Stochastic Volatility Modeling

60 Stochastic volatility modeling

Figure 2.5

:

S

T

for the Euro Stoxx 50 index as a function of

T

(in years) as read

o the market smile (actual) and as given by the power-law benchmark (2.52), for

the smiles of October 4, 2010 (left:

τ

0

= 0.15, γ = 0.37

) and May 16, 2013 (right:

τ

0

= 0.12, γ = 0.52).

The long-maturity value of the

ˆ

Aš is given by (2.81):

R

∞

=

2 −γ

1 −γ

R

∞

= 2.6

for the October 4, 2010 smile and

R

∞

= 3.1

for the May 16, 2013 smile –

we know that R

∞

is only reached for very long maturities.

In the local volatility model, because implied volatilities are a function of

(t, S)

the SSR provides substantial information: it determines both the break-even levels ...