Applications of Malliavin Calculus to Monte Carlo Methods in Finance

Finance Stochast. 3, 391412 (1999)

c Springer-Verlag 1999

Applications of Malliavin calculusto Monte Carlo methods in financeEric Fournie1,2, Jean-Michel Lasry1,3, Jerome Lebuchoux3,4,Pierre-Louis Lions3,4 Nizar Touzi3,4

1 PARIBAS Capital Markets, 10 Harewood Avenue, London NW1 6AA, United Kingdom(e-mail: eric [email protected]; [email protected]; [email protected])2 Laboratoire de Probabilites, Universite Pierre et Marie Curie, 4, Place Jussieu,F-75252 Paris Cedex 5, France3 CEREMADE, Universite Paris IX Dauphine, Place du Marechal de Lattre de Tassigny,F-75775 Paris Cedex 16, France (e-mail: [email protected]; [email protected])4 Consultant Caisse Autonome de Refinancement (CDC group), 33, rue de Mogador,F-75009 Paris, France

Abstract. This paper presents an original probabilistic method for the numericalcomputations of Greeks (i.e. price sensitivities) in finance. Our approach is basedon the integration-by-parts formula, which lies at the core of the theory of vari-ational stochastic calculus, as developed in the Malliavin calculus. The Greeksformulae, both with respect to initial conditions and for smooth perturbations ofthe local volatility, are provided for general discontinuous path-dependent payofffunctionals of multidimensional diffusion processes. We illustrate the results byapplying the formula to exotic European options in the framework of the Blackand Scholes model. Our method is compared to the Monte Carlo finite differenceapproach and turns out to be very efficient in the case of discontinuous payofffunctionals.

Key words: Monte Carlo methods, Malliavin calculus, hedge ratios and Greeks

JEL classification : G13Mathematics Subject Classification (1991):60H07, 60J60, 65C05

1 Introduction

In frictionless markets, the arbitrage price of most financial derivatives (European,Asian, etc. ...) can be expressed as expected values of the associated payoff whichis usually defined as a functional of the underlying asset process.

Manuscript received: July 1997; final version received: September 1998

392 E. Fournie et al.

In this paper, we will show how one can use Malliavin calculus to deviseefficient Monte Carlo methods for these expected values and their differentials.Other applications of Malliavin calculus for numerical figure and risk manage-ment will appear in companion papers.

In order to precise our goal, we need to introduce some mathematical no-tations. The underlying assets are assumed to be given by fX (t); 0 t Tgwhich is a markov process with values in IRn and whose dynamics are describedby the stochastic differential equation

dX (t) = b(X (t)) dt + (X (t)) dW (t) ; (1)

where fW (t); 0 t Tg is a Brownian motion with values in IRn . The coef-ficients b and are assumed to satisfy usual conditions in order to ensure theexistence and uniqueness of a continuous adapted solution of equation (1).

Given 0 < t1 : : : tm = T , we consider the function

u(x ) = IE (X (t1); : : : ;X (tm )) j X (0) = x ; (2)where satisfies some technical conditions to be described later on. In financialapplications, u(x ) describes the price of a contingent claim defined by the pay-off function involving the times (t1; : : : ; tm ). Examples of such contingentclaims include both usual and path dependent options and sophisticated objectssuch as MBS or CMOs. The function u(x ) can then be computed by MonteCarlo methods. However, financial applications require not only to compute thefunction u(x ) but also to compute its differentials with respect to the initialcondition x , the drift coefficient b and the volatility coefficient .

A natural approach to this numerical problem is to compute by Monte Carlosimulation the finite difference approximation of the differentials. To simplifythe discussion, let us specialize it to the case of the Delta, i.e. the derivativewith respect to the initial condition x . Then, one has to compute a Monte Carloestimator of u(x ) and a Monte Carlo estimator for u(x + ") for some small ";the Delta is then estimated by [u(x + ") u(x )]=". If the simulations of the twoestimators are drawn independently, then it is proved in Glynn (1989) that thebest possible convergence rate is typically n1/4. Replacing the forward finitedifference estimator by the central difference [u(x +")u(x ")]=(2") improvesthe optimal convergence rate to n1/3. However, by using common random num-bers for both Monte Carlo estimators, one can achieve the convergence rate n1/2which is the best that can be expected from (ordinary) Monte Carlo methods, seeGlasserman and Yao (1992), Glynn (1989) and LEcuyer and Perron (1994). Animportant drawback of the common random numbers finite difference method isthat it may perform very poorly when is not smooth enough, as for instance ifone computes the delta of a digital or the gamma of European call options.

An alternative method which allows to achieve the n1/2 convergence rateis suggested by Broadie and Glasserman (1996) : for simple payoff functionals, an expectation representation of the Greek of interest can be obtained bysimple differentiation inside the expectation operator; the resulting expectation is

Monte Carlo simulations and Malliavin calculus 393

estimated by usual Monte Carlo methods. An important limitation of this methodis that it can only be applied to simple payoff functionals.

