s 2

r°° i s ex

< P(

——q{u)du)-l\~ Jo

u + s

I Jo \u

r°°

s2q(u)du T + s ) Jo

as s —> oo.

q{u)du •

In order to solve some problems, it is necessary to extend the domain of the Laplace transform of an ME+ distribution to the complex number plane C. We denote by C+ the open upper half plane in C. A function analytic on C+ with positive imaginary part is called a Pick function ([8]). By P(a, b), we denote the subclass of Pick functions which admit an analytic continuation by reflection across the interval (a, b) into the lower half-plane. Pick functions are related to moment generating functions. In order to apply to Laplace transforms, we consider other classes of functions dual to Pick functions : Let P(a,b) = {4>(s)--4>(-s)£P(-b,-a)}.

163

Proposition 3.1. ([5], [8]) In order that a function h belongs to P(0,oo), it is necessary and sufficient that there are a e R, fS < 0 and a measure a on [0,oo) satisfying f (1 + M2)-1a(d«) < oo such that h(s) = a + tSs+ f ( — J[o,co) " + s

^-r)o(du). M2 + r

77ze measure a is called the spectral measure ofh. Since the imaginary part of the principal value Log h(s) (s e C+) for h e P(0,oo) belongs to [-7T,0], we have another representation (exponential representation). Proposition 3.2. ([5], [8]) In order that a nonnegative function h on (0, oo) not identically 0 belongs to P(0, oo), it is necessary and sufficient that there are ay e R and a measurable function q(u) satisfying 0 < q(u) < 1 a.s. such that h(s) = exp(y + I (— Jo u + s

^—)q(u)du\ w +1

fors € C+. This proposition shows that the Laplace transform of an ME+-distribution belongs to P(0, oo) with y = - j£°° u,J+1)q(u)du and a{du) = uG(du). Remark 3.1. For s e C + U (0, oo)

s+a = / e X P ( " f ( ^ - l)du

+1

°Sflt '/« > °'

The imaginary part of the quantity in the braces belongs to [0, n]. Lemma 3.1. Let p e ME+ with exponential representation (3). Then, for s e C+U(0,oo) (s

W

(s)

- |

exp{?/ +

^ ( _ j _ _ j_Mu)

_

l)du]

.f

a = Q

where y = - JT* uru2+i)q(u)du. Tte imaginary part of the quantity in the braces belongs to [-n, n\. It belongs to [0, n], in particular, if a - 0. Remark 3.2. Although the reciprocal j ^ ^ does not belong to the class ME+, it has a representation in Proposition 3.2 and hence j£rn$ belongs to P(0, oo).

164

Lemma 3.2. Let ji e ME+ and let G be its G-measure. Let a > 0. Then for s - -a + ib, ib£ji(s) -» aG(\a}) as b | 0. Proof. Write -£^(s) as „

..

aG([a\)

£u(s) = — ^ - ^ + s + a

C

1

„,, ,

uG(du).

J(0AMa,oo] S + U

Thus, ib£n(s) = aGHa))+ f

b +

*

lb

J\~£uG(du)

The second term in the right hand side of the above equality tends to 0 as biO, since \P-u[(u - a)2 + b2}'1 is bounded in b e (0,1], b2{(u -a)2 + b2\~l [ 0 as b { 0 if u ± a and \bu(u - a)((u - a)2 + b2}~x\ < \bu(u - a)~1\ A \u. u The following result is obtained by Aronszajn and Donoghue Jr. ([1]) under more general setting and hence the proof is complicated. The proof in our restricted setting is simplified in [28]. Theorem 3.5. ([2], [28]) Let n 6 ME+ and let G and q(u)du be its G-measure and Q-measure, respectively. Let a > 0. Then G({a)) > 0 if and only if there is an e > 0 such that (4)

I

^Z^f 1 ^ 0 0 )^") ~ <7(") P " < °°-

Moreover, G({fl)) = e x p | -

I

(^3^--)(1[«,OO)(«)-^(M)W|

Proof. By Lemma 3.1, we have *£m

= ex P {log* + J[ ^ua-~a*+£[«u)~^^

W

for s = -a + ib (a,b > 0). By Lemma 3.2, ib£ii(s) approaches to aG{{a\) as biO. Assume that G([a}) > 0. Then, Aig(ikC/i(s)) -> 0, log\ib£^i(s)\ -»log(«G({fl})) as b -> 0. Here, Arg denotes the principal value of the argument. Thus, C°° a{u-a)-b2

X

2 2 I u) ui(u-a) +b M

1 [ ( M \)

- ^ r -*logG(,fl})

165

and r°°

(5)

h

\

i

u) 1

{u) u

I {u-ar+A« - ™ r ^°

as b i 0. By the monotone covergence theorem, we have

Si

u[{u-aAb^{u)-l^){U¥

^^)(qiu)~i[a'x){u)yu-~o°-

^r Also, we have

Thus ^ r ^ ( ^ ( " ) - l[«,co)(u))rf« = -logG({a}) > -c Hence for 0 < e < a a+e

£ ^(w«>-*«))*'

< oo.

Conversely, assume (4) for some e > 0. Then, by the same argument as above, we have f°° a(u-a)-b2

=

(

\

(?(M)_1[ )(M) rf > 00

r^)( ^ ) " ~ -

Since b|M-fl|{|w-fl|2+b2}_1 < ±, we have (5) by the assumption (4). Therefore, G({o}) = exp J - J

( — - - -)(l[fl,oo)(M) - q(u))duj > 0.

n

Remark 3.3. Lemma 3.2 and Theorem 3.5 are valid for P(0, oo) function with o(du) = uG(du).

166

4.

Convolutions of exponential distributions Let CE{ =

{/I

€ P([0, oo)): £u(s) = fl ak(s +

aky\

k=\

form > 1 andfli,... ,am > 0} and let C E { be the mirror image of CE{. We denote by CE +/ CE_ and CE the closures in the weak convergence sense of CE{, CEl and CE{ * CEl, respectively (CE = Convolutions of Exponential distributions). Theorem 4.1. Let \i e P([0, oo)). Then, \i is a CE+ distribution if and only if ju 6 I and £fi(s) = exp | - y's + f (e~sx - l)n(x)dx\

=exp _/s+

{

{

^s-\)q{u){du)}

X

where, y' > 0, q(u) = £it l[flt;00)(w)/or u > 0 and n(x) = {x~x T.k e~"kX) far x > 0 with (0, oo)-valued finite or infinite sequence [a^] satisfying Lj: 0 ^ 1 < °°Theorem 4.2. Let p. e 'P(R). Then, p. is a CE distribution if and only if p e I and there is an R\{0}-valued finite or infinite non-decreasing sequence {a/tt such that •••
Ef

-2

< oo

and the Levy measure v of p. is represented as _ J (* Lk>o e~"kX)dx for x > 0, v(dx) = (\x\~1Lk
P61ya frequency functions

Definition 5.1. A function f(x) defined on (-co, oo) is said to be a PF r function (a Polya frequency function of order r ) or an r-times positivefunction if for M = 2 , 3 , . . . , r and for x\ < x% • • • < x„, y\ < y2 < • • • < y„,

det (/fe--y;)):=

f(xi ~ y\) f(xi - 1/2) • • • f{x\ - y„) f{x2 - yx) f(x2 - y2) • • • f(x2 - y„) f(xn ~ y\) f(x„ - y2) • • • f{x„ - y„)

>0.

167

Definition 5.2. A function f(x) is said to be a PF (or totally positive ) function if it is r-times positive for any r > 2. PF (or PFr) function is said to be a PF (or PFr) density if it is a probability density. The concept PF-function is considered by Polya et al. and PF r function is introduced by Schoenberg ([19]). PF and PF r densities have sign diminishing property. This fact is used in hypothesis test of number of modes using kernel density estimate ([14]). For an increasing sequence X\ < Xi < • • • < xn, v(x\, ••• ,xn) denotes the number of sign changes of the sequence. For a function / on R, we define w(/) = sup z?(/(*i),...,/(*„)) for all n and all choices of the sequence tatJJL^ Theorem 5.1. Let f be a PF density and let gbea probability density on R. Let h(x) = Jf(x - y)g(y)dy. Then v(h) < v(g). Theorem 5.2. Let fbea PF r density and let gbea probability density on R. Let h(x) = Jf(x - y)g{y)dy. Ifv(g) < r, then v(h) < v(g). Theorem 5.3. A necessary and sufficient condition that a probability density on R be PF is that its Laplace transform is represented as oo

2 2

(p(s) = expjys + a s /2) JJ (ifa + s)-V /fl ' where a1 > 0, y e R, a;- are real and 0 < a 2 /2 + Y,J=\ #72 < °°This theorem and Theorem 4.1 show that the class of PF densities coincides with the class of nondegenerate CE distributions. It is known that a density function is a PF2 density if and only if its logarithm is concave. Example 5.1. Normal densities and exponential densities are PF densities. Rather a nontrivial example is ^ enle-,ii ([11])- A gamma density with parameter (a,p), a > 1, is a PF 2 density. Total positivity is related to Riemann hypothesis. Let C(s) be Riemann's zeta function. Let £(s) = y(s - l)7T5T(±s)C(s) and let S(z) = E,(\ + iz). The function S is written as E(z) = 2 I O(M) cos luzdu, Jo

168

where oo

d>(u) = 4 £ ( 2 n V e 9 " - S^rar5"^-"2™4", n=l

(refer [22]). Riemann hypothesis is equivalent to that | $ | is the Laplace transform of a PF density (Thorin, private communication with Bondesson ([5] p.124), and [16] et al.). 6. Classes CME, CEME and CG Definition 6.1. We say that a probability measure \i on K is a CME distribution if n e I and its Levy measure v is absolutely continuous with density (. represented as e WQ{dU) Hu\Ak^ ~ y

f ry>Q yU

d

°

'

\l-o»,0)e- Q( »)f°ry<0>

where Q is a measure on R \ {0} satisfying (6)

f

\u\~l A \u\-3Q(dy) < oo.

JR\|OI

We denoted this class by B in [29]. This class on [0, oo) is called gcmed (generalized convolutions of mixtures of exponential distributions) in [5] et al. The class CME (= Convolutions of Mixtures of Exponential distributions) is closed under convolution and weak convergence ([29]). We define CEME by the class of convolutions of CE distributions and ME distributions. We denoted this class by CME in [29]. It seems that the name CME is more suitable to the above class in Definition 6.1. We say that a distribution p. belongs to class CG (Convolutions of gamma distributions) if it belongs to the class CME and the Q-measure Q is absolutely continuous and the density is nonincreasing on (-oo, 0) and nondecreasing on (0, oo). This class is called ggc in [5] et al. The class CG coincides with the weak closure of finite convolutions of gamma distributions ([5]). By the nondecreasingness of the density of the Q-measure of CG distribution, we see that every CG distribution is selfdecomposable. It is easy to see the inclusions. L u CE c CG c CEME c CME c I . u ME

169

Example 6.1. Every 1-dimensional stable distribution belongs to CG. Levy measure of an a-stable (0 < a < 2) distribution is written as follows: „, ^

(c + x- a ~ l = c+T(a + l)" 1 C° uae~xudu 1

Ic-lxr"" = c-T(a + I )

-1

a

xu

JH. \u\ e- du

for

x>0,

forx<0

where c+,c- > 0 and c+ + c~ > 0. The CG distribution on [0,oo) (= CG+ distribution ) has a remarkable representation resemble to (3). Theorem 6.1. ([5] p.49) Let p e CG+ with Laplace transform £p(s) = exp[

r 00

I I (—— - 7)q(u)du] Jo u + s u

where a > 0 is nondecreasing, 0 < limu_,oo q(u) = a < <x> and satisfies JQ M-1(j(«)dM < oo, then the density of p. is represented as f(x) = xLX'lh{x) where h is completely monotone. Proof. For the proof of this theorem, the following fact is essential: Let X and Y be random variables with gamma distributions with parameters (ai,/$) and (at2,f}), then (3^7, j^y) and X + Y are independent. If a CG-distributed random variable X is a sum of n independent gamma distributed random variables, then it can be represented as X = c\X\ + C2X2 + ••• + c„X„, where Xi,X2,..-,X n are independent and scale parameter 1 gamma distributed random variables and c\,cr,..., c„ > 0. Let Y = YH=\ ^itThen Y is (a,l)-gamma distributed and Y,Xi/Y,X2/Y,...,X„/Y are independent. Hence, YC1X1/YC2X2/Y,.. .,c„X„/Y are also independent. This shows that X/Y and Y are independent. Hence X is a mixture of gamma distributions with shape parameter a. Taking a limit, we obtain the theorem. • For CME on [0,00) (= CME+) distribution, we have a similar but slightly weaker result. Lemma 6.1. Let pi and p.2 be mixtures ofgamma distribution with shape parameters a and ji, respectively. Then the convolution U\ * p.2 is a mixture of gamma distributions with shape parameter a + /?.

