on the impact of correlation on option prices: a malliavin ...calculate the malliavin derivative of...
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On the impact of correlation on option prices: a Malliavin Calculus approach
E. Alòs (2006): A generalization of the Hull and White formula with applicationsto option pricing approximation. Finance and Stochastics 10 (3), 353-365.
E. Alòs, J. A. León and J. Vives (2007): On the short-time behaviour of theimplied volatility for stochastic volatility models with jumps. Finance and Stochastics 11 (4), 571-589
RESULTS FROM
STOCHASTIC VOLATILITY MODELS
Stochastic volatility models allow us to describe the smiles andskews observed in real market data:
( )*2*2 12
1ttttt dBdWdtrdX ρρσσ −++
−=
0=ρ
Log-price Volatility (stochastic, adapted to the filtrationgenerated by W)
Implied volatility smile Implied volatility skew 0≠ρ
SOME QUESTIONS AND MOTIVATION
How to quantify the impact of correlation on option prices?
What about the term structure?
Option price=option price in the uncorrelated case
(classical Hull and White formula)+ correction due by correlation
We will develop a formula of the form
This result will allow us to describe the impact of the correlation on theoption prices. As an application, we can use it to construct option pricing
approximation formulas, or to study the short-time behaviour of the impliedvolatility for stochastic volatility models with jumps.
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (I)
Calculate the Malliavin derivative of a diffusion process
Use the duality relationship between the Malliavinderivative and the Skorohod integral to develop adequate
change-of-variable formulas for anticipating processes
Here our pourpose is to present the basic concepts onMalliavin calculus that have been used up to now in financial
applications. Basically, we will see how to:
MAIN IDEA: THE FUTURE INTEGRATED VOLATILITY IS AN ANTICIPATING PROCESS
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (II)
Malliavin derivative : definition
( ) ( ) ( )( )[ ]( )Ω×
=
TL
hWhWhWfF n
,0in variablerandom
...,,2
21
( ) [ ]( ) processGaussian
,0, 2 TLhhW ∈
( ) ( ) ( )( ) ( )
[ ]( )Ω×
∂∂=∑
TL
thhWhWhWx
fFD in
it
,0in DerivativeMalliavin
...,,
2
21
(closable operator)
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (III)
Malliavin derivative: examples
[ ] )(1
Scholes)-(Black 2
-rexp
,0
2
t
rSSD
WtS
tttW
r
t
σ
σσ
=
+
=
[ ] )(1 ,0 tWD ssW
t =
( ) ( )
( )[ ] )(1
)(
,0
00
rceYD
UhlenbeckOrnstein
dWecemYmY
trt
tW
r
r
t rttt
−−
−−−
=
−
+−+= ∫
α
αα
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (IV)
Skorohod integral: definition
It is the adjoint of the Malliavin derivative operator:
[ ]( ) ( ) ( )hWhTLh W =⇒∈ δ,02
( )( ) ( ) SFdsuFDEFuE s
T Ws
W ∈= ∫ allfor ,0
δ
Example:
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (V)
Skorohod integral: properties
The Skorohod integral of a process multiplied by a random variable
( )∫∫∫ +=T
sWs
T
ss
T
ss dsuFDdWuFdWFu000
