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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 www.iosrjournals.org 30 | Page 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-by- parts 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 product H . , . . Ω denotes the sample space, P denotes the probability space P. 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 h, for all h, g ϵ H. Remark 2.2. The mapping h W(h) is linear [8]. From the above, we have that . ) ) ( ( ) ( 2 2 ) ( 2 2 H h h W h W P L E 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 L 2 (Ω, G , P). We state some results concerning the Hermite polynomials in order to find the decomposition . Let H x n denote the nth Hermite polynomial, then H x n = , ! 1 2 2 2 2 x n n x n e dx d e n n ≥1 (1) and H x 0 = 1. These hermite polynomials are coefficients of the power expansion in t of the function ) ( exp , 2 2 t tx x t F which can easily be seen by rewriting 2 2 2 1 2 exp , t x x x t F and expanding the function around t = 0.

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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

www.iosrjournals.org 30 | Page

Stochastics Calculus: Malliavin Calculus in a simplest way

1Udoye, 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-by-

parts 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

product H

.,. . Ω denotes the sample space, P denotes the probability space P.

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

gh, for all h, g ϵ H.

Remark 2.2. The mapping h → W(h) is linear [8]. From the above, we have that

.))(()(22

)(

22

HhhWhW PL E 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 xn denote the nth Hermite polynomial, then

H xn =

,!

122

22

x

n

nxn

edx

de

n

n ≥1 (1)

and H x0 = 1. These hermite polynomials are coefficients of the power expansion in t of the function

)(exp,2

2ttxxtF which can easily be seen by rewriting

22

2

1

2exp, tx

xxtF

and expanding the function around t = 0.

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 31 | Page

The power expansion combines with some particular properties of F, that is

tFt

txtx

F

2exp

2

Ftxt

txtxt

F

2exp

2

and

txFt

txtxF

,

2exp,

2

provides the corresponding properties of the Hermite polynomials for n ≥ 1

xHxH nn 1

'

xHxxHxHn nnn 111

xHxH n

n

n 1

This is shown by using induction method:

To show that xHxH nn 1

'

;

Let n = 1, from

H xn =

22

22

!

1x

n

nxn

edx

de

n

we have

'

22

'

22'

1

2222

xxxx

exeedx

dexH

xHHx n 01

' 1

Let n = 2,

'

2

'

2

2

2'

22

2

2

2 22

2

1

2

1

xx

xedx

dee

dx

dexH

xx

.12

1

2

11

'2

'

2222

222

xHxxexee

xxx

Also for n = 3 we have

.12

12

6

1

6

12

2'3

'

23

3

2'

3

22

xHxxxxedx

dexH

xx

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

m;nif

m,nif

,0

,nn!

1XY

YHXH mnE

E

Proof. See the proof of lemma 1.1.1 in [8]

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 32 | Page

have we

mapping oflinearity By the.9 allfor 0thatsuchLet

.,Ω,

ofsubsettotalaformevariablesrandomThe ;hW

hWh

hXeLX

PL

Lh

hW2

2

2

HEG

G

GH

Proof.

2.4Lemma

21,,...2,1 ,,0exp1

1

mmihthWtX ii

m

i

H R,E

mB hWhWXv

v

,...,B

bygiven is of Laplace that theshows (2)Equation

11E

.0Gfor 0 is,That Gset every

for zero bemust measure thezero, is transform theAs .set Borel afor

1

XX

v

G

m

GG.

B

E

R

.1,: variablesrandom by the generated ,Ω, of subspace

linear closed theasn order of choas the define we,1each For

HH

H

hhhWHPFL

n

n

2

n2.5. Definition

.,Ω,

:9 subspaces theof sum

orthogonalinfinity theinto decomposed becan ,Ω, space The .

0 nPL

PL

n

2

n

2

HG

H

G

2.6 Theorem

3 0exp

therefore,00 have we

,0, polynomial e theHermitofn combinatiolinear a as expressed becan

fact that the Using.1///such that allfor 0 that have We

.0 that show We0.n allfor toorthogonal be ,Ω,Let proof.

htWX

nhXW

nrxH

hhhWXH

XPFLX

n

r

n

n

n

2

E

E

E xH

H

H

implies(3)equationthathavewe),,,(ofsubsettotalaform,e

variablesrandomthethatstateswhichabovelemmatheBy1.normofhallforand

2W(h) PLh

t

GH

H

R

that X= 0

iable.smooth var a , and ,..., where

(4) )(),...,(

form the variablerandom all ofset

theby denote we,C and 1nLet growth). polynomial with sderivative

partial denoted (growth polynomialmost at have sderivative its of all and such that

denoted : functions abledifferenti infinitely all of Cset heconsider t We

1

1

SFhh

hWhWfF

Sf

pf

f

n

n

n

p

nn

p

H

R

RRR

Derivative Malliavin The 3.

