Stochastic Volatility Modeling

242 Stochastic volatility modeling

our pricing level

ν

T

(τ)

. Can we hedge this exposure of

R

τ

to the realized volatility

of S

t

?

As we will see shortly, the issues of (a) accounting for the term structure of VS

volatilities in the option price, and (b) hedging the exposure of

R

τ

to the realized

volatility of the underlying are connected.

Let us rst amend (7.49b) to take into account the term structure of VS volatili-

ties. For general term structures the expectations at time

t

of the numerator and

denominator of R

τ

in (7.47) are:

E

t

[U

τ

] = U

t

≡

Q

t

+ (T − t) bσ

2

T

(t)

T

E

t

[bσ

2

T

(τ)] =

1

T − τ

Z

T

τ

ξ

u

t

u = bσ

2

τT

(t)

where

bσ

τT

(t)

is the forward VS volatility at time

t

for the interval ...