The Skorohod integral is an extension ofthe classical Itô integral
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)
THE ANTICIPATING ITÔ’S FORMULA
( ) ( ) ( )
( ) ( )( ) ,''2
1'
'
00
00
∫∫
∫
∇++
+=
t
sss
t
ss
t
ssst
dsuuXFdsvXF
dWuXFXFXF
∫ ∫++=t t
ssst dsvdWuXX0 00
Non necessarily adapted
( ) drvDdWuDuus
rWs
s
rrWsss ∫∫ ++=∇
0022 : where
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VI)
Proof (sketch)
.0 assume wesimplicity of sake For the ≡v
( ) ( ) ( )
( )∑ ∫
∑ ∫
+
+=
+
+
2
0
1
1
''2
1
'
i
ii
i
ii
t
t sst
t
t sstt
dWuXF
dWuXFXFXF
We proceed as in the proof of the classical Itô’s formula
∫t
s dsu0
2
2
1
SOME PRELIMINARIES ON STOCHASTIC CALCULUS FOR ANTICIPATING PROCESSES (VII)
( )( ) ( )∑∫∑∫
∑ ∫++
+
−=
11
1
''
'
i
ii
i
ii
i
ii
t
t stWs
t
t sst
t
t sst
dsuXFDdWuXF
dWuXF
∫t
sss dWuXF0
)('2
1
∫ ∫
t
s
s
rrWss dsudWuDXF
0 0)(''
( )[ ]∫−t
ssWs dsuXFD
0'
AN EXTENSION OF THE HULL AND WHITE FORMULA (I)
[ ]ttTr
t FHeEV )(* −−=
option price payoff
( )TTT XTBSV ν;,=
The Black-Scholes function final conditionimplies that
−= ∫ ds
tTv
T
t
st22 1 σ
Basic idea
Black-Scholespricing formula
Log-price
where
AN EXTENSION OF THE HULL AND WHITE FORMULA (II)
Then
( )( )[ ]tTT
tTtTr
t
FXTBSE
FVEeV
ν;,*
*)(
=
= −−
The classical Hull and White term
( )( )ttt FXtBSE ν;,*
is the option price in the uncorrelated case
com
pare
Then we want to evaluate the difference
( ) ( )ttTT XtBSXTBS νν ;,;, −
We need to construct an adequateanticipating Itô’s formula
process nganticipatian is tν
AN EXTENSION OF THE HULL AND WHITE FORMULA (III)
AN EXTENSION OF THE HULL AND WHITE FORMULA (IV)
Anticipating Itô’s formula
( ) ( ) ( )
( ) ( )
( )( )
( )∫
∫
∫ ∫
∫
∂∂+
∂∂∂+
∂∂+
∂∂+
∂∂+=
−
t
sss
t
ssss
t t
ssssss
t
sstt
dsuYXsx
F
dsuYDYXsyx
F
dYYXsy
FdXYXs
x
F
dsYXss
FYXFYXtF
0
22
2
0
2
0 0
000
,,
,,
,,,,
,,,,0,,
∫=T
t st dsY θ ( ) ∫=− T
s rWss drDYD θ:
additionalterm
AN EXTENSION OF THE HULL AND WHITE FORMULA (V)
( )
−= −−
ttrt
ttrt Y
tTXtBSeXtBSe
1;,;, ν
Main result:
the extension of the Hull and White formula
We apply the above Itô’s anticipating formula to the process
and we obtain
AN EXTENSION OF THE HULL AND WHITE FORMULA (VI)
( ) ( )
( ) ( ) ( )
( ) ( )( ) ( ) ( )
( ) ( )( )∫
∫
∫
∫
−−
∂∂−
−∂∂∂+
−+∂
∂+
∂∂−
∂∂−++
==
−
−−
−
−
−−−
T
ts
ssss
rs
t
sss
ssrs
tss
T
t ssrs
T
t sssssBSrs
ttrt
TTrT
TrT
dssT
XsBS
e
dsYDsT
Xsx
BSe
dZdWXsx
BSe
dsXsBSxx
Le
XtBSeXTBSeVe
ννσν
σ
σν
νσ
ρ
ρρσν
ννσν
νν
22
0
2
*2*
2
222
,,2
1
1,,
2
1,,
,,2
1
,,,,
Black-Scholes differential operator
Cancel
Zero expectation
AN EXTENSION OF THE HULL AND WHITE FORMULA (VII)
( ) ( )( )( ) ( )∫
−−
−−
−∂∂∂+
=t
sss
ssrs
tttrt
tTrT
dsYDsT
Xsx
BSe
FXtBSEeFVEe
0
2
**
)(
1,,
2
,,
σν
νσ
ρ
ν
( ) ( )ssss XsHXsxx
νν ,,:,,2
2
3
3
=
∂∂−
∂∂
s
T
s rWss drD σσ
=Λ ∫
2*
:
( ) ( )( )( )
( ) ( )
Λ+
=
=
∫−−
−−
t
t
ssstsr
ttt
tTtTr
t
FdsXsHeE
FXtBSE
FVEeV
0
*
*
*
,,2
,,
νρν
AN EXTENSION OF THE HULL AND WHITE FORMULA (VIII)
( ) ( )
Λ∫
−−t
t
ssstsr FdsXsHeE
0
* ,,2
νρ
The above arguments do not requiere the volatility to be Markovian.