170

Proof. It is enough to notice that, for x, a, fi, y, 5 > 0,

Jo = xa+f!-1 f «^- 1 (l-u)' , - 1 e- (6 " +(1 - I ' )J,)I dM.

D

Jo Theorem 6.2. ([5]) Let p e CME+ with Laplace transform r

i i (—— - -)q(u)du] Jo u + s u where q > 0, 0 < ess sup u > 0 q(u) = a < oo and satisfies JQ wlq(u)du < oo, then the density f of \i is represented as £p(s) = exp[

J

f-tOC

Ae~AxG(dA)

o where n is the integer satisfying n — \ < a < n and G is a measure on (0, oo) satisfying I Jo

(8)

Ax-nG{A) < T(n)-1.

Moreover, (9)

Ae~AxG{dA) <

I Jo

x-nnne-nT(nY1.

Proof. The probability measure JJ is represented as a convolution of n ME+ distributions. Hence by Lemma 6.1, we have (7). Since > I Jo

f(x)dx

= f if x^e-^dx^AGidA) Al-"G{dA),

= T(n) Jo n

we have (8). Since A e~ (8).

Ax

< x~nnne~n for x > 0 and A > 0, we have (9) by D

In the above theorem, n can not be replaced by a. A counter example is seen in [5]. A fact similar to Theorem 3.3 holds for CME+ distributions as follows.

171

Theorem 6.3. Let p 6 CME+ with Laplace transform r £p(s) = exp[

i i (—— - -)q(u)du] U+S

J0

U

where 0 < q(u) < a for a > 1 and for all large u > 0, limu_,oo q(u) - a, and satisfies fQ u~lq(u)du < oo. Let f be the density of p . Then x1~af(x) is bounded if and only if r°°

I J\

I

-(a - q(u))du < oo. u

Ifq(u) < a u-a.e., then xiO

J0

U(U + 1)

Especially, if p. e G C + , then the right-hand side of the above equality is represented as (11)

nay'expll

(logu)q(du)]. Jo

Proof. Let q\{u) - q(u) Aa and let ^i be a CME distribution with Q-measure qi(u)du. Since a>0, pi has a density g. The quantity sa£pi(s) is written as sa£p\(s)

= exp I ( Uo " +s =

)qi(u)du+ «

-luJuTsjqi{u)du cs

+ I

+

i

r°°

qi(u)du-

I

I -du Ji u \ Pu{a-^u))du s

—

-qi(u)du.

The first and the second terms tend to f1 1

C°° 1

- f u-q\(u)du > -oo and I u- ( a -q\(u))du < oo, Jo Ji respectively, by monotone convergence theorem. The third term minus fourth term is written as

which converges to I J0

T^u - I — —du, W+l J i U(U + 1)

172

as s —* oo by the dominated convergence theorem. By Theorem 6.2, g(x) = Oix'1) as x —> oo. Hence x1~ag(x) is bounded for large x. Note that lim s _ 00 s".£jii(s) exists allowing infinity. Hence, f^° \{a - q\{u))du < oo if and only if hm sa£ui(s) = expj s^°°

— Jo

du) < oo.

H(« + 1 )

Assume that ^°° i(a-(j(«))dw < oo, equivalently, Jj°° \(a-q\(u))du < oo. By Tauberian theorem, we have pi([0, x]) ~ ^x" as x J, 0 where C is the right hand side of (10). We have, by L'hospital's rule, g(x) ~ Cx a_1 as x I 0. This shows (10). We also have that x1_"g(x) is bounded. The CME distribution pi with Q-measure q(u) - q\(u) is written as pdo(dx) + (1 - p)h(x)x. Since a > 1, we have x ^ / W = pxl-«g{x) + (1 - p) f x ^ x - y)/i(y)dy Jo l

< px -"g(x) + (1 - p) f(x - y ) 1 " ^ - y)h{y)dy Jo < SUpX1_
Conversely, assume that xl af(x) is bounded. Since pi has a point mass at the origin, x1~ag(x) is bounded near the origin. Hence xl~"g(x) is bounded. Then s"£pi(s) is bounded. Hence we have JJ £(a - q\{u))du < oo, equivalently, J \(a - q(u))du < oo. (11) is straightforward. • 7.

Unimodality

Definition 7.1. A probability measure p. on R is said to be unimodal if there is a e R such that the distribution function of p is convex on (-oo, a) and concave on (a, oo). Definition 7.2. A probability measure p on R is said to be strongly unimodal if for every unimodal distribution p, the convolution p* pis again unimodal. Theorem 7.1. ([10]) A probability measure ponR is strongly unimodal if and only if it is absolutely continuous with logarithmic concave density (PF2- density). Theorem 7.2. Every CEME distribution is unimodal.

173

Proof. ME-distributions are unimodal with mode 0. CE-distributions are strongly unimodal. Hence CEME-distributions are unimodal. • It is easy to see that every stable distribution is selfdecomposable. Every selfdecomposable distribution is unimodal ([25]). Hence every stable distribution is unimodal. The proof of unimodality of selfdecomposable distributions is not simple. But, since we know that stable distributions are CG-distributions, the proof of unimodality of stable distributions is quite simple as follows. Theorem 7.3. ([26]) Every 1-dimensional stable distribution is unimodal. Proof. 1-dimensional stable distributions belongs to CG and CG is a subclass of CEME. Hence 1-dimensional stable distributions are unimodal by Theorem 7.2. • 8.

Boundedness of transition densities of CME + processes Let {X{t)\ be a Levy process. If the distribution of X(l) is a CME+ distribution, then we call (X(f)} a CME+ process. Assume that, in this section, the distribution of X(l) is absolutely continuous. Namely, J[j \Q{du) = oo for the Q-measure of the distribution of X(l). We consider in this section, under what condition the transition density of (X(f)} is bounded in the space variable x and the supremum in x tends to 0 as time variable t tends to oo. The following result is shown in [24]. Theorem 8.1. Let (X(f)| be a CME+ process with transition density p(t, x). Then for any 0 < to < t\ and XQ > 0, sup

p(t, x) < oo.

1 0

for x e [a, b], torxe[a,b]c-

1. LetO
q(x) =

Let ii € CME+ with Q-measure q(u)du. Then we have ^([0,*]) = p + q(l e~ax) for x > 0 where p = g and q = 1 - p. Hence pn\dx) = pn50(dx) + p„ (x)dx where k k

ka

p«w = £ ( j ) r v k=l

x

'

x

1

e

(k-iy.

v

'

174

Letfk(x) = ^ ^ r - Then, by Stirling formula T(z) ~ V ^ z 2 - 1 ^ - 2 , (z -> oo), l

sup/^W =

= Oik-1'2)ask^

*_*,

co.

The quantity sup^p^x) can be regarded as an integral of supxfk(x) with respect to a binomial distribution {(l)pn~kqk)l-Q- The binomial distribution tends to d^dx) as «—> oo. Hence lim supp„(x) = 0. n-.cc

x

distribution correspond< 2. Let Q(dx) = 5,,(dx) (a > 0) and let /i be a CME+ing to Q. Then /i**(dx) = e- ,/fl 5 0 (^) + pt(x)dx where anxn-\e-ax

(n-1)! •

M—1

In this case, we also have sup pt(x) —> 0 as t —» oo Theorem 8.2. Let ju e CME+. Let -M-l1

? l W

"\0

forxe[a,b], forxe[a,b]c

as.

0 < (fcM < 1fl-s.,JJ°° ^qz(x)dx = 00 anrf f°° ^(l-az{x))dx < 00 whereO
f - » °' °0<X
ax

Proof. Let 1
00

175

where pz is a probability measure corresponding to Qi(dx). Let p"' be a probability measure corresponding to nqi(x)dx. Then for p = | , pn{{dx) = p"50(dx) + p„(x)dx, and supp„(x) —> 0 as n —» 00, where p„(x) = Z'U (lV~kqksi$ff(t,x)

Density f(t,x) of ^(* satisfies

< png(t,x) +sup p„(x) x

< pn suph(x) + sup p„(x) X

-» 0

X

a s « -> 00.

n

Theorem 8.3. Let ^ 6 CME+. Let 0 < g2(x) < 1 a.s., I Ji

-qz{x)dx - 00 and I * Ji

- ( 1 - q2(x))dx < 00.

x

If the Q-measure of p is given by Q(dx) = 5a(dx) + q2(x)dx + Q?,{dx) where a > 0, then dp.1* lim sup —r—(x) = 0. f ^°°o 0. The following Example 8.1 exhibits such an example. Example 8.1. Let p. e ME+ with G-measure G. Assume that G({oo}) = 0 and L uG(du) ~ ar(log^) -2 as x —> 00. Then p is absolutely continuous with respect to Lebesgue measure and p has a density r°°

/(*):=

Hence Jx that

1

ue-uxG(du) ~x-\log-)-2

as x 10. x Jo j ( l - q(u))du < 00 by Theorem 3.3. Integration by parts yields F(x) := \ f(u)du ~ ( - log*) - 1 as x | 0. Jo

176 By Abelian-Tauberian theorem for Laplace transform, this asymptotic relation is equivalent to that

/

\ e~sxF(dx) ~ (- log - ) _ 1 as s —»oo.

Then, (12)

( f

e-sxF(dx)) ~ (logs)"' as s -» oo.

This is equivalent to that F"(x)~(-lo$xytasxiO,

(13)

by Abelian-Tauberian theorem. Hence f*(x) is unbounded in x > 0 for each t > 0. If f'(x) is monotone for all small x, then f*{x) ~ ^t(— log*) - ' - 1 as x I 0. Let

ex [ (

p l ; r b - i)(?(w)rfM]

be the Laplace transform of \i. Then r°° 1 fl(«) , fs I du ~ flog logs ass —> oo J0 M + S U by (12). By Abelian-Tauberian theorem for Stieltjes transform ([4]), this asymptotic relation is equivalent to that t I Jo

u

du ~ flog log xasx —> oo.

If ^ - is monotone for all large u, then q(u) ~ ^-^ as u —> oo. Conversely, assume that [ du ~ log log x as x —> oo. Jo u Then we have (13) by Abelian-Tauberian theorems for Stieltjes transform and Laplace transform. 9. 1-dimensional generalized diffusion processes Let [B(t)} be a one-dimensional Brownian motion and let t{t,x) be its local time. We denote by M the class of right continuous nondecreasing function m on [-00,00] to [-00,00] with m(±oo) = ±00 and m(0—) = 0. For m e At, we define ij = (j(m) by (-l)kj

= inf{(-l)>';r > 0 : {-\)>m{x) = 00}

177 for;' = 1,2 and we define a measure m(dx) on [-00,00] by m(dx)

= dm(x) on

m({£j})

=00

for

((\, £2), j = l,2.

