NEW BOOK ANNOUNCEMENT Probability and Finance: It's Only a Game! by Glenn Shafer and Vladimir Vovk Wiley-Interscience www.wiley.com Probability and Finance is essential reading for anyone who studies or uses probability. Mathematicians and statisticians will find in it a new framework for probability: game theory instead of measure theory. Philosophers will find a surprising synthesis of the objective and the subjective. Practitioners, especially in financial engineering, will lean new ways to understand and sometimes eliminate stochastic models. The first half of the book explains a new mathematical and philosophical framework for probability, based on a sequential game between an idealized scientist and the world. Two very accessible introductory chapters, one presenting an overview of the new framework and one reviewing its historical context, are followed by a careful mathematical treatment of probability's classical limit theorems. The second half of the book, on finance, illustrates the potential of the new framework. It proposes greater use of the market and less use of stochastic models in the pricing of financial derivatives, and it shows how purely game-theoretic probability can replace stochastic models in the efficient-market hypothesis. GLENN SHAFER is Professor in the Graduate School of Management at Rutgers University. He is also the author of The Art of Causal Conjecture, Probabilistic Expert Systems, and A Mathematical Theory of Evidence. VLADIMIR VOVK is Professor in the Department of Computer Science at Royal Holloway, University of London. TABLE OF CONTENTS 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. These lead to upper and lower probabilities for Player II's behavior in the course of the game. 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.