The main contribution of this formula is to describe the effect of thecorrelation as the term
APPLICATIONS TO OPTION PRICING APPROXIMATION (I)
( )( )
( )∫
∫
−=
Λ+
=
T
t tst
t
T
t stt
ttaprox
dsFEtT
FdsEXtH
XtBSV
2**
**
*
1
,,,2
,,
σν
νρν
Consider the approximation
APPLICATIONS TO OPTION PRICING APPROXIMATION (II)
( )( )
enoughregular and
with ,
f
dWdtYmdY
Yf
ttt
ss
αλα
σ
+−=
=
Using Malliavin calculus again we can see that, in the case
( )ααλ
ln12
+≤− CVV aproxt
A similar result was proven by Alòs and Ewald(2008) for the Heston volatility model
APPLICATIONS TO OPTION PRICING APPROXIMATION (III)
Application:
the Stein and Stein model with correlation
( )tt Y=σ
( ) ( )∫
−−− =−+=uts
tt deuFemmsM0
2)(,)( θσ αθα
( ) ( )( )dstsFcsMtT
T
t tt ∫ −+−
= 222* )(1ν
( )
( ) ( )∫∫ ∫
∫ ∫
−+
=
−−
−−
T
t
T
t
T
s tsr
t
s
T
t s
T
s
srr
dssFsTFcdsdrrMesMc
FdsYdreYE
)()()( 2
*
α
α
,αλ=c
APPLICATIONS TO OPTION PRICING APPROXIMATION (IV)
Numerical results
)2.0,0953.0,05.0,2.0,4,100ln,5.0( =======− tt rmXtT σλα
APPLICATIONS TO THE STUDY OF LONG-MEMORY VOLATILITY MODELS (I)
Example: long-memory volatilities
( ) ( )
)(~ where
,~10
212
ss
t
st
Yf
dsst
=
−Γ
= ∫−
σ
σβ
σ β
Assume that (see for example Comte, Coutin and Renault (2003)),
( ) ( )
( )( ) drduur
durTdr
T
s
s
u
T
s r
T
sr
∫ ∫
∫ ∫
Γ−+
−+Γ
=
−
0
21
22
~
,~1
1
σβ
σβ
σ
β
β
APPLICATIONS TO THE STUDY OF LONG-MEMORY VOLATILITY MODELS (II)
( ) ( )∫ ∫ −+Γ
=T
s rWs
T
srWs drDrTdrD 2*2* ~
1
1 σβ
σ β
( ) ( )
( )
( ) ( ) ( ) ( ) ( )
−×
+Γ=
−
+Γ=
Λ
∫ ∫
∫ ∫
∫
−−ts
T
t
T
s rrsr
ts
T
t
T
s rWs
t
T
t s
FdsYfdrYfYferTE
FdsdrDrTE
FdsE
'
1
2
~~1
*
2**
*
αβ
β
βραλ
σσβ
ραλ
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY FOR JUMP-
DIFFUSION MODELS WITH STOCHASTIC VOLATILITY (I)
( )
( ) [ ]TtZdBdW
dstkrxX
t
tsss
t
st
,0,1
2
1
0
2
0
2
∈+−++
−−+=
∫
∫
ρρσ
σλ
In Alòs, León and Vives (2007) we considered the following modelfor the log-price of a stock under a risk-neutral probability Q:
independentAdapted to the
filtration generatedby W
( ) ( )∫ ∞<−=
=
dyek
yf
y νλ
λνλ
11
with
,)( measureLévy and
intensity th Poisson wi Compound
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (II)
( )( )( ) ( )
( ) ( ) ( )( ) ( )
( ) ( )
∂−
−++
Λ∂+
=
∫
∫ ∫
∫
−−
−−
−−
t
T
t ssxtsr
t
T
t R sssstsr
t
t
sssxtsr
tttt
FdsXsBSekE
FdsdyXsBSyXsBSeE
FdsXsGeE
FXtBSEV
νλ
ννν
νρν
,,
,,,,
,,2
,,
0
Hull and White
Correlation
Jumps
Similar arguments as in the previous paper give us thefollowing extension of the Hull and White formula:
( ) ( ) ( ) ( )σσν ,,,,; 2 xtBSxtGtT
Yt xxx
t ∂−∂=−
=
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (III)
After some algebra, we can prove from this expression that:
( ) 0,)( If 22≥−≤
δσ δsrCFDE trs
[ ] ttrttrDD σσ +
↓ =lim
and ( )t
ttt
tt
t kDx
X
I
σλσ
σρ −−→
∂∂ +*
( ) 0,)( If 22 <−≤
δσ δsrCFDE trs
and
( ) ( ) tt
T
t
T
s trs LdrdsFDEtT
σσ δδ
++ →
− ∫ ∫,
2
1( ) ( )
tttt
tt LX
xItT σ
σρ δδ +− −=
∂∂− ,
*
lim
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (IV)
Example 1: classical jump-diffusion models
Assume that the volatility process can be written as
( )( ) ( ) rrrr
rr
dWYrbdrYradY
Yf
,,
,
+==σ
( ) ( )
( ) uusu
r
s
susu
r
srs
dWYDYux
b
YsbduYDYux
aYD
,
,,
∫
∫
∂∂+
+∂∂=
( ) ( )tttt YtbYfD ,'=+σ
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (V)
