On the Sensitivity of Option Prices with respect to the Risk Premium - - PowerPoint PPT Presentation

On the Sensitivity of Option Prices with respect to the Risk Premium for Stochastic Volatility Models

Jorge P . Zubelli jt work with C.Gomez Velez

IMPA-Brazil.

RICAM LINZ October 29, 2008 THANKS Hanna Pikkarainen & Karl Kunish & Wolfgang Runggaldier

Outline

Intro on the Black-Scholes-Merton Model Stochastic Volatility Models Risk Premium and Incomplete Markets Fast Mean Reversion Stochastic Volatility Regimes The Calibration Problem for the Risk Premium The Malliavin Calculus Approach Conclusions

The Black-Scholes price model

asset price dynamics dXt = µXtdt +σXtdWt (1) where Wt is the Brownian Motion. The option price U(x,t) for x = Xt solves the Black-Scholes eq.:

∂U ∂t + σ2x2

2

∂2U ∂x2 + r(x ∂U ∂x − U) =

0, U(x,T;T,K,σ2) = (x − K)+, Note: U = U(x,t;T,K,r,σ2).

Probabilistic Representation for U(x,t)

From Feynman-Kac U(x,t,T,K,σ2) = e−r(T−t)EQ[(XT − K)+|Xt = x]. where the expectation is w.r.t. the unique risk neutral measure Q determined by the model parameters {σ,r,µ} where the asset dynamics takes the form dXt = rXtdt +σXtdWt.

Limitations of Classical Black-Scholes

◮ log-normality of asset prices is not verified by statistical tests ◮ option prices are subjet to the smile effects ◮ volatility of the prices tends fluctuate with time and revert to a

mean value

Figure: Example of Data from IBOVESPA. Index × Vol

0.8 0.85 0.9 0.95 1 1.05 1.1 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 Moneyness K/x Implied Volatility Historical Volatility 9 Feb, 2000 Excess kurtosis Skew

Figure: Implied Volatility - S& P 500 - 2/Fev/2000 - As a function of K/x (moneyness) Current index value: 1411.71 - 2 months for maturity. (From Fouque - IMA talk)

Figure: Implied Volatility Surface- (From Bruno Dupire - IMPA talk)

Smile effect: Empirical remark that calls having different strikes, but

• therwise identical, have different implied volatilities. Before the 1987

crash the graph of I(K) (fix t,X,T) had a U shape with a minimum close to K = X. Since 1987 it is a decreasing function in the range 95% < K/X < 105% then (for K >> X) it bends upwards.

Smile effect: Empirical remark that calls having different strikes, but

• therwise identical, have different implied volatilities. Before the 1987

crash the graph of I(K) (fix t,X,T) had a U shape with a minimum close to K = X. Since 1987 it is a decreasing function in the range 95% < K/X < 105% then (for K >> X) it bends upwards.

Smile effect: Empirical remark that calls having different strikes, but

• therwise identical, have different implied volatilities. Before the 1987

crash the graph of I(K) (fix t,X,T) had a U shape with a minimum close to K = X. Since 1987 it is a decreasing function in the range 95% < K/X < 105% then (for K >> X) it bends upwards.

Stochastic Volatility Models

dXt = µXtdt +σ(Yt)XtdW 1

t ,

dYt = α(m − Yt)dt +β(ρdW 1

t +

• 1−ρ2dW 2

t ),

(2) In the risk neutral measure the dynamics takes the form: dXt = rXtdt +σ(Yt)XtdW 1∗

t ,

dYt = (α(m − Yt)−βΛ(Xt,Yt,t))dt +β(ρdW 1∗

t

+

• 1−ρ2dW 2∗

t ),

(3) where

Λ(t,x,y) = ρµ− r σ(y) +(1−ρ)1/2γ(x,y,t).

