J. Japan Statist. Soc., Vol. 37 (No. 1), pp. 87-104, 2007
Akimichi Takemura and Taiji Suzuki
Abstract. In this expository paper, we illustrate the generality of the game-theoretic probability protocols of Shafer and Vovk (2001) in finite-horizon discrete games. By restricting ourselves to finite-horizon discrete games, we can explicitly describe how discrete distributions with finite support and discrete pricing formulas, such as the Cox-Ross-Rubinstein formula, are naturally derived from game-theoretic probability protocols. Corresponding to any discrete distribution with finite support, we construct a finite-horizon discrete game, a replicating strategy of Skeptic, and a neutral forecasting strategy of Forecaster, such that the discrete distribution is derived from the game. Construction of a replicating strategy is the same as in the standard arbitrage arguments of pricing European options in binomial tree models. However the game-theoretic framework is advantageous because it eliminates the need for any a priori probabilistic assumption.
Key words and phrases: Binomial distribution, Cox-Ross-Rubinstein formula, hypergeometric distribution, lower price, Polya's distribution, probability protocol, replicating strategy, upper price.