15-ArbPric-ContState/Time Page 441 Wednesday, February 4, 2004 1:08 PM

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CHAPTER

15

Arbitrage Pricing:

Continuous-State,

Continuous-Time Models

n the previous chapter we described arbitrage pricing using finite-state

models. In this chapter we describe arbitrage pricing in the continuous-

state, continuous-time setting. There are a number of important conceptual

changes in going from a discrete-state, discrete-time setting to a continuous-

state, continuous-time setting. First, each state of the world has probability

zero. As described in Chapter 6, this precludes the use of standard con-

ditional probabilities for the definition of conditional expectation and

requires the use of filtrations (rather than of information structures) to

describe the propagation of information. Second, the tools of matrix

algebra are inadequate; the more complex tools of calculus and stochas-

tic calculus described in Chapters 4, 8, 9, and 10, respectively, are

required. Third, simple generalizations are rarely possible as many patho-

logical cases appear in connection with infinite sets.

THE ARBITRAGE PRINCIPLE IN CONTINUOUS TIME

Let’s start with the definition of basic concepts. The economy is repre-

sented by a probability space (Ω, ℑ, P) where Ω is the set of possible

states, ℑis the σ-algebra of events, and P is a probability measure. Time

is a continuous variable in the interval [0,T]. Recall from Chapter 6 that

the propagation of information is represented by a filtration ℑt. The lat-

ter is a family of σ-algebras such that ℑt ⊆ ℑs, t < s.

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