## Probability and Finance: It's Only a Game! by Glenn Shafer and Vladimir Vovk

Probability and Finance as a Game: Mathematical probability can be based on a two-person sequential game of perfect information. On each round, Player II states odds at which Player I may bet on what Player II will do next. In statistical modeling, Player I is a statistician and Player II is the world. In finance, Player I is an investor and Player II is a market.

### PART I: PROBABILITY WITHOUT MEASURE

Game theory can handle classical topics in probability (the weak and strong limit theorems). No measure theory is needed. The Historical Context: From Pascal to Kolmogorov. Collectives and Kolmogorov complexity. Jean Ville's game-theoretic martingales. Objective and subjective probability. The Bounded Strong Law of Large Numbers: The game-theoretic strong law for coin-tossing and bounded prediction. Kolmogorov's Strong Law: Classical and martingale forms. The Law of the Iterated Logarithm: Validity and Sharpness. The Weak Laws: Game-theoretic forms of Bernoulli's and De Moivre's theorems. Using parabolic potential theory to generalize De Moivre's theorem. Lindeberg's Theorem: A game-theoretic central limit theorem. The Generality of Probability Games: The measure-theoretic limit theorems follow easily from the game-theoretic ones.

### PART II: FINANCE WITHOUT PROBABILITY

The game-theoretic framework can dispense with the stochastic assumptions currently used in finance theory. It uses the market, instead of a stochastic model, to price volatility. It can test for market efficiency with no stochastic assumptions. Game-Theoretic Probability in Finance: The game-theoretic Black-Scholes theory requires the market to price a derivative that pays a measure of market volatility as a dividend. Discrete Time: The game-theoretic treatment can be made rigorous and practical in discrete time. Continuous Time: Using non-standard analysis, we can pass to a continuous limit. The Generality of Game-Theoretic Pricing: In the continuous limit, it is easy to see how interest and jumps can be handled, and how the dividend-paying derivative can be replaced by derivatives easier to market. American Options: Pricing American options requires a different kind of game. Diffusion Processes: They can also be represented game-theoretically. The Game-Theoretic Efficient-Market Hypothesis: Testing it using classical limit theorems. Risk versus return.