In this paper, using Malliavin calculus we will show that all the differentialsof interest can be expressed as

IE (X (t1); : : : ;X (tm )) j X (0) = x

; (3)

where is a random variable to be determined later on. Therefore, the requireddifferential can be computed numerically by Monte Carlo simulation and theestimator achieves the n1/2 usual convergence rate. An important advantageof our differential formula is that the weight does not depend on the payofffunction .

While the aim of this paper is to design efficient numerical scheme, let uspoint out a theorical aspect of the formulations (3) that our use of Malliavincalculus leads us to set up. As it is well known, risk neutral probability is thetechnical tool by which one introduces observed market prices in a given model :this is done in practice through a calibration process, i.e. the computation of theArrow-Debreu prices over future various states of the world. Hence asset pricescan be written

price = IEQ0pay-offs

;

where price is todays value of the contingent claim, IEQ0 is the expected valueunder the risk neutral probability Q0, and the discounted pay-offs are the futurecontingent cash amounts. Hedging is trying to protect the portfolio against atleast some of the possible changes. But changes in market will come throughthe calibration process as changes of the risk neutral probability Q . So marginalchanges of Q will lead to new prices according to

variation of prices = new price old price;= IEQ

pay-offs

IEQ0 pay-offs ;= IEQ0

pay-offs ;

where is =

dQ dQ0dQ0 :

Now suppose that the probability Q lies within a parametrized family (Q), = (1; : : : ; n ). In the typical diffusion case studied in this paper Q is parame-terized by the drift and the volatility functions which may be specified in someparameterized family. Then the marginal moves of the market can be assessedthrough the derivatives

@

@i(price) = IEQ0

pay-offs i

: (4)

where G = dQdQ0 and i =Gi

, i.e i is the logarithmic derivative of Q at Q0 inthe i direction. Our use of Malliavin calculus helps to set up the formula (4)and other various formulas based on the various derivatives or primitives of the


pay offs. But even if in many cases, it might be analytically easier to start withthe formula of derivative before (4), our opinion is that (4) is likely to be a morefundamental hedging formula than other ones.

Finally let us observe that the case of stochastic interest rates is easily acco-modated in the framework of this paper by working under the so-called forwardmeasure or by extending the state space to include the additional state variableexp

R t0 r(u)du .

The paper is organized as follows. We first present in Sect. 2 a few basics ofMalliavin calculus. Then, in Sect. 3, we derive the formulae for various differ-entials which correspond to the quantities called greeks in Finance. These caseshave to be seen as an illustration of a general method which can be adaptedand applied to all other pratical differentials. Finally, Sect. 4 is devoted to somenumerical examples and further comments on the operational use of our method.

2 A primer of Malliavin calculus for finance

This section briefly reviews the Malliavin calculus and presents the efficient rulesto use it in financial examples (see Nualart [9] for other expositions).

The Malliavin calculus defines the derivative of functions on Wiener spaceand can be seen as a theory of integration by parts on this space. Thanks tothe Malliavin calculus, we can compute the derivatives of a large set of randomvariables and processes (adapted or not to the filtration) defined on the Wienerspace. We present the following notations which shall be used in the rest of thepaper.

Let fW (t); 0 t Tg be a n-dimensional Brownian motion defined on acomplete probability space (;F ;P ) and we shall denote by fFtg the augmen-tation with respect to P of the filtration generated by W . Let C be the set ofrandom variables F of the form :

F = fZ 1

0h1(t)dW (t); : : : ;

Z 10

hn (t)dW (t); f 2 S (IRn )

where S (IRn ) denotes the set of infinitly differentiable and rapidly decreasingfunctions on IRn and h1; : : : ; hn 2 L2( IR+). For F 2 C , the Malliavinderivative DF of F is defined as the process fDtF , t 0g of L2( IR+) withvalues in L2(IR+) which we denote by H :

DtF =nXi=1

@f@xi

Z 10

h1(t)dW (t); : : : ;Z 10

hn (t)dW (t)hi (t); t 0 a:s:

We also define the norm on C

kFk1,2 =(IE(F 2)1/2 + IE(Z 1

0(DtF )2dt)

1/2;


Then ID1,2 denotes the banach space which is the completion of C with respectto the norm k k1,2. The derivative operator D (also called the gradient operator)is a closed linear operator defined in ID1,2 and its values are in L2( IR+).

The next result is the chain rule for the derivation.

Property P1. Let : IRn ! IR be a continuously differentiable function withbounded partial derivatives and F = (F1; : : : ;Fn ) a random vector whose com-ponents belong to ID1,2. Then (F ) 2 ID1,2 and :

Dt(F ) =nXi=1

@

@xi(F )DtFi ; t 0 a:s:

In the case of a Markov diffusion process, the Malliavin derivative operator isclosely related to the derivative of the process with respect to the initial condition.

