Robust control approach to option pricing, including transaction costs

We adopt the robust control, or game theoretic, approach of [5] to option pricing. In this approach, uncertainty is described by a restricted set of possible price trajectories, without endowing this set with any probability measure. We seek a hedge against every possible price trajectory. In the absence of transaction costs, the continuous trading theory leads to a very simple differential game, but to an uninteresting financial result, as the hedging strategy obtained lacks robustness to the unmodeled transaction costs. (A feature avoided by the classical Black and Scholes theory through the use of unbounded variation cost trajectories. See [5].) We therefore introduce transaction costs into the model. We examine first the continuous time model. Its mathematical complexity makes it beyond a complete solution at this time, but the partial results obtained do point to a robust strategy, and as a matter of fact justify the second part of the paper. In that second part, we examine the discrete time theory, deemed closer to a realistic trading strategy. We introduce transaction costs into the model from the outset and derive a pricing equation, which can be seen as a discretization of the quasi variational inequality of the continuous time theory. The discrete time theory is well suited to a numerical solution. We give some numerical results. In the particular case where the transaction costs are null, we recover our theory of [5], and in particular the Cox Ross and Rubinstein formula when the contingent claim is a convex function of the terminal price of the underlying security. ∗i3s, University of Nice-Sophia Antipolis, and CNRS, France †Corrected December 1st 2001