Stochastic Volatility Modeling

Local-stochastic volatility models 473

If we expand around a constant volatility

σ

0

, rather than a deterministic

volatility

σ(t)

we get – replacing

σ(t)

,

bσ

2

t

and

bσ

T

with

σ

0

– expression (2.48),

page 46:

S

LV

T

=

1

T

Z

T

0

t

T

α(t)dt (12.42)

•

Likewise,

δS

SV

t

is generated by the perturbation at order one in volatility of

volatility in a two-factor forward variance model with an initial variance curve

given by:

ξ

τ

0

= σ

2

(τ)

We can readily recycle expression (8.54), page 329, of the ATMF skew in the

two-factor model:

S

SV

T

=

να

θ

bσ

3

T

T

2

Z

T

0

dt σ(t)

Z

T

t

du σ

2

(u)

h

(1−θ)ρ

SX

1

e

−k

1

(u−t)

+θρ

SX

2

e

−k

2

(u−t)

i

(12.43)

where

bσ

T

=

q

1

T

R

T

0

ξ

t

0

dt =

q

1

T

R

T

0

σ

2

(t)dt

. Expanding around a constant

volatility σ

0

leads to the ...