Schoutens - BSDE’s and Feynman-Kac Formula for L ́evy Processes with Applications in Finance

7/30/2019 Schoutens - BSDEs and Feynman-Kac Formula for L evy Processes with Applications in Finance

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BSDEs and Feynman-Kac Formula for Levy

Processes with Applications in Finance

David Nualart

Universitat de Barcelona

Gran Via de les Corts Catalanes, 585

E-08007 Barcelona

Spain

Email: [email protected]

Wim Schoutens

K.U.Leuven Eurandom

Celestijnenlaan 200 B

B-3001 Leuven

Belgium

Email: [email protected]

June 15, 2001

Running head: BSDEs for Levy Processes

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Abstract: In this paper we show the existence and uniqueness of a

solution for backward stochastic differential equations driven by a Levy pro-cess with moments of all orders. The results are important both from amathematical point of view as in finance: An application to Clark-Oconeand Feynman-Kac formulas for Levy processes is presented. Moreover, theFeynman-Kac formula and the related Partial Differential Integral Equation(PDIE) provide us an analogue of the famous Black-Scholes partial differen-tial equation and is used for the purpose of option pricing in a Levy market.

AMS Subject Classification: 60J30, 60H05Keywords: Backward Stochastic Differential Equations, BSDE, Levy

Processes, Orthogonal Polynomials, Option Pricing

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1 Introduction

The first paper concerned with Backward Stochastic Differential Equations(BSDEs) is the paper Bismut (1973), where he introduced a non-linearRicatti BSDE and showed existence and uniqueness of bounded solutions.Pardoux and Peng (1990) considered general BSDEs and this paper was thestarting point for the development of the study of these equations. On theother hand, BSDEs have important applications in the theory of mathe-matical finance, especially, they play a major role in hedging and non-linearpricing theory for imperfect markets (see El Karoui and Quenez (1997)).

One can consider a BSDE driven by a Brownian motion as a nonlineargeneralization of the integral representation theorem for square integrablemartingales. Then it is natural to extend these kind of equations to thecase of Levy processes, that is, processes with independent and stationaryincrements. We recall that a Levy process consists of three stochastically in-dependent parts: a purely deterministic linear part, a Brownian motion anda pure-jump process. In Situ (1997) BSDEs driven by a Brownian motionand a Poisson point process are studied. Ouknine (1998) considers BSDEsdriven by a Poisson random measure. In both papers the main ingredientis the integral representation of square integrable random variables in termsof a Poisson random measure (see Jacod (1979)).

In Nualart and Schoutens (2000) a martingale representation theorem forLevy processes satisfying some exponential moment condition was proved.The purpose of this paper is to use this martingale representation result to

establish the existence and uniqueness of solutions for BSDEs driven by aLevy process of the kind considered in Nualart and Schoutens (2000). Theresults are important both from a mathematical point of view as in finance.This is illustratated in the applications. The resulting Clark-Ocone and theFeynman-Kac formulas are fundamental ingredients in the build up of anMalliavin calculus for Levy process. Moreover, the Feynman-Kac formulaand the related Partial Differential Integral Equation (PDIE) also have animportant application in finance: they provide us an analogue of the famousBlack-Scholes partial differential equation and is used for the purpose ofoption pricing in a Levy market.

The paper is organized as follows. Section 2 contains some preliminaries

on Levy processes. Section 3 contains the main result on BSDEs driven byLevy processes. In Section 4 we have included some applications of BSDEsdriven by Levy processes to the Clark-Ocone, the Feynman-Kac formulas,and option pricing in a Levy market. Finally, in the appendix one can finddetailed proofs of the main results.

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2 Preliminaries

Let X = {Xt, t 0} be a Levy process defined on a complete probabilityspace (, F, P). That is, X is a real-valued process starting from 0 withstationary and independent increments and with cadlag trajectories. It isknown that Xt has a characteristic function of the form

E

eiXt

= exp

iat 1

222t + t

R

eix 1 ix1{|x| 0, and is a measure on R with R

(1 x2)(dx) < .We will assume that the Levy measure satisfies for some > 0

(,)c e

|x|(dx) < ,

for every > 0. This implies that the random variables Xt have moments ofall orders. Moreover, it will assure us the existence of the below mentionedPredictable Representation, which we will use in our proofs. We refer toSato (2000) or Bertoin (1996) for a detailed account on Levy processes.

