Sensitivity analysis and density estimation using the

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

“F ζ

8>>><>>>:E

»f ′

´ ∂

∂ζF ζ

`F ζ+h

´˜− E

`F ζ−h

´˜2h

Expectations can be computed by the Monte Carlo method:

E [F ] ' F1 + · · ·+ Fn

where F1, . . . , Fn is a random sample of F .

Application (1): sensitivity analysis in finance - Greeks

Price process:dSζ

= r(Sζt )dt + σ(Sζ

t )dMt , Sζ0 = x .

F ζ = ST and ζ ∈ x , r , σ, T , ....

Payoff function:

f (x) = (x − K)+

1 1.5 2 2.5 3 3.5 4

f (x) = 1[K ,∞)(x).

1 1.5 2 2.5 3 3.5 4

Option price:

E [f (SζT )].

Greeks:

ζ = x : Delta

ζ = σ: Vega

ζ = r : Rho

ζ = T : Theta.

Price process:dSζ

t )dMt , Sζ0 = x .

F ζ = ST and ζ ∈ x , r , σ, T , ....

Payoff function:

f (x) = (x − K)+

1 1.5 2 2.5 3 3.5 4

f (x) = 1[K ,∞)(x).

1 1.5 2 2.5 3 3.5 4

Option price:

E [f (SζT )].

Greeks:

ζ = x : Delta

ζ = σ: Vega

ζ = r : Rho

ζ = T : Theta.

Price process:dSζ

t )dMt , Sζ0 = x .

F ζ = ST and ζ ∈ x , r , σ, T , ....

Payoff function:

f (x) = (x − K)+

1 1.5 2 2.5 3 3.5 4

f (x) = 1[K ,∞)(x).

1 1.5 2 2.5 3 3.5 4

Option price:

E [f (SζT )].

Greeks:

ζ = x : Delta

ζ = σ: Vega

ζ = r : Rho

ζ = T : Theta.

Price process:dSζ

t )dMt , Sζ0 = x .

F ζ = ST and ζ ∈ x , r , σ, T , ....

Payoff function:

f (x) = (x − K)+

1 1.5 2 2.5 3 3.5 4

f (x) = 1[K ,∞)(x).

1 1.5 2 2.5 3 3.5 4

Option price:

E [f (SζT )].

Greeks:

ζ = x : Delta

ζ = σ: Vega

ζ = r : Rho

ζ = T : Theta.

Price process:dSζ

t )dMt , Sζ0 = x .

F ζ = ST and ζ ∈ x , r , σ, T , ....

Payoff function:

f (x) = (x − K)+

1 1.5 2 2.5 3 3.5 4

f (x) = 1[K ,∞)(x).

1 1.5 2 2.5 3 3.5 4

Option price:

E [f (SζT )].

Greeks:

ζ = x : Delta

ζ = σ: Vega

ζ = r : Rho

ζ = T : Theta.

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

“F ζ

”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]

Eˆ1[ζ−h,ζ+h](F )

Example: F =R T

0e−rtdNt .

1.5 2 2.5 3 3.5 4 4.5 5 5.5

h=1h=0.1

h=0.01

Let F ζ = F − ζ and f (x) = 1(−∞,0].

Density of F :

φF (ξ) : =∂

∂ζP(F ≤ ζ) =

∂ζE

ˆ1(−∞,0](F − ζ)

∂ζE

“F ζ

”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]

Example: F =R T

0e−rtdNt .

1.5 2 2.5 3 3.5 4 4.5 5 5.5

h=1h=0.1

h=0.01

Let F ζ = F − ζ and f (x) = 1(−∞,0].

Density of F :

φF (ξ) : =∂

∂ζP(F ≤ ζ) =

∂ζE

ˆ1(−∞,0](F − ζ)

∂ζE

“F ζ

”i' E [f (F − (ζ + h))]− E [f (F − (ζ − h))]

Example: F =R T

0e−rtdNt .

1.5 2 2.5 3 3.5 4 4.5 5 5.5

h=1h=0.1

h=0.01

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)

hf ′

“F ζ

”∂ζF

»Dw [f (F ζ)]

DwF ζ∂ζF

–= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–.

f ′(F ζ) =Dw

ˆf (F ζ)

˜DwF ζ

wF 〉.

3 Main argument:

∂ζE

hf (F ζ)

hf ′

“F ζ

”∂ζF

»Dw [f (F ζ)]

DwF ζ∂ζF

–= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–.

f ′(F ζ) =Dw

ˆf (F ζ)

˜DwF ζ

wF 〉.

3 Main argument:

∂ζE

hf (F ζ)

hf ′

“F ζ

”∂ζF

»Dw [f (F ζ)]

DwF ζ∂ζF

–= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–.