Remark: In (3) we have a market price of volatility risk

Some references

J-P .Fouque, G.Papanicolaou, and K-R.Sircar. Derivatives in financial markets with stochastic volatility. Cambridge University Press, Cambridge, 2000. J-P .Fouque, G.Papanicolaou, R.Sircar, and K.Solna. Multiscale stochastic volatility asymptotics. Multiscale Model. Simul. 2(1):22–42 (electronic), 2003. Y.Achdou, B.Franchi, and N.Tchou. A partial diferential equation connected to option pricing with stochastic volatility: regularity results and discretization. Math. Comp., 74(251), 2005. Y.Achdou and N.Tchou. Variational analysis for the Black and Scholes equation with stochastic volatility. M2AN Math. Model. Numer. Anal., 36(3), 2002.

The Pricing Equation for Stochastic Volatility Models

The european call price U(x,y,t,T,K) satisfies Ut + σ(y)2x2 2 Uxx−r(xUx − U)+ρβxσ(y)Uxy + β2 2 Uyy+

(α(m − y)−βΛ(y))Uy = 0 ,

U(x,y,T) = (x − K)+ , (4) where,

Λ(y) = ρµ− r σ(y) +

• 1−ρ2γ(y),and

0 ≤ t < T,x > 0,y ∈ R. We will assume that γ = γ(y).

Figure: Solution of the Heston Model (Using the software PREMIA)

Interpretation of the risk premium γ

dU(t,Xt,Yt)

= µ− r σ(y)

• xσ(y)∂U

∂x +βρ∂U ∂y

• + rU +γβ
• 1−ρ2 ∂U

∂y

• dt

+

• xσ(y)∂U

∂x +βρ∂U ∂y

• dW ∗

1,t +β

• 1−ρ2 ∂U

∂y dW ∗

2,t

Note:

◮ Small changes in the vol-vol β leads to an infinitesimal change on

the price amplified by a factor γ.

◮ If ρ2 = 1 it does not appear (but eq. degenerates). ◮ Even if ρ = 0 it is present. ◮ The risk premium has to be determined from market data

MAIN PROBLEM: Find the functional derivative of the price w.r.t γ.

Interpretation of the risk premium γ

dU(t,Xt,Yt)

= µ− r σ(y)

• xσ(y)∂U

∂x +βρ∂U ∂y

• + rU +γβ
• 1−ρ2 ∂U

∂y

• dt

+

• xσ(y)∂U

∂x +βρ∂U ∂y

• dW ∗

1,t +β

• 1−ρ2 ∂U

∂y dW ∗

2,t

Note:

◮ Small changes in the vol-vol β leads to an infinitesimal change on

the price amplified by a factor γ.

◮ If ρ2 = 1 it does not appear (but eq. degenerates). ◮ Even if ρ = 0 it is present. ◮ The risk premium has to be determined from market data

MAIN PROBLEM: Find the functional derivative of the price w.r.t γ.

Different Directions...

• 1. Asymptotic Approach: Consider the asymptotic behavior when

some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS).

• 2. IP: Inverse Problems Identification of the Risk Premium.

We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map:

Γ : Λ − → P(t,x,y;T,K)

Different Directions...

• 1. Asymptotic Approach: Consider the asymptotic behavior when

some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS).

• 2. IP: Inverse Problems Identification of the Risk Premium.

We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map:

Γ : Λ − → P(t,x,y;T,K)

Different Directions...

• 1. Asymptotic Approach: Consider the asymptotic behavior when

some of the parameters goes to infinity... Fouque, Papanicolaou, Sircar,Solna (See the book by FPS).

• 2. IP: Inverse Problems Identification of the Risk Premium.

We initiated a study along this direcion in a recent PhD thesis of my student Cesar Gomez Considered the map:

Γ : Λ − → P(t,x,y;T,K)

Asymptotic Approach

The Equation in Dimensionless Variables

We work with the dimensionless variables t=rt, f = √ rf, x = x/K, and drop the hats altogether.

