Stochastic Volatility Modeling

Linking static and dynamic properties of stochastic volatility models 367

•

The SSR is dierent than

1

and is related to

γ

: for long maturities

R

T

→ 2−γ

The decay of the ATMF skew of market smiles is consistent with Type II, but

what about the SSR? From the expression of the SSR in (9.3) we dene the realized

SSR as:

R

r

T

=

P

i

ln

S

i+1

S

i

(bσ

T,i+1

− bσ

T,i

)

P

i

S

T,i

(ln

S

i+1

S

i

)

2

(9.22)

where

bσ

T,i

(resp.

S

T,i

) is the ATMF implied volatility (resp. ATMF skew) at time

i

for residual maturity T .

R

r

T

for

T = 2

years is shown in Figure 9.3 for the Euro Stoxx 50 and S&P 500

indexes. We have used for simplicity ATM rather than ATMF volatilities.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

0.0

0.5

1.0 ...