sensitivity analysis and density estimation using the
TRANSCRIPT
Sensitivity analysis and density estimation using the Malliavincalculus
Nicolas Privault
December 17, 2005
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Sensitivity analysis and Monte Carlo method
Consider (F ζ)ζ a family of random variables depending on a parameter ζ.
∂
∂ζE
hf
“F ζ
”i=
8>>><>>>:E
»f ′
`F ζ
´ ∂
∂ζF ζ
–
'E
ˆf
`F ζ+h
´˜− E
ˆf
`F ζ−h
´˜2h
Expectations can be computed by the Monte Carlo method:
E [F ] ' F1 + · · ·+ Fn
n,
where F1, . . . , Fn is a random sample of F .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (1): sensitivity analysis in finance - Greeks
Price process:dSζ
t
Sζt
= r(Sζt )dt + σ(Sζ
t )dMt , Sζ0 = x .
F ζ = ST and ζ ∈ x , r , σ, T , ....
Payoff function:
f (x) = (x − K)+
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
x
f (x) = 1[K ,∞)(x).
0
0.5
1
1.5
2
1 1.5 2 2.5 3 3.5 4
x
Option price:
E [f (SζT )].
Greeks:
ζ = x : Delta
ζ = σ: Vega
ζ = r : Rho
ζ = T : Theta.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (1): sensitivity analysis in finance - Greeks
Price process:dSζ
t
Sζt
= r(Sζt )dt + σ(Sζ
t )dMt , Sζ0 = x .
F ζ = ST and ζ ∈ x , r , σ, T , ....
Payoff function:
f (x) = (x − K)+
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
x
f (x) = 1[K ,∞)(x).
0
0.5
1
1.5
2
1 1.5 2 2.5 3 3.5 4
x
Option price:
E [f (SζT )].
Greeks:
ζ = x : Delta
ζ = σ: Vega
ζ = r : Rho
ζ = T : Theta.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (1): sensitivity analysis in finance - Greeks
Price process:dSζ
t
Sζt
= r(Sζt )dt + σ(Sζ
t )dMt , Sζ0 = x .
F ζ = ST and ζ ∈ x , r , σ, T , ....
Payoff function:
f (x) = (x − K)+
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
x
f (x) = 1[K ,∞)(x).
0
0.5
1
1.5
2
1 1.5 2 2.5 3 3.5 4
x
Option price:
E [f (SζT )].
Greeks:
ζ = x : Delta
ζ = σ: Vega
ζ = r : Rho
ζ = T : Theta.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (1): sensitivity analysis in finance - Greeks
Price process:dSζ
t
Sζt
= r(Sζt )dt + σ(Sζ
t )dMt , Sζ0 = x .
F ζ = ST and ζ ∈ x , r , σ, T , ....
Payoff function:
f (x) = (x − K)+
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
x
f (x) = 1[K ,∞)(x).
0
0.5
1
1.5
2
1 1.5 2 2.5 3 3.5 4
x
Option price:
E [f (SζT )].
Greeks:
ζ = x : Delta
ζ = σ: Vega
ζ = r : Rho
ζ = T : Theta.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (1): sensitivity analysis in finance - Greeks
Price process:dSζ
t
Sζt
= r(Sζt )dt + σ(Sζ
t )dMt , Sζ0 = x .
F ζ = ST and ζ ∈ x , r , σ, T , ....
Payoff function:
f (x) = (x − K)+
0
0.5
1
1.5
2
2.5
3
1 1.5 2 2.5 3 3.5 4
x
f (x) = 1[K ,∞)(x).
0
0.5
1
1.5
2
1 1.5 2 2.5 3 3.5 4
x
Option price:
E [f (SζT )].
Greeks:
ζ = x : Delta
ζ = σ: Vega
ζ = r : Rho
ζ = T : Theta.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (2): density estimation
Let F ζ = F − ζ and f (x) = 1(−∞,0].
Density of F :
φF (ξ) : =∂
∂ζP(F ≤ ζ) =
∂
∂ζE
ˆ1(−∞,0](F − ζ)
˜=
∂
∂ζE
hf
“F ζ
”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]
2h=
Eˆ1[ζ−h,ζ+h](F )
˜2h
.
Example: F =R T
0e−rtdNt .
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
h=1h=0.1
h=0.01
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (2): density estimation
Let F ζ = F − ζ and f (x) = 1(−∞,0].