For t 0, let Ft denote the -algebra generated by the family of randomvariables {Xs, 0 s t} augmented with the P-null sets of F. Fix atime interval [0, T] and set L2T = L

2(, FT, P). We will denote by P thepredictable sub--field ofFT B[0,T]. First we introduce some notation:

Let H2T denote the space of square integrable and Ftprogressivelymeasurable processes = {t, t [0, T]} such that

||||2 = ET

0|t|2dt

< .

M2T will denote the subspace of H2T formed by predictable processes. H2T(l2) and M2T(l2) are the corresponding spaces ofl2valued processes

equipped with the norm

||||2

= ET

0

i=1

|(i)

t |2

dt

.

Set H2T = H2T M2T(l2).

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Following Nualart and Schoutens (2000) we define for every i = 1, 2,...

the so-called power-jump processes {X(i)t , t 0} and their compensated

version {Y(i)t = X(i)t E[X(i)t ], t 0}, also called the Teugels martingales,as follows

X(1)t = Xt and X

(i)t =

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and

E[X1] = a +{|z|1} z(dz).

In the case

R

|z|v(dz) < , assuming a = {|z|


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f is uniformly Lipschitz in the first two components, i.e., there existsC > 0 such that dt dP a.s., for all (y1, z1) and (y2, z2) in Rl

2

|f(t, y1, z1) f(t, y2, z2)| C(|y1 y2| + z1 z2l2) .

L2T.

If (f, ) satisfies the above assumptions, the pair (f, ) is said to bestandard data for the BSDE. A solution of the BSDE is a pair of processes,{(Yt, Zt), 0 t T} H2T M2T(l2) such that the following relation holdsfor all t [0, T]:

Yt = +T

t f(s, Ys, Zs)ds

i=1

Tt Z

(i)s dH

(i)s . (2)

Note that the progressive measurability of{(Yt, Zt), 0 t T} implies that(Y0, Z0) is deterministic.

A first key-result concerns the existence and uniqueness of solution ofBSDE:

Theorem 1 Given standard data(f, ), there exists a unique solution(Y, Z)which solves the BSDE (2).

The proof can be found in the Appendix, as the proof of the continuousdependency of the solution on the final data and the function f.

Theorem 2 Given standard data (f, ) and (f, ), let (Y, Z) and (YZ)be the unique adapted solutions of the BSDE (2) corresponding to (f, ) and(f, ). Then

E

T0

|Ys Ys|2 +

i=1

|Z(i)s Z(i)s |2

ds

C

E[| |2] + ET

0|f(s, Ys, Zs) f(s, Ys, Zs|2ds

.

4 ApplicationsSuppose our Levy process Xt has no Brownian part, i.e. Xt = at + Lt,where Lt is pure jump process with Levy measure (dx).

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4.1 Clark-Ocone Formula and Feynman-Kac Formula

Let us consider the simple case of a BSDE where f = 0, and the terminalrandom variable is a function of XT, that is,

dYt =

i=1

Z(i)t dH

(i)t ; YT = g(XT)

or equivalently

Yt = g(XT)

i=1

Tt

Z(i)s dH(i)s , (3)

where E(g(XT)2) < . Let = (t, x) be the solution of the following PDIE

(Partial Differential Integral Equation) with terminal value g:

t(t, x) +

R

((t, x + y) (t, x) x

(t, x)y) (dy) + a

x(t, x) = 0,

(T, x) = g(x), (4)

where a = a +{|y|1} y(dy). Set

(1)(t,x,y) = (t, x + y) (t, x) x

(t, x)y. (5)

The following result is a version of the Clark-Ocone formula for functionsof a Levy process. Again the proof can be found in the Appendix.

Proposition 3 Suppose that is a C1,2 function such that x and2x2

are bounded by a polynomial function of x, uniformly in t, then the uniqueadapted solution of (3) is given by

Yt = (t, Xt)

Z(i)t =

R

(1)(t, Xt, y)pi(y)(dy) for i 2,

Z(1)t =

R

(1)(t, Xt, y)p1(y)(dy) +

x(t, Xt)(

R

y2(dy))1/2,

where = (t, x) is the solution of the PDIE (4) and (1)(t,x,y) is given by

(5).

