Fundamental theorem of arbitrage price theory

When fundamental theorem of arbitrage pricing theory ( English fundamental theorem of asset pricing ), there are two important statements from the financial mathematics , which are used in numerous financial models to value financial options application. They provide necessary and sufficient conditions as to whether arbitrage opportunities exist in the market model and whether the market is complete .

The fundamental theorem consists of two parts, which are referred to as the first and second fundamental theorem of arbitrage pricing theory. The first part says that in a financial market model there is no arbitrage if there is a martingale measure equivalent to the market measure . Freedom from arbitrage is the characteristic of a market in which it is not possible to make a profit without risk . The statement of the second fundamental theorem of arbitrage price theory is that a market model is complete if and only if there is exactly one equivalent martingale measure. This also means that every complete market is free of arbitrage. A market means complete if it is possible to replicate every derivative with other financial instruments . ${\displaystyle P}$ ${\displaystyle Q}$

Many of the market models considered in financial mathematics are arbitrage-free and complete, for example the Black-Scholes model or the Cox-Ross-Rubinstein model . In general, it makes sense to demand in every financial market model that this is arbitrage-free, although in reality arbitrage opportunities exist for a short time. Since a financial market model is the basis for the risk-neutral valuation of derivatives, completeness is often required.

literature

• Nicholas H. Bingham, Rüdiger Kiesel: Risk-Neutral Valuation . Springer Publishing House, London 2004.