Stochastic Volatility Modeling

50 Stochastic volatility modeling

Let us consider the special case of the ATMF volatility, that is the implied volatility

for a strike equal to the forward: x

K

= 0. We get:

dbσ

KT

d ln S

0









K=F

T

=

1

T

Z

T

0



1 −

t

T



α(t)dt

This formula quanties how the implied volatility for a xed strike equal to the

forward

F

T

moves when the spot moves. It resembles equation (2.48) except the

weight is

1 −

t

T

rather than

t

T

. This is natural, as when

S

0

moves while

K

stays

xed, for

t = T

only the value

σ(T, S = K)

contributes to formula (2.32), thus

α(t = T ) is immaterial.

Symmetrically, for calculating how the implied volatility changes with strike

K

for a xed spot

S

0

, knowledge of

α(t)

for ...