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

76 CHAPTER 2. THE CALIBRATION PROBLEM
a Lévy process under a consistent minimal entropy martingale measure and no analytic results
similar to Theorem 2.7 are available for the constrained case. For example, let {X t } t≥0 be a
real-valued Lévy process on (A, ν, γ) such that for every t,
X t = N ′ t − N ′′
t , (2.17)
where N ′ and N ′′ are independent Poisson processes with intensity 1 under P . It follows from
Proposition 1.5 that the set of Lévy processes equivalent to P and satisfying the martingale
condition (1.2) contains all processes under which X has the form (2.17) where N ′ and N ′′ are
still independent and have respective intensities λ ′ = λ and λ ′′ = eλ for some λ > 0. The set of
all equivalent martingale measures under which X remains a Lévy process is thus parametrized
by one parameter λ.
If one allows X to be an additive process (process with independent
increments, continuous in probability) under the new measure, the class of equivalent martingale
measures is much larger: it follows from Theorem IV.4.32 in [54] that N ′ and N ′′ can now
have variable (but deterministic) intensities λ ′ (t) = λ(t) and λ ′′ (t) = eλ(t) for some function
λ : [0, T ∞ ] → (0, ∞). It is clear that many market data sets (involving several maturities) can
be reproduced by an equivalent martingale measure, under which X is an additive process but
not by a martingale measure under which X is a Lévy process, which implies that under the
consistent minimal entropy martingale measure X will not be a Lévy process.
Stutzer [91] suggests a three-step algorithm for numerical evaluation of derivative prices
under the CMEMM in a model with a single time horizon T . This method allows to compute
prices of European options with maturity T , consistent with prices of market-quoted European
options with the same maturity. First, possible asset price values at time T , {P h } N h=1
and the
corresponding probabilities ˆπ(h) are estimated nonparametrically using a histogram estimator
from the historical return values. Second, one needs to find the probabilities π ∗ (h), that satisfy
the martingale constraint and the pricing constraints and have the smallest relative entropy with
respect to ˆπ. In this simplified one period model the martingale condition reduces to a single
constraint that the discounted expectation of stock price under π ∗ be equal to its present value.
The European options expiring at T can then be priced using these martingale probabilities π ∗ .
In this paper, Stutzer suggests an information theoretic rationale for using the relative
entropy (also called Kullback-Leibler information criterion) for pricing and calibration. From
a Bayesian point of view, the historical prices constitute a prior information about the future