( ) ( )( )),('1
lim *tt
tt
ttT YtbYfkx
x
I ρλσ
+−=∂∂
→
( ) ( ) ( )r
srr
t
trtr dWecemYmY −−−−
∫+−+= αα α2
If Y is an Ornstein-Uhlenbeck process of the form
( ) ( )( )tt
tt
tT Yfckxx
I'2
1lim * αρλ
σ+−=
∂∂
→
(this agrees with the results in Medvedev andScaillet (2004))
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VI)
Example 2: fractional stochastic volatility models with H>1/2
Assume that the volatility process can be written as
0=+ttD σ
( ) ( ) ( ) ( )∫
−−−− +−+==r
t
Hr
srtrtrrr dWecemYmYYf αα ασ 2;
( )∫ ∫
−
−−−−r
t s
r
s
Hur dWdusueH 3
1
)(2
1 α
( )t
tt
tT
kx
x
I
σλ−=
∂∂
→*lim
That is, the at-the-money short-datedskew slope is not affected by the
correlation in this case
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VII)
Example 3: fractional stochastic volatility models with H<1/2
Assume that the volatility process can be written as
( ) ( ) ( ) ( )∫
−−−− +−+==r
t
Hr
srtrtrrr dWecemYmYYf αα ασ 2;
( ) ( )( )
( )s
Hr
t
sr
r
t s
r
s
Hsrur
dWsre
dWdusueeH
2
1
3
1
)(
)(2
1
−−−
−−−−−
−+
−−
−
∫
∫ ∫
α
αα
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (VIII)
( )
.as0
'2
)(
1
2
12
tT
FYfcdrdsD
tT
E t
T
t
T
s trWs
H
→→
−−
∫ ∫−+ασ
( ) ( ) ( )ttt
tHtT Yfcx
X
ItT '2lim *
2
1
αρ−=∂∂
− −→
That is, the introduction of fractional components withHurst index H<1/2 in the definition of the volatilityprocess allows us to reproduce a skew slope of order
( )2
1, −>− δδtTO
More similar to the ones observed in empirical data (see Lee (2004))
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (IX)
Example 4: Time-varying coefficients
(Fouque, Papanicolaou, Sircar and Solna (2004))
Assume that the volatility process can be written as
( )
( ) ( ) ( ) ( )
( ) 0,)(
,2
2
1
>−=
+∫−+=
=
+−
−−−
∫
εα
α
σ
ε
αα
sTs
dWescemYmY
Yf
r
t rsrdss
tr
rr
r
t
Next maturity date
APPLICATIONS TO THE STUDY OF THE SHORT-TIME BEHAVIOUR OF THE IMPLIED VOLATILITY (X)
( )( )
( )
tT
Yfc
drdsFDEtT
t
T
t
T
s trs
→
+
+−+
−∫ ∫
−+
as zero totends
2
'
2/1
1
2/1
1
1
2
12
εερ
σε
In this case, the short-date skew slope of theimplied volatility is of the order
( ) ε+−− 2
1
tTO
Then
BIBLIOGRAPHY
E. Alòs (2006): A generalization of the Hull and White formula with applicationsto option pricing approximation. Finance and Stochastics 10 (3), 353-365.
E. Alòs, J. A. León and J. Vives (2007): On the short-time behaviour of theimplied volatility for stochastic volatility models with jumps. Finance and Stochastics 11 (4), 571-589
E. Alòs and C. O. Ewald (2008): Malliavin Differentiability of the Heston Volatilityand Applications to Option Pricing. Advances in Applied Probability 40 (1), 144-162.
F. Comte, L. Coutin and E. Renault (2003): Affine fractional stochastic volatilitymodels with application to option pricing. Preprint.
J. P. Fouque, G. Papanicolau, K. R. Sircar and K. Solna (2004): Maturity Cyclesin Implied Volatilities. Finance and Stochastics 8 (4), 451-477.
R. Lee (2004): Implied volatility: statics, dynamics and probabilistic interpretation. Recent advances in applied probability. Springer.
A. Medvedev and O. Scaillet (2004): A simple calibration procedure of stochasticvolatility models with jumps by short term asymptotics. Discussion paper HEC, Gèneve and FAME, Université de Gèneve.