Property P2. Let fX (t), t 0g be an IRn valued Ito process whose dynamics aredriven by the stochastic differential equation :

dX (t) = b(X (t)) dt + (X (t)) dW (t) ;

where b and are supposed to be continuously differentiable functions withbounded derivatives. Let fY (t), t 0g be the associated first variation processdefined by the stochastic differential equation :

dY (t) = b0(X (t))Y (t) dt +nXi=1

0i (X (t))Y (t) dW i (t); Y (0) = In ;

where In is the identity matrix of IRn , primes denote derivatives and i is thei -th column vector of . Then the process fX (t), t 0g belongs to ID1,2 and itsMalliavin derivative is given by :

DsX (t) = Y (t)Y (s)1(X (s))1fstg; s 0 a:s:

Hence, if 2 C 1b (IRn ) then we haveDs (XT ) = r (XT )Y (T )Y (s)1(X (s))1fsTg; s 0 a:s:

and also

DsZ T0

(Xt ) dt =Z Ts

r (Xt )Y (t)Y (s)1(X (s)) dt a:s:

The divergence operator (also called Skorohod integral) associated with thegradient operator D exists. The following integration by parts formula definesthis divergence operator.

Property P3. Let u be a stochastic process. Then u 2 Dom() if for any 2 ID1,2,we have


IE(< D; u >H ) := IE(Z 10

Dt u(t) dt) C (u) kk1,2:

If u 2 Dom(), we define (u) by:IE( (u)) = IE(< D; u >H ) for any 2 ID1,2 :

The stochastic process u is said to be Skorohod integrable if u 2 Dom(). Oneof the most important properties of the divergence operator is that its domainDom() contains all adapted stochastic processes which belong to L2( IR+);for such processes, the divergence operator coincides with the Ito stochasticintegral.

Property P4. Let u be an adapted stochastic process in L2( IR+). Then, wehave:

(u) =Z 10

[u(t)] dW (t) ;

Moreover, if the random variable F is FTadapted and belongs to ID1,2 thenfor any u in dom(), the random variable Fu will be Skohorod integrable. Wehave the following property.

Property P5. Let F be an FTadapted random variable which belongs to ID1,2then for any u in dom() we have:

(Fu) = F (u) Z T0

DtF u(t) dt :

Finally, we recall the Clark-Ocone-Haussman formula.

Property P6. Let F be a random variable which belongs to ID1,2. Then we have

F = IE(F ) +Z T0

IE(DtF j Ft ) dW (t) a:s:

The latter property shows that the Malliavin derivative provides an identificationof the integrator in the (local) martingale representation Theorem in a Brown-ian filtration framework, which plays a central role in financial mathematics.Therefore, in frictionless markets, the hedging portfolio is naturally related tothe Malliavin derivative of the terminal payoff.

3 Greeks

We assume that the drift and diffusion coefficients b and of the diffusion pro-cess fX (t), 0 t Tg are continuously differentiable functions with boundedLipschitz derivatives in order to ensure the existence of a unique strong solution.Under the above assumptions on the coefficients b and and using the theoryof stochastic flows, we may choose versions of fX (t), 0 t Tg which are


continuously differentiable with respect to the initial condition x for each (t ; !)2 [0;T ] (see e.g. Protter 1990, Theorem 39 p250). We denote by fY (t),0 t Tg the first variation process associated to fX (t), 0 t Tg definedby the stochastic differential equation :

Y (0) = In (1)

dY (t) = b0(X (t))Y (t)dt +nXi=1

0i (X (t))Y (t)dWi (t) (2)

where In is the identity matrix of IRn , the primes denote derivatives and i is thei -th column of . Moreover, we need another technical assumption.

Assumption 3.1 The diffusion matrix satisfies the uniform ellipticity condition :9" > 0; (x )(x ) " j j2 for any ; x 2 IRn :

Since b0 and 0 are assumed to be Lipschitz and bounded, the first vari-ation process lies in L2( [0;T ]), see e.g. Karatzas and Shreve (1988)Theorem 2.9 p289, and therefore Assumption 3.1 insures that the process1(X (t))Y (t); 0 t T} belongs to L2( [0;T ]). Moreover, if the func-

tion is bounded then the process f1(X (t)), 0 t Tg will belong toL2( [0;T ]) and 1 is a bounded function.

