Nonlinear Option Pricing

276 Calibration of Local Stochastic Volatility Models to Market Smiles

show that (see e.g., [11], Chapter 6)

lim

t→0

σ

BS

(t, f) = lim

t→0

ln

f

f

0

R

f

f

0

dx

x

√

σ(t,x)

2

E

Q

[a

2

t

|f

t

=x]

Our specification of σ(0, f) has been obtained by using the approxima-

tion E

Q

[a

2

t

|f

t

= x] ≈ α

2

. A better approximation can be found in [11]

(see Remark 6.8).

2. Solve the linear two-dimensional Fokker-Planck equation on this interval

[t

k

, t

k+1

]:

∂

t

p(t, f, a) =

1

2

∂

2

f

(σ(t, f)

2

f

2

a

2

p(t, f, a)) +

1

2

∂

2

a

(σ(a)

2

p(t, f, a))

+ ρ∂

fa

(σ(a)σ(t, f )fp(t, f, a)) − ∂

a

(b(a)p(t, f, a))

and store the vector p(t

k+1

, ·, ·) evaluated on the two-dimensional space

grid.

3. Compute the local volatility at time t

k+1

:

σ(