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.
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