Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
IOSR Journal of Mathematics (IOSR-JM) e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 30-41 www.iosrjournals.org Stochastics Calculus: Malliavin Calculus in a simplest way 1 Udoye, Adaobi Mmachukwu, 2. Akoh, David, 3. Olaleye, Gabriel C. Department of Mathematics, University of Ibadan, Ibadan. Department of Mathematics, Federal Polytechnic, Bida Department of Mathematics, Federal Polytechnic, Bida. Abstract: We present the theory of Malliavin Calculus by tracing the origin of this calculus as well as giving a simple introduction to the classical variational problem. In the work, we apply the method of integration-byparts technique which lies at the core of the theory of stochastic calculus of variation as provided in Malliavin Calculus. We consider the application of this calculus to the computation of Greeks, as well as discussing the calculation of Greeks (price sensitivities) by considering a one dimensional Black-Scholes Model. The result shows that Malliavin Calculus is an important tool which provides a simple way of calculating sensitivities of financial derivatives to change in its underlying parameters such as Delta, Vega, Gamma, Rho and Theta. I. Introduction The Malliavin Calculus also known as Stochastic Calculus of Variation was first introduced by Paul Malliavin as an infinite-dimensional integration by parts technique. This calculus was designed to prove results about the smoothness of densities of solutions of stochastic differential equations driven by Brownian motion. Malliavin developed the notion of derivatives of Wiener functional as part of a programme for producing a probabilistic proof of the celebrated Hörmander theorem, which states that solutions to certain stochastic differential equations have smooth transition densities. Classical variational problems are problems that deal with selection of path from a given family of admissible paths in order to minimize the value of some functionals. The calculus of variation originated with attempts to solve Dido’s problem known as the isoperimetric problem. An infinite dimensional differential calculus on the Wiener space, known as Malliavin Calculus, was initiated by Paul Malliavin (1976) with the initial goal of giving conditions insuring that the law of a random variable has a density with respect to Lebesgue measure, as well as estimates for this density and its derivative. Malliavin Calculus looks forward to finding the derivative of the functions of Brownian motion which will be referred to as Malliavin derivative. We will highlight the theory of Malliavin Calculus. In what follows, H is a real separable Hilbert space with inner . , . H . Ω denotes the sample space, P denotes the probability space P. product II. The Wiener Chaos Decomposition Definition 2.1. A stochastic process W = {W(h), h ϵ H }defined in a complete probability space (Ω, F, P) is called an isonormal Gaussian process if W is a centered Gaussian family such that E (W(h)W(g)) = h, g Remark W (h) 2 2.2. 2 L ( P) The mapping E(W (h) ) h 2 H h 2 H for all h, g ϵ H. → W(h) is linear [8]. From the above, we have that . Let G be the σ-field generated by the random variables {W(h), h ϵ H }, the main objective of this part is to find a decomposition of L2(Ω, G , P). We state some results concerning the Hermite polynomials in order to find the decomposition . Let H n x denote the nth Hermite polynomial, then 1n e x2 2 H n x = n! dn dx n x e 2 2 , n ≥1 (1) and H 0 x = 1. These hermite polynomials are coefficients of the power expansion in t of the function F t , x exp (tx t2 2 ) which can easily be seen by rewriting x2 1 2 F t , x exp x t 2 2 and expanding the function around t = 0. www.iosrjournals.org 30 | Page Stochastics Calculus: Malliavin Calculus in a simplest way The power expansion combines with some particular properties of F, that is F t2 t exp tx tF x 2 F t2 x t exp tx x t F t 2 and t F x, t exp tx F x,t 2 2 provides the corresponding properties of the Hermite polynomials for n ≥ 1 H n' x H n1 x n 1H n1 x xH n x H n1 x H n x 1 H n x This is shown by using induction method: n To show that H n x H n1 x ; Let n = 1, from ' 1n e x2 2 H n x = n! x e 2 2 dn dx n we have ' x2 x2 d x2 x2 ' 2 2 2 H1 x e e e x e 2 dx ' x 1 H n1 H 0 x ' 2 x 1 x 2 d xe 2 e 2 2 dx ' Let n = 2, 1 x2 d 2 x22 e H x e 2 2 2 dx ' 2 ' x2 1 x2 x2 2 e 2 e 2 x e 2 2 ' 1 1 x 2 ' x H x . 