Now by taking expectations we derive that the solution (t, x) to ourPDIE (4) equation has the stochastic representation

(t, x) = E[g(XT)|Xt = x].

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This is an extension of the classical Feynman-Kac Formula.

If

R |y|(dy) < , and we take a = {|y|


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Suppose that = (t, x) satisfies the following PDIE:

t(t, x) +

R

(1)(t,x,y)(dy) + a x

(t, x) (7)

+f

t, (t, x) ,

(i)(t, x)

i=1

= 0,

(T, x) = g(x).

where as in the previous section, we define (1)(t,x,y) by (5),

(1)(t, x) =

R

(1)(t,x,y)p1(y)(dy) +

x(t, x)(

R

y2(dy))1/2, (8)

and for i 2(i)(t, x) =

R

(1)(t,x,y)pi(y)(dy). (9)

Proposition 4 Suppose that is a C1,2 function such that x and2x2 are

bounded by a polynomial function of x, uniformly in t. Then the (unique)adapted solution of (6) is given by

Yt = (t, Xt)

Z(i)t =

R

(1)(t, Xt, y)pi(y)(dy) for i 2,

Z(1)t =

R

(1)(t, Xt, y)p1(y)(dy) +

x(t, Xt)(

R

y2(dy))1/2.

where = (t, x) is the solution of the PDIE (7) and (1)(t,x,y) is given by(5).

Notice that taking expectations we get

(t, x) = E[g(XT)|Xt = x]+

E

Tt

f

s, (s, Xs) ,

(i)(s, Xs)

i=1

ds|Xt = x

.

Example: Consider again the very special case where we have a Poissonprocess Nt with E[Nt] = t. Set Xt = Nt t. Then the PDIE (7) reducesto

((t, x + 1) (t, x)) x

(t, x) +

t(t, x)+

f(t, (t, x) , (t, x + 1) (t, x)) = 0, (10)(T, x) = g(x).

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And we derive the nonlinear Feynman-Kac Formula:

(t, x) = E[g(XT)|Xt = x]

+E

Tt

f(s, (s, Xs), (s, Xs + 1) (s, Xs)) ds|Xt = x

.

4.3 Option Pricing

Assume a market consisting of one riskless asset (the bond) with price pro-cess given by Bt = e

rt, where r is compound interest rate, and one riskyasset (the stock), with price process:

St = S0 exp(Xt),

where Xt is a Levy process. Denote by P(dx) the probability measure ofX1.

In the last two decades several particular choices for non-Brownian Levyprocesses where proposed. Madan and Seneta [16] have proposed a Levyprocess with variance gamma distributed increments. We mention also theHyperbolic Model proposed by Eberlein and Keller (1995). In the sameyear Barndorff-Nielsen (1995) proposed the normal inverse Gaussian Levyprocess. Recently the CMGY model was introduced in Carr et al. (2000).Finally, we mention the Meixner model (see Grigelionis (1999) and Schoutens(2001)). All models give a much better fit to the data and lead to an

improvement with respect to the Black-Scholes model.We recall the density f, the cumulant generating function K, the drifta, and the Levy measure , for the Meixner Process {Mt, t 0}, for whichwe will illustrated the method.

PMeix(dx)

dx= fMeixner(x; ,,,) =

(2 cos 2 )2 e

(x)

(+ i(x) )2(2 )

,

KMeixner(; ,,,) = + 2

log cos

2 log cos +

2

,

aMeixner(,,,) = + tan

2 2

1

sinh xsinh x

dx,

Meixner(dx; ,,,) = e

x

x sinh xdx,

where > 0, < < , R, and > 0.

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From the form of the cumulant generating function one easily deduces

that the density at any time t can be calculated by multiplying the param-eters and by t for both cases.

