Download Document 8686749

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
 1n 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 n1 x 
n  1H n1 x  xH n x  H n1 x

H n  x    1 H n x 
This is shown by using induction method:
n
To show that H n x   H n1 x  ;
Let n = 1, from
'
 1n 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 n1  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
vB  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
  EFW (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 nk  , 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 nk   f n
n1
1, 2
This gives F  D

 lim lim  nn! f nk   f n j 
L  
k  j 
2
2
n
n1
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 n1  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 gx1 ,..., xn   x1 ,..., xn we have
m
n
Dt F  
W ( Ai1 )...1Aij ...W ( Aim )  mI m1  f m (. , t ) . □
a
i1...im
j 1 i1,...,im1
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 EF \ FA    I n f n1A 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

EF 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, DFG   GDF 
 E u F  u, DF G.
□
EG 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    DF 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  EF    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


EDt F Ft   E  nI n1  f n ., t  Ft  dW t 
0
 n1

  nEI  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 
EDt F Ft  
0
n1
0

0,t 
n

T 

n1

( . ) dW t 



n1
n1
 nn  1! In1 f n ., t X0,tn-1  dW t    n!In  f n .   I n  f n 
n1

  I n  f n   I 0  f 0   F  EF .
□
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:
dBt   rBt 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   ii 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  Bt1St  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  GuD 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 us ds   EF u   E  F  ut 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 0T
(15)
The weight is given by
weight =
6.3 Computation of Gamma
WT
.
S 0T
 

 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 WT  1  Dt ST dt      ST WT  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 02T

T




6.4 Computation of Vega


V0 E Q e  rT ST 



Q rT
E e  S 0 exp  







 E Q e  rT ST  ST
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