I know the fundamental theorem of asset pricing in discrete time which says that there are no arbitrage opportunities if and only if there exists a equivalent martingale measure. As far as I understand this equivalence is not true for the general setting; i.e. existence of equivalent martingale measure implies no arbitrage but not the other way around.

Can anyone please state the general fundamental theorem of asset pricing?


1 Answer 1


Let's begin with a preliminary definition.

Definition No free lunch with vanishing risk (NFLVR). We say that a process $S$ satisfies NFLVR if there does not exist a sequence of admissible integrands $H^n$ such that, if $f_n = (H^n \cdot S)_\infty$, then $f_n^- \to 0$ uniformly and $f_n \to f_0$ where $f_0$ is a $[0,\infty]$-valued random variable satisfying $P(f_0 > 0 ) > 0$.

Note that existence of arbitrage implies there is a free-lunch with vanishing risk. The general statement of the first fundamental theorem of asset pricing can now be stated:

Theorem Let $S$ be a bounded $\mathbb{R}$-valued semi-martingale. There is an equivalent martingale measure for $S$ if and only if $S$ satisfies NFLVR.

For a proof of this theorem see the book The Mathematics of Arbitrage by Delbaen and Schachermayer, or their original paper A general version of the fundamental theorem of asset pricing. For a counterexample to "no arbitrage $\implies$ existence of an equivalent martingale measure", see On the fundamental theorem of asset pricing with an infinite state space by Back and Pliska.


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