12
The Existence of Absolutely Continuous Local
Martingale Measures (1995)
Abstract.We investigate the existence of an absolutely continuous martingale
measure. For continuous processes we show that the absence of arbitrage for gen-
eral admissible integrands implies the existence of an absolutely continuous (not
necessarily equivalent) local martingale measure. We also rephrase Radon-Nikod ́ym
theorems for predictable processes.
12.1 Introduction
In Chap. 9 we showed that for locally bounded finite dimensional stochastic
price processesS, the existence of an equivalent (local) martingale measure,
sometimes called risk neutral measure, is equivalent to a property called no
free lunch with vanishing risk(NFLVR). We also proved that if the set of
(local) martingale measures contains more than one element, then necessarily,
there are non-equivalent absolutely continuous local martingale measures for
the processS. We also gave an example, see Example 9.7.7, of a process that
does not admit an equivalent (local) martingale measure but for which there
is a martingale measure that is absolutely continuous. The example moreover
satisfies the weaker property of no-arbitrage with respect to general admissible
integrands. We were therefore led to investigate the relationship between the
two properties, the existence of an absolutely continuous martingale measure
(ACMM)and the absence of arbitrage for general admissible integrands(NA).
From an economic viewpoint a local martingale measureQ, that gives zero
measure to a non-negligible event, sayF, poses some problems. The price of
the contingent claim that pays one unit of currency subject to the occurrence
of the eventF, is given by the probabilityQ[F]. SinceF is negligible for
this probability, the price of the commodity becomes zero. In most economic
[DS 95a] The Existence of Absolutely Continuous Local Martingale Measures.An-
nals of Applied Probability, vol. 5, no. 4, pp. 926–945, (1995).
∗Part of this research was supported by the European Community Stimulation
Plan for Economic Science contract Number SPES-CT91-0089.