3.1 Variations in the drift coefficientIn this section, we allow the payoff function to depend on the whole samplepath of the process fX (t), 0 t Tg. More precisely, let be some functionmapping the set C [0;T ] of continuous functions on the interval [0;T ] into IRand satisfying

IE(X (:))2 < 1 : (3)

Next, consider the perturbed process fX (t), 0 t Tg defined bydX (t) = b(X (t)) + "(X (t)) + (X (t))dW (t) ; (4)

where " is a small real parameter and is a bounded function from [0;T ] IRninto Rn . To simplify notations, we shall denote by fX (t), 0 t Tg the non-perturbed process corresponding to " = 0. We also introduce the random variable

Z (T )=exp"Z T0< 1(X (t)); dW (t) >"

2

2

Z T0

k1(X (t))k2dt: (5)

From the boundedness of 1, we have that IE[Z (T )] = 1 for any " > 0 sincethe Novikov condition is trivially satisfied. Now, consider the expectation

u(x ) = IE (X (:))jX (0) = x : (6)The following result then gives an expression of the derivative of u(x ) withrespect to " in " = 0.


Proposition 3.1 The function " 7! u(x ) is differentiable in " = 0, for any x 2 IRn ,and we have :

@

@"u(x )

=0

= IE(X (:))

Z T0< 1(X (t)); dW (t) >

X (0) = x:

Proof. Since IE[Z (T )] = 1, the probability measure Q defined by its Radon-Nikodym derivative dQ=dP = Z (T ) is equivalent to P and we have :

u(x ) = IEQ Z (T )(X (:)) X (0) = x ;where Z (T ) = exp

h" R T0 22 R T0 k1(X (t))k2dti

and fW (t), 0 t Tg is defined by W (t) = W (t) + " R t0 1(X (t))dt is aBrownian motion under Q, by the Girsanov Theorem. Let us observe that thejoint distribution of (X (:);W (:)) under Q coincides with the joint distributionof (X (:);W (:)) under P and therefore :

u(x ) = IE Z (T )(X (:)) j X (0) = x :Now, let us notice that we have

1"(Z (T ) 1) =

Z T0

Z (t) < 1(X (t)); dW (t) >

so that

1"(Z (T ) 1) !

Z T0< 1(X (t)); dW (t) > in L2:

Therefore, by the Cauchy-Schwarz inequality and using (3), we get : 1 (u(x ) u(x )) IE h(X (:)) R T0 < 1(X (t)); dW (t) >i

K IE

1 (Z (T ) 1)

R T0 <

1(X (t)); dW (t) >2

where K is a constant. This provides the required result.

Remark 3.1 The same kind of arguments as in the previous proof can be usedto obtain similar expressions for higher derivatives of the expectation u(x ) withrespect to " in " = 0 as a weighted expectation of the same functional; the weightsbeing independent of the payoff functional.

Remark 3.2 The result of Proposition 3.1 does not require the Markov feature ofthe process fX (t), 0 t Tg. The arguments of the proof go on even if b, and are adapted processes.


3.2 Variations in the initial condition

In this section, we provide an expression of the derivatives of the expectationu(x ) with respect to the initial condition x in the form of a weighted expectationof the same functional. The payoff function is now a mapping from (IRn )m intoIR with

E (X (t1); : : : ;X (tm ))2

< 1:

for a given integer m 1 and 0 < t1 : : : tm T , where IEx (:) = IE(:jX (0) =x ). The expectation of interest is

u(x ) = IEx [(X (t1); : : : ;X (tm ))] ; (7)We shall denote by ri the partial derivative with respect to the i -th argumentand we introduce the set m defined by :

m =

a 2 L2([0;T ]) j

Z ti0

a(t) dt = 1 8i = 1; : : : ;m

Proposition 3.2 Under Assumption 3.1, for any x 2 IRn and for any a 2 m, wehave :

ru(x ) = IEx(X (t1); : : : ;X (tm ))

Z T0

a(t)[1(X (t))Y (t)]dW (t): (8)

Proof. (i) Assume that is continuously differentiable with bounded gradient;the general case will be proved in (ii) by density argument. We first prove thatthe derivative of u(x ) with respect to x is obtained by differentiating inside theexpectation operator. Indeed, since is continuously differentiable, we have that

1khk(Xx (t1); : : : ;Xx (tm )) (Xx+h (t1); : : : ;Xx+h (tm ))

1khk

Pmi=1 ri (X (t1); : : : ;X (tm ))Y (ti ); h

(9)converges to zero a.s. as h goes to zero. The second term of the last expressionis uniformly integrable in h since the partial derivatives of the payoff function are assumed to be bounded. Denoting by h the first term, it is easily seen that :

k hk MkXj=1

Xx (tj ) Xx+h (tj )khk ;

where M is a uniform bound on the partial derivatives of . The uniform inte-grability of the right hand side term of the last inequality follows from Protter(1990, p246) and implies the uniform integrability of (9) which then convergesto zero in the sense of the L1() norm, by the dominated convergence Theorem.This proves that :

ru(x ) = IEx"

mXi=1

ri (X (t1); : : : ;X (tm ))Y (ti )#:


Now, by Property P2, the process fX (t); 0 t Tg belongs to ID1,2. Besides,one can easily check that for all i 2 f1; : : : ;mg and for all t 2 [0;T ] we haveDtX (ti ) = Y (tj )Y (t)1(t) 1fttig. This shows that :

