Game theoretic analysis of incomplete markets: emergence of probabilities, nonlinear and fractional Black-Scholes equations

TL;DR

This paper introduces a game-theoretic framework for option pricing in incomplete markets, deriving risk-neutral probabilities without stochastic models, leading to nonlinear and fractional Black-Scholes equations.

Contribution

It develops a pure game-theoretic approach to option pricing that handles incomplete markets and path-dependent payoffs, bypassing stochastic assumptions.

Findings

Risk-neutral probabilities emerge from robust control evaluation.

The method accommodates various market rules and transaction costs.

Continuous limit yields nonlinear and fractional Black-Scholes equations.

Abstract

Expanding the ideas of the author's paper 'Nonexpansive maps and option pricing theory' (Kibernetica 34:6 (1998), 713-724) we develop a pure game-theoretic approach to option pricing, by-passing stochastic modeling. Risk neutral probabilities emerge automatically from the robust control evaluation. This approach seems to be especially appealing for incomplete markets encompassing extensive, so to say untamed, randomness, when the coexistence of infinite number of risk neutral measures precludes one from unified pricing of derivative securities. Our method is robust enough to be able to accommodate various markets rules and settings including path dependent payoffs, American options and transaction costs. On the other hand, it leads to rather simple numerical algorithms. Continuous time limit is described by nonlinear and/or fractional Black-Scholes type equations.