Stochastics Calculus: Malliavin Calculus in a simplest way

IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728,p-ISSN: 2319-765X, Volume 6, Issue 3 (May. - Jun. 2013), PP 30-41
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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
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
)(
2
2
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 =
  ,
!
1 22
22







 
x
n
nxn
e
dx
d
e
n
n ≥1 (1)
and H  x0 = 1. These hermite polynomials are coefficients of the power expansion in t of the function
  )(exp, 2
2
t
txxtF  which can easily be seen by rewriting
    






2
2
2
1
2
exp, tx
x
xtF
and expanding the function around t = 0.

The power expansion combines with some particular properties of F, that is
tF
t
txt
x
F









2
exp
2
   Ftx
t
txtx
t
F









2
exp
2
and
   txF
t
txtxF 





 ,
2
exp,
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
!
1
x
n
nxn
e
dx
d
e
n
we have
   
'
22
'
22'
1
2222



























xxxx
exee
dx
d
exH
 xHHx n 01
'
1  
Let n = 2,
 
'
2
'
2
2
2'
2
2
2
2
2 22
2
1
2
1


































 xx
xe
dx
d
ee
dx
d
exH
xx
   .1
2
1
2
1
1
'2
'
2222
222
xHxxexee
xxx



















Also for n = 3 we have
       .1
2
1
2
6
1
6
1
2
2'3
'
2
3
3
2'
3
22
xHxxxxe
dx
d
exH
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
,n
n!
1
XY
YHXH mn
E
E
Proof. See the proof of lemma 1.1.1 in [8]

 
   
 
   
   
  havewe
mappingoflinearityBy the.9allfor0thatsuchLet
.,Ω,
ofsubsettotalaformevariablesrandomThe ;hW
hWh
hXeLX
PL
Lh
hW2
2
2



HEG
G
GH
Proof.
2.4Lemma
   21,,...2,1,,0exp
1
1 












mmihthWtX ii
m
i
HR,E
       mB hWhWXv
v
,...,B
bygivenisofLaplacethat theshows(2)Equation
11E
  .0Gfor0is,ThatGsetevery
forzerobemustmeasurethezero,istransformtheAs.setBorelafor
1 

XX
v
G
m
GG.
B
E
R
□
     .1,:variablesrandomby thegenerated,Ω,ofsubspace
linearclosedtheasnorderofchoasthedefinewe,1eachFor


H
H
H
hhhWHPFL
n
n
2
n2.5.Definition
 
 
  .,Ω,
:9subspacestheofsum
orthogonalinfinitytheintodecomposedbecan,Ω,spaceThe.
0 n
PL
PL
n
2
n
2
HG
H
G



2.6Theorem
 
   
 
  
     30exp
therefore,00havewe
,0,polynomialetheHermitofncombinatiolinearaasexpressedbecan
fact thattheUsing.1///such thatallfor0thathaveWe
.0thatshowWe0.nallfortoorthogonalbe,Ω,Letproof.





htWX
nhXW
nrxH
hhhWXH
XPFLX
n
r
n
n
n
2
E
E
E xHH
H
implies(3)equationthathavewe),,,(ofsubsettotalaform},e{
variablesrandomthethatstateswhichabovelemmatheBy1.normofhallforand
2W(h)
PLh
t
GH
H

 R
that X= 0 □
 
 
 
iable.smooth vara,and,...,where
(4))(),...,(
formthevariablerandomallofset
thebydenotewe,Cand1nLetgrowth).polynomialwithsderivative
partialdenoted(growthpolynomialmostathavesderivativeitsofallandsuch that
denoted:functionsabledifferentiinfinitelyallofCsetheconsider tWe
1
1
SFhh
hWhWfF
Sf
pf
f
n
n
n
p
nn
p






H
R
RRR
DerivativeMalliavinThe3.
 
H.



