Stochastic Volatility Modeling

252 Stochastic volatility modeling

We now revert to the continuous-time framework. The quadratic variation of

U

t

over [0, T ] now acquires an extra contribution and reads :

Z

T

0

4R

2

τ



T − τ

T



2

ν

2

T

(τ) dτ + N



∆t

T



2

(2 + κ) (7.69)

where

N =

T

∆t

is the number of returns in the interval

[0, T ]

. Dividing now (7.69)

by T provides the following amended expression for σ

2

e

which supersedes (7.56a):

σ

2

e

=

1

T

Z

T

0

4



T − τ

T



2



bσ

2

τT

(0)

bσ

2

T

(0)



2

ν

2

T

(τ) dτ +

2 + κ

NT

(7.70)

where

N

is the number of returns over

[0, T ]

:

N =

T

∆t

. The second piece in (7.70) is

generated by the intrinsic variance of the variance estimator itself: as expected, its

relative contribution to σ

2

e

is largest for ...