Given our market model, let G(ST) = F(XT) denote the payoff ofthe derivative at its time of expiry T. In case of the European call withstrike price K, we have G(ST) = (ST K)+ or equivalently F(XT) =(S0 exp(XT) K)+. According to the fundamental theorem of asset pric-ing (see Delbaen and Schachermayer 1994) the arbitrage free price Vt of thederivative at time t [0, T] is given by

Vt = EQ[er(Tt)G(ST)|Ft],

where the expectation is taken with respect to an equivalent martingale

measure Q(dx) and F = {Ft, 0 t T} is the natural filtration ofX = {Xt, 0 t T}. An equivalent martingale measure is a probabilitymeasure which is equivalent (it has the same null-sets) to the given (histor-ical) probability measure and under which the discounted process {ertSt}is a martingale. Unfortunately for most models, in particular the more re-alistic ones, the class of equivalent measures is rather large and often coversthe full no-arbitrage interval. In this perspective the Black-Scholes model,where there is an unique equivalent martingale measure, is very exceptional.Models with more than one equivalent measures are called incomplete.

Our Levy model is such an incomplete model. Following Gerber and Shiu(1994) and Gerber and Shiu (1996), we can, by using the so-called Esscher

transform, easily find at least one equivalent martingale measure, which wewill use in the sequel for the valuation of derivative securities. The choiceof the Esscher measure may be justified by a utility maximizing argument(see Gerber and Shiu (1996)).

Let K be the cumulant generating function of X under the measureP(dx), and let be the solution of K( + 1) K() = r. Then, we de-fine the risk-neutral measure Q(dx) as the probability measure with theRadon-Nykodym derivative with respect to P(dx) given by Q(dx)/P(dx) =exp(x K()).

For our Meixner-example, the parameters for the Esscher transformsare easily found; explicit values for can be found in Schoutens (2001) or

Grigelionis (1999). In the Meixner case one only has to shift to + toget the density under the measure Q(dx). This means that under the risk-neutral measure Q(dx) our process Mt is again a Meixner process. In allsuch cases where the underlying process is a Levy process in the risk-neutralworld and the price Vt = V(t, Mt) at time t of a given derivative satisfies

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some regularity conditions (i.e. V(t, x) C(1,2)), the function V(t, x) canalso be obtained by solving a partial differential integral equation (PDIE)with a boundary condition:

a

xV(t, x) +

tV(t, x) +

+

V(t, x + y) V(t, x) y

xV(t, x)

Q(dy)

= rV(t, x)

V(T, x) = F(x) ,

where Q(dy) is the Levy measure of the risk-neutral distribution Q(dx).This PDIE is the analogue of the famous Black-Scholes partial differentialequation and follows from the above Feynman-Kac formula for Levy Pro-cesses. In the Meixner case, it is clear that:

Q(dx) = dexp((a + b)x/a)

x sinh(x/a)dx.

and a = aMeixner(, + ,,).

Appendix: Proofs of the Results

Proof of Theorem 1:

We define a mapping from H2T into itself such that (Y, Z) H2T is asolution of the BSDE if and only if it is a fixed point of . Given ( U, V) H

2T, we define (Y, Z) = (U, V) as follows:

Yt = E

+

Tt

f(s, Us, Vs)ds|Ft

, 0 t T,

and {Zt, 0 t T} is given by the martingale representation of Nualartand Schoutens (2000) applied to the square integrable random variable

+

T0

f(s, Us, Vs)ds,

i.e.,

+

T

0f(s, Us, Vs)ds = E

+

T

0f(s, Us, Vs)ds

+

i=1

T0

Z(i)s dH(i)s .

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Taking the conditional expectation with respect to Ft in the last identityyields

Yt +

t0

f(s, Us, Vs)ds = Y0 +

i=1

t0

Z(i)s dH(i)s ,

from which we deduce that

Yt = +

Tt

f(s, Us, Vs)ds

i=1

Tt

Z(i)s dH(i)s

and we have shown that (Y, Z) H2T solves our BSDE if and only if it is afixed point of .

Next we prove that is a strict contraction on H2

T equipped with thenorm

(Y, Z) =

E

T0

es

|Ys|2 +

i=1

|Z(i)s |2

ds

1/2,

for a suitable > 0. Let (U, V) and (U, V) be two elements ofH2T and set(U, V) = (Y, Z) and (U, V) = (Y, Z) . Denote (U , V) = (U U, V V) and (Y , Z) = (Y Y, Z Z) .

