DIFFERENTIAL GAME-THEORETIC THOUGHTS ON OPTION PRICING AND TRANSACTION COSTS

Abstract

This paper investigates some differential game applications to option pricing mechanisms and related problems. Two players, an investor and "Nature", play a zero-sum game. The usual uncertainty modelling (log-normality for instance) in systems describing the price evolution of stocks is replaced by "Nature", a player who counteracts the investor as much as possible. A relationship between a restricted version of the Black-Scholes and the Hamilton-Jacobi-Bellman partial differential equations is given. This paper, is a first step to possibly solve various option pricing problems (with constraints and/or transactions costs for instance) by means of the available numerical software for optimal control problems. In the second part of the paper, another model, now with three players, is considered. The third player is the bank interested in maximising its own profits by choosing the right formula for transaction costs. Thus a three-person nonzero-sum game, with a special kind of Stackelberg information structure, results. Some simple examples hint in the direction that the bank will be a clear winner.