1 2 Also for n = 3 we have x2 x2 1 2 d 3 2 ' H 3 x e e 6 dx 3 ' 1 x 2 x x 3 ' 1 x 2 1 H 2 x . 6 2 Lemma 2.3. Let X, Y be two random variables with joint Gaussian distribution such that E (X) = E (Y) = 0 and E (X2) = E (Y2) = 1. Then for all m,n≥0, we have 0, E H n X H m Y 1 E XY n , n! if n m; if n m, Proof. See the proof of lemma 1.1.1 in [8] www.iosrjournals.org 31 | Page Stochastics Calculus: Malliavin Calculus in a simplest way Lemma 2.4 The random variables e W h ; h H form a total subset of L2 G L2 Ω, G , P . Proof. Let X L2 G such that E XeW h 0 for all h H 9. By thelinearity of mapping h W h we have m E X exp t1W h 0, ti R, hi H , i 1,2,...m, m 1 i 1 Equation (2) shows that the Laplace of v is given by vB E X 1B W h1 ,..., W hm 2 for a Borel set B R m . As the transform is zero, the measure v must be zero for □ every set G G.That is, E X 1G 0 for G G X 0. Definition2.5. For each n 1, we define H n the choas of order n as the closed linear subspace of L2 Ω, F , P generated by therandom variables H n W h : h H , h H 1. Theorem 2.6. The space L Ω, G , P can be decomposed into the infinity orthogonal 2 sum of the subspaces H n 9 : L2 Ω, G , P Hn . n 0 proof. Let X L Ω, F , P be orthogonal to H n for all n 0. We show that X 0. 2 We have that E XH n W h 0 for all h H such that / h // H 1. Using the fact that x n can be expressed as a linear combinatio n of theHermite polynomial H r x ,0 r n, we have E XW h 0 n 0 therefore, n 3 E X exp tW h 0 t R and for all h H of norm1. By the lemma above which states that the random variables { e W(h) , h H} form a totalsubset of L2 (, G , P), we have that equation (3) implies that X= 0 □ 3. The Malliavin Derivative We consider t he set C p R n of all infinitely differentiable functions f : R n R denoted such that f and all of its derivatives have at most polynomial growth ( p denoted partial derivatives with polynomial growth). Let n 1 and f C p R n , we denote by S the set of all random variable the form F f W (h1 ),..., W (h n ) (4) where h1 ,..., hn H and F S , a smooth variable. Definition3.1. The derivative of random variable F S is define as DF i f W (h1 ),..., W (h n ) hi . n (5) i o where the derivative is a mapping DF : H. www.iosrjournals.org 32 | Page Stochastics Calculus: Malliavin Calculus in a simplest way Lemma 3.2 Let F F and h H then E DF , h Proof . Suppose that h H EFW (h). (6) 1. Due to the linearity of the scalar product on W, H there exists an orthonomal family of H,e1 ,...en such that h e1 , such that that we can write F as F f W (e 1 ),..., W (e n ) for a suitable function f C R n . Let x be the multivaria te density of the p standard normal distribution, that is 1 n 2 xi . 2 i 1 We have by classical integratio n by parts formular x 2 2 exp n E DF , h H Rn 1 f x x x1dx n f x ( x) xdx R E FW (e1 ) E FW (h) . This completes the proof. □ Proposition 3.3. (Nualart,2006). Let : R m R be a continously diferentiable function partial with bounded Suppose F F1 ,..., Fm is a random derivatives. vector with components in D . Then ( F ) is in D 1, p 1, p and m D ( F ) i ( F ) DF i . i 1 The proof of proposition 3.3 is similar to the proof of proposition 1.20, pp. 13 Of (Nualart, 2009). The chain rule can be extended to the case of a Lipschitz function. Theorem 3.4 (Closability). Assume F L2 P and Fk D1, 2 , k 1,2,...such that 1. 