Applying Itos formula from s = t to s = T, to es (Ys Ys )2, it followsthat

et

Yt Yt 2 = Tt

es

Ys Ys2 ds2

Tt

es

Ys Ys

d(Ys Ys )

T

tesd[Y Y, Y Y]s. (11)

We have

d(Yt Yt ) = (f(t, Ut, Vt) f(t, Ut, Vt ))dt

i=1

Z(i)t Z(i)t dH

(i)t ,

d

Y Y, Y Yt

=

i=1

j=1

Z(i)t Z(i)t

Z(j)t Z(j)t

d[H(i), H(j)]t,

H(i), H(j)

t

= ij t.

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Hence, taking expectations in (11), we have

E

et

Yt Yt2

+

i=1

E

Tt

es

Z(i)s Z(i)s2

ds

= ET

tes

Ys Ys

2ds

+2E

Tt

es

Ys Ys

f(s, Us, Vs) f(s, Us, Vs )

ds

.

Using the fact that f is Lipschitz with constant C yields

E

et

Yt Yt2

+ i=1

E

T

tes(

Z(i)s Z(i)s

2 ds

ET

tes(Ys Ys)2ds

+2CE

T

tes

Ys Ys|Us Us| +

i=1

|V(i)s V(i)s |2 ds

.

If we now use the fact that for every c > 0 and a, b R we have that2ab ca2 + 1c b2 and (a + b)2 2a2 + 2b2, we obtain

E

etYt Yt 2 +

i=1

ET

tes

Z(i)s Z(i)s

2 ds

(4C2 )ET

tes

Ys Ys 2 ds

+1

2E

Tt

es

|Us Us|2 +

i=1

|V(i)s V(i)s |2

ds

.

Taking now = 4C2 + 1, and noting that etE[(Yt Yt )2] 0, we finallyderive

E

T

tes

Ys Ys 2 ds + i=1

E

T

tes (Z(i)s Z(i)s )2ds

12

E

Tt

es

|Us Us|2 +

i=1

|V(i)s V(i)s |2

ds

,

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that is,

(Y, Z)2 12 (U, V)2 ,from which it follows that is a strict contraction on H2T equipped with thenorm if = 4C2 +1. Then has a unique fixed point and the theoremis proved.

Proof of Theorem 2:

Applying Itos formula from s = t to s = T, to (Ys Ys )2, it follows that

YT Y

T2 Yt Y

t

2= 2

T

t Ys Y

s d(Ys Y

s )

+

T

td[Y Y, Y Y]s.

Taking expectations and using the relations

d(Yt Yt ) = f(t, Yt, Zt) f(t, Yt, Zt)dt

i=1

Z(i)t Z(i)t

dH

(i)t

d[Y Y, Y Y]t =

i=1

j=1

Z(i)t Z(i)t

Z(j)t Z(j)t

d[H(i), H(j)]t,

H(i), H(j)

t

= ij t,

we have

E[

Yt Yt2

] +

i=1

E

Tt

Z(i)s Z(i)s 2 ds

= E[

2]+2E

Tt

Ys Ys

f(s, Ys, Zs) f(s, Ys, Zs)

ds

.

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Using the Lipschitz property of f, and computations similar to those of

the proof of Theorem 1 we obtain

E[Yt Yt 2] + 12 E

Tt

i=1

|Z(i)s Z(i)s |2ds

E[| |2] + (1 + 2C + 2C2)ET

t|Ys Ys|2ds

+E

Tt

|f(s, Ys,Zs) f(s, Ys,Zs|2ds

.

Then by Gronwalls inequality the result follows.

Lemma 5 Leth : [0, T]R R be a random function measurable withrespect to P B

R

such that

|h(s, y)| as(y2 |y|) a.s., (12)

where{as, 0 s T} is a nonnegative predictable process such thatE[T0 a

2sds]

Schoutens - BSDE’s and Feynman-Kac Formula for L ́evy Processes with Applications in Finance

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Transcript of Schoutens - BSDE’s and Feynman-Kac Formula for L ́evy Processes with Applications in Finance