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2.2. SELECTION USING RELATIVE ENTROPY 69
algorithm used for numerical optimization may stop in one of these local minima, leading to a
much worse calibration quality than that of the true solution.
Figure 2.3 illustrates this effect in the (parametric) framework of the variance gamma model
(1.18). The left graph shows the behavior of the calibration functional in a small region around
the global minimum. Since in this model there are only three parameters, the identification
problem is not present, and a nice profile appearing to be convex can be observed. However,
when we look at the calibration functional on a larger scale (κ changes between 1 and 8),
the convexity disappears and we observe a ridge (highlighted with a dashed black line), which
separates two regions: if the minimization is initiated in the region (A), the algorithm will
eventually locate the minimum, but if we start in the region (B), the gradient descent method
will lead us away from the global minimum and the required calibration quality will never be
achieved.
B
15000
x 10 5
2
1.8
10000
5000
1.6
1.4
1.2
1
A
0.8
0.25
0
0.2
0.15
0.1
0.05
κ
0
min
0.14
0.16
0.18
σ
0.2
0.22
0.24
0.6
8
7
6
5
4
κ
3
2
1
0
0.1
0.15
σ
0.2
Figure 2.3: Sum of squared differences between market prices (DAX options maturing in 10
weeks) and model prices in the variance gamma model (1.18) as a function of σ and κ, the third
parameter being fixed. Left: small region around the global minimum. Right: error surface on
a larger scale.
2.2 Selection of solutions using relative entropy
When the option pricing constraints do not determine the exponential Lévy model completely
(this is for example the case if the number of constraints is finite), additional information may