∂P ∂t + 1

2 (f(y))2 x2 ∂2P

∂x2 +ρνxf(y) ∂2P ∂x∂y + ν2

2

∂2P ∂y2 +

• x ∂P

∂x − P

• +
• ε−1(m − y)−νΛ(t,x,y)

∂P ∂y = 0,

(5) P(T,x,y) = h(x), with T = rTE,

ε = rα−1

and

ν = r−1/2β.

(6)

ν:

Called the adimensional vol-vol

Interpretation

• 1. ε = r/α is the adimensional inverse of the rate of mean-reversion
• 2. Small ε means FAST MEAN-REVERSION
• 3. ν = r−1/2β is the adimensional vol-vol
• 4. Caveat: Fouque-Papanicolaou-Sircar (FPS ) work with the vol-vol
• ν = νε1/2/

√

2 If the volatility is fast mean-reverting, then the market will see, to leading order, an effective constant volatility plus small corrections: This is modeled by subsuming that the mean reversion time ε := r/α is small as compared to the other time scale

PDEs and the Problem

Asymptotic Behavior of the Solutions to the Equation:

∂P ∂t + 1

2 (f(y))2 x2 ∂2P

∂x2 +ρνxf(y) ∂2P ∂x∂y + ν2

2

∂2P ∂y2 +

• x ∂P

∂x − P

• +
• ε−1(m − y)−νΛ(t,x,y)

∂P ∂y = 0,

(7) P(t = T,x,y) = h(x) (Final Condition.) under the regimes:

• 1. ν2 ∼ ε−1 (Fouque, Papanicolaou, Sircar)
• 2. ν2 ≪ ε−1 (M. Souza & JPZ)

PDEs and the Problem

Asymptotic Behavior of the Solutions to the Equation:

∂P ∂t + 1

2 (f(y))2 x2 ∂2P

∂x2 +ρνxf(y) ∂2P ∂x∂y + ν2

2

∂2P ∂y2 +

• x ∂P

∂x − P

• +
• ε−1(m − y)−νΛ(t,x,y)

∂P ∂y = 0,

(7) P(t = T,x,y) = h(x) (Final Condition.) under the regimes:

• 1. ν2 ∼ ε−1 (Fouque, Papanicolaou, Sircar)
• 2. ν2 ≪ ε−1 (M. Souza & JPZ)

Main Idea (Fouque, Papanicolaou, Sircar)

◮ Consider the correct solution as a perturbation of the B-S price:

Pε = P0 +ε1/2P1 +εP2 +ε3/2P3 +O(ε2)

◮ Show P0 is actually the Black-Scholes price with an effective

volatility (homogeneization!!!)

◮ Obtain an expression for the correction ◮ Under certain regimes and reasonable asumptions: correction

can be computed solving a forced B-S equation with parameters that can be calibrated from the implied volatility curve.

Variogram’s mean with the fitted curve

Computed Parameters

• ν2

f = β2

2α = ν2ε 2

α

566.4

ν2ε

0.4084 These results imply a mean-reversion rate of 1.3 days. Note: It is not a priori clear whether we are in the FPS vol-vol regime

• r in a moderately small vol-vol regime.

The Scales and Regimes: ν = β/r1/2

ε = r/α

Recall: dXt = µXtdt +σtXtdWt

σt = f(Yt)

dYt = α(m − Yt)dt +βd Zt

ν = β/r 1/2 ε = r/α

Asymptotic Approach

Assumptions and Distinguished Limits

with Fouque-Papanicolaou-Sircar (FPS ) assume that:

◮ f is bounded away from zero and from above; ◮ Λ is independent of x; (this can actually be proved)

The following scale regimes for ν in (7):

• 1. ν2 ∼ ε−1 corresponds to the scaling considered in FPS and

leads to a balance between the terms

ν2

2

∂2P ∂y2 and ε−1(m − y)∂P ∂y .

