Stochastic Volatility Modeling

118 Stochastic volatility modeling

expression (3.19) of

R

T

we can see that the coecient of

S ln S

in the portfolio

e

q(T

2

−T

1

)

R

T

2

−R

T

1

vanishes: our hedge has vanishing gamma and vanishing theta

as well.

Is bσ

T

1

T

2

well-dened?

The answer is yes. Implied volatilities of

S ln S

payos of maturities

T

1

and

T

2

satisfy the convex order condition (3.3) because of the convexity of the

S ln S

payo.

The

S ln S

payo falls in the class of payos considered in Section 2.2.2.2, page 32,

for which (a) an implied volatility can be dened, (b) the convex order condition

(3.3) holds.

When there are no cash-amount dividends, implied volatilities

bσ

T

of

S ln S

payos are expressed directly ...