# 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.
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.

This page is maintained by
Vladimir Vovk.
Last modified on 20 December 2001