Stochastic Volatility Modeling

292 Stochastic volatility modeling

We start from expression (8.29) of vanilla option prices derived in Section 8.4 of

Chapter 8 and the resulting expression of the ATMF skew at order one in volatility

of volatility (8.32), page 319:

S

T

=

1

2bσ

3

T

Z

T

0

T − t

T



dln S

t

dbσ

2

T

(t)



0

dt

dt (7.112)

where the

0

susbcript means the covariation is evaluated in the unperturbed state,

that is with forward variances equal to their values at

t = 0

and with the instan-

taneous volatility of

S

t

read o the initial term structure of VS volatilities. In our

context, expanding at order one in volatility of volatility corresponds to expanding

at order one both in ν and (β − 1).

We calculate ...