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

100 CHAPTER 3. NUMERICAL IMPLEMENTATION
model from the time series of asset returns. Since, under the conditions of Theorem 2.14,
the calibration procedure yields a martingale probability equivalent to the prior, the choice
of historical probability as prior ensures the absence of arbitrage opportunity involving stock
and options in the calibrated model. Estimation of exponential Lévy models is not discussed
here and interested reader is referred to [27, Chapter 7].
Specific exponential Lévy models
are discussed in [57] (Merton model), [79] (generalized hyperbolic model) and [21] (CGMY or
tempered stable model). [9] and [12] treat more general jump-diffusion type models.
To ensure the stability of calibrated Lévy measures over time, one can systematically choose
the calibrated exponential Lévy model of the previous day as today’s prior. This choice guarantees
that the current calibrated Lévy measure will be altered in the least possible way, to
accommodate the arrival of new option pricing information.
Alternatively, the prior can simply correspond to a model that seems “reasonable” to the
user (e.g. an analyst). In this case our calibration method may be seen as a way to make the
smallest possible change in this model to take into account the observed option prices.
When the user does not have such detailed views, it should be possible to generate a reference
measure P automatically from options data. To do this we choose an auxiliary model Q(θ) (e.g.
Merton model) described by a finite-dimensional parameter vector θ and calibrate it to data
using the least squares procedure:
θ ∗ = arg min
θ
‖C Q(θ) − C M ‖ 2 w (3.9)
It is generally not a good idea to recalibrate this parametric model every day, because in this
case the prior will completely lose its stabilizing role. On the contrary, one should try to find
typical parameter values for a particular market (e.g.
averages over a long period) and fix
them once and for all. Since the objective function in (3.9) is usually not convex, a simple
gradient descent procedure may not give the global minimum. However, the solution Q(θ ∗ ) will
be corrected at later stages and should only be viewed as a way to regularize the optimization
problem (2.4) so the minimization procedure at this stage need not be precise.
Theorem 3.2 shows that the calibrated solutions are continuous with respect to the prior,
that is, small changes in the prior process induce small changes in the solutions.
To assess
empirically the influence of finite changes in the prior on the result of calibration, we have
carried out two series of numerical tests. In the first series of tests, Lévy measure was calibrated