Here [f\, (2Y is the complement of [€\, (2Y Let (0 = f e(t,x)m(dx). JR

Define a stochastic process {X(f), CI by X(f) = B(0_1(O) and the life time C = inf{f > 0 : X(0 = h or €2] if {f * 0, = 00 otherwise. This process is a strong Markov process with state space Em = (supp m)\^x/2) and is called the generalized diffusion process corresponding to the function m (see [12]). The measure restricted to (€i,h) is called the speed measure of the process (X(t)|. For y € Em, we define the hitting time of y by Ty = infff > 0 : X(t) = y\if (} * 0, = 00 otherwise. If \tj\ < 00 and (j e Em, where Em is the closure of Em in R, then we define T(. by y by limy_,T, ry for / = 1,2, respectively. We denote by Em the set with €j(j = 1,2) adjoined to Em whenever \tj\ < 00 and (j e Em. UPX(TV

< 00) > 0

for x in Em and y in Em, we define Hxy(dt) = PX(TV

6 df|Ty < 00).

We denote by Hgd = {pxyidy) :x&EmiyeEm,x±y,m&

M\,

the class of conditionnal hitting time distributions of generalized diffusion processes. In [27] (Theorem 1), the following characterization is obtained. Proposition 9.1. In order that a probability measure p on TR+ belongs to Hgtt, it is necessary and sufficient that there are a CE+ distribution \i\ with Laplace transform £.^i\(s) = Yl £s and an ME+ distribution p.2 with ^2([0|) = 0 such that H = p-i * P-2 and (a,) is either empty or a strictly increasing (finite or infinite) sequence and the spectral measure a o/(sXjU2(s))_1 has a positive point mass at at for each i.

178

This proposition shows that Hgd is a proper subclass of CEME+. We can restate this result in terms of Q-measure using Theorem 3.5 in Section 3. Theorem 9.1. ([28]) In order that a probability measure p. on K+ belongs to H ^ , it is necessary and sufficient that its Laplace transform is represented as r

i i (— )(<ji(u) + q2(u))du, Jo u +s u where q\ and q2 satisfy the following conditions: 1. (a)qi = 0 or (b) q\ is a non-decreasing step function with step size l,qi(0) = 0 and jump points {aj) of q\ satisfy £,• a~l < oo (hence, 0 is not a jump point). £p(s) = exp

2. 0 < q2(u) < 1, j0 \q2(u)du < oo, J[°° ±q2(u)du = oo. 3. In case (b) in 1, q2 satisfies

£

i+ei

. -e,

1 U

(q2(u) - l{0Mj)(u))du < oo

a

i

for [£j\ with aj > e;- > 0. Proof. Apply Theorem 3.5 to the reciprocal of the expression of sX^(s) in Lemma 3.1. D This theorem shows that every stable distribution ju with Laplace transform

^(S) = 6 X P I

{

v^s-\)cuadu

belongs to Hjrf. Here, c > 0 and 0 < a < 1. \-stable (a - \) distribution is the hitting time distribution of Brownian motion. It is not known what kind of generalized diffusions correspond to other stable distributions. Gamma distribution does not belong to Hgj if and only if the shape parameter a is greater than 1. 10.

Levy processes appearing in mathematical finance Recently, various types of Levy processes, namely VG (Variance Gamma), NIG (Normal Inverse Gaussian), GIG (Generalized Inverse Gaussian), GH (Generalized Hyperbolic) processes often appear in Mathematical Finance literature as a model of stock price. Also, stationary processes of Ornstein Uhlenbeck type are used as volatility processes in stochastic volatility models (refer [3] and references therein). The class of stationary distributions of the statinary one-dimensional processes of Ornstein Uhlenbeck type coincides with the class of selfdecomposable distributions on

179

1R ([18]). We show that the above classes (VG, NIG, GIG, GH) of processes (or distributions) belongs to the class CG or the class of selfdecomposable distributions. Let p(x; p, 6) be the density of the positive p/2-s table distribution with Laplace transform e-sxp(x;p,5)dx = e-6W2. Jo Boyarchenko and Levendorskii ([7]) called a probability measure on [0, oo) with density p(x;p,5,y) = e6y,'p(x;p,5)e-Wx, y > 0 a Tempered Stable distribution and denoted TS(p,5,y) ([7]). Its Laplace transform is written as e*(s>, where
and the Levy density is given by 6p2P'2-1

/2-1

i,2x

They ([6]) called an infinitely divisible distribution p n E a KoBoL distribution of order p < 2 if it is infinitely divisible with the Levy measure v(dx) = M - P - M c - e ^ l , ^ * ) + c+e-A*xllI>0)(x)}dx where c± > 0 and A± > 0. This shows that the Tempered Stable distribution is a one sided KoBoL distribution of order p/2. KoBoL distribution is a CG distribution with the density of its Q-measure q(u) = {T(p + l j r ^ c - l w + A_|''1, H< _ A .,(M) + c+(u - A+)f'l|„>A+l(")}-

KoBoL distribution with p = 0 is called VG distribution. The use of VG distribution in finance is proposed by Madan and Seneta [13]. In [15], KoBoL distribution is called tilted stable distribution and the name "tempered stable" is used for an infinitely divisible distribution with Levy measure Ixr^Mlf-^o)^) I

e-xuq(u)du + 1(0,M)W I

e~xuq(u)du\lx,

where a e (0,2) and the measure Q(du):- q(u)du on R\{0} satisfies (6). The meaning of "tilted" is explained in [3].

180

We denote by K\ the modified Bessel function of the third kind with index A. Let Yt = fit + Bt where (Bf) is a Brownian motion. Let {Zt} be a subordinator generated by TS(p,5, y)-distribution independent of {Yt}. Then the characteristic function of the subordination Xt = Yz, is of the form e^ (z) , where (j)(z) = (|z2 - ipz). It is rewritten as (14)

4>(z) = 5[(a2 - fY12

- {a2 - (jS + iz)2)f/2]

where a = ^y2 + /32. Since the 1-dimensional distributions of a subordinated process of a Brownian motion with drift by a selfdecomposable subordinator is selfdecomposable ([17]), the distribution of Xt is selfdecomposable for each t > 0. Adding ipz to (14) and then letting p = 1, we get the NIG distribution with characteristic function exp (ifjz + 5[(a2 - 0 2 ) 1/2 - (a2 - (0 + iz) 2 ) 1/2 ]l The transition density is given by

where = (1 + lyl2)1'2. A probability measure on (0, oo) is said to be a GIG distribution if it has a density (y/S) A

2KA(yS)

x A _ 1 e x p ( - i ( 6 2 / ; c + y 2 *))

where the parameters satisfy b > 0, y > 0 for A < 0, 6 > 0, y > 0 for A = 0, 6 > 0, y > 0 for A > 0. Halgreen [9] and, Shanbhag and Sreehari [20] showed that GIG distribution is a CG distribution. A probability measure with characteristic function

\a2-(p

+ iz)2)

KA(6 V alpha2 - p)

is called GH distribution. It is absolutely continuous with respect to Lebesgue measure and the density is represented by KA-I/2- GH distributions are obtained by the subordination of Brownian motion with drift

181

b y GIG subordinator. They ([9], [20]) p r o v e d that G H distributions are selfdecomposable b y s h o w i n g that the one dimensional distributions of a subordinated process of Brownian motion w i t h drift b y CG subordinator are selfdecomposable. Sato's result ([17]) is its extension. We remark that NIG distribution is a G H distribution w i t h A = -\. References 1. N. Aronszajn and W. F. Donoghue Jr., On exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math., 5 (1956), 321-388. 2. N. Aronszajn and W. F. Donoghue Jr., A supplement to the paper on exponential representations of analytic functions in the upper half-plane with positive imaginary part, J. Analyse Math., 12 (1964), 113-127. 3. O. E. Barndorff-Nielsen and N. Shephard, Modelling by Levy processes for financial econometrics, in Levy Processes Theory and Applications, Birkhauser (2001), 283-318. 4. N. H. Bingham, C. M. Goldie and J. L. Teugels, "Regular Variation", Cambridge University Press (1987), Cambridge. 5. L. Bondesson, Generalized gamma convolutions and related classes of distributions and densities, Lecture Notes in Statistics, 76 (1992) Springer-Verlag, New York. 6. S. Boyarchenko and S. Levendorskii, Perpetual American options under Levy processes, SIAM J. Corral Optim. 40 (2002), 1663-1696. 7. S. Boyarchenko and S. Levendorskii, "Non-Gaussian Merton-Black-Scholes Theory", Advanced series of statistical science & applied probability Vol. 9, World Scientific (2002), New Jersey-London-Singapore-Hong Kong. 8. W. F. Donoghue, Jr., "Monotone matrix functions and analytic continuation", Springer 1974, Berlin Heidelberg New York. 9. C. Halgreen, Self-decomposability of the generalized inverse Gaussian and hyperbolic distributions, Z. Wahrsch. Verw. Gebiete 47 (1979) 13-17. 10. I. A. Ibragimov, On the composition of unimodal distributions, Theor. Probability Appl. 1 (1956) 255-260. 11. S. Karlin, "Total Positivity", Vol. 1, Stanford Univ. (1968), Stanford. 12. S. Kotani and S. Watanabe, Krein's spectral theory of strings and generalized diffusion processes, Functional Analysis in Markov Processes (M. Fukushima, ed.), Lecture Notes in Mathematics, 923 (1982), 235-259, Springer, Berlin Heidelberg New York. 13. D. B. Madan and E. Seneta, The VG model for share market returns, J. Business 63 (1990), 511-524. 14. E. Mammen, J. S. Marron and N. I. Fisher, Some asymptotics for multimodality tests based on kernel density estimates. Probab. Theory Relat. Fields 91 (1992), 115-132. 15. J. Rosinski, Tempered stable processes, Mini-proceedings: 2nd MaPhysto Conference on Levy Processes Theory and Applications (2002), Aarhus University. 16. B. Roynette et M. Yor, Couples de Wald indefiniment divisibles Examples lies

182

17. 18. 19. 20. 21. 22. 23.

24. 25. 26. 27. 28.

29.

a la fonction gamma d'Euler et a la fonction zeta de Riemann, To appear in Ann. Inst. Fourier. K. Sato, Subordination and selfdecomposability, Statistics & Probability Letters 54 (2001) 317-324. K. Sato and M. Yamazato, Stationary processes of Ornstein-Uhlenbeck type, Lecture Notes in Math., 1021 Springer (1983) 541-551. I. J. Schoenberg, On Polya frequency functions I. The totally positive functions and their Laplace transforms, Journal d'Analyse Math., 1 (1951), 331-374. D. N. Shanbhag and M. Sreehari, An extension of Goldie's result and further results in infinite divisibility, Z. Wahrsch. Verw. Gebiete 47 (1979) 19-25. F. W. Steutel, "Preservation of infinite divisibility under mixing and related topics", Mathematical Center Tracts 33 (1970), Matematisch Centrum, Amsterdam. E. C. Titchmarsh, "The zeta-function of Riemann", Hafner (1972), New York. N. Tsilevich, A. Vershik and M. Yor, Distinguished properties of the gamma process, and related topics, Pre-publication du Laboratoire de Probabilites et Modeles Aleatoires No. 575 (2000). S. Watanabe, K. Yano and K. Yano, A density formula for the law of time spent on the positive side of one-dimensional diffusion processes, preprint. M. Yamazato, Unimodality of infinitely divisible distribution functions of class L. Ann. Probability 6 (1978), 523-531. M. Yamazato, On strongly unimodal infinitely divisible distributions, Ann. Probability 10 (1982), 589-601. M. Yamazato, Hitting time distributions of single points for 1-dimensional generalized diffusion processes, Nagoya Math. J. 119 (1990), 143-172. M. Yamazato, Characterization of the class of hitting time distributions of 1-dimensional generalized diffusion processes, Proc. 6th Japan-USSR Symposium on Probability Theory and Mathematical Statistics, (1992) 422-428, World Scientific, Singapore-New Jersey-London-Hong Kong. M. Yamazato, On subclasses of infinitely divisible distributions on R related to hitting time distributions of 1-dimensional generalized diffusion processes, Nagoya Math. J. 127 (1992), 175-200.

