Mathematical Finance Seminar
May 25, 2006 , 5:30 PM to 7:00 PM
Rama Cont, CNRS and Ecole Polytechnique
A probabilistic approach to model calibration
The inverse problem of constructing an option pricing model (or risk–neutral
rocess) compatible with a set of given market prices of options, known in
finance as the model calibration problem, has been treated in the
literature either in a parametric setting, which involves solving
non-convex ill-posed optimization problems, or in a nonparametric setting,
using relative entropy minimization, which makes it difficult to handle
arbitrage constraints.
We propose a new approach to the model calibration problem, which avoids
these pitfalls. Starting from a prior distribution on model parameters and
a set of observed option prices, we propose a probabilistic construction
of an arbitrage–free pricing rule consistent with these observed option
prices. We describe a feasible numerical algorithm for computing prices
under the calibrated model and characterize the limit behavior of the
algorithm. Our algorithm can be seen as an arbitrage-free extension to the
continuous-time setting of Avellaneda et al.'s Weighted Monte Carlo
algorithm.
Unlike many calibration methods in the literature which involve numerical
minimization of non-convex criteria, our construction only involves the
unconstrained minimization of a convex function, easily performed with
gradient-based methods.
Using convex duality, we show that the result of our computation
has a natural interpretation in terms of minimization of “model risk”. We
also show that the resulting risk-neutral measure is asymptotically
related to a class of Bayesian estimators for the posterior distribution
of model parameters given observed option prices.
As a by-product, our method yields a posterior distribution for the price
of an exotic, given observed prices of vanilla options. We apply the
method to the case of a knock out option on an index.