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Zverev O. V., Shelemekh E. A. European option pricing on an incomplete market as an antagonistic game. Izvestiya of Saratov University. Mathematics. Mechanics. Informatics, 2026, vol. 26, iss. 1, pp. 132-138. DOI: 10.18500/1816-9791-2026-26-1-132-138, EDN: WNIFEO
European option pricing on an incomplete market as an antagonistic game
We describe in detail the stochastic multi-step game corresponding to the European option pricing problem on an incomplete market with discrete time and a finite number of assets, without transaction costs and trading restrictions. Recurrent Bellman-type relations for the upper and lower guaranteed values of the game are given. The equivalence of the following statements is established: the market model is arbitrage-free; there are option seller's portfolios delivering minimum in the Bellman-type relations; there is a super-hedging portfolio for the European option. The game with an arbitrage-free market model results in an equilibrium. Based on this statement, we propose to construct a super-hedging portfolio via a seller's game strategy. Examples of analytical European option pricing on an incomplete market with a finite support and numerical pricing of an option for gold are given.
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