9
A General Version of the Fundamental
Theorem of Asset Pricing (1994)
9.1 Introduction ............................................
A basic result in mathematical finance, sometimes called thefundamental the-
orem of asset pricing(see [DR 87]), is that for a stochastic process (St)t∈R+,
the existence of an equivalent martingale measure isessentiallyequivalent
to the absence of arbitrage opportunities. In finance the process (St)t∈R+
describes the random evolution of the discounted price of one or several finan-
cial assets. The equivalence of no-arbitrage with the existence of an equivalent
probability martingale measure is at the basis of the entire theory of “pricing
by arbitrage”. Starting from the economically meaningful assumption that
Sdoes not allow arbitrage profits (different variants of this concept will be
defined below), the theorem allows the probabilityPon the underlying proba-
bility space (Ω,F,P) to be replaced by an equivalent measureQsuch that the
processSbecomes a martingale under the new measure. This makes it possi-
ble to use the rich machinery of martingale theory. In particular the problem
of fair pricing of contingent claims is reduced to taking expected values with
respect to the measureQ. This method of pricing contingent claims is known
to actuaries since the introduction of actuarial skills, centuries ago and known
by the name of “equivalence principle”.
The theory of martingale representation allows to characterise those assets
that can be reproduced by buying and selling the basic assets. One might get
the impression that martingale theory and the general theory of stochastic
processes were tailor-made for finance (see [HP 81]).
The change of measure fromPtoQcan also be seen as a result of risk
aversion. By changing the physical probability measure fromPtoQ,one
can attribute more weight to unfavourable events and less weight to more
favourable ones.
[DS 94] A General Version of the Fundamental Theorem of Asset Pricing.Mathema-
tische Annalen, vol. 300, pp. 463–520, Springer, Berlin, Heidelberg, New York (1994).