Y (ti ) =Z T0

DtX (ti ) a(t)1(t)Y (t) dt 8a 2 m

ru(x ) = IEx"Z T

0

mXi=1

ri (X (t1); : : : ;X (tm ))DtX (ti ) a(t)1(t)Y (t)dt#

and by the chain rule Property P2, we obtain :

ru(x ) = IExZ T

0Dt(X (t1); : : : ;X (tm ))a(t)1(t)Y (t) dt

Now, for a function a fixed in m , we define the fF (t)g adapted process fv(t),0 t Tg by :

v(t) = a(t)1(X (t))Y (t);which belongs to L2( [0;T ]) by Assumption 3.1. Then,

ru(x ) = IExZ T

0Dt(X (t1); : : : ;X (tm )) v(t)dt

and the result follows from a direct application of the Malliavin integration byparts, see Property P3.(ii) We now consider the general case 2 L2. Since the set C1K of infinitelydifferentiable functions with compact support is dense in L2, there exists a se-quence (n )n C1K converging to in L2. Let un (x ) = IE [n (X (t1); : : : ;X (tm ))]and

"n (x ) = IE [n (X (t1); : : : ;X (tm )) (X (t1); : : : ;X (tm ))]2 :First it is clear that

un (x ) ! u(x ) for all x 2 IRn : (10)Next denote by g(x ) the function on the right hand-side of (8). Applying (i) tofunction n and using Cauchy-Schwartz inequality, we see that :

jrun (x ) g(x )j "n (x ) (x );

where (x ) = IEhR T

0 a(t)[1(X (t))Y (t)]dW (t)i2. By a continuity argument of

the expectation operator, this proves that :

supx2K

jrun (x ) g(x )j "n (x ) (x ) for some x 2 K ;

where K is an arbitrary compact subset of IRn which provides :


run (x ) ! g(x ) uniformly on compact subsets of IRn : (11)From (10) and (11), we can conclude that function u is continuously differentiableand that ru = g.

3.3 Variations in the diffusion coefficient

In this section, we provide an expression of the derivatives of the expectation u(x )with respect to the diffusion coefficient in the form of a weighted expectationof the same functional. As in the previous section, the coefficients b and defining the diffusion process fX (t), 0 t Tg are assumed to be continuouslydifferentiable and with bounded derivatives. Also, the payoff function is assumedto be path dependent and has finite L2 norm. We start by introducing the set ofdeterministic functions

m =

(a 2 L2([0;T ]) j

Z titi1

a(t) dt = 1; for i = 1 : : :m):

Let : IRn ! IRnn be continuously differentiable function with boundedderivatives. The function and the function are assumed to satisfy the follow-ing condition.

Assumption 3.2 The diffusion matrix + " satisfies the uniform ellipticity con-dition for any " :

9 > 0; ( + ")(x )( + ")(x ) j j2 for any ; x 2 IRn :In order to evaluate the Gateaux derivative of the expectation u(x ) with respectto the diffusion matrix in the direction , we consider the process fX (t),0 t Tg defined by :

X (0) = xdX (t) = b(X (t))dt + (X (t)) + "(X (t)) dW (t): (12)

We also introduce the IRn valued variation process of the process with respect to" :

Z (0) = 0ndZ (t) = b0(X (t))Z (t)dt + (X (t))dW (t)

+nXi=1

[0i + "0i ](X (t))Z (t)dWi (t); (13)

where 0n is the zero column vector of IRn . As in the previous section, we simplyuse the notation X (t), Y (t) and Z (t) for X 0(t), Y 0(t) and Z 0(t). Next, considerthe process f(t), 0 t Tg defined by :


(t) = Z (t)Y 1(t); 0 t T a:s: (14)This process satisfies the following regularity assumption.

Lemma 3.1 The process f(t); 0 t Tg belongs to ID1,2.The process fY 1(t); 0 t Tg satisfies

Y 1(0) = In

dY 1(t) = Y 1(t)"b0(X (t)) +

nXi=1

0i (X (t))

2# dtY 1(t)

nXi=1

0i (X (t))dW i (t):

By Lemma 2.2.2 p104 in Nualart [9], the process fY 1(t); 0 t Tg belongsto ID1,2. We also prove by the same argument that the process fZ (t); 0 t Tg is in ID1,2 . The required result follows from a direct application of theCauchy-Schwarz inequality.