H.

:DF mapping a is derivative thewhere

(5) .)(),...,(DF

as define is variablerandom of derivative The

1

oi

i in

n

hhWhWf

SF3.1. Definition

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 33 | Page

)6(.)(,

then andLet

hFWhDF

hF

EE

H

HF 3.2 Lemma

)(),...,(

as can write

that wesuch that ,such that ,... offamily orthonomalan exists there

on W,product scalar theoflinearity the toDue .1 that Suppose .

1

11

n

n

eWeWfF

F

ehee

h

H,

HProof

.2

1exp2

is that on,distributi normal standard

theofdensity temultivaria thebe Let .Cfunction suitable afor

1

22

n

i

i

n

n

p

x

f

x

xR

.)()(

)(,

formular partsby n integratio classicalby have We

1

11

hFWeFW

xdxxxfdxxxxfhDFn n

EE

ER R

H

proof. thecompletes This

Proposition 3.3. (Nualart,2006). blediferentiaycontinousla be: RR mLet

function with bounded partial derivatives. randomais,...,Suppose 1 mFFF

andins)(Then.incomponentswithvector ,1,1 pp iF DD

.)()(1

m

i

i

i DFFFD

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).

such that,...2,1,F and Assume 2,1

k

2 kPLF D

PLinkFFk

2,,.1

.Lin converges .2 2

1

PFD

kkt 2,1FThen D and inkFDFD tkt ,, P2L .

Proof. Let

00

,....2,1, and n

k

nn

n

knn kfIFfIF

By (1) we have n

n

k

n Linkff 2,, for all n.

By (2) we have

kjFDFDffnnPLjtkt

Ln

j

n

k

nn

,,0!2

2

1

2

2

Hence, by Fatou Lemma,

1

22

1

.0!limlim!lim 22

nL

j

n

k

njkLn

k

n

nk

nn ffnnffnn

This gives 2,1DF , in,, kFDFD tkt PL2

.

Proposition 3.5. [8] Suppose 2,1DF is a square integrable random variable

With a decomposition given above, we have

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 34 | Page

)7(.),(.1

1 tfnIFD n

n

nt

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 mm fIF for a symmetric function mf , then, by applying

i

n

i

ni hhWhWfDF

0

1 )(),...,(

to nnini xxxxAWAWg ,...,,...,g with )(),...,( 111 we have

m

j

n

imi

mmimAijiimit tfmIAWAWaFD1 1,...,1

11...1 .),(.)(...1)...(

Theorem 3.6. Let F be a square integrable random variable denoted by

00

.1\FE then ,Let .n

n

AnnA

n

nn fIAfIF FΒ

Proof. Assume that nn fIF such that nf is a function in nE . We also

assume that the kernel nf is of the form nBB BBn

,..., with 1 1...1 being mutually

disjoint sets of finite measure. Through the linearity of W and the properties of the

conditional expectation we have

n

i

c

iin ABWABWEBWBWEFE1

1 \)()(/)()...(\AAA

FFF

.1 ...1 ABABn nI

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 an2L -space of the form ,,2

BTL , where µ is a finite atomless

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

H hhWW ),( associated with the Hilbert space H . We assume that W is defined on a complete

probability space ,, PF, 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 H;2 L defined

on the dense subset . of 2 L1,2D

Definition 4.1. The adjoint of the operator D denote by δ is an unbounded operator

satisfiesand;on 2HL

The domain of δ denoted as Domδ is the set of H -valued square integrable random variable

H,2 Lu where

2,1

2allfor , DE FFcuDF

H such that the c is a constant depending on u.