Density of F :
φF (ξ) : =∂
∂ζP(F ≤ ζ) =
∂
∂ζE
ˆ1(−∞,0](F − ζ)
˜=
∂
∂ζE
hf
“F ζ
”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]
2h=
Eˆ1[ζ−h,ζ+h](F )
˜2h
.
Example: F =R T
0e−rtdNt .
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
h=1h=0.1
h=0.01
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Application (2): density estimation
Let F ζ = F − ζ and f (x) = 1(−∞,0].
Density of F :
φF (ξ) : =∂
∂ζP(F ≤ ζ) =
∂
∂ζE
ˆ1(−∞,0](F − ζ)
˜=
∂
∂ζE
hf
“F ζ
”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]
2h=
Eˆ1[ζ−h,ζ+h](F )
˜2h
.
Example: F =R T
0e−rtdNt .
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
h=1h=0.1
h=0.01
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
1 Dw a derivation operator acting on random variables:
f ′(F ζ) =Dw
ˆf (F ζ)
˜DwF ζ
.
2 D∗w the adjoint of Dw :
〈F , DwG〉 = E [FDwG ] = E [GD∗wF ] = 〈G , D∗
wF 〉.
3 Main argument:
∂
∂ζE
hf (F ζ)
i= E
hf ′
“F ζ
”∂ζF
ζi
= E
»Dw [f (F ζ)]
DwF ζ∂ζF
ζ
–= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
1 Dw a derivation operator acting on random variables:
f ′(F ζ) =Dw
ˆf (F ζ)
˜DwF ζ
.
2 D∗w the adjoint of Dw :
〈F , DwG〉 = E [FDwG ] = E [GD∗wF ] = 〈G , D∗
wF 〉.
3 Main argument:
∂
∂ζE
hf (F ζ)
i= E
hf ′
“F ζ
”∂ζF
ζi
= E
»Dw [f (F ζ)]
DwF ζ∂ζF
ζ
–= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
1 Dw a derivation operator acting on random variables:
f ′(F ζ) =Dw
ˆf (F ζ)
˜DwF ζ
.
2 D∗w the adjoint of Dw :
〈F , DwG〉 = E [FDwG ] = E [GD∗wF ] = 〈G , D∗
wF 〉.
3 Main argument:
∂
∂ζE
hf (F ζ)
i= E
hf ′
“F ζ
”∂ζF
ζi
= E
»Dw [f (F ζ)]
DwF ζ∂ζF
ζ
–= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
Proposition
Assume that DwF ζ 6= 0, a.s. on ∂ζFζ 6= 0, ζ ∈ (a, b). We have
∂
∂ζE
hf
“F ζ
”i= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–, ζ ∈ (a, b). (1)
No differentiability assumption on f , compare with:
∂
∂ζE
hf
“F ζ
”i= E
hf ′
“F ζ
”∂ζF
ζi.
Independence on the bandwidth parameter h, compare with:
∂
∂ζE
hf
“F ζ
”i'
Eˆf
`F ζ+h
´˜− E
ˆf
`F ζ−h
´˜2h
.
The weight
W := D∗w
„∂ζF
ζ
DwF ζ
«is independent of f .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
Proposition
Assume that DwF ζ 6= 0, a.s. on ∂ζFζ 6= 0, ζ ∈ (a, b). We have
∂
∂ζE
hf
“F ζ
”i= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–, ζ ∈ (a, b). (1)
No differentiability assumption on f , compare with:
∂
∂ζE
hf
“F ζ
”i= E
hf ′
“F ζ
”∂ζF
ζi.
Independence on the bandwidth parameter h, compare with:
∂
∂ζE
hf
“F ζ
”i'
Eˆf
`F ζ+h
´˜− E
ˆf
`F ζ−h
´˜2h
.
The weight
W := D∗w
„∂ζF
ζ
DwF ζ
«is independent of f .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
Proposition
Assume that DwF ζ 6= 0, a.s. on ∂ζFζ 6= 0, ζ ∈ (a, b). We have
∂
∂ζE
hf
“F ζ
”i= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–, ζ ∈ (a, b). (1)
No differentiability assumption on f , compare with:
∂
∂ζE
hf
“F ζ
”i= E
hf ′
“F ζ
”∂ζF
ζi.
Independence on the bandwidth parameter h, compare with:
∂
∂ζE
hf
“F ζ
”i'
Eˆf
`F ζ+h
´˜− E
ˆf
`F ζ−h
´˜2h
.