2. Fk F , k , in L2 P Dt Fk k 1 converges in L2 P . Then F D1, 2 and Dt Fk Dt F , k , in L2 P . Proof. Let F f , k , in L for all n. I n f n and Fk I n f nk , k 1,2,.... n 0 k n By (1) we have f n 0 2 n n By (2) we have nn! n 1 k j fn fn 2 Dt Fk Dt F j L2 n 2 L2 P 0, j, k Hence, by Fatou Lemma, lim k nn! f nk f n n1 1, 2 This gives F D lim lim nn! f nk f n j L k j 2 2 n n1 2 , Dt Fk Dt F , k , in L P . 2 0. L2 n □ Proposition 3.5. [8] Suppose F D is a square integrable random variable With a decomposition given above, we have 1, 2 www.iosrjournals.org 33 | Page Stochastics Calculus: Malliavin Calculus in a simplest way Dt F nI n1 f n (. , t ) . (7 ) n 1 Proof. We prove the statement for a simple function (see proposition 1.2.7 of [8]). Using a simple function, the general case results from the density of the simple function. Let F I m f m for a symmetric function f m , then, by applying n DF i f W (h1 ),...,W (hn ) hi i 0 to g W ( Ai1 ),...,W ( Ain ) with gx1 ,..., xn x1 ,..., xn we have m n Dt F W ( Ai1 )...1Aij ...W ( Aim ) mI m1 f m (. , t ) . □ a i1...im j 1 i1,...,im1 Theorem 3.6. Let F be a square integrable random variable denoted by n 0 n 0 F I n f n . Let A Β , then EF \ FA I n f n1A n . Proof. Assume that F I n f n such that f n is a function in E n . We also assume that the kernel f n is of the form 1B1...Bn with B1 ,..., Bn being mutually disjoint sets of finite measure. Through the linearity of W and the properties of the conditional expectation we have n E F \ FA E W ( B1 )...W ( Bn ) / FA E W ( Bi A) W ( Bi Ac ) \ FA i 1 I n 1 B1 A... Bn A . □ IV. The Divergence Operator In this section, we consider the divergence operator, defined as the adjoint of the derivative operator. If the underlying Hilbert H is an L -space of the form L T , B , , where µ is a finiteatomless measure, we interpret the divergence operator as a stochastic integral and call it Skorohod integral because in the Brownian motion case it coincides with the generalization of the ItÔ stochastic integral to anticipating integrands. We first introduce the divergence operator in the framework of Gaussian isonormal process W W (h), h H associated with the Hilbert space H . We assume that W is defined on a complete 2 probability space 2 , F,P, and that F is generated by W. We shall consider results from[1], [8] [ and [12]. We note that the derivative operator D is a closed and unbounded operator with values in L ; H defined 2 on the dense subset D of L . Definition 4.1. The adjoint of the operator D denote by δ is an unbounded operator 1,2 2 on L2 ; H and satisfies The domain of δ denoted as Domδ is the set of u L , H where H -valued square integrable random variable 2 E DF , u H c F for all F D1, 2 2 such that the c is a constant depending on u. Let u ϵ Domδ, then δ(u) is an element of L characteri zed by 2 EF u E DF , u H (8) each for F D1, 2 . From equation (8) above, E(δ(u)) = 0 if u ϵ Domδ and F = 1. We see SH to be the class of smooth elementary elements of the form n u Fj h j (9) j 1 www.iosrjournals.org 34 | Page Stochastics Calculus: Malliavin Calculus in a simplest way H . Applying integration-by-parts such that F j are smooth random variables and the h j are elements of formula, we deduce that u ϵ Domδ and n u F jW (h j ) DFj h j j 1 j 1 H . (10) Theorem 4.2. Let F ϵ D1,2 and u ϵ Domδ such that Fu L ; H . Then Fu Domδ and we have the equality 2 Fu F u DF , u belongs to H provided the right hand side is square integrable. Proof. Let G be any smooth random variable, we obtain E u, DFG GDF E u F u, DF G. □ EG Fu E DG, Fu H H H Lemma 4.3. D F h n Let D u : h j j 1 form u where u S H , the class of smooth elementary processes of the h j n F h , F S , h H , we have that the following commutativity relationship holds j 1 j j D h u u, h H D hu . This is true from the following: u F jW h j DFj h j n j 1 j 1 H . yields D h u DF jW (h j ) D DFj , h j n j 1 n = F j 1 = u, h n h, h j j H H ,h H D h F jW (h j ) D D h F j , h