Stochastic Volatility Modeling

302 Stochastic volatility modeling

I

The natural hedge instrument is the VS for the residual maturity. A simple

model (SM) is built by specifying the dynamics of U

t

, dened by:

U

t

=

Q

t

+ (T − t) bσ

2

T

(t)

T

Q

t

is the quadratic variation at

t

, and

bσ

T

(t)

is the VS volatility at

t

for maturity

T

.

U

t

has no drift.

I Assuming a lognormal dynamics for U

t

given by:

dU

t

U

t

= 2R

t

T − t

T

ν

T

(t) dW

t

yields the price of an option on realized variance, in the form of a simple Black-

Scholes formula, where R

t

=

bσ

2

T

(t)

U

t

has been take equal to 1.

P (t, U ) = P

BS

(t, U, σ

e

, T )

σ

2

e

=

1

T − t

Z

T

t

4



T − τ

T



2

ν

2

T

(τ) dτ

where ν

2

T

(τ) is the volatility at τ of a VS volatility of maturity T .

I

Numerical ...