*J. Japan Statist. Soc.,* Vol. 37 (No. 1), pp. 87-104, 2007

## Game-Theoretic Derivation of Discrete Distributions and Discrete
Pricing Formulas

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.

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