The Mathematics of Arbitrage

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In 1973 F. Black and M. Scholes published their pathbreaking paper [BS 73] on option pricing. The key idea attributed to R. Merton in a footnote of the Black-Scholes paper is the use of trading in continuous time and the notion of arbitrage.

The simple and economically very convincing “principle of noarbitrage” allows one to derive, in certain mathematical models of financial markets (such as the Samuelson model, [S 65], nowadays also referred to as the “Black-Scholes” model, based on geometric Brownian motion), unique prices for options and other contingent claims.

This remarkable achievement by F. Black, M. Scholes and R. Merton had a profound effect on financial markets and it shifted the paradigm of dealing with financial risks towards the use of quite sophisticated mathematical models. It was in the late seventies that the central role of no-arbitrage arguments was crystallised in three seminal papers by M. Harrison, D. Kreps and S. Pliska ([HK 79], [HP 81], [K 81]) They considered a general framework, which allows a systematic study of different models of financial markets.

The Black-Scholes model is just one, obviously very important, example embedded into the framework of a general theory. A basic insight of these papers was the intimate relation between no-arbitrage arguments on one hand, and martingale theory on the other hand.

This relation is the theme of the “Fundamental Theorem of Asset Pricing” (this name was given by Ph. Dybvig and S. Ross [DR 87]), which is not just a single theorem but rather a general principle to relate no-arbitrage with martingale theory.

Loosely speaking, it states that a mathematical model of a financial market is free of arbitrage if and only if it is a martingale under an equivalent probability measure; once this basic relation is established, one can quickly deduce precise information on the pricing and hedging of contingent claims such as options.

In fact, the relation to martingale theory and stochastic integration opens the gates to the application of a powerful mathematical theory

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