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

3.6. NUMERICAL AND EMPIRICAL TESTS 123
5
4.5
4
true
calibrated with λ 0
=2
calibrated with λ 0
=1
3.5
3
2.5
2
1.5
1
0.5
0
−0.5 0 0.5
Figure 3.7: Levy densities calibrated to option prices generated from Kou model, using two
different initial measures with intensities λ = 1 and λ = 2.
good precision (for both values of σ 0 that we took, the standard deviation is less than 0.5% in
implied volatility units).
The calibrated Lévy densities for two different values of prior diffusion volatility σ 0 are
shown in the right graph of Figure 3.8: a smaller value of the volatility parameter leads to a
greater intensity of small jumps.
Here we observe once again the redundancy of volatility and small jumps, this time in an
infinite intensity context. More precisely this example shows that call option prices generated
from an infinite intensity exponential Lévy model can be reproduced with arbitrary precision
using a jump-diffusion with finite jump intensity. This leads us to conclude that given a finite
(but realistic) number of option prices, the shape of implied volatility skews/smiles does not
allow to distinguish infinite activity models like variance gamma from jump diffusions.
3.6.2 Empirical properties of implied Lévy measures
To illustrate our calibration method we have applied it to a data set spanning the period from
1999 to 2001 and containing daily prices of options on the DAX (German stock market index)
for a range of strikes and maturities.
Figure 3.10 shows the calibration quality for three different maturities. Note that here
each maturity has been calibrated separately (three different exponential Lévy models). These