Proposition 3.3 Under Assumption 3.2, for any a in m we have :@

@"u(x )

=0

= Ex(X (t1); : : : ;X (tm ))

(1(X )Y a (T )

where

a (t) =mXi=1

a(t) ((ti ) (ti1)) 1fti1ttig

and where (1(X )Y a (T )

is the Skorohod integral of the anticipating process

1(X (t))Y (t) a (T ) ; 0 t T}:

Proof. Proceeding as in the proof of Proposition 3.2, it is clear that it suffices toprove the result for continuously differentiable function with bounded deriva-tive; the general result follows from a density argument as in (ii) of the proofof Proposition 3.2. We first prove that the derivative of u(x ) with respect to "is obtained by differentiating inside the expectation operator. Considering " asa degenerate process, we can apply Theorem 39 p250 in Protter (1990) whichensures that we can choose versions of fX (t), 0 t Tg which are contin-uously differentiable with respect to " for each (t ; !) 2 [0;T ] . Since iscontinuously differentiable, we prove by the same arguments that we have in thesense of the L1 norm:

@

@"u(x )

=0

= Ex"

mXi=1

ri (X (t1); : : : ;X (tm ))Z (ti )#: (15)

Using Property P2, we have DtX (ti ) = Y (tj )Y (t)1(t) 1fttig for any i 2f1; : : : ;mg and for any t 2 [0;T ]. Hence, for all i 2 f1; : : : ;mg we have


Z T0

DtX (ti )1(t)Y (t) a (T ) dt =Z ti0

Y (ti ) a (T ) dt (16)

= Y (ti )iX

k=1

Z tktk1

a(t) ((tk ) (tk1)) dt!

Since a belongs to m , the right-hand side of (16) can be simplified in Y (ti )(ti )which is equal to Z (ti ) according to the definition (14). This shows that :

@

@"u(x )

=0

= Ex"Z T

0

mXi=1

ri (X )DtX (ti )1(X (t))Y (t) a (T )dt#(17)

Using again Property P2, the expression (17) of the derivative of the expectationu(x ) can be rewritten in

@

@"u(x )

=0

= IExZ T

0Dt(X (t1); : : : ;X (tm ))1(X (t))Y (t) a (T ) dt

Finally, we define the fFTg adapted process fu(t), 0 t Tg by :v(t) = 1(X (t))Y (t) a (T );

Since the process f1(X (t))Y (t); 0 t Tg belongs to L2( [0;T ]) andis fFtg adapted and since we have proved in Lemma 3.1 that a (T ) is in ID1,2(recall that a is a deterministic function) and is fFTg adapted, we can apply theProperty P5. It follows that the Skorohod integral of the product process v exists.More precisely, we have

(v) = a (T )Z T0

[1(X (t))Y (t)] dW (t) Z T0

Dt a (T )1(X (t))Y (t) dt

Then, we can apply the Malliavin integration by parts property to obtain therequired result.

Remark 3.3 The same kind of arguments as in the proof of Proposition 3.3 (resp.Proposition 3.2) can be used to obtain similar expressions for higher derivativesof the expectation u with respect to " in " = 0 (resp. with respect to the initialcondition) as a weighted expectation of the same functional; the weights beingindependent of the payoff functional.

Remark 3.4 We can also extend our results to the case of a payoff function which is a function of the mean value of the process fX (t); 0 t Tg. We givethe formula for the derivative with respect to the initial condition in dimensionone. The function u is defined by

u(x ) = IEx

Z T0

X (t)dt


In this case, we have

u 0(x ) = IEx"

Z T0

X (t)dt

2Y 2(t)(X (t))

Z T0

Y (s)ds1!#

4 Numerical experiments

This section presents some simple examples which illustrate the results obtainedin the previous sections.

We consider the famous Black and Scholes model, i.e. a one dimensionalmarket model which consists of a risky asset S and a non-risky one with deter-ministic instantaneous interest rate r(t). Let (;F ;Q ; (Ft ); ( Wt ) be a standardWiener process on IR. Then, it is well known, under mild conditions on the coef-ficients of the SDE driving the price process, that there exists a unique equivalentprobability measure P such that the Pdynamic of the price process is

dS (t)S (t) = r(t) dt + dW (t); S0 = x : (18)

In this framework, most problems of pricing contingent claims are solved bycomputing the following mathematical expectation :

u(0; x ) = IE[eR T0

r(t) dt(S 0,x (T ))] (19)

where is a payoff functional.In practice, the hedging of the contingent claim requires the computation of

the Greeks, i.e. the derivatives of the value function u , @u@x

,

@2u

@x 2,

@u

@, etc. By

using the general formulae developed in the previous section, we are able tocompute analytically the values of the different Greeks without differentiatingneither the value function nor the payoff functional.