On Stochastic Differential Equations Driven by Symmetric Stable Processes of Index a Hiroya Hashimoto 1 , Takahiro Tsuchiya2 and Toshio Yamada3

23

'Sanwa Kagaku Kenkyusho Co.,LTD, Department of Mathematical Sciences, Ritsumeikan University

1.

Introduction In the first part of the present paper, famous Tanaka's equation is discussed in the case of symmetric stable processes. Then some important uniqueness results in one-dimensional case will be reviewed. The second part is devoted to obtain some results concerning comparison problems using Lamperti's method. Marcus integral plays an essential role to formulate comparison theorems. In the last part, a sufficient condition which guarantees the pathwise uniqueness in d-dimensional case, will be proposed. 2.

On uniqueness problems: One dimensional case We consider following stochastic differential equations driven by a symmetric stable process of index a. (1)

dXt = a{XtJ)dZt

(2)

dXt = a(t,Xt-)dZt

where Zt is a symmetric stable process of index a of which characteristic function is given by E[etiZ']=e-M't,

l
As is well known, famous Tanaka's example shows that the weak uniqueness does not imply the pathwise uniqueness in the case of SDE driven by a Brownian motion. We will mention that Tanaka type argument is still applicable to the case of the equation with respect to a symmetric stable process of index a. Theorem 2.1. Consider the equation (3)

X t = f sgn(X s _)dZ s , Jo 183

184

where SgnW =

\\, (-l,

ifx > 0 ifx<0

Then, the weak uniqueness holds for solutions to (3), but the pathwise uniqueness fails. To prove Theorem 2.1, the following theorem by Rosiriski and Woyczynski [11] plays essential roles. Theorem 2.2. Let F be an ft := a(Zs; s oo,a.s., when u —» oo.

Consider the inverse O/T and Qt: T~\t) = inf(« : T(U) >t\

Qt = 7v.(o

Then the time changed stochastic integral (4)

Z,=

FsdZs Jo is a Qtadapted symmetric stable process of index a. x We have also,

f

(5)

|

Fs dZs = Z T( (),

a.s.

Jo

[Proof of the Theorem 2.1] Let Xt be a solution to (3). Then, we observe that

(«):= f

T(W) := J | sgn(Xf-) \a dt = u —> oo,as u —> oo. Jo Jo Then by the Theorem 2.2, Xt is a symmetric stable process of index a with respect to TT-\(t) = Ti • So, any solution to (3) has the same law. Then the solution to the equation (3) is unique in the weak sense. We will observe that the weak existence of a solution to (3). Let Xt be a symmetric stable process of index a with X0 = 0. Then, Zt=

f sgn(Xs_)dXs Jo

185 is also a symmetric stable of index a and Xt satisfies Xt = f sgn(Xs_)sgn(Xs_)dXs = f sgn(Xs_)rfZs Jo Jo This means the existence of a solution to (3) in the weak sense. Now, let Xt be a solution to the equation (3): Xitt== |f sgn(Xs_) _)rfZs. Jo Then (-X f ) satisfies -Xt=

f sgn(-Xs_)<2Zs Jo So, (-Xt) is also a solution to (3). The pathwise uniqueness fails for the solution to (3). • When is the solution to (1) or (2) is pathwise unique? If the coefficients are Lipschitz continuous, the Picard iteration method works very well and it proves the pathwise uniqueness for the solution to (1) or (2). In one dimensional case much weaker conditions suffice for uniqueness for SDE's with respect to one dimensional Brownian motion. For example (see [12]), a sufficient condition for pathwise uniqueness is that f p~2(u) du = oo where p is the modulus of continuity: I a(x) - o(y) \< p(\ x - y |). In view of the above result, one would hope that analogous weaker conditions would suffice for the pathwise uniqueness to the solution to (1) or (2). The following condition is due to Komatsu [7]. (See also Bass [2]). Theorem 2.3. Suppose that \o(x)-a(y)\a
Vx,VyeR\

where the modulus of continuity p is a continuous increasing function with p(0) = 0 such that p x(x)dx-

oo, V e > 0 . Jo Then, for all XQ € R1 the solution to the equation dXt = a(Xt-)dlt, is pathwise unique.

Xo = x0,

186

Concerning to the solution to the equation (2) of which coefficient is time inhomogeneous, following Nagumo type modification to the condition in the above theorem is proposed in [6]. Theorem 2.4. Suppose that |g(f,s)-a(t,y)|"
Vx,\/yeR\

Vfe[0,oo),

where the function p is continuous increasing with p(0) = 0 such that I p-\x)dx

= oo,

Ve>0,

and where the function h(t) is non-negative continuous such that I fr 1 (s)ds
Vte[0,oo).

Then for all x0 e R1, the solution to the equation dXt = a{t,Xt-)dZt,

Xo = x0

is pathwise unique. Remark 2.1. The conditions given in the Theorem 2.2 and Theorem 2.3 are sharp and best possible in some sense. 3.

Lamperti's method Given b(x) satisfying the Lipschitz condition: Vx.VyeR1,

\b(x)-b(y)\<M\x-y\

the equation driven by an one dimensional Brownian motion: (6)

Xt=x0

+ Bt+ I

b(Xs)ds

Jo

can be solved very simply using the following inequalities Dn{t) := max | xf +1) - X<"> |< f |ft(X
< M [ D„. Jo

-\(s)ds,

187

where X<0) = x0 and X<"> = x0 + B, + j£ KXt^ds. Now we make a change of scale x —> y = f(x) with / e C2(RX). Let Yt be Y[ = / ( X t ) . Then Ito's formula implies (7)

Y, = Y0 + f a(Ys)ds + f 5(Ys)ds Jo

Jo

where, (a) < ? ( / ) = / ' (b) b(f) = f'b + f"/2. Lamperti's idea is to construct the solution to the equation (7) by solving (a) and (b) for / and b. Standard Picard's iteration procedure applies to a wider class of coefficients, but Lamperti's method is simpler, because it uses neither the martingale inequality nor Borel-Cantelli lemma. (See Lamperti [9] and also McKean [10].) Just as in the same way as in one dimensional Brownian motion case, the following equation concerning a symmetric stable process of index a can be solved: Xt = X0 + Zt+

(8)

\ b(Xs)ds. Jo

Make a change of scale Yt - f(Xt) with increasing / e C^R 1 ). Then in the language of Marcus integral Yt satisfies (9)

Yf = Y 0 + f

f'(Xs-)0dXs,

Jo

or (10)

Yt = Y0 + f f'(f-\Ys-))0dZs Jo

+ f Jo

f'{f-\Ys-))b{f-\Ys-))ds,

where <> means the Marcus integral. Remark 3.1. For the precise definition of Marcus integral, see KurtzPardoux-Protter [8] and also Applebaum [1]. In the case where

f

Xf = X0 + Z ( + I Jo

b(Xs)ds,

188

Marcus integral: f (p(Xs-)OdXs = f (p(Xs-)OdZs+ f

+ £ 0<s
w h e r e <£>' =
4.

Some comparison results In the present section Zt stands for a symmetric stable process of index

a. In view of comparison results for solutions to SDE's concerning a one dimensional Brownian motion, one would hope that that solutions to SDE's driven by a symmetric stable process of index a: dXf = a(Xt-) dZi + bi(Xt-) dt,

x{f = *,-, i = 1,2,

enjoy comparison properties under the conditions such that x\ < x2, b\ < b2 and a satisfies the same condition as in Theorem 2.3. But, unfortunately the next example would suggest that in our case, comparison problems for solutions would take much more complicated aspects. Example 4.1. Consider the following two equations: (11)

Xf = Xj+ f X® dZs,

i = 1,2.

Jo

Then the solutions to (11) are given by Dolean-Dade as follows; (12)

Xf = Xi exp(Z < )[[]( 1 + AZ *) exp(-AZ s )]. s<(

Consider the following series of random times T„, n = 0,1,2,..., such that to = 0 , TI = infjs; AZS < -1},- • -T„ = inf{s;s > T„_I, AZS < -1}.

189

Under the condition x\ <xi, we observe that x f > < Xf : T2„ < t < T2n+i, n = 0,1,2,... XJ1' > Xf> : T 2 „ + 1 < t < T2„+2, n = 0,1,2,.... In the followings of the section, we will apply Lamperti's method to obtain some comparison results. Consider the following SDE's: (13)

X<° = X<° +Zt+

f bj(Xs)ds, j = 1,2 Jo

where b\ satisfies the Lipschitz condition such that \bi(x)-bi(y)\
V^VyeR1

As we have seen in the section 3, solutions to (13) can be constructed by simple way. We have two comparison lemmas. Lemma 4.1. (Weak comparison lemma) Suppose that for the equations (13) (i) X® < X<2) fl.s., and (ii) h(x)
VxeR 1 .

Then, P(X| 1 ) <X) 2 ) , f > 0 ) = l holds. Lemma 4.2. (Strong comparison lemma) Suppose that for the equations (13) (i) X ^ < X<2) a.s., and (ii) h(x)
VxeR 1 .

Then, P(X| 1 ) <Xp ) , t > 0) = 1 holds.

190

Let feC^R1) with/'>0. Make a change of scale Y, = f(X\). Marcus equation:

rf = Yf+ ff
(14)

Then Yy satisfies following

f f(f-HY?-)Mf-HYf_))ds Jo

= f a(Y(j)_)0dZs+ f Gi(Y®.)ds, Jo Jo

with (a) d(f)=f

and

(b) 6i(f) = f'k Suppose that the coefficients a and b\ are given such that the equations (a) and (b) can be solved for / e C3(R1) / ' > 0 and &,-. Then we obtain following comparison results for solutions to Marcus equations (14). Theorem 4.1. (Comparison theorem for Marcus equations) Suppose that h(x) < b2(x), Vx 6 R\

and Y^ < Y<jVs.,

Then P(Yj 1)
and Yf < Yfa.s.,

Then P(Y[ 1 )
191

5. Pathwise uniqueness: d-dimensional case Let Zj be a d-dimensional symmetric stable process of index a, (1 < a<2): E[exp((6Z ( ))] = e x p H k n Consider the following stochastic differential equation driven by Zt: (15)

Y t = Y 0 + f d(V)rfZ s , Jo

or rf

(16)

Y " = Y" + > Jo ,

ti

a,*(Ys_) dZf,

i=\

d.

Assumption 5.1. Assume that the coefficient matrix a = [ait] satisfies Oik(x) = 5ika(x), x e Rd Under the Assumption 5.1, we have the following theorem. Theorem 5.1. Suppose that the continuous function p defined on [0, oo) with p(0) = 0 is increasing and such that: G{u) := is concave and

p"{u^)u^

6

Hi, rr du Jo G(u)) Let p be the modulus of continuity:

oo / Ve>0.

| a ( x ) - a ( y ) | < p ( | x - y | ) , Vx,VyeR d . Then for every y0 e Rd, the solution to Yf = Y<;) + f a(Ys_)dZf, Jo is pathwise unique.

i = \,...,d.

