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Game‐Theoretic Probability in Finance

In this introductory chapter, we sketch our game-theoretic approach to some basic topics of finance theory. We concentrate on how the Black-Scholes method for pricing and hedging a European option can be made purely game-theoretic, but begin with an examination of the apparently random behavior of stock-market prices, and we conclude with a discussion of informational efficiency. The usual derivation of the Black-Scholes formula for the price of an option relies on the assumption that the market price S(t) of the underlying security S follows a diffusion process. As we explain in the first two sections of this chapter, this stochastic assumption is used in two crucial ways: Taming the Market The diffusion model limits the wildness of fluctuations in S(t). This is the celebrated √ dt effect: the change in S(t) over an increment of time of positive length dt has the order of magnitude (dt) 1/2. This is wild enough, because (dt) 1/2 is much larger than dt when dt is small, but one can imagine much wilder fluctuations—say fluctuations of order (dt) 1/3. Averaging Market Changes The diffusion model authorizes the use of the law of large numbers on a relatively fine time scale. The model says that relative changes in S(t) over nonoverlapping time intervals are independent, no matter how small the intervals, and so by breaking a small interval of time [t 1 , t 2 ] into many even smaller intervals, we can use the law of large numbers to replace certain effects by their theoretical mean values. Our purely game-theoretic approach does not need the diffusion model for either of these purposes.