j H j 1 D hu . H □ Remark 4.4. Let h ϵ H and F ϵ D .Then Fh belongs to domain of δ and the is true h,2 following equality Fh FW h D h F . Theorem 4.5 (The Clark-Ocone formula). Let F D1, 2 and W is a one-dimensional Brownian motion on [0,1]. Then F EF E Dt F F t dWt . T 0 Proof. Suppose that F I f , n 1,2,... (see[9] and Proposition 1.3.8 of n 0 n n [12], we have T 0 T EDt F Ft E nI n1 f n ., t Ft dW t 0 n1 nEI f ., t F dW t T = 0 n 1 n 1 n t Thus, www.iosrjournals.org 35 | Page Stochastics Calculus: Malliavin Calculus in a simplest way T 0 nI f ., t .X T EDt F Ft 0 n1 0 0,t n T n1 ( . ) dW t n1 n1 nn 1! In1 f n ., t X0,tn-1 dW t n!In f n . I n f n n1 I n f n I 0 f 0 F EF . □ n 1 Note. The Clark-Ocone formula and its proof can be found in [7], [8] and [10]. V. A model of a financial market The Black-Scholes Model. We consider a market consisting of a non-risky asset (Bank account) B and a risky asset (stock) S. Let the process of the risky asset be given by S (t ) S0 exp ( 2 2 )t W (t ) (11) W W (t ) : t 0, T is a Brownian motion defined on a complete probabilityspace , F , P and Ft , t 0, T is a filtration generated by Brownian motion σt denote the volatility process, µ where is the mean rate of return, and are all assumed to be constant. The price of the bond B(t) and the price of the stock S(t) satisfies the differential equations: dBt rBt dt B(0) = 1 and dS (t ) S (t )(dt dW (t )) S(0) = 0, S(t) = St We have that B(t ) Bt e rt (12) where r denotes the interest rate and it is a nonnegative adapted process satisfying T 0 rt dt a.s. , F which is equivalent to P. Definition 5.1. Let Q be a probability measure on (Local) martingale measure (or a non-risky probability 1 rt Q is called equivalent measure) if the discounted price process St Bt St e St , t 0, T is a local martingale under Q. We note that: A process X is sub-martingale (respectively, a super-martingale) if and only if Xt =Mt+At (respectively, Xt =M t- At) where, M is a local martingale and A is an increasing predictable process. t 0 for all t 0, T and T0 s ds 2 Suppose a.s. where ii Fi . We define the process 1 t t Z t exp s dWs s 2 0 0 2 ds which is positive local martingale. If 1 T 2 T E exp s dWs s dt 1 2 0 0 dQ Z T is a then the process ZT is a martingale and measure Q such that dP probability measure, equivalent to P, such that under Q,, the process t ~ Wt Wt s ds 0 is a Brownian motion. ~ In terms of the process Wt , the price process can be expressed as www.iosrjournals.org 36 | Page Stochastics Calculus: Malliavin Calculus in a simplest way t t ~ 2 St S0 exp (rs 2 )ds s dW . 0 0 Thus, the discounted prices from a local martingale: ~ ~ 1 t t St Bt1St S 0 exp s dWs s2 ds . 2 0 0 Definition 5.2. A derivative is contract on the risky asset that produces a payoff H at maturity T. The payoff is an FT -measurable nonnegative random variable H. Some fact about filtration: Filtrations are used to model the flow of information over time. Ft has occurred or not. E[X│ FT ] of a random variable X At time t, one can decide if the event A ϵ Taking conditional expectation the expectation on the basis of all information available at time t. Proposition 5.3. (Girsanov theorem) There exist a probability Q absolutely continuous with respect to P such that t Q T 1 P that is, Wt u s ds has 0 dQ only if E(ξ1) =1 and in this case 1 . dP the law of Brownian motion under means Q) taking if and Proof. See Proposition 4.1.2, pp. 227 of [8] Remark 5.4. The probability P o T-1 is absolutely continuous with respect to P. Proposition 5.5 [8] Suppose that F,G are two random variables such that F ϵ D1,2. Let u be an H –valued random variable such that DuF = u DF , u H 0 almost surely and -1 Gu(D F) ϵ Domδ. Then, for any differentiable function f with bounded derivative we have E f ( F )G E f ( F ) H ( F , G), where H F , G Gu( Du F ) 1 . Proof. By chain rule, we have Du f ( F ) f F Du F . By the duality relationship, equation (8), we obtain u E f F GuD F = E f F Gu D F E f F G E D u f F D u F 1 G E D f F , u D u F 1 u 1 . 