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A general version of the fundamental theorem of asset pricing. (English) Zbl 0865.90014
The fundamental theorem of asset pricing is that for a stochastic process \(S=(S_t)_{t\in\mathbb{R}}\), the existence of an equivalent martingale measure is essentially equivalent to the absence of arbitrage opportunities. The present paper focuses on the term essentially in the above statement. The general idea underlying the no arbitrage condition is that there should be no trading strategy \(H\) for the process \(S\) such that the final payoff described by the stochastic integral \((H.S)_\infty\) is a nonnegative function strictly positive with positive probability which has been termed as the no free lunch with vanishing risk (NFLVR). The major result of this paper is contained in the following: Let \(S\) be a bounded real valued semi-martingale. There is an equivalent martingale measure for \(S\) if and only if \(S\) satisfies NFLVR. The fact that NFLVR guarantees the existence of an equivalent martingale measure for \(S\) allows wide applicability of martingale theory. Several versions of the no free lunch condition are also introduced and their relationship studied.

MSC:
91G99 Actuarial science and mathematical finance
91B24 Microeconomic theory (price theory and economic markets)
60G44 Martingales with continuous parameter
60H30 Applications of stochastic analysis (to PDEs, etc.)
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