Y0 = y 0

Remark 5.1. For examples, the functions such that 1

p(u) = u,

p(u) = u((a - 1) log - ) « , w p(u) = u((a - l)log -)«(log(a - 1) log - ) - , . . . ,

satisfy the conditions imposed on p in the theorem 5.2. Concerning the equation driven b y a d-dimensional Brownian motion: Xf' ) =X<' ) +

f o(Xs)dB?, Jo

i= l

d,

the functions such that p{u) = u,

p(u) = M(log-)5,

p(«) = M(log - ) 5 ( l 0 g l 0 g - ) 5 , . imply the p a t h w i s e uniqueness. (See [13].)

References 1. D. Applebaum. Levy Processes and Stochastic Calculus. Cambridge Univ. Press (2004). 2. R. F. Bass. Stochastic differential equations driven by symmetric stable processes. Seminaire de Probability XXXVI (2003), 302-313. 3. R. F. Bass, K. Burdzy and Z. Q. Chen. Stochastic differential equations driven by stable processes for which pathwise uniqueness fails, Stock. Proc. their Appl. vol. I l l (2004), 1-15. 4. R. F. Bass. Stochastic differential equations with jumps. Probability Surveys, vol. 1 (2004), 1-19. 5. C. Doleans-Dade. Quelques applications de la formule de changement de variable pour les semimartingales. Z. Wahr. vol. 16 (1970), 181-194. 6. H. Hashimoto. On stochastic differential equations driven by symmetric stable processes—uniqueness and comparison problems (in Japanese), Master thesis (2004), Ritsumeikan University. 7. T. Komatsu. On the pathwise uniqueness of solutions of one-dimensional stochastic differential equations of jump type. Proc. Japan Acad., vol. 58, Ser. A, no. 8 (1982), 353-356. 8. T. G. Kurtz, E.Pardoux and P. Protter. Storatonovich stochastic differential equations driven by general semimartingales. Ann. Inst. H. Poicare. vol. 31, no. 2 (1995), 351-377. 9. J. Lamperti. A simple construction of certain diffusion processes. Jour. Math. Kyoto Univ., vol. 4 (1964), 161-170. 10. H. P. McKean, Jr. Stochastic Integrals. Academic Press (1969). 11. J. Rosinski and W. A. Woyczynski. On Ito stochastic integration with respect to p-stable motion: Inner clock, integrability of sample paths,double and multiple integrals. Ann. Prob., vol. 14 (1986), 271-286.

193 12. T. Tsuchiya, On the pathwise uniqueness of solutions of stochastic differential equations driven by multi-dimensional symmetric a stable class, submitted. 13. T. Yamada and S. Watanabe. On the uniqueness of solutions of stochastic differential equations. Jour. Math. Kyoto Univ., vol. 11 (1971), 155-167. 14. S. Watanabe and T. Yamada. On the uniqueness of solutions of stochastic differential equations II. ibid., vol. 11 (1971), 553-563.

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Martingale Representation Theorem and Chaos Expansion Shinzo Watanabe Ritsumeikan University 1.

Introduction Given a filtration F = [Tt] (i.e. an increasing family of sub a-fields of events), a martingale representation theorem is concerned with a representation of F-martingales as stochastic integrals by basic martingales. In the case of Broivnian filtration, of which we shall discuss in Section 2, a result is well-known as Itd's representation theorem which states that every square-integrable martingale can be represented as a stochastic integral by the path of Brownian motion. This theorem was first found by Ito ([I]) as a corollary of his theory on Wiener chaos expansion of L2-Wiener functionals and plays an important role in the problem of financial markets. In this respect, we would quote the following remark by Daniel Stroock in page 180 of [S]: "In fact, it (ltd's representation theorem) shares with ltd's formula responsibility for the widespread misconception in the financial community that ltd is an economist." In this introduction, we will review such a theorem and discuss its idea in the simplest case of a random walk. Let {£kh=i,2,... be a coin tossing sequence, i.e., i.i.d. sequence with P(E,k = 1) = P(£,k = -1) = \- It is also called an i.i.d. sequence of random signs. If we set A

(0, n=0 " - \ £ i + --. + £„,n = l , 2 , . . .

X = (X„)„=o,i,... is a simple random walk on Z starting from the origin. Since £,k = Xjt - Xk-i,k = 1,2,..., X = (X„) and {&} generate the same filtration F = \T„\n=ox..., where r0 = {@,Q), T„=a[X1,---fXtt]

= aU1,- ••,£„], « = 1,2,....

In the following, we take and fix N e Z++(:= { n e Z | n > 0 }) and consider the time up to N; N is called the maturity in financial problems. As usual, a family Y = (Y„)O
196

process if Yn is ^-measurable for every 0 < n < N, and a family $ = (®k)ik is Tk-r measurable for every 1 < k < N. Given an F-predictable process <5 = (®k)i
M ~iMo' "

"=0

1 Mo + LU **(** - X H ) , 1 < « < N

and call it the martingale transform ofX, more precisely, [M 0 ,0]-martingale transform of X. We denote this as M = MQ +
M„=M0 + Y, ®k(Xk - Xk-i), n = 1,2, • • • ,N k=l

by the constant Mo and an F-predictable process <E> = (<S>k)i
k=l,---.N.

Note that Mo = E(MN) by the martingale property. Here is a simple proof: Since Mk - Mk-\ is ^-measurable, there exists a function dk(h, •••, ik) on the product space {-1, l}k such that Mk-Mk-i=dk(Zu---,Zk)By the martingale property, E(Mk -M*_iin-i)

= \dk(Zi.•••,Zk-i,

1) + \dk{Zx,• • •,ik-x,-1) = 0

and hence, we have dk{l\, ••• , E,k-\, 1) = ~ 4 ( £ i , • • •, &-i, -1). This implies, by setting <S>k = <4(£i, • • •, &-i, 1), that dk(E,i, ••-,&) = *»• &. Then, n

N

M„=M0 + J^iMk - Mjt-i) = M0 + YJ ®k • Zkk=\

k=\

Obviously O = (Ojt)i<jtkThis completes the proof.

197

We will give still two different proofs; it may look too much exaggerated, as the proof given above is so simple. However, the ideas of the following proofs can be applied to the case of continuous time: indeed, they provide us with two different prototypes of the proof in continuous time, as we shall see in Section 2. Proof A. A real "F^-measurable random variable Y has a representation as Y = /(£!, • • • , £N) by a function f{i\, •••, iN) on the product space {-1,1)N, and the totality of such Y forms a 2 N -dimensional (real) Hilbert space ( s R2N) with the L2 -inner product (V,Z) = £(YZ). A orthonormal basis (ONB) {Hs| is given, indexed by a subset S c (1,2, • • • ,N(, as Ss := I I £;,

S c {1,2, • • • ,N],

with the convention

SJEI.

Take any Y and consider the orthogonal expansion of Y - £(Y) by the orthonormal system (Ss; S + 0}. Writing each Ss as Es\(it| • lk where k is the largest element in S, we immediately have the following expansion N

Y - £(Y) = \ \ O/t -E,k where


Define an F-martingale Z by a martingale transform Z = E(Y) +
4>t = Et(M t -M t _i)-^|!Ft-i],

k = l,---,N.

Define an F-martingale M = (M„) by the [Mo, <5]-martingale transform of X. Then we have (1.2)

d>k = (Mk - M n ) • & = E[Mk - M H ) • Zk\Tk-i].

Therefore, for an F-martingale L = M - M, we see by (1.1) and (1.2) that E[(Lk -Lfc_i) • ZtVn-i] = E[(Lk -LH) • (X* - X t - O m - i ] = 0, k =

l,...,N.

By applying the next lemma, we can conclude that L = 0, implying M = M, which completes the proof. D Lemma 1.1. If an F-martingale L - (L„) satisfies that (1.3)

L0 = 0 and E[(Lk - L H ) • (X, - X H ) | f M ] = 0, k = 1,. . . , N ,

then, L = 0.

198

Proof. Every ^-measurable random variable is bounded and hence, we can take a constant c > 0 such that max|L;v| < l/(2c). Then, noting E(LN) = E(L0) = 0,1 + cLN > \ and E(l + cLN) = 1. Define a new probability Q on (Q, TN) by setting Q(A) = E[(l+cLN)-lA],

AeTN.

Then X = (X„) is an F-martingale under Q. Indeed, we have E«2[(X„ - X„-i)|r„-i] = (1 + cLn^EKl + cLN)(X„ - X„_i)|!F„_i] = (1 + cL^y'EKl + cln){Xn - Xn^Wn-x] = c(l + cLn-xYlE[Ln{Xn - X„_i)r„_i] = c(l + c L ^ i ) " ^ £(Z*-I*_i)(X„-X f I _ 1 )|:F B -i = c(l + cL,,-!)"1 £

E[(Lt - L,_1)(X„ - X„.iWn-i] = 0,

/t=i

because E[(Ln - Ln-i)(X„ - X„_i)|r„-i] = 0 by (1.3), and E[(Lk - Lk-i)(Xn X^Wn-i] = {Lk-Ur.i)E[{X„-Xn-1)\T„-i\ = Owhenfc < n. SinceX„-X„_i takes values 1 or - 1 only, this implies that Q(X„ - X„_i = l|f„-i) = Q(X„ - X„_i = - l | r „ - i ) = | , that is, X = (X„) is also a symmetric random walk under Q. We have thus deduced that P = Q on (Q, TM), which implies that LN = 0 and hence Ln=E[LN\T„] = 0 for allO
(L,M)n = YJ £KL* - i*-i)(A*fc - M ^ W k - i ] ,

n = 1, • • • ,N.

fc=i

We say that two martingales are orthogonal if (L, M) = 0. Now, Lemma 1.1 can be stated as follows: An F-martingales L which is orthogonal to the basic martingale X = (X„) must be a constant. Remark 1.2. The notion of martingale transforms has been introduced by D. L. Burkholder ([B]). In continuous time, a corresponding notion is that of stochastic integrals.

199

Example 1.1. We consider the case where above random signs {£jtl are replaced by more complicated i.i.d. random variables. For a positive integer L > 3, let E = {a\,• • • ,aL] where a\,• • • ,ai are different real numbers. Let E, be an £-valued random variable such that P(£ = a,) > 0 for all /, and E(£) = 0. We denote o2 = E(£2). We consider an i.i.d. sequence £i. • - •, &v such that & = £,, k = 1,..., N, and define the process X = (X„)O<«
y_f0, «= 0 " ~\LUMSk),n = lt...,NThen, Y® are all F-martingales and, obviously, aY^ = X. It is easy to see that (Y®, Y(/,))„ = 6 y .n, 0 < n < N for every /,/' = 1,...,L - 1. We can show by the same idea as above that every F-martingale M = (M„) with Mo = 0 has a representation as a sum of martingale transforms by these martingales in which F-predictable integrands 0 ( , ) = {^>f)\
M^EJE^.CYf-Y^)), n = 0,...,N. ;=i U=i

;

Thus, we see that the martingale representation by a single basic martingale X is impossible. In this example, we consider S„ = eaKn+^n, (a, 0; constants), as a stock price in a financial market. We choose constants so that (S„)O<„
200

market is complete; an equivalent martingale measure is unique, which is a key in Proof B. 2. The Case of Continuous Time In continuous time, we have several technical difficulties in defining corresponding notions such as predictability, martingale transforms which are called in this case as stochastic integrals, and so on. We would summarize here some of basic notions, cf. e.g., [IW], [KK], [P], [RW], [RY]. 2.1 Fundamental notions The time parameter set T is usually the interval [0, <x>); we often take it to be an finite interval [0, T] for some T > 0. A filtration F = {TtIteT is an increasing family of sub cr-fields of events. We assume the usual condition unless otherwise stated: (i) The probability space (Q, T, P) is complete and every Tt contains all P-null sets. (ii) F is right-continuous, that is, Tt+(:= Cie>oTt+£) = Tt for every t e T. We introduce the following notations. Mi = { M = (M(); a square-integrable F - martingale, Mo = 0 } M2,ioc = ( M = (M(); a locally square-integrable F - martingale, Mo = 0 } M^ = { M e M2 111-» M, is continuous, a.s. j ^2,/oc = I M e Mzjoc I f i-» M/ is continuous, a.s. | A stochastic process X = (Xt(a))) is a function of (t, w) e T x Q . It is called t-adapted if Xt is ^-measurable for every t e T. We identify two processes X = (Xt(w)) and Y = (Y,{co)) and write X = Y, if P{(cj\3t e T, Xt(co) * Yt(b))\) = 0. A real process O = (<5f(a>)) is called a simple f'-predictable process if it is given, by a sequence 0 = So < Si < • • • < s„ of times and bounded real Ts,^-measurable random variables fu as (2.1)

®t(a>) = Zl=0fi((u)l{Sl_uSi](t).