1 G H □ We recall that in Malliavin Calculus, Integrating by part is T T E Ds F us ds EF u E F ut dW t . 0 0 VI. Applications Greeks are used for risk management purposes referred to as hedging in financial mathematics. Finite difference methods have been used to find the sensitivities of options by the use of Monte-Carlo methods, but the speed of convergence is not so fast very close to the discontinuities. The use of Malliavin Calculus provides a better way to calculate the greeks, both in terms of simplicity and speed of convergence. Thus, this method provides a good solution when the payoff function is strongly discontinuous. A greek can be defined as the derivative of a financial quantity with respect to a parameter of the model. List of Greeks include: Delta: = The derivative with respect to the price of the underlying; ∆: = VS . www.iosrjournals.org 37 | Page Stochastics Calculus: Malliavin Calculus in a simplest way : Gamma: =Second derivative with respect to the price of the underlying; 2V S 2 Vega: = Derivative with respect to the volatility; v: V Rho: = Derivative with respect to interest rate; : Vr Theta: = Derivative with respect to time; : Vt Greeks are useful in studying the stability of the quantity under variations of the chosen parameters. If the price of an option is calculated using the measure Q as V E Q e r T t x where Ф(x) is the payoff function; the greek will be calculated under the same simulation together with the price. The equation gives Greek E e x .W where W is a random variable called Malliavin weight. Malliavin Calculus is a special tool for calculating sensitivities of financial derivatives to change in its underlying parameter. We now discuss the model of a financial market and the computation of the greeks. 6.1 Computation of the Greeks We consider the Hilbert space which are constant on compact interval. We Assume that W = {W(h),hϵ H} denotes an isonormal Gaussian process associated with the Hilbert space H. Let W be defined on a complete probability space (Ω, F, P) and let F be generated by W . Q r T t Remark 6.1.1 The following observation will be important for the application of proposition 5.5. If u is deterministic. Then, for Gu(DuF)-1 ϵ Domδ it suffices to say that Gu(DuF)-1 ϵ D1,2 as this implies that Gu DF , u 1 H D1, 2 Dom . If u = DF, then the conclusion of Proposition 5.5 is written as GDF . E f F G E f F 2 DF H We consider an option with payoff H such that E Q(H2) < ∞. We have that rs ds Vt E e t H F t . The price at t = 0 gives V0 E Q e rT H . Suppose T Q represent one of the parameters of S0, σ, r. Let H f F . Then dF V0 e rT E Q f F d . (13) From Proposition 5.5, we have dF V0 e rT E Q f F H F , d . (14) If f is not smooth, then (14) provides better result in combination with Monte-Carlo Simulation than (13). We note that the following: In the calculation of the greeks, differentiation can be done before finding the expectation, this will still give the same result. T Dt ST dS dW t , w ST . The Malliavin derivative of From proposition 5.5 DuF must not be zero in order to make sure that exist. Let S t S0 exp ( 2 2 1 Du F )t W (t ) , t 0, T ; then, www.iosrjournals.org 38 | Page Stochastics Calculus: Malliavin Calculus in a simplest way ST exp ( S 0 2 2 )T W (T ) ST S0 In calculating the greeks, we assume that S t S0 exp ( 2 2 )t W (t ) , t 0, T except otherwise stated. 