:DFmappingaisderivativethewhere
(5).)(),...,(DF
asdefineisvariablerandomofderivativeThe
1
oi
i in
n
hhWhWf
SF3.1.Definition

    )6(.)(,
thenandLet
hFWhDF
hF
EE 

H
HF3.2Lemma
 
 )(),...,(
ascan write
that wesuch that,such that,...offamilyorthonomalanexiststhere
on W,productscalartheoflinearitythetoDue.1thatSuppose.
1
11
n
n
eWeWfF
F
ehee
h



H,
H
Proof
   
    .
2
1
exp2
isthaton,distributinormalstandard
theofdensitytemultivariathebeLet.Cfunctionsuitableafor
1
2
2 










n
i
i
n
n
p
x
f
 x
xR
       
   .)()(
)(,
formularpartsbynintegratioclassicalbyhaveWe
1
11
hFWeFW
xdxxxfdxxxxfhDF n n
EE
E
R R

   H
proof.thecompletesThis □
Proposition 3.3. (Nualart,2006). blediferentiaycontinouslabe: RR m
Let 
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,FandAssume 2,1
k
2
 kPLF D
 PLinkFFk
2
,,.1 
   .Linconverges.2 2
1



PFD kkt
2,1
FThen D and inkFDFD tkt ,,   P2
L .
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
   
   




 kjFDFDffnn
PLjtkt
Ln
j
n
k
n
n
,,0!
2
2
1
2
2


Hence, by Fatou Lemma,
 
 
   
 







1
22
1
.0!limlim!lim 22
n
L
j
n
k
n
jkLn
k
n
n
k
nn ffnnffnn 
This gives
2,1
DF , in,,  kFDFD tkt  PL2
. □
Proposition 3.5. [8] Suppose
2,1
DF is a square integrable random variable
With a decomposition given above, we have

  )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 ,...,,...,gwith)(),...,( 111  we have
   

m
j
n
imi
mmimAijiimit tfmIAWAWaFD
1 1,...,1
11...1 .),(.)(...1)...( □
Theorem 3.6. Let F be a square integrable random variable denoted by
     






00
.1FEthen,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
,...,with1 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 ABWABWEBWBWEFE
1
1 )()(/)()...( AAA FFF
    .1 ...1 ABABn n
I  □
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
2
L -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
L1,2
D
Definition 4.1. The adjoint of the operator D denote by δ is an unbounded operator
  satisfiesand;on 2
HL
 The domain of δ denoted as Domδ is the set of H -valued square integrable random variable
 H,2
 Lu where
  2,1
2
allfor, 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(, H
uDFuF EE 
.foreach 2,1
DF
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
jjhFu
1
(9)

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 hDFhWFu
1 1
.)(
H
 (10)
Theorem 4.2. Let F ϵ D1,2
and u ϵ Domδ such that  .;2
H LFu Then Fu belongs to
Domδ and we have the equality
    H
uDFuFFu , 
provided the right hand side is square integrable.
Proof. Let G be any smooth random variable, we obtain
       HH
GDFFGDuFuDGFuG  ,, EEE 
   ., GDFuFu H
 E □
Lemma 4.3. Let     ,where:
1


n
j
jj
hh
SuhFDuD H 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 hDFhWFu
1 1
.
H

yields
    

n
j
jjjj
h
hhDFDhWFDuD
1
,,)(
HH

=     

n
j
n
j
jj
h
jj
h
jj hFDDhWFDhhF
1 1
,)(,
HH
=  ., uDhu h
H
□
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).
WandLet 2,1
DF is a one-dimensional Brownian motion on [0,1]. Then
   
T
ttt dWFDEFF
0
.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 tdWtfnIEFDE
0
1
1
0
., FF
=       



T
n
tnn tdWtfInE
0
1
1 ., F
Thus,

  
T
tt FDE
0
F        




T
n
nn tdWtfnI
0
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
t dtr
0
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
s0and,0allfor0  a.s.
processthedefineWe.where i
ii F


 







  
t t
ssst dsdWZ
0 0
2
2
1
exp 
which is positive local martingale. If
1
2
1
exp
0 0
2












  
T T
sss dtdW E
then the process ZT is a martingale and measure Q such that TZ
dP
dQ
 is a
probability measure, equivalent to P, such that under Q,, the process

t
stt dsWW
0
~

is a Brownian motion.
In terms of the process tW
~
, the price process can be expressed as
     
 
         









1 1
1
0
1
.!.,!1
n n
nnnnnn
T
n
fIfntdWtfnn IXI 1-n
t0,

.
~
)(exp
0 020
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 dsuWPTQ
0
1
is,that has the law of Brownian motion under Q) if and
only if E(ξ1) =1 and in this case .1
dP
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 Du
F = 0, H
uDF almost surely and
Gu(Du
F)-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; ∆: = S
V


.

