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

120 CHAPTER 3. NUMERICAL IMPLEMENTATION
0.5
0.45
simulated
calibrated
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simulated
calibrated
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0.35
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0.25
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6 7 8 9 10 11 12 13 14
0.1
6 7 8 9 10 11 12 13 14
Figure 3.4: Calibration quality (implied volatilities as a function of strike prices) for Kou’s jump
diffusion model. Left: no perturbations. Right: 1% perturbations were added to the data
The Lévy density used in the tests was asymmetric with the left tail heavier than the
right one (α 1 = 1/0.07 and α 2 = 1/0.13). The jump intensity λ was equal to 1 and the last
parameter p was chosen such that the Lévy density is continuous at x = 0. The option prices
were computed using the Fourier transform method of Section 1.4. The maturity of the options
was 5 weeks and we used 21 equidistant strikes ranging from 6 to 14 (the spot being at 10). To
capture tail behavior it is important to have strikes quite far in and out of the money. Merton’s
jump diffusion model [71] was used as prior.
After generating sets of call option prices from Kou’s model, in the first test, the algorithm
described in Section 3.5.3 was applied to these prices directly, and in the second test, to model
the presence of data errors, random perturbations of order of 1% of implied volatility were
added to the simulated prices. As can be observed from Figure 3.4, in both cases the accuracy
of calibration at the level of implied volatilities is satisfying with only 21 options.
Figure 3.5 compares the non-parametric reconstruction of the Lévy density to the true Lévy
density which, in this case, is known to be of the form (1.17). It can be observed that even in
presence of data errors we retrieve successfully the main features of the true density with our
non-parametric approach. The only region in which we observe a detectable error is near zero:
very small jumps have a small impact on option prices. The gradient of option’s time value
given by Equation (3.32) vanishes at zero (i.e. for m = m 0 ) which means that the Lévy density