- Fundamental theorem of arbitrage-free pricing
In a general sense, the fundamental theorem of arbitrage/finance is a way to relate
arbitrageopportunities with risk neutral measures that are equivalent to the original probability measure.
The fundamental theorem in a finite state market
In a finite state market, the fundamental theorem of arbitrage has two parts. The first part relates to existence of a risk neutral measure, while the second relates to the uniqueness of the measure (see Harrison and Pliska):
#The first part states that there is no arbitrage if and only if there exists a risk neutral measure that is equivalent to the original probability measure.
#The second part states provided absence of arbitrage, a market is complete if and only if there is a unique risk neutral measure that is equivalent to the original probability measure.
The fundamental theorem of pricing is a way for the concept of arbitrage to be converted to a question about whether or not a risk neutral measure exists.
The fundamental theorem in more general markets
When stock price returns follow a single
Brownian motion, there is a unique risk neutral measure. When the stock price process is assumed to follow a more general semi-martingale (see Delbaen and Schachermayer), then the concept of arbitrage is too strong, and a weaker concept such as No free lunch with vanishing riskmust be used to describe these opportunities in an infinite dimensional setting.
Arbitrage pricing theory
*M. Harrison and S. Pliska, (1981), Martingales and Stochastic integrals in the theory of continuous trading. Stoch. Proc. & Appl., Vol. 11, pp. 215–260.
*F. Delbaen, W. Schachermayer, (1994), A General Version of the Fundamental Theorem of Asset Pricing. Math. Annalen, Vol. 300, pp. 463–520.
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