The smallest a-field P on T x Q with respect to which all simple Fpredictable processes are measurable, is called the predictable a-field with respect the filtration F, or simply, F-predictable a-field. A process <1> = (Ot(o))) is called T-predictable if it is P-measurable. Every left-continuous F-adapted process is F-predictable. A real process A = (At(co)) for which t 1-* At is right-continuous and increasing (in the wide sense) is called an increasing process, and a real process which can be represented as a difference of two increasing processes

201

is called a process of bounded variation. The following fact holds: Given M,N e Mz,ioc, there exists a unique F-predictable process of bounded variation <M, N) = ((M, N)t) such that the process (MtNt - (M, N)t) is an F-local martingale. We denote (M,M) simply by (M). (M) is an F-predictable increasing process. Furthermore, if, at least, one of M and N is continuous, then (M, N) is a continuous process. Let the samplewise total variation of s e [0, t] H-> (M, N ) S be denoted by \\(M,N)\\t. Then it holds that \\{M,N)\\t < V(M>, • », a.s. Hence, if M,Ne M2, then E [||(M,N)||t] < E[M 2 ] 1/2 E[N 2 ] 1/2 < oo for every t e T. Given M e Mi,ioc, we denote by L2,i0c{M) {£i{M)) the totality of real F-predictable processes <J> - (Of) such that JT |Os|2d(M)s < oo, a.s. for every t e T, (resp. E [jTf |d>s|2d<M)s] < oo for every £ € T). 1 Then, for given M 6 Al2>c and <& e £.i,i0C(M), there exists a unique N 6 At^/oc such that the following holds for any L e Mijoc: (N,L)t=

Jo

<S>sd{M,L)s a.s.

for every t 6 T.

We call this N = (N() the stochastic integral of €> by M and denote it by c N, = £<£>sdMsr or simply, by N = f®dM. f®dM e Mc2Joc if M € lM l,loc'21 and / O d M e M2 if <5 € £ 2 (M). If <5 is given by a simple F-predictable process as given by (2.1), then <£ e -£2>c(M) for every M 6 A t ^ and JOdM is given by the following Riemann sum: Joo

sdMs = \ J,k_x I E£o MM,M - MSi) + fk(Mt - MSt), sk
sk+l •

When a right-continuous process X = (Xt) is such that M x € M2joc where M x is defined by M x = Xt - Xo, then the stochastic integral J<5rfMx is also denoted by J OdX. We call it also the stochastic integral of O by X. When an F-predictable process O = (Of) is given by <J>( =l[a>t] where a e T U {oo} is an F-stopping time, the process (Xo + J0 OsdXs)teT coincides with the stopped process X" = (XtAa) ofX by o. To realize the idea of Proof B above in the continuous time case, we introduce the following notion. We consider the case of Mi, the case of Mi,ioc can be discussed similarly by an obvious modification. For M\, M2 e Mi, we say that Mi and M2 are orthogonal, and denote this by Mi±M2, 'Two 0 , W e £2Joc{M) are identified and denoted by 0> = W if £ \<&s - ^ s | 2 d<M) s = 0, a.s. for every t.

202

if <Mi,M2> = 0. If Mi±M 2 and
M = L" = 1 JO/dM, + N and N1.M;,

I' = 1 , . . . , M .

<5, is an F-predictable process satisfying (M, Mj)t = jC (<5,)sd(M;)s, i =

\,...,n. 2.2 The case of Brownian nitrations Let W = (W() be the d-dimensional standard Brownian motion (Wiener process) such that W0 = 0. Let F w = {%] be the natural filtration, that is, ft is the a-field generated by Ws, 0 < s < t (and P-null sets). Then it is known that it satisfies the usual condition. F w is called the (d-dimensional) Brownian filtration. Denoting Wt = (W}, • • •, Wf) by its components, W = (WJ) € Mj(F w ), i = \,...,d. This system of martingales satisfies (2.3) (Wi,Wj)t = 5i,jt, so that, in particular, <W,->, = t, i,j = l,...,d. Hence, -CijodW')(:= £2,ioc) = ( - (4>t); F w -predictable and f
and £ 2 (W')(:= £2) = {* = (<&,); F w -predictable andE

<&ds

< 00, a.s. Vf ]

Jo

For * = ($( = (Oj.. ..,«&?)) e (£2,/oc)d, the stochastic integral J
Jo

" J o

The martingale representation theorem for the Brownian filtration F w asserts that every martingale in At2,;0C(Fw) can be represented by such a 2

cf. Footnote 1.

203

stochastic integral. This results implies, in particular, that every martingale with respect to the Brownian nitration is necessarily continuous. Namely, we have Theorem 2.1. (i) For every M e M2,i0c(¥w), (^2(F W ))/ there exist unique ® 6 JJIJIOC (resp. £2),i = !,•••,d,such that (2.4)

M = f d>dW = 1 1 J /
In particular, it holds that (2.5)

A W F W ) = M2M(¥W)

and

M 2 (F W ) =

M2(¥w).

(ii) Every ¥w-locally square integrable martingale X = (Xt) can be represented in the form (2.6)

X, = X 0 + j £ 3>sdWs

by some

0 be arbitrary but fixed. Let Y e L2(T™). Then there exists a unique O = ($') € (X.2)d such that (2.7)

Y = E[Y] + / 0 T O s dW s .

Proof. The proof is based on the well-known Wiener chaos expansion (the expansion by multiple Wiener-Ito integrals) as will be given by Theorem 2.4 below. (2.8)

Y = £[Y] + i : = 1 / . . . /0(*!, • • • ,tn)dW(h) • -dW(tn).

Here /„ 6 L2 (A£° -> (Rd)®"), A<"> := {(tlr ••• , t„)\0 < h < ... < t„
204

a\ = (a'j) e R d ,..., a„ - (a'n) e Rd, Aa\ • • -a„ is a real number defined by d

d

'1=1

i'»=i

Setting(f^ih,• • •,^-i))'"'"''"" 1 = (f„(h,•••,f„_i,t)t-"J-lJ,

*l = /I'(0 + E f • • • f

define

&(*!'''' ' f«-i W i ) • • -dW(t„-i).

Then O' = (<&|) has a version as an F-predictable process so that <J> = (<&') € (£2)d and (2.7) follows from (2.8). • The proof of Theorem 2.1 (i) follows at one from Theorem 2.2. Proof B. By Prop. 2.1, it is enough to show that every N e Mz(Fw) which is orthogonal to every W, i = \,...,d must be 0. When NT e U°{T^), we can choose c > 0 such that 1 + CNT is bounded from above and below by positive constants, so that a probability Q on (Q, T™) can be defined which is equivalent to P with density 1 +CNT- We deduce that any stochastic integral fdW for O e Li is in Al2(Fw) with respect to Q. In particular, Wt and WJWj - Syf are all Q-martingales. This implies, by Levy's martingale characterization of Wiener process, that W = (W) is also a d-dimensional Wiener process with respect to Q, that is, P = Q on ( G , ^ ^ ) . Hence, 1 + cNT = 1, that is, NT = 0, a.s. Then Nt = E(NT\Tt) = 0, a.s. for 0 < t < T. In the general case of unbounded N, the proof can be reduced to the bounded case by approximation (cf. e.g. [IW], p. 82, [RY], p. 210). • 2.3 The Clark-Bismut-Ocone Formula and Wiener chaos expansion. The Clark-Bismut-Ocone Formula is concerned with an expression of the integrand O in the ltd representation (2.7). In discussing this, we apply some functional analysis (Malliavin calculus) on Wiener process so that it is convenient to set up the Wiener process canonically: We take, as our basic probability space (Q,T), the path space Wo(Rd) := { w, [0,T] B t •-> w(t) 6 R d , continuous, w(0) = 0 }, which is a Banach space with the usual maximum norm, endowed with the d-dimensional Winer measure Pw on the u-field T of P w -measurable sets. The natural filtration F w = {Ttw} is defined as usual, so that T = TTW. x(t,w) := w{t), w e W 0 (R d ), is the canonical realization of d-dimensional Wiener process. Let H c Wo(Rd) be the Cameron-Martin subspace; H=

h e W 0 (R d ) | h(t) = J

h'(s)ds, h' e L2 ([0, T] -» R d ) J , ||fe||H = \\h'\\Lz.

205

A P w -measurable function on Wo(Rd) is called a Wiener functional. The Malliavin calculus is a differential and integral calculus for Wiener functionals (cf. e.g. [IW], [M]). Typical differential operators are, the gradient operator or Gross-Malliavin-Shigekawa operator D which sends a real Wiener functional to an H-valued Wiener functional, its dual operator or Skorohod operator D* and the Ornstein-Uhlenbeck operator L = -D*D. D is defined formally, for a Wiener functional F = (F(w)), by

(DF, h)H = lim(F(w + eh) - F(w))/e,

heH.

e-»0

For a real separable Hilbert space E, we denote by LP(E), 1 < p < oo, the usual LP-space of E-valued Wiener functionals. In the Malliavin calculus, a family of Sobolev-type spaces D^(E), 1 < p < oo, k = 0,1, • • •, is introduced; roughly, F e D^E) if F 6 U{E) and DkF e U(E ® H®k). When E = R, lf(E) and D^(E) are denoted simply by U and D£, respectively. Let F e D^. Then DF e L2(H) and we denote DtF = (DF)'(t), that is, (DF,h)H = £(DtF,h'(t))Kidt, for every heH. The Clark-Bismut-Ocone formula may be stated as follows:

Theorem 2.3. Let F e D^(c L2). Then we can define a version
(2.9)

F = E[F] + JTT E(DtF\Ttw)dWt.

A proof based on the Wiener chaos expansion will be given below.

Example 2.1. Let d = 1. For the canonical representation of onedimensional Wiener process w(t) = x(t,w), we define m(t) = m(t,w) := max0<s( = E(DtF\ftw), 0 < t < T, is given by the following formula; letting r(t) = r(t, w) := m(t) - w(t),

206

{w)=

exp

exp

[2r(t)-x]2 2(T -1)

*< ir^\S^( {-^o}df x -±(x + w(t),m(t))

x(^

+ ^](x

=

rfx

+ w(t),y + w(t))

exp

xr V5^( f^o}-

exp

[2r(t)-x]2

2(7-0

X -^(x + w(t),m{t))

xl-£

+A x

+ w(t),V + w(t))

x(-£

+ -J-\(x + w(t),y + w(t)).