Proposition

Assume that DwF ζ 6= 0, a.s. on ∂ζFζ 6= 0, ζ ∈ (a, b). We have

∂ζE

“F ζ

”i= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–, ζ ∈ (a, b). (1)

No differentiability assumption on f , compare with:

∂ζE

“F ζ

”i= E

hf ′

“F ζ

”∂ζF

Independence on the bandwidth parameter h, compare with:

∂ζE

“F ζ

`F ζ+h

´˜− E

`F ζ−h

´˜2h

The weight

W := D∗w

„∂ζF

DwF ζ

«is independent of f .

Proposition

∂ζE

“F ζ

”i= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–, ζ ∈ (a, b). (1)

∂ζE

“F ζ

”i= E

hf ′

“F ζ

”∂ζF

∂ζE

“F ζ

`F ζ+h

´˜− E

`F ζ−h

´˜2h

The weight

W := D∗w

„∂ζF

DwF ζ

Proposition

∂ζE

“F ζ

”i= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–, ζ ∈ (a, b). (1)

∂ζE

“F ζ

”i= E

hf ′

“F ζ

”∂ζF

∂ζE

“F ζ

`F ζ+h

´˜− E

`F ζ−h

´˜2h

The weight

W := D∗w

„∂ζF

DwF ζ

Proposition

∂ζE

“F ζ

”i= E

“F ζ

”D∗

„∂ζF

DwF ζ

«–, ζ ∈ (a, b). (1)

∂ζE

“F ζ

”i= E

hf ′

“F ζ

”∂ζF

∂ζE

“F ζ

`F ζ+h

´˜− E

`F ζ−h

´˜2h

The weight

W := D∗w

„∂ζF

DwF ζ

Wiener case [FLL+99], [FLLL01], [Ben02]

Mt = Bt is a Brownian motion:

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

= σ(Sζt )dBt + r(Sζ

t )dt, Sζ0 = x .

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn ) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn ).

δ(wF ) := D∗wF , and δ coincides with the Ito stochastic with respect to w :

δ(w) =

wsdBs .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t )dt, Sζ0 = x .

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn ) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn ).

δ(w) =

wsdBs .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t )dt, Sζ0 = x .

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn ) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn ).

δ(w) =

wsdBs .

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1

t )dt, Sζ0 = x .

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn ) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn ).

δ(w) =

wsdBs .

Delta - first variation process

In [FLL+99] the relations

f ′(F ζ) =Dt

ˆf (F ζ)

˜DtF ζ

, 0 ≤ t ≤ T , a.s.,

DtSxT =

∂xSxT

∂xSxt

σ(Sxt ), 0 ≤ t ≤ T , a.s.,

are used, cf. [Nua95]. This gives

∂xSxT f ′(Sx

T ) =∂xS

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

= E»Z T

wt∂xS

σ(Sxt )

Dt f (SxT )dt

–= E

»f (Sx

„1[0,T ]w

∂xSx

σ(Sx)

«–= E

»f (Sx

wt∂xS

σ(Sxt )

–= E

»f (Sx

Poisson case

Mt = Nt is a Poisson process with jump times T1, T2, T3, . . .,

0 1 2 3 4 5 6 7 8 9 10

w ∈ C1c ((0,∞)), and

DwF = −∞Xn=1

1NT =n

∂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

· · ·Z T

fn(t1, . . . , tn)dt1 · · · dtn.

Poisson case

0 1 2 3 4 5 6 7 8 9 10

w ∈ C1c ((0,∞)), and

DwF = −∞Xn=1

1NT =n

∂fn∂xi

(T1, . . . , Tn),

for F of the form

F = f01NT =0 +∞Xn=1

1NT =nfn(T1, . . . , Tn).

Recall that

∞Xn=1

· · ·Z T

fn(t1, . . . , tn)dt1 · · · dtn.

Poisson case

0 1 2 3 4 5 6 7 8 9 10

w ∈ C1c ((0,∞)), and

DwF = −∞Xn=1

1NT =n

∂fn∂xi

(T1, . . . , Tn),

for F of the form

F = f01NT =0 +∞Xn=1

1NT =nfn(T1, . . . , Tn).

Recall that

∞Xn=1

· · ·Z T

fn(t1, . . . , tn)dt1 · · · dtn.

By standard integration by parts we have, under the boundary conditionw(0) = w(T ) = 0:

E [DwF ] = −e−λTmX

· · ·Z T

w(tk)∂k fn(t1, . . . , tn)dt1 · · · dtn

= e−λTmX

· · ·Z T

fn(t1, . . . , tn)nX

w(tk)dt1 · · · dtn

k=N(T )Xk=1

35 = E

w(t)dN(t)

Next, letting

D∗wG = G

w(t)dN(t)− DwG

we get

E [GDwF ] = E [Dw (FG)− FDwG ] = E

w(t)dN(t)− DwG

«–= E [FD∗

Asian options and reserve processes [KP04], [PW04]

Price process: Sζt = Sζ

0 ert(1 + σ)Nt .

Asian options: F ζ =1

Sζt dt.

W∆ =−1

0B@1−R T

t dtR T

0wtdNtR T

t−dNt

t dtR T

0wt (wt + rwt)Sζ

t−dNt“R T

t−dNt

Density of reserve processes: F =

e(T−t)rdNt .