The weight
W := D∗w
„∂ζF
ζ
DwF ζ
«is independent of f .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
Proposition
Assume that DwF ζ 6= 0, a.s. on ∂ζFζ 6= 0, ζ ∈ (a, b). We have
∂
∂ζE
hf
“F ζ
”i= E
»f
“F ζ
”D∗
w
„∂ζF
ζ
DwF ζ
«–, ζ ∈ (a, b). (1)
No differentiability assumption on f , compare with:
∂
∂ζE
hf
“F ζ
”i= E
hf ′
“F ζ
”∂ζF
ζi.
Independence on the bandwidth parameter h, compare with:
∂
∂ζE
hf
“F ζ
”i'
Eˆf
`F ζ+h
´˜− E
ˆf
`F ζ−h
´˜2h
.
The weight
W := D∗w
„∂ζF
ζ
DwF ζ
«is independent of f .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Wiener case [FLL+99], [FLLL01], [Ben02]
Mt = Bt is a Brownian motion:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B(t)
t
dSζt
Sζt
= σ(Sζt )dBt + r(Sζ
t )dt, Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn ) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn ).
δ(wF ) := D∗wF , and δ coincides with the Ito stochastic with respect to w :
δ(w) =
Z ∞
0
wsdBs .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Wiener case [FLL+99], [FLLL01], [Ben02]
Mt = Bt is a Brownian motion:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B(t)
t
dSζt
Sζt
= σ(Sζt )dBt + r(Sζ
t )dt, Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn ) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn ).
δ(wF ) := D∗wF , and δ coincides with the Ito stochastic with respect to w :
δ(w) =
Z ∞
0
wsdBs .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Wiener case [FLL+99], [FLLL01], [Ben02]
Mt = Bt is a Brownian motion:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B(t)
t
dSζt
Sζt
= σ(Sζt )dBt + r(Sζ
t )dt, Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn ) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn ).
δ(wF ) := D∗wF , and δ coincides with the Ito stochastic with respect to w :
δ(w) =
Z ∞
0
wsdBs .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Wiener case [FLL+99], [FLLL01], [Ben02]
Mt = Bt is a Brownian motion:
-0.2
0
0.2
0.4
0.6
0.8
1
1.2
1.4
1.6
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
B(t)
t
dSζt
Sζt
= σ(Sζt )dBt + r(Sζ
t )dt, Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn ) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn ).
δ(wF ) := D∗wF , and δ coincides with the Ito stochastic with respect to w :
δ(w) =
Z ∞
0
wsdBs .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Delta - first variation process
In [FLL+99] the relations
f ′(F ζ) =Dt
ˆf (F ζ)
˜DtF ζ
, 0 ≤ t ≤ T , a.s.,
and
DtSxT =
∂xSxT
∂xSxt
σ(Sxt ), 0 ≤ t ≤ T , a.s.,
are used, cf. [Nua95]. This gives
∂xSxT f ′(Sx
T ) =∂xS
xt
σ(Sxt )
Dt f (SxT ), 0 ≤ t ≤ T , a.s., (2)
which implies ifR T
0wsds = 1:
Delta =∂
∂xE [f (Sx
T )] = Eˆ∂xS
xT f ′ (Sx
T )˜
= E»Z T
0
wt∂xS
xt
σ(Sxt )
Dt f (SxT )dt
–= E
»f (Sx
T )δ
„1[0,T ]w
∂xSx
σ(Sx)
«–= E
»f (Sx
T )
Z T
0
wt∂xS
xt
σ(Sxt )
dBt
–= E
»f (Sx
T )BT
σxT
–.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Poisson case
Mt = Nt is a Poisson process with jump times T1, T2, T3, . . .,
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
N(t)
t
w ∈ C1c ((0,∞)), and
DwF = −∞Xn=1
1NT =n
nXi=1
∂fn∂xi
(T1, . . . , Tn),
for F of the form
F = f01NT =0 +∞Xn=1
1NT =nfn(T1, . . . , Tn).
Recall that
E [F ] = f0e−λT + e−λT
∞Xn=1
1
n!
Z T
0
· · ·Z T
0
fn(t1, . . . , tn)dt1 · · · dtn.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Poisson case
Mt = Nt is a Poisson process with jump times T1, T2, T3, . . .,
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
N(t)
t
w ∈ C1c ((0,∞)), and
DwF = −∞Xn=1
1NT =n
nXi=1
∂fn∂xi
(T1, . . . , Tn),
for F of the form
F = f01NT =0 +∞Xn=1
1NT =nfn(T1, . . . , Tn).
