The Mathematics of Financial Modelingand Investment Management

15-ArbPric-ContState/Time Page 457 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Continuous-State, Continuous-Time Models 457

T

E ∫θθθθudSπ u = 0

0

and therefore any self-financing trading strategy is a martingale. We can

thus write

θθθθ π π

0 S 0 = E[θθθθTST ]

If

π π π π

θθθθTST ≥ 0 then θθθθ 0 S 0 ≥ 0 and if θθθθTST > 0 then θθθθ 0 S 0 > 0

which shows that there cannot be any arbitrage.

We have now stated that the existence of state-price deflators ensures

the absence of arbitrage. The converse of this statement in a continuous-

state, continuous-time setting is more delicate and will be dealt with later.

We will now move on to equivalent martingale measures.

EQUIVALENT MARTINGALE MEASURES

In the previous section we saw that if there is a regular state-price deflator

then there is no arbitrage. A state-price deflator transforms every price pro-

cess and every self-financing trading strategy into a martingale. We will

now see that, after discounting by an appropriate process, price pro-

cesses become martingales through a transformation of the real probability

measure into an equivalent martingale measure.^4 This theory parallels the

theory of equivalent martingale measures developed in the discrete-state,

discrete-time setting. First some definitions must be discussed.

Given a probability measure P, the probability measure Q is said to

be equivalent to P if both assign probability zero to the same events,

that is, if P(A) = 0 if and only if Q(A) = 0 for every event A. The equiva-

lent probability measure Q is said to be an equivalent martingale mea-

(^4) The theory of equivalent martingale measures was developed in the following arti-
cles: J.M. Harrison and S.R. Pliska, “A Stochastic Calculus Model of Continuous
Trading: Complete Markets,” Stochastic Process Application 15 (1985), pp. 313–
316; J.M. Harrison and S.R. Pliska, “Martingales and Stochastic Integrals in the
Theory of Continuous Trading,” Stochastic Process Application 11 (1981), pp. 215–
260 and, J.M. Harrison and D.M. Kreps, “Martingales and Arbitrage in Multiperiod
Securities Markets,” Journal of Economic Theory 20 (June 1979), pp. 381–408.