Stochastic Volatility Modeling

426 Stochastic volatility modeling

We know from Chapter 8 that, at order two in volatilities of volatilities, the

smile is determined by the spot/variance and variance/variance covariance functions

µ (t, u, ξ) =

hd ln S

t

dξ

u

t

i

dt

and

ν (t, u, u

0

, ξ) =

hdξ

u

t

dξ

u

0

t

i

dt

. We then focus on these

objects and dene cross spot/factor and factor/factor correlations so that cross-

covariance functions are related simply to their diagonal counterparts.

11.2.1 A homogeneous basket

Consider the case a basket of

n

assets, each driven by the two-factor model

of Section 7.4 with identical parameters

ν, θ, k

1

, k

2

, ρ

X

1

X

2

(previously noted

ρ

12

)

, ρ

SX

1

, ρ

SX

2

. For the sake of readability ...