Stochastic Volatility Modeling

Local volatility 45

(2.37): for

t = 0

, the density

ρ

σ

0

vanishes unless

S = S

0

, while for

t = T

the dollar

gamma S

2

d

2

P

σ

0

dS

2

vanishes, unless S = K.

In (2.42), the largest weight is obtained for

y = 0

: this singles out a path for

ln(S)

which is a straight line starting at

ln(S

0

)

for

t = 0

and ending at

ln(K)

at

t = T

.

Replacing the integral over

y

by the value for

y = 0

would give an approximation

of bσ

KT

as a uniform average of the local volatility along this line:

bσ

KT

'

1

T

Z

T

0

σ



t, F

t

e

t

T

ln

K

F

T



dt (2.43)

Summing over values of

y 6= 0

, includes other paths in the average, with their

ends pinned down at S

0

at t = 0 and at K for t = T by the factor

p

σ

2

0

(T − t) t.

However appealing expression ...