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

78 CHAPTER 2. THE CALIBRATION PROBLEM
probability measure on a finite set of paths chosen beforehand; it does not allow to reconstruct
a process on the original space Ω. Second, the martingale condition is not imposed in this
approach (because this would correspond to an infinite number of constraints). As a result,
derivative prices computed with the weighted Monte Carlo algorithm may contain arbitrage
opportunities, especially when applied to forward start contracts.
Nguyen [77] studies the convergence of the above method by Avellaneda et al. when the
number of paths tends to infinity as well as the stability of the calibration procedure, minimization
criteria other than relative entropy, possible ways to impose the (approximate) martingale
condition and many other issues related to this calibration methodology. Statistical properties
of weighted Monte Carlo estimators are also studied in [46].
Chapter 8 of [77] introduces an interesting theoretical approach to minimal entropy calibration,
that is related to the present work and is worth being discussed in more detail. Starting
with a very general prior jump-diffusion model P for the stock price S t of the form
∫
dS t = S t− [b(t, S t− )dt + σ(t, S t− )dW t + Φ(t, S t− , z)(Π(dt, dz) − π(dt, dz))], (2.18)
z∈R
where Π is a homogeneous Poisson random measure with intensity measure π(dt, dz) = dt ×
ρ(dz), and the coefficients satisfy some regularity hypotheses not listed here, Nguyen suggests
to find a martingale measure Q ∗ which reproduces the observed option prices correctly and has
the smallest relative entropy with respect to P in a subclass M ′ of all martingale measures on
(Ω, F). This subclass M ′ contains all martingale measures Q K under which S t has the same
volatility σ and the compensator of Π is given by K(t, X t− , z)π(dt, dz) with
K ∈ K H := {K(t, x, z) : [0, T ] × R × R → (0, ∞) Borel with | log K| ≤ log(H)φ 0 }
for some φ 0 ∈ L 2 (ρ) positive with ‖φ 0 ‖ ∞ ≤ 1. In other words, the object of calibration here is
the intensity of jumps of a Markovian jump diffusion.
The fact that the set
M ′ := { Q K : K ∈ K H}
is convex enables Nguyen to use classical convex analysis methods to study the calibration
problem. In particular, he proves the existence and uniqueness of the solution of the (penalized)
calibration problem under the condition that the coefficients appearing in (2.18) are sufficiently
regular and the drift b and option price constraint C belong to some neighborhood of (r, C r ),