Let u ϵ Domδ, then δ(u) is an element of byzedcharacteri2 L

)8(,

HuDFuF EE

.foreach 2,1DF

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

j

jjhFu1

(9)

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 35 | Page

such that jF are smooth random variables and the jh are elements of H . Applying integration-by-parts

formula, we deduce that u ϵ Domδ and

n

j j

jjjj hDFhWFu1 1

.)(H

(10)

Theorem 4.2. Let F ϵ D1,2 and u ϵ Domδ such that .;2HLFu Then Fu belongs to

Domδ and we have the equality

HuDFuFFu ,

provided the right hand side is square integrable.

Proof. Let G be any smooth random variable, we obtain

HHGDFFGDuFuDGFuG ,, EEE

., GDFuFu

H E

Lemma 4.3. Let , where:1

n

j

jj

hh SuhFDuDH

the class of smooth elementary processes of the

form ,,,1

n

j

jj hSFhFu H we have that the following commutativity relationship holds

., uDhuuD hh

H

This is true from the following:

n

j j

jjjj hDFhWFu1 1

.H

yields

n

j

jjjj

h hhDFDhWFDuD1

,,)(HH

=

n

j

n

j

jj

h

jj

h

jj hFDDhWFDhhF1 1

,)(,HH

= ., uDhu hH

Remark 4.4. Let h ϵ H and F ϵ Dh,2.Then Fh belongs to domain of δ and the following equality

is true

.FDhFWFh h

Theorem 4.5 (The Clark-Ocone formula).

Wand Let 2,1DF is a one-dimensional Brownian motion on [0,1]. Then

T

ttt dWFDEFF0

.FE

Proof. Suppose that

0

,...2,1,n

nn nfIF (see[9] and Proposition 1.3.8 of

[12], we have

T

n

tnn

T

tt tdWtfnIEFDE0

1

10

., FF

=

T

n

tnn tdWtfInE0

1

1 ., F

Thus,

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 36 | Page

T

tt FDE0

F

T

n

nn tdWtfnI0

1

1 ).(..,t0,

X

1

00 .n

nn FfIfI FE

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

)()(exp)(20

2

tWtStS (11)

where TttWW ,0:)( is a Brownian motion defined on a complete probabilityspace

TtP t ,0, and ,, FF is a filtration generated by Brownian motion σt denote the volatility process, µ

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:

dttrBtdB

B(0) = 1

and

))()(()( tdWdttStdS

S(0) = 0, S(t) = St

We have that

)12()( rt

t eBtB

where r denotes the interest rate and it is a nonnegative adapted process satisfying

T

tdtr0

a.s.

Definition 5.1. Let Q be a probability measure on F, which is equivalent to P. Q is called equivalent

(Local) martingale measure (or a non-risky probability

measure) if the discounted price process TtSeSBS t

rt

ttt ,0,1 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.

Suppose dsTt T

t

2

s0 and ,0 allfor 0

a.s.

processthedefineWe.wherei

ii F

t t

ssst dsdWZ0 0

2

2

1exp

which is positive local martingale. If

12

1exp

0 0

2

T T

sss dtdW E

then the process ZT is a martingale and measure Q such that TZdP

dQ is a

probability measure, equivalent to P, such that under Q,, the process

t

stt dsWW0

~

is a Brownian motion.

In terms of the process tW~

, the price process can be expressed as

1 1

10

1

.!.,!1n n

nnnnnn

T

n

fIfntdWtfnn IXI 1-n

t0,

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 37 | Page

.~

)(exp0 0

20

2

t t

sst WddsrSS

Thus, the discounted prices from a local martingale:

.2

1~exp

~

0 0

2

0

1

t t

sssttt dsWdSSBS

Definition 5.2. A derivative is contract on the risky asset that produces a payoff

H at maturity T. The payoff is an TF -measurable nonnegative random variable H.

Some fact about filtration:

Filtrations are used to model the flow of information over time.

At time t, one can decide if the event A ϵ tF has occurred or not.

Taking conditional expectation E[X TF ] of a random variable X means taking

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

st dsuWPTQ0

1 is,that has the law of Brownian motion under Q) if and

only if E(ξ1) =1 and in this case .1dP

dQ

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 = 0, H

uDF almost surely and

Gu(DuF)-1 ϵ Domδ. Then, for any differentiable function f with bounded

derivative we have

,),()()( GFHFfGFf EE

where

.)(, 1 FDGuGFH u

Proof. By chain rule, we have

.)( FDFfFfD uu

By the duality relationship, equation (8), we obtain

H

GFDuFfDGFDFfDGFf uuu 11,EEE

= .11

FDGuFfFDGuFf uu EE

We recall that in Malliavin Calculus, Integrating by part is

.00

TT

s tdWtuFuFdssuFD EEE

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; ∆: =SV .

