Processus de Lévy en Finance - Laboratoire de Probabilités et ...

3.6. NUMERICAL AND EMPIRICAL TESTS 129
50
30−day ATM options
450−day ATM options
45
40
35
30
25
20
15
2/01/1996 2/01/1997 2/01/1998 30/12/1998
Figure 3.13: Implied volatility of at the money European options on CAC40 index.
type of time dependence is therefore more complicated than simply a time dependent intensity.
A second major difficulty arising while trying to calibrate an exponential Lévy model is the
time evolution of the smile. Exponential Lévy models belong to the class of so called “sticky
moneyness” models, meaning that in an exponential Lévy model, the implied volatility of an
option with given moneyness (strike price to spot ratio) does not depend on time. This can
be seen from the following simple argument.
In an exponential Lévy model Q, the implied
volatility σ of a call option with moneyness m, expiring in τ years, satisfies:
e −rτ E Q [(S t e rτ+Xτ − mS t ) + |F t ] = e −rτ σ2
rτ+σWτ −
E[(S t e 2 τ − mS t ) + |F t ]
Due to the independent increments property, S t cancels out and we obtain an equation for
the implied volatility σ which does not contain t or S t . Therefore, in an exp-Lévy model this
implied volatility does not depend on date t or stock price S t . This means that once the smile
has been calibrated for a given date t, its shape is fixed for all future dates. Whether or not this
is true in real markets can be tested in a model-free way by looking at the implied volatility
of at the money options with the same maturity for different dates. Figure 3.13 depicts the
behavior of implied volatility of two at the money options on the CAC40 index, expiring in
30 and 450 days. Since the maturities of available options are different for different dates, to
obtain the implied volatility of an option with fixed maturity T for each date, we have taken
two maturities, present in the data, closest to T from above and below: T 1 ≤ T and T 2 > T .