15-ArbPric-ContState/Time Page 467 Wednesday, February 4, 2004 1:08 PM
Arbitrage Pricing: Continuous-State, Continuous-Time Models 467θ 0 S 0 < 0 and θTST ≥ 0, or θ 0 S 0 ≤ 0 and θTST > 0The previous arguments extend to this case. An equivalent martingale
measure for the pair (D,S) is defined as an equivalent probability mea-
sure Q such that the Radon-Nikodym derivativeξ = dQ---------
dPhas finite variance and the process G = S + D is a martingale. Under these
conditions, the following relationship holds:T s- ru ud T ∫ –ruud
Q ∫
St = Et et + ∫ e t dDs
tIMPLICATIONS OF THE ABSENCE OF ARBITRAGE
We saw that the existence of an equivalent martingale measure or of
state-price deflators implies absence of arbitrage. We have also seen
that, in the absence of arbitrage, markets are complete if and only if
there is a unique equivalent martingale measure.
In a discrete-state, discrete-time context we could establish the com-
plete equivalence between the existence of state-price deflators, equiva-
lent martingale measures and absence of arbitrage, in the sense that any
of these conditions implies the other two. In addition, the existence of a
unique equivalent martingale measure implies absence of arbitrage and
market completeness.
In the present continuous-state context, however, absence of arbi-
trage implies the existence of an equivalent martingale measure and of
state price deflators only under rather restrictive and complex technical
conditions. If we want to relax these conditions, the condition of
absence of arbitrage has to be slightly modified. These discussions are
quite technical and will not be presented in this chapter.^5(^5) See F. Delbaen and W. Schachermayer, “The Fundamental Theorem of Asset Pric-
ing for Unbounded Stochastic Processes,” Mathematische Annalen 312, no. 2 (Oc-
tober 1999), pp. 215–250 and F. Delbaen and W. Schachermayer, “A General
Version of the Fundamental Theorem of Asset Pricing,” Mathematische Annalen
300, no. 3 (November 1994), pp. 463–520.