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 38 | Page

Gamma: =Second derivative with respect to the price of the underlying;

2

2

:S

V

Vega: = Derivative with respect to the volatility; Vv:

Rho: = Derivative with respect to interest rate;rV:

Theta: = Derivative with respect to time;tV :

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

xeV tTrQ E

where Ф(x) is the payoff function; the greek will be calculated under the same

simulation together with the price. The equation gives

Greek WE .xe tTrQ

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 .

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 ., 2,11DomuDFGu

D

H

If u = DF, then the conclusion of Proposition 5.5 is written as

.2

HDF

GDFFfGFf EE

We consider an option with payoff H such that EQ(H2) < ∞. We have that

.

t

dsrQ

t HeEV

T

ts

F The price at t = 0 gives .0 HeV rTQ E Suppose

represent one of the parameters of S0, σ, r.

Let .FfH Then

.0

d

dFFfe

V QrT E (13)

From Proposition 5.5, we have

.,V0

d

dFFHFfe QrT E (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.

The Malliavin derivative of ., TdW

dS

Tt SwtSD T

From proposition 5.5 DuF must not be zero in order to make sure that FDu

1

exist.

Let ;,0,)()(exp20

2

TttWtStS then,

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 39 | Page

0

2

)()(exp2

0

S

ST TTWTS

S

In calculating the greeks, we assume that

TttWtStS ,0,)()(exp20

2

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 ,rTeE

.)(

0000

TT

rT

tT

rTT

rT

SSS

e

S

SSe

S

Se

S

V

QQ

Q

EEE

From proposition 5.5, by letting u =1, F = ST , and G = ST, we obtain

T T

TTttT

u TSdtSdtSDSD0 0

.

From the condition in the above remark, we have

T

TT

TtTT

W

T

dW

TdtSDS

0

1

0.

1

As a result,

.0

TT

QrT

WSTS

e

E

(15)

The weight is given by

weight = .0 TS

WT

6.3 Computation of Gamma

00

2

0

2

2

0

0

2

S

S

S

SSe

S

Se

S

V TTT

rTQT

rTQ

EE

.2

2

0

2

0

TT

QrT

TT

rTQ SSS

e

S

SSe

EE

We now apply the Malliavin derivative property in order to eliminate “/”. Suppose

is Lipschitz, let G = ,2

TS F = ST and u = 1.

From proposition 5.5 we have

T

DST

ST

SdtSDS TT

TT

TtT

1

,1

1

0

2

TT

TTdW

dSdW

TS

0

1,

1

TTT

TT

dtS

T

WS

0

.1

T

WSS

T

WS T

TTT

T

Thus,

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 40 | Page

.12

T

WSSSS T

TT

Q

TT

Q

EE

Using Malliavin property, we now eliminate the remaining “/”.

From proposition 5.5, taking .1T

W

TTSG F = ST and u = 1 gives

T

T

W

T

T

TtT

W

TTS

SdtSDS TT

111

1

0

11

TT

W

T TSS T

TT

W

TT

W TT

11

2222

T

TTT

dW

TDW

TW

02222

1,

1

T TT

TT

W

T

dt

T

dWW

0 0 2222

.1222

2

2222

T

W

TT

W

T

W

T

T

T

WW TTTTT

As a result,

.1

1222

2

T

W

TT

WS

T

WSS TT

T

QTTT

Q

EE

This implies that

.12

2

0

TT

T

QrT

WT

WS

TS

e

E (16)

6.4 Computation of Vega

T

rTQ SeV E0

T

rTQ WTSe20

2

expE

TS

T

rTQ SeE

.TWSSe TTT

QrT E

Applying proposition 5.5, let G = ST (WT – σT), F = ST and u = 1 we have

T

TW

TS

TWSdtSDTWS T

T

TTT

TtTT

1

0

11

T

W

T

W TT

T

TT dWT

DWT

W0

1,

1

T T

Tt

T WT

dt

T

dWW

0 0

TT

T WT

TWW

T

T

T

W

122

Stochastics Calculus: Malliavin Calculus in a simplest way

www.iosrjournals.org 41 | Page

.12

TT

T

QrT WT

WSe

E (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.