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

1.4. PRICING EUROPEAN OPTIONS 47
with α + < 2 and α − < 2. This rich parametric form of the Lévy measure is probably sufficient
for most applications.
The third approach is to specify the density of increments of the process at a given time
scale, say ∆, by taking an arbitrary infinitely divisible distribution. Generalized hyperbolic
processes (see [36–38]) can be constructed in this way. In this approach it is easy to simulate
the increments of the process at the same time scale and to estimate parameters of the distribution
if data are sampled with the same period ∆, but, unless this distribution belongs to
some parametric class closed under convolution, we do not know the law of the increments at
other time scales. Also, given an infinitely divisible distribution, one may not know its Lévy-
Khintchine representation, so it may not be easy to see whether the corresponding Lévy process
has a Gaussian component, finite or infinite jump intensity, etc.
1.4 Pricing European options in exp-Lévy models via Fourier
transform
In this section we present a method, adapted from [23], for pricing European call options in exp-
Lévy models using Fourier transform and, in particular, the Fast Fourier transform algorithm
[29]. We suggest several improvements to the original procedure and give a rigorous analysis of
truncation and discretization errors. These results are of independent interest and will also be
used for the numerical solution of the calibration problem in Chapter 3. Similar analysis has
recently appeared in [63].
Let {X t } t≥0 be a Lévy process satisfying the martingale condition (1.2). To compute the
price of a call option
C(k) = e −rT E[(e rT +X T
− e k ) + ], (1.21)
we would like to express its Fourier transform in log strike in terms of the characteristic function
Φ T (v) of X T and then find the prices for a range of strikes by Fourier inversion. However we
cannot do this directly because C(k) is not integrable (it tends to 1 as k goes to −∞). The
key idea is to instead compute the Fourier transform of the (modified) time value of the option,
that is, the function
z T (k) = e −rT E[(e rT +X T
− e k ) + ] − (1 − e k−rT ) + . (1.22)