Game theoretic analysis of incomplete markets: emergence of probabilities, 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.

42 pages

References in corpus (5)

• Power laws, Pareto distributions and Zipf's law
• Coupled continuous time random walks in finance
• Fractional Fokker-Planck Equations for Subdiffusion with Space-and-Time-Dependent Forces
• Correlated continuous time random walks
• Generalized Continuous-Time Random Walks (CTRW), Subordination by Hitting Times and Fractional Dynamics

Cited by in corpus (2)

• Controlled Continuous Time Random Walks and fractional Hamilton Jacobi Bellman equations
• On some models of many agent systems with competition and cooperation