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

122 CHAPTER 3. NUMERICAL IMPLEMENTATION
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Figure 3.6: Lévy measure calibrated to option prices simulated from Kou’s jump diffusion model
with σ 0 = 10%. Left: σ = 10.5% > σ 0 . Right: σ = 9.5% < σ 0 .
The stability of the algorithm with respect to initial conditions can be tested by altering
the starting point of the optimization routine and examining the effect on the output. As
illustrated in Figure 3.7, the entropy penalty removes the sensitivity observed in the non-linear
least squares algorithm (see Figure 2.2 and Section 2.1). The only minor difference between
the two calibrated measures is observed in the neighborhood of zero i.e. the region which, as
remarked above, has little influence on option prices.
Variance gamma model In a second series of tests we examine how our method performs
when applied to option prices generated by an infinite intensity process such as the variance
gamma model. We assume that the user, ignoring that the data generating process has infinite
jump intensity, chooses a (misspecified) prior which has a finite jump intensity (e.g. the Merton
model).
We generated option prices for 21 equidistant strikes between 0.6 and 1.4 (the spot being at
1) using the variance gamma model [67] with no diffusion component and applied the calibration
algorithm using a Merton jump diffusion model as prior to these prices. The parameters of the
variance gamma process (cf Equation (1.18)) were σ = 0.3, θ = −0.2 and k = 0.04. All the
options were maturing in five weeks. The left graph in Figure 3.8 shows that even though the
prior is misspecified, the calibrated model reproduced the simulated implied volatilities with