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

Chapter 2
The calibration problem and its
regularization
This chapter lays the theoretical foundations of our calibration method. Discretization of the
regularized calibration problem and numerical implementation of the calibration algorithm are
addressed in the next chapter.
The calibration problem consists, roughly speaking, of finding a risk-neutral exponential
Lévy model consistent with market prices of traded options {C M (T i , K i )} i∈I for some index set
I. In other words, the solution of the calibration problem is a probability Q on the path space
(Ω, F), such that (X, Q) is a Lévy process, satisfying the martingale condition (1.2) and such
that the option prices, computed using (1.14) are in some sense close to market prices C M .
Suppose first that the market data C M are consistent with the class of exponential Lévy
models. This is for example the case when the true model underlying market data is an exponential
Lévy model, but this is not the only situation where the above is true: many models
may give the same prices for a given set of European options. For instance, it is easy to construct,
using Dupire’s formula, a local volatility model that gives the same prices, for a set of
European options, as a given exp-Lévy model. Denoting by L the set of all probabilities Q
such that (X, Q) is a Lévy process and by M the set of probabilities Q such that {e Xt } t≥0 is a
Q-martingale, the calibration problem assumes the following form:
57