Stochastic Volatility Modeling

332 Stochastic volatility modeling

observed smiles, in particular with respect to the term structure of the ATMF skew.

This is especially important when pricing cliquets: see the discussion in Section 3.1.

Typically equity index smiles display a term structure of the ATMF skew that is well

approximated by a power law with an exponent usually around

1

2

– see examples in

Figure 6.5.

The values for

ρ

SX

1

,

ρ

SX

2

in Table 8.2 are such that they generate a term structure

for the ATMF skew that is approximately a power law with exponent

1

2

, with

bσ

0.95F

T

− bσ

1.05F

T

= 3% for T = 1 year. This is illustrated in Figure 8.5.

0%

1%

2%

3%

4%

5%

6%

0 1 2 3 4 5

Order one

Power ...