Stochastic Volatility Modeling

262 Stochastic volatility modeling

date of the

i

th future, and

T − t = ∆ = 30

days, and use the notation

bσ

VIX,T

i

for

the settlement value of the VIX future expiring at

T

i

. As indicated in Section 5.2, the

payo −2 ln S

T

is replicated by a continuous density

2

K

2

of vanilla options, which

yields:

bσ

2

VIX,T

i

(T

i

) =

2e

r∆

∆



Z

F

T

i

+∆

T

i

0

P

K,T

i

+∆

mkt,T

i

dK

K

2

+

Z

∞

F

T

i

+∆

T

i

C

K,T

i

+∆

mkt,T

i

dK

K

2



(7.83)

where

P

KT

mkt,t

(resp.

C

KT

mkt,t

) is the (discounted) market price at time

t

of a put (resp.

call) option of strike

K

, maturity

T

on the S&P 500 and

F

T

i

+∆

T

i

is the forward of

the S&P 500 index for maturity

T

i

+ ∆

observed at

T

i

. In the formula actually used

by the CBOE, the integrals above are discretized ...