6.2 Computation of Delta We discuss delta of European options. Suppose H depends only on the price of the stock at maturity time T, that is, H = Φ(ST). Let the price of the stock at time 0 be given by E e rT , St e rT Q V E e ST Q rT E e ST E (ST ) ST . S 0 S 0 S S 0 0 Q rT From proposition 5.5, by letting u =1, F = ST , and G = ST, we obtain T T 0 0 D u ST Dt St dt ST dt TST . From the condition in the above remark, we have 1 1 T dW WT . T 0 T T ST Dt ST dt T 0 As a result, e rT Q E ST WT . S 0T (15) The weight is given by weight = 6.3 Computation of Gamma WT . S 0T rT 2V0 2 E Q e rT ST Q e ST ST E 2 2 S S 0 S 0 0 S E e rT ST T S 0 Q 2 ST S 0 e rT E Q ST ST2 . S 02 We now apply the Malliavin derivative property in order to eliminate “/”. Suppose is Lipschitz, let G = ST2 , F = ST and u = 1. From proposition 5.5 we have 1 T 1 S 1 2 DST , ST Dt ST dt T ST 0 T T T T ST 0 dST 1 1 dW , T dW T ST T S dt WT T 0 T T ST W WT ST ST T 1 . T T Thus, www.iosrjournals.org 39 | Page Stochastics Calculus: Malliavin Calculus in a simplest way W E Q ST ST2 E Q ST ST T 1 . T Using Malliavin property, we now eliminate the remaining “/”. From proposition 5.5, taking G ST 1 . F = ST and u = 1 gives WT T ST WT 1 Dt ST dt ST WT 1 0 1 T T ST T WT T WT 2T 2 1 TST 1 TST 1T 1 WT 2T 2 1 T T dW 1 1 WT 2 2 DWT , 2 2 0 T T T T dW T W dt WT 2 2 2 2 T 0 T 0 T T W2 WW W W T 1 T2 T2 2 2 T 2 T 2 2 T . T T T T T T As a result, W2 W 1 W E Q ST ST T 1 E Q ST 2 T 2 2 T . T T T T This implies that WT2 1 e rT Q E S W T T . S 02T T 6.4 Computation of Vega V0 E Q e rT ST Q rT E e S 0 exp E Q e rT ST ST e rT (16) 2 2 T W T E ST ST WT T . Q Applying proposition 5.5, let G = ST (WT – σT), F = ST and u = 1 we have 1 ST WT T W T T TST T ST WT T Dt ST dt T 0 W W T 1 T 1 T T T 1 1 WT DWT , dW 0 T T T dW T dt t WT WT 0 T 0 T W 2T 1 W2 T T WT WT T T T www.iosrjournals.org 40 | Page Stochastics Calculus: Malliavin Calculus in a simplest way e rT WT2 1 E ST WT . T Q (17) The above formulas still hold by means of an approximate procedure although the function and its derivative are not Lipschitz. The important thing is that should be piecewise continuous with jump discontinuities and with a linear growth. The formulas are applicable in the case of European call option ( (x) = (x - K)+) and European put option ( (x) = (K - x)+), or a digital ( (x) = 1x>K). References [1] Bally, V., Caramellio, L. & Lombardi, L. (2010). An introduction to Malliavin Calculus and its application to finace. Lambratoire d’analysis et de Mathématiques. Appliquées, Uviversité Paris-East, Marne-la-Vallée. [2] El-Khatilo, Y. & Hatemi, A.J. (2011). On the Price Sensitivities During Financial crisis. Proceedings of the World Congress on Engineering Vol I. WCE 2011, July 6-8, London, U.K. [3] Fox, C. (1950). An Introduction to Calculus of Variations. Oxford University Press. [4] Malliavin, P. (1976). Stochastic Calculus of variations and hypoelliptic operators. In Proceedings of the International Symposium on Stochastic differential Equations (Kryto) K.Itô edt. Wiley, New York, 195-263 [5] Malliavin, P. (1997). Stochastic Analysis. Bulletin (New Series) of American Mathematical Society, 35(1), 99-104, Jan. 1998. [6] Malliavin, P. & Thalmaier, A. (2006). Stochastic Calculation of Variations in Mathematical Finance. Bulletin (New Series) of the American Mathematics Society, 44(3), 487-492,July 2007. [7] Matchie, L. (2009). Malliavin Calculus and Some Applications in Finance. African Institute for Mathematical Sciences (AIMS), South Africa. [8] Nualart, D. (2006). The Malliavin Calculus and Related Topics: Probability and its Applications. Springer 2nd edition. [9] Nualart, D.G. (2009). Lectures on Malliavin Calculus and its applications to Finance.University of Paris. [10] Nunno, D.G. (2009). Introduction to Malliavin Calculus and Applications to Finance,Part I. Finance and Insurance, Stochastical Analysis and Practicalmethods. Spring School, Marie Curie. [11] Schellhorn, H. & Hedley, M. (2008). An Algorithm for the Pricing of Path-Dependent American options using Malliavin Calculus. Proceedings of World Congress on Engineering and Computer Science (WCECS 2008). San Francisco, U.S.A. [12] S chröter, T. (2007). Malliavin Calculus in Finance. A Special Topic Essay.St. Hugh’s College, University of Oxford. www.iosrjournals.org 41 | Page