Robust Control Approach to Option Pricing: A Representation Theorem and Fast Algorithm

Abstract

The so-called interval model for security prices, together with a robust control approach, allows one to construct a consistent theory of option pricing, including discrete time trading and arbitrary transaction costs. In this context, a new representation theorem for the viscosity solution of the relevant Isaacs (differential) quasi-variational inequality leads to simple formulas and fast numerical algorithms to compute a hedging portfolio. We argue that in spite of a less satisfactory market model, the overall theory is not much less realistic than the classical Black and Scholes theory but rather only that it shifts from the portfolio model to the market model the place where the model is violated when sudden large price changes occur on the market. As such, and subject to a more detailed validation, the new theory might be the basis of a possible alternative as a normative theory whenever transaction costs or discrete time trading are the main concerns.