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

114 CHAPTER 3. NUMERICAL IMPLEMENTATION
the user to supply the function value and its gradient and tries, iteratively, to build up a good
approximation of the inverse Hessian matrix of the functional being optimized. In our numerical
examples we used the LBFGS implementation by Jorge Nocedal et al. [20]. The algorithm is
typically initialized with the prior Lévy measure.
Since gradient-based methods only allow to find one local minimum of the objective function
and the calibration functional (3.26) is not convex, there is no guarantee that the BFGS algorithm
will converge to its true global minimum. However, starting the optimization procedure
from different initializers, we have empirically observed (see Section 3.6.1) that in presence of
even a small regularization, the minimizers do not depend on the starting point of the minimization
algorithm and that using a gradient-based optimization procedure produces an acceptable
calibration quality.
In the rest of this section we show how to compute numerically the functional (3.26) and
its gradient. For simplicity, from now on we will suppose that the prior Lévy process has a
non-zero diffusion part (A > 0).
3.5.1 Computing the calibration functional
Substituting the discrete expressions (3.1) and (3.2) into formula (3.25), we obtain:
Ĵ α (Q) =
N∑
w i (ĈQ (T i , K i ) − C M (T i , K i )) 2
i=1
⎛
⎞
+ α ⎝ A M−1
2A 2 + ∑
bP + (e x j
− 1)q j
⎠
j=0
2
M−1
∑
+ α (q j log(q j /p j ) + 1 − q j ) , (3.27)
j=0
where b P = γ P − ∫ |x|≤1 xνP (dx) is the drift of the prior process. The last two terms of the
above equation can be evaluated directly using a finite number of computer operations, we will
therefore concentrate on the first term.
The approximate option prices ĈQ (T i , K i ) are computed using the Fourier transform algorithm
of Section 1.4 as follows: for each maturity date, present in the data, prices are first
computed on a fixed grid of strikes and then interpolated to obtain prices at strikes K i corresponding
to this maturity date. The number of Fourier transforms is thus determined by the
number of maturity dates present in the data, which is typically smaller than 10.
To use the Fourier transform method, the first step is to compute the characteristic exponent