• 2. ν2 ≪ ε−1, IJTAF 2007 (M.Souza-JPZ)
• 3. Work in progress ν2 ≫ ε−1

Figure: IBOVESPA Price Surface- (jt w/ C. Alves and M. Souza)

Figure: Implied Vol. Surface of IBOVESPA (jt w/ C. Alves and M. Souza)

• Inv. Probl Approach

Related to the Calibration Problem

for simplicity ρ = 0. Inverse Problem Given the function U(x,y0,t,K,T) for x and y0 find the risk premium γ. Ut + σ(y)2x2 2 Uxx + β2 2 Uyy+

(α(m − y)−βγ(y))Uy − r(xUx − U) = 0 ,

U(x,y,T) = (x − K)+ . (8)

Consider the operator

Γ : X ∩ C(R ) → Y,

(9)

γ → U(x,y0,t,K,T).

(10) We want to study:

◮ Sensitivity of the prices U(x,y0,t,K,T), w.r.t. γ. ◮ Invertibility of the operator Γ. ◮ Practical problem of identifying γ.

Our results on Γ

• 1. Γ is Fr´

echet differentiable w.r.t γ

• 2. Γ is analytic in the sense that it can be expressed as a

convergent series of multilinear operators The result is based on results of Achdou, Franchi e Tchou. Y.Achdou, B.Franchi, and N.Tchou. A partial diferential equation connected to option pricing with stochastic volatility: regularity results and discretization. Math. Comp., 74(251), 2005. Y.Achdou and N.Tchou. Variational analysis for the Black and Scholes equation with stochastic volatility. M2AN Math. Model. Numer. Anal., 36(3), 2002.

Our results on Γ

• 1. Γ is Fr´

echet differentiable w.r.t γ

• 2. Γ is analytic in the sense that it can be expressed as a

convergent series of multilinear operators The result is based on results of Achdou, Franchi e Tchou. Y.Achdou, B.Franchi, and N.Tchou. A partial diferential equation connected to option pricing with stochastic volatility: regularity results and discretization. Math. Comp., 74(251), 2005. Y.Achdou and N.Tchou. Variational analysis for the Black and Scholes equation with stochastic volatility. M2AN Math. Model. Numer. Anal., 36(3), 2002.

Sensitivity Formulae

Proposition (1st expression for ∂Γ/∂γ.)

Assuming sufficiently smooth σ(y) and γ(y), we have that

∂Γ ∂γ (γ)[h](x,y,t) =

e−r(T−t) RR T

t (u − K)+φ¯ y(u,v,T;¯

x,¯ y,s)h(¯ y)φ(¯ x,¯ y,s;x,y,t)d¯ yd¯ xdsdvdu. where φ(u,v,s;x,y,t0) is the joint density of the diffusions dXt = rXtdt +σ(Yt)XtdW 1

t ,

Xt0 = x, dYt = (α(m − Yt)−βγ(Yt))dt +βdW ∗

t ,

Yt0 = y. DRAWBACK: We have to solve numerically for φ and compute φy

Sensitivity Formulae

Proposition (1st expression for ∂Γ/∂γ.)

Assuming sufficiently smooth σ(y) and γ(y), we have that

∂Γ ∂γ (γ)[h](x,y,t) =

e−r(T−t) RR T

t (u − K)+φ¯ y(u,v,T;¯

x,¯ y,s)h(¯ y)φ(¯ x,¯ y,s;x,y,t)d¯ yd¯ xdsdvdu. where φ(u,v,s;x,y,t0) is the joint density of the diffusions dXt = rXtdt +σ(Yt)XtdW 1

t ,

Xt0 = x, dYt = (α(m − Yt)−βγ(Yt))dt +βdW ∗

t ,

Yt0 = y. DRAWBACK: We have to solve numerically for φ and compute φy

Malliavin Calculus Applied to this Context

Problem: How to compute the sensitivity of prices w.r.t. the different functional parameters that enter the pricing formula? Long literature. E.G.:

• 1. E.Fourni´

e, J-M.Lasry, J.Lebuchoux, and P-L.Lions.Applications of Malliavin calculus to Monte-Carlo methods in finance II. Finance