 Gamma: =Second derivative with respect to the price of the underlying;
2
2
: S
V



 Vega: = Derivative with respect to the volatility; 

 V
v:
 Rho: = Derivative with respect to interest rate; r
V


:
 Theta: = Derivative with respect to time; t
V


 :
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(Du
F)-1
ϵ Domδ it suffices to say that
Gu(Du
F)-1
ϵ D1,2
as this implies that   ., 2,11
DomuDFGu 

DH
 If u = DF, then the conclusion of Proposition 5.5 is written as
     .2

















H
DF
GDF
FfGFf EE
We consider an option with payoff H such that EQ
(H2
) < ∞. We have that
  .




 

t
dsrQ
t HeEV
T
t
s
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
dF
Ffe
V QrT
E (13)
From Proposition 5.5, we have
  .,
V0
















 



d
dF
FHFfe 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 Du
F must not be zero in order to make sure that FDu
1
exist.
 Let      ;,0,)()(exp 20
2
TttWtStS   
then,

  0
2
)()(exp 2
0
S
ST T
TWT
S
S



 
 In calculating the greeks, we assume that
     TttWtStS ,0,)()(exp 20
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  ,rT
eE
      .)(
0000
TT
rT
t
T
rTT
rT
SS
S
e
S
S
Se
S
Se
S
V


















QQ
Q
EE
E
From proposition 5.5, by letting u =1, F = ST , and G = ST, we obtain
  
T T
TTttT
u
TSdtSdtSDSD
0 0
.
From the condition in the above remark, we have
 
















 T
T
T
TtT
T
W
T
dW
T
dtSDS
0
1
0
.
1


As a result,
  .
0
TT
Q
rT
WS
TS
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
S
Se
S
Se
S
V TT
T
rTQT
rTQ
E
E
    .2
2
0
2
0
TT
Q
rT
T
T
rTQ
SS
S
e
S
S
Se 



















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
DS
T
S
T
S
dtSDS TT
T
T
TtT




1
,
1
1
0
2



















 
T
T
T
TdW
dS
dW
T
S
0
1
,
1


T
TT
T
T
dtS
T
W
S
0 


.1






T
W
SS
T
W
S T
TT
T
T

Thus,

     .12















T
W
SSSS T
TT
Q
TT
Q

EE
Using Malliavin property, we now eliminate the remaining “/”
.
From proposition 5.5, taking  .1 T
W
T
T
SG  F = ST and u = 1 gives
    

















 T
T
W
T
T
TtT
W
T
TS
SdtSDS TT

 
1
11
1
0
   1
1

 TT
W
T TSS T
 
     TT
W
TT
W TT

 11
2222 







T
TT
T
dW
T
DW
T
W
02222
1
,
1


  
T T
T
T
T
W
T
dt
T
dW
W
0 0 2222

.
1
222
2
2222 






T
W
TT
W
T
W
T
T
T
WW TTTTT

As a result,
    .
1
1 222
2



























T
W
TT
W
S
T
W
SS TT
T
QT
TT
Q

EE
This implies that
  .
12
2
0
















T
T
T
Q
rT
W
T
W
S
TS
e

E (16)
6.4 Computation of Vega
  









T
rTQ
SeV E0
    

 




T
rTQ
WTSe 20
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
TWS
dtSDTWS T
T
TT
T
TtTT






1
0
 11 



 












T
W
T
W TT






 
T
TT dW
T
DW
T
W
0
1
,
1


  
T T
T
t
T W
T
dt
T
dW
W
0 0 






 TT
T
W
T
TW
W
T
T
T
W

122

  .
12














 
T
T
T
QrT
W
T
W
Se

 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
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