In particular, if f(x, y) = g(x), then

and, if fix, y) = giy), then

A proof can be deduced from the following facts: If F(w) = m(T), then F e D j and Df = l[o,T](0/ where T is a unique random time in [0, T] such that W(T) = mij). The condition T > Hs equivalent to the condition that miT) > m(t). If, for each 0 < t < T, a continuous path wf is defined by w+(s) = w(f + s) - a;(0,0 < s < T - t, then w(T) = wit) + x(T - \.,w\), and miT) = mit)l[m(t)=m{T)] + iwit) + miT - t, < ) ) l[m{t)<m(T)\ = mit)l[m(T-t.w{)
+ m{T - t, W+t)) l[m(T-t,w!)>r(t)]-

207

We use the independence of Ttw and w+t, and also the following well-known formula for the joint distribution of w{t) and m(t), cf. [IM], / 2 P(w(t) 6 dx,m{t) 6 dy) = 1 [xo] • J^expl

f

[2y-x]2l ——

\(2y-x)dxdy

The next example is concerned with a stochastic differential equation (SDE). Consider the following SDE for a diffusion process X = (X") on Rm with the initial point x = (xa): 3 (2.10) dXf = of(Xt)dwi(t) + ba(Xt)dt,

X£=x",

a =

l,...,m.

Here, coefficients of and b" are assumed to be smooth with bounded first order derivatives. Then a solution X* = (X*(w)) exists, unique in pathwise, so that X* defines an ^ w -measurable Wiener functional with values in Rm. Let Pt be the transition operator: Ptf(x) = E[/(X*)], acting on C(Rm). Then, if / 6 C2(Rm), Ptf e C2(Rm), and (t, x) ^ P,f(x) is C1 in t and C 2 in x. Here, C(Rm) = {/; real, continuous on Rm having limits as \x\ —» oo), and Ck(Rm) = {/ e C(Rm); derivatives up to the order k are in C(Rm)}. Then, u(t,x) := Ptf(x), f e C2(Rm), satisfies the heat equation: (2.11)

f=L«,

u\t=0 = f,

where L is a second order differential operator given by (2.12)

L=\aaP(x)j^+b"(x)£

with

a<%) = LU of{x)a%x). 2

Example 2.2. Consider the case when F 6 L is given by F(w) = /(X*) where / e C2(Rm). Then the ltd representation (2.7) for F is given by (2.13)

F(w) = PTf(x) + /0T

%gf-(X*)o?(X*)dtf(t).

It is well-known that F e D 2 and, by (2.13), we see that the Clark-BismutOcone density E ((D,F)'|Tt) is given by

£((DtFy\T>) = ^ A x > ? ( X * ) ,

i = 1,...,d-

This follows at once from the Ito formula: If u(t,x) := Ptf(x), then u(T-t,X*)-u(T,x) = JT ^ ( T - s , X s X P O < M s ) + J [ ( - ^ + LM) (T - s,X*)ds. 3

We omit the summation sign by following the usual convention.

208

Noting that u satisfies (2.11), (2.13) follows readily by letting t —> T. Wiener chaos expansion. Consider the space L2 = L2(W0(Rd),TTvv,Pvv) of square-integrable Wiener functionals. Then we have its Wiener chaos decomposition into sum of mutually orthogonal subspaces C}1. , n = 0 , 1 , . . . Lz = ®LZoCf,

(2.14)

C<°>=R.

Cp' is one-dimensional, consisting of all constant functionals. d p for n > 1 is usually described by Ito's multiple Wiener integrals (cf. [1-1]) and, as is remarked in this paper, a multiple Wiener integral may be defined by an iteration of It&'s stochastic integrals as follows: Let, as above, A^n) := {(fi,• ••,tn)\Q < 11 < ... < tn < T\ c [0,T]" and 2 L (A^n)) be the usual L2 -space of real functions / on A^' such that mlHA?):=J---ljf(hr--/tn)\2dt1.--dtn
=

(/''"• '•<'«)

u

6

Lz (A(Tn) -+ (Rd)®"),

so that /''"-''''<

2

=

n

(f ™Hti, • • • , t„)) e L (A) for every (h, •••,«„) e {1, ••• , d\ . We define J (n) (/) e L2 by the following iterated ltd stochastic integrals:4 l(n\f){w) =

I

{/"{'"{/2/,1''"''"(fl'"'

>in)dwi^tA--\dwi"-^tn-x)\dwi''{tn).

As in (2.8), the right-hand side is denoted simply by

{... f

J

JAM

f(h,---,tn)dw(h)--dw(tn).

It holds that (2.15)

||/(">(/)||L2 = ll/ll L 2 ( A ? u ( R , r ) .

Theorem 2.4. For each n, the subspace Cj is isometrically isomorphic to L2 (A£° -> (Rd)®") by the correspondence /„(/) ^ / , so that every F e L2 has 4

We omit the summation sign for i\, • • • ,in-

209

an orthogonal expansion by uniquely determined f„ e L2 ( A ^ -> (Rd)®"), n = 1,2,...,: (2.16)

F = E(F) + I ~ x / . . . JA,„ /„(ti, • • •, tn)dw{h) • • • dw{tn).

We have by (2.15) that (2.17)

IIFII^Ed^ + L ^ i H / J ^ ^ ^ .

Example 2.3. (Veretennikov-Krylov expansion [VK]) We consider the SDE (2.10) and consider the same F(w) = /(X*) as in Example 2.2. Let the transition operator Pt, t e [0,T], be defined as in Example 2.2 by Ptf(x) = E[f{Xxt)\, and define the operator Q\, t e [0,T], i = 1, • • • ,d, by

= aia(x)^-(x),

mx)

feC2(Rm).

Take / 6 C2(Rm). Then a smoothness of Ptf is guaranteed as in Example 2.2: If a\x is nondegenate so that the operator L is elliptic, we may take any / e C(Rm). Then the Wiener chaos expansion of F(w) = /(X*), for a fixed x e Rm, is given by the following Veretennikov-Krylovformula: E(F) = Prf(x) and, for n = 1,2,..., fr'-iti,

- , * „ ) = ptl ( Q U (• • • (QT-J)

•••)) (*)•

A proof can be given by applying the following formula successively: If g e C 2 (R m ), then

= E IfQJ_,S<X?)M0 QU

g(Xxs) - Psg(x) = V

for every 0 < s < T.

This formula is obtained from the ltd formula applied to u(t, Xx(t)), t e [0, s], where u(t, x) = Ps^tg(x), for each fixed 0 < s < T. Proof of the Clark-Bismut-Ocone formula (Theorem 2.3). It is enough to show this when F is in the n-th order Wiener chaos, i.e. F £ Of, for n > 1. So we assume that F = J . . . L,> /(fi, • • •, f„)dw(fi) • • • dw(t„), f e L2 (A£° -* (Rd)®"). Then, for each 0 < s < T and i = 1,... ,d, (DF,)' in the (n - l)-th order Wiener chaos given by (f'(s), (DFS)' = I j ^

^j,,(^

when n = 1 ^ tn-Jdwih) • • • dw(tn-i), when « > 1 '

210

where, if n > 1, f** e L2 (A
{ff-^itu-,^ n = Yu V , A ] ( s ) • / ( '<''" A ->' , '' , ^'-'''»- l ) (£ 1 , • • • , tk-l,S, h r - ,

tn-l)

with the conventions t0 = 0 and tn = T. Then,

£[(DsF)'rsw] JA'" -1 '

J

= [... f J

_1)

• •,tn-lrs)dwHh)• ••dwi^(tn-l),

f^~^{h,•

JA<"

because, if (h, • • •, f«-i) e A'" -1 ', that is, if 0 < fi < • • • < f„_i < s, then (f''') (/,, '"' / " M) {tu...

^ tni)

= f{h,-,i^,i){hf

...t

tn_us)

We can now conclude that J

E [(DSF)''FSW] M s ) = J

E [DsFr s w ]
•

2.4 The case of one-dimensional diffusion processes. Let = -oo < / < r < oo be given and fixed. A most general regular diffusion process X = (X(f)) on the interval / = [/, r] is determined by a canonical scale s(x), a speed measure dm(x) and, when the boundary / or r is regular, a Feller boundary condition on the boundary: s(x) is a strictly increasing continuous function on 1° = (I, r), dm(x) is an everywhere positive Radon measure on 1° = (/, r), and / (r), is regular if -oo < s(/+) + m((l, c]), (resp. s(r-) + m[c,r) < oo). Here (and below), we fix a point c such that I < c < r. The local generator of X is given by a generalized second-order differential operator ^ fs, (cf. [IM], [I]). By changing the coordinate by the scale function, we may and do assume from now that s(x) = x, unless otherwise stated. If X(t) starting at a point in 1° cannot hit a boundary in a finite time, we may delete the boundary from the state space I. For simplicity, we assume that every boundary point is a trap5. That is, we assume X(t) = X(m,?) for all 5

cf. Remark 2.1 below for the case of more general boundary conditions.

211

t > ma when ntg < oo, where we set ma = inf{ t | X(t) = a } for a e I and mg -ntt A Writ is well-known that, if X(0) = a, a £ I", X(t) is obtained from one-dimensional Brownian motion by a time change: Let W(t) be a one-dimensional Wiener process with W(0) = 0, Ba(t) = a + W(t) and l(t,x) := lim ei0 (4e) _1 j£ l{x-£fX+e)(Ba(s))ds be the local time of B" = (B"(t)). Define an increasing process A(t) by Upl{t,x)dm(x), \ oo,

H t<m» if t > m°

where mf is the hitting time to the boundaries / and r for Ba = (B"(t)) defined similarly as above. Let t i-» xt be the right-continuous inverse of 11-> A(t). Then, X(t) = Bfl(r,). We fix a e I". Let F x = {Ttx} be the natural filtration of X = (X(f)) with X(0) = a. From the above time change expression, we see that X = (X(f)) is a local martingale. Theorem 2.5. Every M - (M(f)) e Al2,;0c(Fx) has the following representation M(t)=

f<&(s)rfX(S), Jo

O = W))

e X2,/oc(X).

Proof. Let C(I) be the space of all real continuous functions on I = [I, r\; I is compact even in the case of unbounded boundaries. The resolvent operator GA, A > 0, on C(I) is defined, as usual, by G^f(b) = E [£° e-Atf(Xb{t))dt], where Xb = (Xb(f)) denote the diffusion starting at b 6 I. The generator L of the diffusion is defined (independently of A) by L = A - G"1 with domain D{L) = GA(C(I)). Then, for u e D(L), Mu(t) = u{Xb{t)) - u(b) JT Lu(Xb(s))ds e M2(Fxb), so that the semimartingale decomposition of u(Xb(t)) is given by (2.18)

u(Xb(t)) = u(b) + Mu(t) + JT' Lu(Xb(s))ds.

On the other hand, u e !D(L) is absolutely continuous on 1° and the Radon-Nikodym derivative u' is a function of bounded variation. Therefore, we can apply an extension of Itd's formula (Ito-Tanaka formula, cf. e.g. [RW], [RY]) to obtain

X

tAmf

p u'(Bb(s))dBb(s)+

I

l(tAtnf,x)du'(x).

212

Then, by the time change t H-> T ( , u{Xb(t)) - u{b) = f ' u'{Bb(s))dBb(s) + f Jo Ji«

l(i,,x)du'(x),

and we have (cf. [IW], p. 102) that £' u'(Bb(s))dBb(s) = j£ u'(Xb(s))dXb(s). Comparing this with the semimartingale decomposition (2.18), we readily see that M"(£) = jf u'(Xb(s))dXb(s), which coincides with jf u'(X(s))dX(s) when we take b = a. We have shown in [KW] that [Mu;u € D(L)\ generates the space A^F**); any M € At2(Fx"), which is orthogonal to every M", u e £>(L), must be zero. Theorem follows at once from this. • We would ask the following question: When is the local martingale X = X(t) with X(0) = a e 1° a true martingale? In order to answer this, we first introduce the well-known classification of boundary points. Define <*(1) J

I I

a(r) = I I J

dm(x)dy,

^(1) = I I

Jl
J

dm(x)dy,

jj(r) = I I

Jc<x
dxdm(y)

Jl
dxdm(y).