0w(t)dN(t) +

0e−rtw(t)(rw(t)− w(t))dN(t)R T

0w(t)e−rtdN(t)

0w(t)er(T−t)dX (t)

0 ert(1 + σ)Nt .

Sζt dt.

W∆ =−1

0B@1−R T

t dtR T

0wtdNtR T

t−dNt

t dtR T

0wt (wt + rwt)Sζ

t−dNt“R T

t−dNt

e(T−t)rdNt .

0w(t)dN(t) +

0w(t)e−rtdN(t)

0 ert(1 + σ)Nt .

Sζt dt.

W∆ =−1

0B@1−R T

t dtR T

0wtdNtR T

t−dNt

t dtR T

0wt (wt + rwt)Sζ

t−dNt“R T

t−dNt

e(T−t)rdNt .

0w(t)dN(t) +

0w(t)e−rtdN(t)

Density estimation - finite differences

φF (y) =∂

ˆ1(−∞,0](F − y)

Eˆ1[y−h,y+h](F )

1.5 2 2.5 3 3.5 4 4.5 5 5.5

h=1h=0.1

h=0.01

Density estimation - Malliavin method

φF (y) =∂

ˆ1(−∞,0](F − y)

˜= −E

»1(−∞,0](F − y)D∗

«–.

1.5 2 2.5 3 3.5 4 4.5 5 5.5

Malliavin methodExact value

Localization [FLL+99], [KHP02]

Consider the decomposition

1[0,∞) = f + g ,

where g is C1:

-2 -1.5 -1 -0.5 0 0.5 1 1.5 2

f(x)g(x)

We have

dyE [1[0,∞)(F − y)] =

„F − y

«–+

„F − y

«–= E

»D∗

„F − y

«–+

»1F>yf

′„

F − y

«–.

Density estimation - localized Malliavin method

φF (y) = −E

„F − y

«D∗

«–− 1

»1F>yf

′„

F − y

«–.

1.5 2 2.5 3 3.5 4 4.5 5 5.5

h=1h=0.2

h=0.01

Optimization: f (x) = e−x , x ≥ 0, h = ‖W ‖−1L2(Ω)

Density estimation - finite differences

e−rtdNt .

2.5 3 3.5 4 4.5 5 5.5

Density estimation - Malliavin method

e−rtdNt .

2.5 3 3.5 4 4.5 5 5.5

Density estimation - localized Malliavin method

e−rtdNt .

2.5 3 3.5 4 4.5 5 5.5

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:

NtXk=1

Zk , t ∈ R+, (3)

(Sxt )t∈R+ a jump-diffusion price process:8>><>>:

= r(Sxt )dt + σ1(S

xt )dBt + σ2(S

xt−)dXt ,

Sx0 = x .

NtXk=1

Zk , t ∈ R+, (3)

= r(Sxt )dt + σ1(S

xt )dBt + σ2(S

xt−)dXt ,

Sx0 = x .

NtXk=1

Zk , t ∈ R+, (3)

= r(Sxt )dt + σ1(S

xt )dBt + σ2(S

xt−)dXt ,

Sx0 = x .

NtXk=1

Zk , t ∈ R+, (3)

= r(Sxt )dt + σ1(S

xt )dBt + σ2(S

xt−)dXt ,

Sx0 = x .

NtXk=1

Zk , t ∈ R+, (3)

= r(Sxt )dt + σ1(S

xt )dBt + σ2(S

xt−)dXt ,

Sx0 = x .

Price process:

= rdt + σ1dBt + σ2(dNt − λdt), Sζ0 = x .

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).

Vega2, European options:

∂σ2E [f (Sσ2

T )] = E

»f (Sσ2

1 + σ2− λT

«–.

Derivation with respect to absolutely continuous jump amplitudes [BMM].

Price process:

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).

∂σ2E [f (Sσ2

T )] = E

»f (Sσ2

1 + σ2− λT

«–.

Price process:

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).

∂σ2E [f (Sσ2

T )] = E

»f (Sσ2

1 + σ2− λT

«–.

Price process:

w ∈ L2(R+) and

Dw f (Bt1 , . . . , Btn , T1, . . . , Tn) =nX

wsds∂f

∂xi(Bt1 , . . . , Btn , T1, . . . , Tn).

∂σ2E [f (Sσ2

T )] = E

»f (Sσ2

1 + σ2− λT

«–.

Vega2 as a function of K and T (finite differences, 10e4 samples, h=0.01)

0.4 0.8

1.2 1.6

40 60 80 100 120 140 160 180 200K

0.4 0.8

1.2 1.6

40 60 80 100 120 140 160 180 200K

Vega2 as a function of K and T (Malliavin method, 10e4 samples, h=0.01)

0.4 0.8

1.2 1.6

40 60 80 100 120 140 160 180 200K

0.4 0.8

1.2 1.6

40 60 80 100 120 140 160 180 200K

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