A guaranteed deterministic approach to superhedging: most unfavorable scenarios of market behaviour and moment problem

Sergey N. Smirnov

Matematičeskaâ teoriâ igr i eë priloženiâ, Tome 12 (2020) no. 3, pp. 50-88.

Résumé

A guaranteed deterministic problem setting of super-replication with discrete time is considered: the aim of hedging of a contingent claim is to ensure the coverage of possible payout under the option contract for all admissible scenarios. These scenarios are given by means of a priori given compacts, that depend on the prehistory of prices: the increments of the price at each moment of time must lie in the corresponding compacts. The absence of transaction costs is assumed. The game-theoretical interpretation implies that the corresponding Bellman-Isaac equations hold, both for pure and mixed strategies. In the present paper, we propose a two-step method of solving the Bellman equation arising in the case of (game) equilibrium. In particular, the most unfavorable strategies of the “market” can be found in the class of the distributions concentrated at most in $n+1$ point, where $n$ is the number of risky assets.

Détail

Keywords: guaranteed estimates, deterministic price dynamics, super-replication, absence of arbitrage opportunities, Bellman-Isaacs equations, mixed strategies, game equilibrium, no trading constraints, risk-neutral measures.
Mots-clés : option, arbitrage

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