In this Black and Scholes framework, the tangent process Y follows, Pa.s.,the stochastic differential equation

dYt = r(t)Yt dt + Yt dWt ; Y0 = 1

and so, we have xYt = St ;8 0 t T ;P a:s:In our first example, we consider a functional which depends only on the

terminal value ST of the risky asset, the so called European case. First, we cancompute easily an extended rho, i.e. the directional derivative of the function ufor a perturbation r(t) of the yield r(t). As was shown in the previous sections,it is a trivial application of the Girsanov Theorem. We have the following result

rhor(t) = IEe

R T0

r(t) dt(ST )

Z T0

r(t)St

dWt

IEZ T

0r(t) dt e

R T0

r(t) dt(ST )

:


For the delta, i.e. the first derivative w.r.t. the initial condition x , we have

to compute an Ito stochastic integralZ T0

a(t) YtSt

dWt where a must satisfy,R T0 a(t) dt = 1. A trivial choice is a(t) = 1T ;80 t T . Then we get theformula

@u

@x(0; x ) = IE

e

R T0

r(t) dt(ST ) WT

xT

:

A straightforward computation, using again the integration-by-parts formula,gives for the gamma (the second derivatives w.r.t. the price) the following for-mula

@2u

@x 2(0; x ) = IE

e

R T0

r(t) dt(ST ) 1

x 2T

W 2TT

WT 1

;

where we also chose a(t) = 1=T .For the vega, the derivative w.r.t. the volatility parameter , direct application

of the formula developed in the previous section again with a(t) = 1=T , provides :

@u

@(0; x ) = IE

e

R T0

r(t) dt(ST )

W 2TT

WT 1

:

To illustrate these formulae, we consider the case of a European digital optionwhose payoff function at time T of the form (x ) = 1[a,b](x ). We compute thevalues of the previous derivatives with a standard quasi Monte Carlo numericalprocedure based on the use of low discrepancy sequences. More precisely, wecompute the values of the Greeks delta, gamma, vega for a digital option withpayoff function (x ) = 1[100,110](x ) with parameters values x = 100, r = 0:1, = 0:2, T = 1 year.

As a second example, we present an application of the integration-by-partsformulas by computing the Greeks for an exotic option. We consider the case ofan asian option with payoff of the form (R T0 St dt). In the Black and Scholesmodel, a straightforward calculus using the formula given in Remark 3.4 givesfor the delta

@u

@x(0; x ) = IE

"e

R T0

r(t) dt(Z T0

St dt)

2x

R T0 Yt dWtR T0 Yt dt

+1x

!#:

As a third example, we are able to extend our result to more complicatedpayoff depending for example on the mean and terminal values of the underlyingasset, like (ST ;

R T0 St dt). Let us define, an asian barrier in option with payout

(x ; y) = 1fyBg(x K )+. We obtain for the delta the following formula

@u

@x(0; x ) = IE

e

R T0

r(t) dt

ST ;Z T0

St dt(G)

;

where G is the random process


-0.0017

-0.0016

-0.0015

-0.0014

-0.0013

-0.0012

-0.0011

-0.001

-0.0009

-0.0008

-0.0007

-0.0006

0 2000 4000 6000 8000 10000

"delta.res"-0.001335

Fig. 1. Delta for a digital option with pay-off 1[100,110] with x = 100, r = 0.1, = 0.2, T = 1 year.We use low discrepency Monte Carlo generation.

-0.00045

-0.00044

-0.00043

-0.00042

-0.00041

-0.0004

-0.00039

-0.00038

-0.00037

-0.00036

-0.00035

-0.00034

0 2000 4000 6000 8000 10000

"gamma.res"-0.0003887

Fig. 2. Gamma for a digital option with pay-off 1[100,110] with x = 100, r = 0.1, = 0.2, T = 1year. We use low discrepency Monte Carlo generation.


-0.9

-0.88

-0.86

-0.84

-0.82

-0.8

-0.78

-0.76

-0.74

-0.72

-0.7

-0.68

0 2000 4000 6000 8000 10000

"vega.res"-0.777853

Fig. 3. Vega for a digital option with pay-off 1[100,110] with x = 100, r = 0.1, = 0.2, T = 1 year.We use low discrepency Monte Carlo generation.

0.3

0.35

0.4

0.45

0.5

0.55

0.6

0.65

0.7

0.75

0.8

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

"delta.res"0.724

Fig. 4. Delta for a asian option with pay-off (R T0 Ss ds K )+ with x = 100, r = 0.1, = 0.2, T = 1

year, K = 100. We use standard Monte Carlo generation.


G(s) = (a + s) YsSs

+ (b + s) 2Y2s

SsR T0 Su du

with

a =2 < s > 1

(2 < s > 1)2 + (2 < s2 > 1)2

=4 < s2 > 2

(2 < s > 1)2 + (2 < s2 > 1)2

b = 12

< s2 > + < s > 1

(2 < s > 1)2 + (2 < s2 > 1)2 = = 0

and < s >=R T0 uSu duR T0 Su du

and < s2 >=R T0 u

2Su duR T0 Su du

.

A trivial computation in the case of the standard Wiener process (S = W )with T = 1 gives (G) = 4W1 6

R 10 s dWs . Further analysis shows this G is

optimal in the sense that it minimizes on L2 the variance of the random variableWT ;

R T0 Wt dt

(G) as we will prove in a forthcoming paper.