Recall that
E [F ] = f0e−λT + e−λT
∞Xn=1
1
n!
Z T
0
· · ·Z T
0
fn(t1, . . . , tn)dt1 · · · dtn.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Poisson case
Mt = Nt is a Poisson process with jump times T1, T2, T3, . . .,
0
1
2
3
4
5
6
7
8
0 1 2 3 4 5 6 7 8 9 10
N(t)
t
w ∈ C1c ((0,∞)), and
DwF = −∞Xn=1
1NT =n
nXi=1
∂fn∂xi
(T1, . . . , Tn),
for F of the form
F = f01NT =0 +∞Xn=1
1NT =nfn(T1, . . . , Tn).
Recall that
E [F ] = f0e−λT + e−λT
∞Xn=1
1
n!
Z T
0
· · ·Z T
0
fn(t1, . . . , tn)dt1 · · · dtn.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Integration by parts
By standard integration by parts we have, under the boundary conditionw(0) = w(T ) = 0:
E [DwF ] = −e−λTmX
n=1
λn
n!
Z T
0
· · ·Z T
0
nXk=1
w(tk)∂k fn(t1, . . . , tn)dt1 · · · dtn
= e−λTmX
n=1
λn
n!
Z T
0
· · ·Z T
0
fn(t1, . . . , tn)nX
k=1
w(tk)dt1 · · · dtn
= E
24F
k=N(T )Xk=1
w(Tk)
35 = E
»F
Z T
0
w(t)dN(t)
–.
Next, letting
D∗wG = G
Z T
0
w(t)dN(t)− DwG
we get
E [GDwF ] = E [Dw (FG)− FDwG ] = E
»F
„G
Z T
0
w(t)dN(t)− DwG
«–= E [FD∗
wG ].
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Asian options and reserve processes [KP04], [PW04]
Price process: Sζt = Sζ
0 ert(1 + σ)Nt .
Asian options: F ζ =1
T
Z T
0
Sζt dt.
W∆ =−1
xσ
0B@1−R T
0Sζ
t dtR T
0wtdNtR T
0wtS
ζ
t−dNt
+
R T
0Sζ
t dtR T
0wt (wt + rwt)Sζ
t−dNt“R T
0wtS
ζ
t−dNt
”2
1CA .
Density of reserve processes: F =
Z T
0
e(T−t)rdNt .
Wy =
R T
0w(t)dN(t) +
R T
0e−rtw(t)(rw(t)− w(t))dN(t)R T
0w(t)e−rtdN(t)
rR T
0w(t)er(T−t)dX (t)
.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Asian options and reserve processes [KP04], [PW04]
Price process: Sζt = Sζ
0 ert(1 + σ)Nt .
Asian options: F ζ =1
T
Z T
0
Sζt dt.
W∆ =−1
xσ
0B@1−R T
0Sζ
t dtR T
0wtdNtR T
0wtS
ζ
t−dNt
+
R T
0Sζ
t dtR T
0wt (wt + rwt)Sζ
t−dNt“R T
0wtS
ζ
t−dNt
”2
1CA .
Density of reserve processes: F =
Z T
0
e(T−t)rdNt .
Wy =
R T
0w(t)dN(t) +
R T
0e−rtw(t)(rw(t)− w(t))dN(t)R T
0w(t)e−rtdN(t)
rR T
0w(t)er(T−t)dX (t)
.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Asian options and reserve processes [KP04], [PW04]
Price process: Sζt = Sζ
0 ert(1 + σ)Nt .
Asian options: F ζ =1
T
Z T
0
Sζt dt.
W∆ =−1
xσ
0B@1−R T
0Sζ
t dtR T
0wtdNtR T
0wtS
ζ
t−dNt
+
R T
0Sζ
t dtR T
0wt (wt + rwt)Sζ
t−dNt“R T
0wtS
ζ
t−dNt
”2
1CA .
Density of reserve processes: F =
Z T
0
e(T−t)rdNt .
Wy =
R T
0w(t)dN(t) +
R T
0e−rtw(t)(rw(t)− w(t))dN(t)R T
0w(t)e−rtdN(t)
rR T
0w(t)er(T−t)dX (t)
.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - finite differences
φF (y) =∂
∂yE
ˆ1(−∞,0](F − y)
˜'
Eˆ1[y−h,y+h](F )
˜2h
.