• Stoch. 5(2), 2001.
• 2. Nizar Touzi
• 3. Christian-Oliver Ewald / Elisa Alos / Aihua Zhang

Malliavin Calculus Applied to this Context

Problem: How to compute the sensitivity of prices w.r.t. the different functional parameters that enter the pricing formula? Long literature. E.G.:

• 1. E.Fourni´

e, J-M.Lasry, J.Lebuchoux, and P-L.Lions.Applications of Malliavin calculus to Monte-Carlo methods in finance II. Finance

• Stoch. 5(2), 2001.
• 2. Nizar Touzi
• 3. Christian-Oliver Ewald / Elisa Alos / Aihua Zhang

Main Idea

Assume that X = {X α

t }0≤t≤T is a diffusion that depends on a

parameter α. We want to compute lim

δ→0

1

δα

• E[f(X α+δα

T

)]−E[f(X α

T )]

• ,

However, in many cases we cannot differentiate under the expectation

Main Idea

Assume that X = {X α

t }0≤t≤T is a diffusion that depends on a

parameter α. We want to compute lim

δ→0

1

δα

• E[f(X α+δα

T

)]−E[f(X α

T )]

• ,

However, in many cases we cannot differentiate under the expectation

Main Idea

Assume that X = {X α

t }0≤t≤T is a diffusion that depends on a

parameter α. We want to compute lim

δ→0

1

δα

• E[f(X α+δα

T

)]−E[f(X α

T )]

• ,

However, in many cases we cannot differentiate under the expectation

A Brief Intro to Malliavin

Reference: Premia man pages

Main Idea: Variational calculus on Wiener spaces. Context: Hilbert space (H,.|.), a complete probability space

(Ω,A,P), and an Isometry

W : H → L2

Ω,A,P;B

• Rd

,Rd

s.t. W (h) is a normal centered random variable defining the Gaussian process

(W (h), h ∈ H)

In our case H = L2

[0,T],B ([0,T]),dt;B

• Rd

,Rd

and we define Wt = W

• 1[0,t]
• P − a.e.

Define by

B = σ(Wt,0 ≤ t ≤ T)

the σ-algebra generated by (Wt,0 ≤ t ≤ T).

A Brief Intro to Malliavin

Reference: Premia man pages

Main Idea: Variational calculus on Wiener spaces. Context: Hilbert space (H,.|.), a complete probability space

(Ω,A,P), and an Isometry

W : H → L2

Ω,A,P;B

• Rd

,Rd

s.t. W (h) is a normal centered random variable defining the Gaussian process

(W (h), h ∈ H)

In our case H = L2

[0,T],B ([0,T]),dt;B

• Rd

,Rd

and we define Wt = W

• 1[0,t]
• P − a.e.

Define by

B = σ(Wt,0 ≤ t ≤ T)

the σ-algebra generated by (Wt,0 ≤ t ≤ T).

Malliavin

Context L2 (Ω,B,P)

Basic variables: W (h) =

d

∑

i=1

W i (hi) =

d

∑

i=1

Z T hi (t)dW i

t

Wiener polynomials: The space P is the set f

• W
• h1

,...,W (hn)

• where f : Rn → R is polynomial and

hj ∈ L2

[0,T],B ([0,T]),dt;B

• Rd

,Rd ,

1 ≤ j ≤ n Note that the space of Wiener polynomials is dense in L2 (Ω,B,P).