JKJ Jc<x
In the following, we state the classification in the case of the boundary / only; the case of the boundary r can be stated similarly by replacing <J(Z) and /i(/) with a(r) and ji(r), respectively. Definition 2.1. The boundary / is called (i)

regular if a(l) < oo and /i(/) < oo,

(ii)

exit if a(l) < oo and ji(l) = oo,

(iii)

entrance if a{l) = oa and \i(l) < oo,

(iv)

natural if a{l) = oo and ^i(Z) = oo.

The following criteria are sometimes more practical: • / is regular if and only if -oo < / and m((l, c]) < oo. • / is exit if and only if m((l,c]) - oo and J( m((x,c])dx < oo. • / is entrance if and only if / = -oo and J^ |x|dm(x) < oo.

213

The following result is due to S. Kotani ([K]): Theorem 2.6. X = (X(f)) (with X 0 = a e 1°) is a martingale if and only if the both boundaries I and r are not entrance. In other words, an existence of entrance boundary, and only this, can destroy the true martingale character of the local martingale X. Before giving a proof, we discuss some examples. Example 2.4. (Two-dimensional Bessel diffusion process.) A twodimensional Bessel diffusion process, that is, the radial motion of a twodimensional Brownian motion, is a diffusion X = (X(f)) on (0, oo) with the generator ^ = l ( i E + JEj- A canonical scale is given by s(x) = Iogx, so that, choosing this as the coordinate, we are considering a diffusion Y = (Y(t)) °n {-oo, oo) with the canonical scale s(y) = y and the speed measure dm(y) = 2ezvdy, i.e. the generator given by L = \e~2^^. By the above criterion, the boundary -oo is entrance and the boundary oo is natural. Then, by the theorem, the local martingale Y(t)(= log(X(f))) is not a true martingale. This is a little surprising, as the all moments of Y(t) are finite. Example 2.5. (Positive diffusions.) Motivated by a problem in mathematical finance, Delbaen and Shirakawa (cf. [DS]) studied the following diffusion process X = (X(f)) on [0, oo) which is given by a solution of the following SDE: dXt = o(Xt)dWt, X0 = 1. The coefficient a(x) is assumed to be 0 when x < 0, and is strictly positive and continuous for x > 0. Then a solution exists, unique in the law sense, until it reaches the origin and, we assume that it is stopped as soon as the origin is reached in a finite time. This is equivalent to considering a diffusion process X = (X(f)),X(0) = 1, on [0, oo), that is, / = 0 and r = oo, with the scale function s(x) = x and the speed measure dm(x) = -^pdx. The origin is assumed to be a trap when it can be reached in a finite time. According to the above definition and criteria, the origin 0 is regular if JT -^sdx < oo, exit if J^ -^dx = oo but Jo Hffidx < °°> and natural if JQ -^dx = oo. It cannot be entrance. As for the boundary point oo, it is either entrance or natural, and it is so according as

J T w?dx < °° ° r r

«x?dx=°°-

It is known (cf. [IM], [IW, p. 450]) that X reaches the origin in a finite time a.s. if the origin is either regular or exit, equivalently, if JQ -jr^dx < oo, and X cannot reach the origin in a finite time, if the origin is natural, equivalently, if JQ -^dx = oo.

214

Also, by the theorem above, we can conclude that X(t) is a true martingale if and only if °o is natural, equivalently, if f -£*dx = oo. In [DS], these results are rediscovered by a method based on the RayKnight Theorem for Brownian local times. Proof of Theorem 2.6. First, we note that X = (X(f)) is a martingale if and only if Xb(t) is integrable and E[Xb(t)] = b for any t > 0 and b e I". This is a simple consequence of the Markov property of the diffusion. By taking the Laplace transform, these conditions can be equivalently stated in terms of the resolvent kernel: Let, as above, GAf(x) = jj°° e~ME [f(X*(t))] dt, f e C(I). Then, GA/(X) = J^ G\(x,dy)f(y) by the resolvent kernel G\(x,dy). X = (X(t)) is a martingale if and only f G\{x,dy)\y\ < oo and \.G\(x,dy)y = j , for every A > 0 and x e 1°. Let u\(x) be a positive and increasing solution of the homogeneous equation Au - Lu = 0 on 1°, where L = ^ ^ is the local generator of X. When I is regular, we impose the condition that u(l+) = 0. Then u\ exists and it is unique up to a multiplicative constant. Similarly, let U2(x) be a positive and decreasing solution of the homogeneous equation Au - Lu - 0 on 1° with the condition u(r-) = 0 when r is regular. Then U2 exists and it is unique up to a multiplicative constant. The following property of functions U\ and «2 is a key in the proof, cf. [IM], [I]; a nice and detailed proof can be found in the latter reference. We state it for the increasing solution u\ and its derivative u'x: For the decreasing solution u2 and —u'v it can be stated similarly by exchanging the role of boundaries / and r. •

U\ (/+) > 0 if and only if / is entrance; so, if / is regular, exit or natural, then wi(/+) = 0.

•

Wi(r-) < oo if and only if r is regular or exit; so, if r is entrance or natural, U\{r—) - oo.

•

u[(l+) > 0 if and only if / is regular or exit; so, if / is entrance or natural, u \ (/+) = 0.

•

Uj(r-) < oo if and only if r is regular or entrance; so, if r is exit or natural, u'x(r-) = oo.

Define d,lW=(MH'ifrisreSularorexit' (0, if r is entrance or natural,

215

and similarly

ch2(x) = { 3 % '

if H s r e g u l a r

\ 0,

° r exit'

if / is entrance or natural.

We define the A-order Green function g\{x, y), x, y e 1°, by («iW»2(y)

[{x y

./ g\{x, y) = I m(w y)M2(j)

-'

where h(A) = «J(x)M2(y) — a i W u ^ i ) is the Wronskian of «i and «2 (which is independent of x). Then the resolvent kernel G^(x,dy), x e 1°, dy c J, is given by 4>i(x) , co). Now we can show that §{G\(x,dy)\y\ < oo, and the following relation:holds: (2.19)

J[GAMy)y=f + ^ g . « 2 ( x ) - ^ g . « 1 W .

For this, we need the following lemma: Lemma 2.1. (1) I

ytti{y)dm{y)

I x L , u\{y)dm(y) - ^ p + "' +A*~ , ;/ / is regular or exit, x f; , u\{y)dm(y) - ^ p + ^f^, i/ / is entrance or natural. (2)

l

yui(y)dm(y)

J(x,r) x

I(x,r) U^y)dm(y)

x

u2(x)

+ ^

+

u'2(r-){r-x)

A

It r) "2(y)^m(y) + A^ ~~ ^A~^'

.

r if r is regular or exit, if

r ?s

entrance or natural.

Note also that w' (/+) = ——— U2(l+)

if / is regular or exit,

216

and uJr-) =

:—r if r is regular or exit. «i(r-) Proof of Lemma 2.1. Suppose that I is regular or exit. Then «i(Z+) = 0 and we have u\{x) = I u[(z)dz = I dz\u[(l+)+ Jl

Jl

Since u\ is a solution of Au-Lu

I

l

dwi(y) .

J(l,z]

= 0, we have du'^y) = Au\{y)dm(y). Hence,

Ui(x) = u[(l+)(x -1) + A I I

dz u\(y)dm(y)

J Jl
= u[(l+)(x -1) + A I

{x -

y)Ul(y)dm(y)

J(lx]

= u[(l+)(x -1) + Ax I

u\{y)dm(y) - A I

J(l,x]

yu\(y)dm(y).

J(l,x]

From this, we obtain yui(y)dm(y) = x J(l,x)

m(y)dm(y) J(l,x)

— + A

. A

The proof in the case when / is entrance or natural is similar: We note, in this case, that w(i(Z+) > 0, («(i(/+) > 0 if and only if / is entrance), and u[(l+) = 0 always. The proof in the case of the right boundary point r is the same if the role of / and r are exchanged. • We have [gA(x,y)\y\dm(y)=^ Jl«

f \y\u2(y)dm(y) + ^ f \y\ux{y)dm{y) n A \ > J(x,r) \ > J(U]

n A

and, by noting J[a g(x, y)dm{y) < 1/A and using Lemma 2.1, we can estimate this as 0(1 +1*|). We have also, gA(x, y)ydm{y) = jj±-

J

yu2(y)dm(y) + ~ - J

yUl(y)dm(y).

Substituting the expressions in Lemma 2.1 to the integrals in the right-hand side, we can obtain (2.19). From (2.19), we see that G\(x,dy)y = x/A, x e I", if and only if Ui(l+)u2(x) - u2{r-)u\(x) = 0 on 1°. We can easily deduce that this holds if

217 and only if «i(/+) = U2(r-) = 0. This holds if and only if neither / nor r is entrance. This completes the proof of Theorem 2.6. • Remark 2.1. W h e n / or r is regular, w e a s s u m e d it to be a trap for our diffusion. Of course, there is a variety of possible b o u n d a r y behaviors and X(t) is generally not a local martingale, any more. References DS. F. Delbaen and H. Shirakawa, No arbitrage condition for positive diffusion price processes, Asia-Pacific Financial Markets 9 (2002), 159-168. IW. N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, Second Edition, North-Holland/Kodansha, Amsterdam/Tokyo, 1988. 1-1. K. Ito, Multiple Wiener Integral, /. Math. Soc. Japan, 3 (1951), 157-169. 1-2. K. Ito, Kakuritu Katei II (Stochastic Processes, II), Iwanami Shoten, Tokyo, 1957 (in Japanese); English Translation by Yuji Ito, Yale University, 1961. IM. K. Ito and H. P. McKean, Jr., Diffusion Processes and Their Sample Paths, Springer, Berlin, 1965, Second Printing 1974, in Classics in Mathematics, 1996. KK. G. Kallianpur and R. L. Karandikar, Introduction to Option Pricing Theory, Birkhauser, Boston/Basel/Berlin, 2000. K. S. Kotani, On a condition that one-dimensional diffusion processes are martingales, 2003. KW. H. Kunita and S. Watanabe, On square integrable martingales, Nagoya Math. J. 30 (1967), 209-245. M. P. Malliavin, Stochastic Analysis, Springer, Berlin, 1997. P. P. Protter, Stochastic Integration and Differential Equations, A New Approach, Springer Verlag, Berlin/Heidelberg/New York, 1990. RW. L. C. G. Rogers and D. Williams, Diffusion, Markov Processes, and Martingales, Vol. 2, ltd Calculus, John Wiley & Sons, Chichester/New York/Brisbane/Toronto/ Singapore, 1987 S. D. W. Stroock, Markov Processes from K. Itd's Perspective, Annals of Mathematical Studies 155 (2003), Princeton University Press, Princeton/Oxford. VK. A. Ju. Veretennikov and N. V. Krylov, On explicit formulas for solutions of stochastic differential equations, Math. USSR Sbornik 29 (1976), 239-256.

Based around recent lectures given at the prestigious Ritsumeikan conference, the t u t o r i a l

and

expository articles contained in this volume are an essential guide for practitioners and graduates alike who use stochastic calculus in finance. Among the eminent papers are: • Harmonic Analysis Methods for Nonparametric Estimation of

=;

>

>

=! )

Volatility: Theory and Applications by E. Barucci, P. Malliavin, and M. E. Mancino • Hedging of Credit Derivatives in Models with Totally Unexpected Default by T. R. Bielecki, M. Jeanblanc, and M. Rutkowski • Martingale

Representation

Theorem and Chaos Expansion by S. Watanabe

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