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
h=1h=0.1
h=0.01
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - Malliavin method
φF (y) =∂
∂yE
ˆ1(−∞,0](F − y)
˜= −E
»1(−∞,0](F − y)D∗
w
„1
DwF
«–.
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
Malliavin methodExact value
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Localization [FLL+99], [KHP02]
Consider the decomposition
1[0,∞) = f + g ,
where g is C1:
0
0.2
0.4
0.6
0.8
1
1.2
-2 -1.5 -1 -0.5 0 0.5 1 1.5 2
x
f(x)g(x)
We have
d
dyE [1[0,∞)(F − y)] =
d
dyE
»f
„F − y
h
«–+
d
dyE
»g
„F − y
h
«–= E
»D∗
w
„1
DwF
«f
„F − y
h
«–+
1
hE
»1F>yf
′„
F − y
h
«–.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - localized Malliavin method
φF (y) = −E
»f
„F − y
h
«D∗
w
„1
DwF
«–− 1
hE
»1F>yf
′„
F − y
h
«–.
0
0.2
0.4
0.6
0.8
1
1.2
1.5 2 2.5 3 3.5 4 4.5 5 5.5
Den
sity
y
h=1h=0.2
h=0.01
Optimization: f (x) = e−x , x ≥ 0, h = ‖W ‖−1L2(Ω)
.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - finite differences
F =
Z T
0
e−rtdNt .
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - Malliavin method
F =
Z T
0
e−rtdNt .
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Density estimation - localized Malliavin method
F =
Z T
0
e−rtdNt .
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
0.2
0.3
0.4
r
2.5 3 3.5 4 4.5 5 5.5
y
0
0.5
1
1.5
2
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
(Bt)t∈R+ a standard Brownian motion,
(Nt)t∈R+ is a Poisson process with intensity λ > 0,
(Zk)k≥1 an i.i.d. sequence of random variables with probability distributionν(dx),
(Xt)t∈R+ a compound Poisson process with Levy measureµ(dy) = λν(dx). and finite intensity λ:
Xt =
NtXk=1
Zk , t ∈ R+, (3)
(Sxt )t∈R+ a jump-diffusion price process:8>><>>:
dSxt
Sxt
= r(Sxt )dt + σ1(S
xt )dBt + σ2(S
xt−)dXt ,
Sx0 = x .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
(Bt)t∈R+ a standard Brownian motion,
(Nt)t∈R+ is a Poisson process with intensity λ > 0,
(Zk)k≥1 an i.i.d. sequence of random variables with probability distributionν(dx),
(Xt)t∈R+ a compound Poisson process with Levy measureµ(dy) = λν(dx). and finite intensity λ:
Xt =
NtXk=1
Zk , t ∈ R+, (3)
(Sxt )t∈R+ a jump-diffusion price process:8>><>>:
dSxt
Sxt
= r(Sxt )dt + σ1(S
xt )dBt + σ2(S
xt−)dXt ,
Sx0 = x .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
(Bt)t∈R+ a standard Brownian motion,
(Nt)t∈R+ is a Poisson process with intensity λ > 0,
(Zk)k≥1 an i.i.d. sequence of random variables with probability distributionν(dx),
(Xt)t∈R+ a compound Poisson process with Levy measureµ(dy) = λν(dx). and finite intensity λ:
Xt =
NtXk=1
Zk , t ∈ R+, (3)
(Sxt )t∈R+ a jump-diffusion price process:8>><>>:
dSxt
Sxt
= r(Sxt )dt + σ1(S
xt )dBt + σ2(S
xt−)dXt ,
Sx0 = x .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
(Bt)t∈R+ a standard Brownian motion,
(Nt)t∈R+ is a Poisson process with intensity λ > 0,
(Zk)k≥1 an i.i.d. sequence of random variables with probability distributionν(dx),
(Xt)t∈R+ a compound Poisson process with Levy measureµ(dy) = λν(dx). and finite intensity λ:
Xt =
NtXk=1
Zk , t ∈ R+, (3)
(Sxt )t∈R+ a jump-diffusion price process:8>><>>:
dSxt
Sxt
= r(Sxt )dt + σ1(S
xt )dBt + σ2(S
xt−)dXt ,
Sx0 = x .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
(Bt)t∈R+ a standard Brownian motion,
(Nt)t∈R+ is a Poisson process with intensity λ > 0,
(Zk)k≥1 an i.i.d. sequence of random variables with probability distributionν(dx),
(Xt)t∈R+ a compound Poisson process with Levy measureµ(dy) = λν(dx). and finite intensity λ:
Xt =
NtXk=1
Zk , t ∈ R+, (3)
(Sxt )t∈R+ a jump-diffusion price process:8>><>>:
dSxt
Sxt
= r(Sxt )dt + σ1(S
xt )dBt + σ2(S
xt−)dXt ,
Sx0 = x .