Malliavin

Definition

For f

• W
• h1

,...,W (hn)

• a Wiener poly. we define its Malliavin

derivative as the d-dimensional stochastic process Df

• W
• h1

,...,W (hn)

• =
• Dtf
• W
• h1

,...,W (hn)

• ,t ∈ [0,T]
• by setting

Dtf

• W
• h1

,...,W (hn)

• =
• n

∑

i=1

∂f ∂xj

• W
• h1

,...,W (hn)

• hj

i (t)

• 1≤i≤d

So Df

• W
• h1

,...,W (hn)

• ∈ L2

[0,T]×Ω,B ([0,T])⊗B,dt ⊗ P;Rd,B

• Rd

Malliavin

Proposition

The operator D extends as a closed unbounded operator with domain

D1,2 where the norm F1,2 =

• E
• |F|2

+

Z T E

• DtF2

Rd

• dt

1

2

is finite. The following chain rule holds: Dϕ

• F 1,...,F m

=

m

∑

i

∂ϕ ∂xi

• F 1,...,F m

DF i

Malliavin

The Skorohod Integral

It is defined as the adjoint to the operator D

Definition

The Skohorod integral δ is defined as the adjoint of D. from L2

[0,T]×Ω,B ([0,T])⊗B,dt ⊗ P;Rd,B

• Rd

to L2 (Ω,B,P), whose domain dom(δ) of processes s.t. L2

[0,T]×Ω,B ([0,T])⊗B,dt ⊗ P;Rd,B

• Rd
• 1. u ∈ dom(δ) iff ∃c s.t

∀F ∈ D1,2,

• E

Z T

0 DtF|utRd dt

• ≤ cE
• F 2 1

2

• 2. δ(u) is defined by the duality relation

∀F ∈ D1,2,

E (Fδ(u)) = E

Z T

0 DtF|utRd dt

Malliavin

Crucial Fact

If u is an addapted process w.r.t. the filter Ft = σ(Ws;0 ≤ s ≤ t) in L2

[0,T]×Ω,B ([0,T])⊗B,dt ⊗ P;Rd,B

• Rd

.

then it belongs to the domain of δ and

δ(u) =

d

∑

i=1

Z T ui

tdW i t

So, for addapted processes the Skorohod integral agrees with the Ito integral. IMPORTANT: The above ideas can be used to ccompute derivatives

• f solutions to SDEs w.r.t. parameters.

Malliavin

Crucial Fact

If u is an addapted process w.r.t. the filter Ft = σ(Ws;0 ≤ s ≤ t) in L2

[0,T]×Ω,B ([0,T])⊗B,dt ⊗ P;Rd,B

• Rd

.

then it belongs to the domain of δ and

δ(u) =

d

∑

i=1

Z T ui

tdW i t

So, for addapted processes the Skorohod integral agrees with the Ito integral. IMPORTANT: The above ideas can be used to ccompute derivatives

• f solutions to SDEs w.r.t. parameters.

Message from the works of Fourni` e et al.

The derivative ∂E[f(X α+δα

T

)]/∂α can be computed by considering

weight π(α) and computing E[π(α)f(X α

T )] where π(α) depends on the

so-called first-variation process. In what follows we explain this in the case of the Black-Scholes model.

Message from the works of Fourni` e et al.

The derivative ∂E[f(X α+δα

T

)]/∂α can be computed by considering

weight π(α) and computing E[π(α)f(X α

T )] where π(α) depends on the

so-called first-variation process. In what follows we explain this in the case of the Black-Scholes model.

Examples from Fourni´ e et al.

To compute the sensitivity ∆:

∂U/∂x = E

• e−r(T−t)h(XT)π
• where

π =

WT−t

σXt(T − t)

To compute the second derivative w.r.t. x:

∂2U/∂x2 = E

• e−r(T−t)h(XT)π
• where

π =

1

σX 2

t (T − t)

• W 2

T−t

σ(T − t) − WT−t − 1 σ

The Hull-White Formula

In the Black-Scholes model with time varying volatility: dXs = rXsds +σ(s)XsdWs, XT = xe

R T

0 (r− σ2(s) 2

)ds+

R T

0 σ(s)dWs,

UBS(x,t,T,K,S) = e−r(T−t)E[(XT − K)+|X0 = x], where S = R T

0 σ2(s)ds

Now consider the stochastic volatility case dXs = rXsds +σ(Ys)XsdWs, XT = xe

R T

t (r− σ2(Ys) 2

)ds+

R T

t σ(Ys)dW 1 s ,

U(x,y,t,T,K) = e−r(T−t)E[E[(XT − K)+|Yt≤s≤T]] = E[UBS[ξT](x,t)],

with ξT =

Z T

t

σ2(Ys)ds.