0.2

0.25

0.3

0.35

0.4

0.45

0.5

0 2000 4000 6000 8000 10000 12000 14000 16000 18000 20000

"delta.res"0.465

Fig. 5. Delta for a complex option with pay-off 1{R 10

Ws dsB}(W1 K )+. We use standard Monte

Carlo generation.

At this stage, we wish to observe that the Malliavin integration-by-partswhich yields the above formulae, creates weights which involve powers of, say,the Brownian motion. These global weights in fact may slow down Monte


Carlo simulations and we now suggest a cure for this difficulty. The idea is tolocalize the integration-by-parts around the singularity.

In order to be more specific, let us consider the delta of a call option in theBlack and Scholes model, i.e.

@

@xIEe

R T0

r(t) dt (ST K )+

= IEe

R T0

r(t) dt 1(ST>K ) YT

= IEe

R T0

r(t) dt (ST K )+ WTxT

:

The term (ST K )+WT is very large when WT is large and has a largevariance. The idea to solve this difficulty is to introduce a localization aroundthe singularity at K . More precisely, we set for > 0

H(s) = 0; if s K ;=

s (K )2

; if K s K + ;= 1; if s K +

and G(t) =R t

1 H(s) ds; F(t) = (t K )+ G(t). Then, we observe that wehave

@

@xIEe

R T0

r(t) dt (ST K )+

=

@

@xIEe

R T0

r(t) dt G(ST )+@

@xIEe

R T0

r(t) dt F(ST )

= IEe

R T0

r(t) dt H(ST )YT+ IE

e

R T0

r(t) dt F(ST ) WTxT

:

Notice that F vanishes for s K and for s K + and thus F(ST )WTvanishes when WT is large.

A similar idea can be used for all the Greeks. For example, we have for thegamma

@2

@x 2IEe

R T0

r(t) dt (ST K )+

= IEe

R T0

r(t) dtK (ST )Y 2T

= IEe

R T0

r(t) dt I(ST )Y 2T

+IEe

R T0

r(t) dt F(ST ) 1x 2T

W 2TT

WT 1

where I(t) = 12 1jtK j


Fig. 6. Gamma of a call option computed by global and localized Malliavin like formula. Theparameters are S0 = 100, r = 0.1, = 0.2, T = 1,K = 100 and = 10 (localization parameter). Weuse low discrepency sequences.

N = 10 000 exact MCFD MCMALLDelta call 0.725747 0.725639 0.725660 (loc.)Gamma call 0.016660 0.015330 0.016634 (loc.)Vega call 33.320063 33.250709 33.267145 (loc.)Delta digital -0.001335 -0.003167 -0.001335Gamma digital -0.000389 +0.099532 -0.000389Vega digital -0.777516 -0.542902 -0.778695Delta average call 0.649078 0.660177 0.654369 (loc.)

We conclude the paper by presenting a benchmark comparing Monte Carlosimulations based on the finite difference approximation of the Greeks and ourlocalized Malliavin calculus approach. The finite difference scheme is the fol-lowing : set u(x ; ) = IE (ST )jS0 = x, we have the approximations

delta = u(x + h; ) u(x h; )2h

gamma = u(x + h; ) 2u(x ; ) + u(x h; )h2sigma = u(x ; + ) u(x ; )

2

We compare the values obtained by those two methods for a given number (10000) of Brownian trajectories with the exact values. Of course, we use the sameBrownian trajectories for the different initial conditions x + h; x ; x h which


0.01

0.011

0.012

0.013

0.014

0.015

0.016

0.017

0.018

0.019

0.02

0 1000 2000 3000 4000 5000 6000 7000 8000 9000 10000

Finite differenceLocalized Malliavin formula

Exact value = 0.0166Exact value + 1%Exact value - 1%

Fig. 7. Gamma of a call option computed by finite difference and localized Malliavin like formula.The parameters are S0 = 100, r = 0.1, = 0.2, T = 1,K = 100 and = 10 (localization parameter).We use low discrepency sequences.

0.62

0.63

0.64

0.65

0.66

0.67

0.68

0 5000 10000 15000 20000 25000 30000 35000 40000 45000 50000

Finite differenceGlobal Malliavin formula

Localized Malliavin formulaExact value = 0.649

Exact value - 1%Exact value + 1%

Fig. 8. Delta of an average call option computed by finite difference, global and localized Malliavinlike formula. The parameters are S0 = 100, r = 0.1, = 0.2, T = 1,K = 100 and = 10 for thelocalization parameter. We use pseudo random sequences.


gives a natural variance reduction to the finite difference method; see also thediscussion in the introduction. Figures 7 and 8 give an idea of the number ofpaths required in order to achieve a given precision of 1%.

Acknowledgements. Most of this work was done while J.M. Lasry was Chairman of Caisse Autonomede Refinancement.

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Applications of Malliavin Calculus to Monte Carlo Methods in Finance

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