A theoretical framework for the pricing of contingent claims in the presence of model uncertainty

Abstract

The aim of this work is to evaluate the cheapest superreplication price of a general (possibly path-dependent) European contingent claim in a context where the model is uncertain. This setting is a generalization of the uncertain volatility model (UVM) introduced in by Avellaneda, Levy and Paras. The uncertainty is specified by a family of martingale probability measures which may not be dominated. We obtain a partial characterization result and a full characterization which extends Avellaneda, Levy and Paras results in the UVM case. Introduction.Our purpose is to set a framework for dealing with model uncertainty in mathematical finance and to handle the pricing of contingent claims in this context.Let (S t ) t∈[0,T ] be a real-valued process which stands for an asset price. Usually, it is assumed that the set P m of the equivalent probabilities under which S is a martingale is not empty. This is a sufficient condition to preclude pure gambling strategies that never fail and win with a positive probability.In the classical example of the Black-Scholes model, S solves the linear stochastic differential equation dS t = S t dB t , where

Martingale representation theorem for theG-expectation

“…One of them is the study of financial problems with uncertainty about the volatility. This important problem was also considered earlier by Denis & Martini [3]. Motivated by this application, Denis & Martini developed an almost pathwise theory of stochastic calculus.…”

“…3 Quasi-sure stochastic analysis of Denis & Martini [3] Let P be a probability measure on (Ω, F) so that the canonical process B is a martingale. Then, the quadratic variation process B t of B under P exists.…”

“…Notice that when a is positive definite, as required in Denis and Martini [3], we do not need cI d in the lower bound. Also, the constant c = c(P) may be different for each measure.…”

“…Also, the constant c = c(P) may be different for each measure. Denis and Martini [3] define the following.…”

“…A brief introduction to this theory is provided in Section 2 below. Denis & Martini [3] also construct a similar structure of quasi-sure stochastic analysis. However, they use a quite different approach which utilizes the set P of all probability measures P so that the canonical map in the Wiener space is a martingale under P and the quadratic variation of this martingale lies between a ≤ a.…”