Now consider a small variation

dXt = rXtdt +σ(Yt)XtdW 1

t ,

X0 = x, dYt = (α(m − Yt)−βγ(Yt)+εh(Yt))dt +βdW ∗

t ,

Y0 = y.

E[UBS[ξε

T](x,t)],

Using Girsanov and differentiating w.r.t. ε we get

E Z T

t

h(Ys)dW ∗

s

• UBS[ξT](x,t)
• = E[

Z T

t (D∗ sUBS[ξT])h(Ys)ds].

where D∗

s is the Malliavin derivative w.r.t. W ∗

Proposition (alternative expression for ∂Γ/∂γ)

The operator ∂Γ/∂γ is given by

∂Γ ∂γ (γ)[h](y) =

−EQγ

• ∂UBS

∂S (ξT)

R T

t

• R T

s 2σ(Yr)σ′(Yr)e R r

s f ′(Yu)dudr

• h(Ys)ds
• Yt = y
• ,

(11) where

UBS(x,K,r,τ,S) =

• xΦ(d1)− Ke−rτΦ(d2)

(S > 0) max(x − Ke−rτ,0) (S = 0) Φ(z) =

1

√

2π Z z

−∞

e− x2

2 dx.

ξT =

Z T

t σ2(Ys)ds

f(y) = α(m − y)−βγ(y)

We can write

−E ∂UBS ∂S (ξT)

Z T

t

e−θs(ηT −ηs)h(Ys)ds

• =

Z Z T

t

∂UBS ∂S (ξT )e−θs(ηs −ηT )h(Ys)Ψ(ξT ,ηT ,T;θs,ηs,Ys,s;y,t)d(θs,ηs,Ys,s,ηT ,ξT ).

(12) Note:

∂UBS ∂S (x,K,τ,S) =

x 2

√

2πS exp(−(ν+ rτ)2 2S

− (ν+ rτ)

2

− S

8 ) where ν = log(X/K) and τ = T − S

where

ηs := 2

Z s

t σ(Yr)σ′(Yr)eθr dr,

θs :=

Z s

t

f ′(Yr)dr,

ξT =

Z T

t σ2(Ys)ds,

f(y) = α(m − y)−βγ(y).

Let L be the generator of such diffusions

L = β2 2

∂2 ∂y2 + f(y) ∂ ∂y + 2σ′(y)σ(y)eθ ∂ ∂η + f ′(y) ∂ ∂θ +σ2(y) ∂ ∂ξ. Setting

¯ ξ = (ξ,η,θ,y),

• eq. (12) can be interpreted as

∂Γ ∂f =

ZZ T

t [U1(¯

ξ)η− U2( ¯ ξ)]e−θh(y)d¯ ξ.

where U1 e U2 solve the problem

∂U ∂s + Ly,η,θ,ξU = 0, for s < T,

with final condition U1(¯

ξ,T) = ∂UBS ∂S (ξ),

U2(¯

ξ,T) = ∂UBS ∂S (ξ)θ.

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...

Conclusions

◮ The mapping Γ : γ −

→ U[γ] is smooth and analytic in appropriate

spaces.

◮ We can compute the functional derivative using Malliavin

techniques

◮ This derivative can be implemented numerically in an efficient

way even for non-smooth payoffs using Monte-Carlo techniques

◮ Next step is to use this implementation to calibrate the model by

means of a Landweber technique.

γk+1 = γk − ∂Γ ∂γ

∗

[γk](Γ[fk]−

U), where U is the vector of quoted prices.

◮ Lots of interesting questions:

◮ Theoretical: Identifiability, degree of ill-posedness, Prof. Hofmann

benchmark analysis

◮ Practicla: Implementation,Regularization How much can we

reconstruct ...