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
Price process:
dSζt
Sζt
= rdt + σ1dBt + σ2(dNt − λdt), Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).
Vega2, European options:
∂
∂σ2E [f (Sσ2
T )] = E
»f (Sσ2
T )BT
σ1T
„NT
1 + σ2− λT
«–.
Derivation with respect to absolutely continuous jump amplitudes [BMM].
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
Price process:
dSζt
Sζt
= rdt + σ1dBt + σ2(dNt − λdt), Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).
Vega2, European options:
∂
∂σ2E [f (Sσ2
T )] = E
»f (Sσ2
T )BT
σ1T
„NT
1 + σ2− λT
«–.
Derivation with respect to absolutely continuous jump amplitudes [BMM].
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
Price process:
dSζt
Sζt
= rdt + σ1dBt + σ2(dNt − λdt), Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).
Vega2, European options:
∂
∂σ2E [f (Sσ2
T )] = E
»f (Sσ2
T )BT
σ1T
„NT
1 + σ2− λT
«–.
Derivation with respect to absolutely continuous jump amplitudes [BMM].
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Jump diffusion model [DJ], [DP04], [BMM]
Price process:
dSζt
Sζt
= rdt + σ1dBt + σ2(dNt − λdt), Sζ0 = x .
w ∈ L2(R+) and
Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX
i=1
Z ti
0
wsds∂f
∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).
Vega2, European options:
∂
∂σ2E [f (Sσ2
T )] = E
»f (Sσ2
T )BT
σ1T
„NT
1 + σ2− λT
«–.
Derivation with respect to absolutely continuous jump amplitudes [BMM].
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Vega2 as a function of K and T (finite differences, 10e4 samples, h=0.01)
0.4 0.8
1.2 1.6
2
T
40 60 80 100 120 140 160 180 200K
0
2
4
6
8
10
0.4 0.8
1.2 1.6
2
T
40 60 80 100 120 140 160 180 200K
0
2
4
6
8
10
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
Vega2 as a function of K and T (Malliavin method, 10e4 samples, h=0.01)
0.4 0.8
1.2 1.6
2
T
40 60 80 100 120 140 160 180 200K
0
2
4
6
8
10
0.4 0.8
1.2 1.6
2
T
40 60 80 100 120 140 160 180 200K
0
2
4
6
8
10
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus
E. Benhamou.
Smart Monte Carlo: various tricks using Malliavin calculus.Quant. Finance, 2(5):329–336, 2002.
M.-P. Bavouzet-Morel and M. Messaoud.
Computation of Greeks using Malliavin’s calculus in jump type market models.Preprint, 2004.
M.H.A. Davis and M.P. Johansson.
Malliavin Monte Carlo Greeks for jump diffusions.Stochastic Processes and their Applications.
V. Debelley and N. Privault.
Sensitivity analysis of European options in jump diffusion models via the Malliavin calculus on Wiener space.Preprint, 2004.
E. Fournie, J.M. Lasry, J. Lebuchoux, P.L. Lions, and N. Touzi.
Applications of Malliavin calculus to Monte Carlo methods in finance.Finance and Stochastics, 3(4):391–412, 1999.
E. Fournie, J.M. Lasry, J. Lebuchoux, and P.L. Lions.
Applications of Malliavin calculus to Monte-Carlo methods in finance. II.Finance and Stochastics, 5(2):201–236, 2001.
A. Kohatsu-Higa and R. Pettersson.
Variance reduction methods for simulation of densities on Wiener space.SIAM J. Numer. Anal., 40(2):431–450, 2002.
Y. El Khatib and N. Privault.
Computations of Greeks in markets with jumps via the Malliavin calculus.Finance and Stochastics, 4(2):161–179, 2004.
N. Privault and X. Wei.
A Malliavin calculus approach to sensitivity analysis in insurance.Insurance Math. Econom., 35(3):679–690, 2004.
Nicolas Privault Sensitivity analysis and density estimation using the Malliavin calculus