The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 430 Wednesday, February 4, 2004 1:08 PM

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430 The Mathematics of Financial Modeling and Investment Management

ARBITRAGE PRICING IN A DISCRETE-TIME,

CONTINUOUS-STATE SETTING

Let’s now discuss the discrete-time, continuous-state setting. This is an

important setting as it is, for example, the setting of the Arbitrage Pric-

ing Theory (APT) Model that we will discuss later in this chapter.

As in the previous discrete-time, discrete-state setting, we use the

probabilistic concepts developed in Chapter 6. The economy is repre-

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

states, ℑ is the σ-algebra of events (formed, in this continuous-state set-

ting, by a nondenumerable number of events), and P is a probability

function. As the number of states is infinite, the probability of each state

is zero and only events, in general, formed by nondenumerable states,

have a finite probability. There are only a finite number of dates from 0

to T. Recall from Chapter 6 that the propagation of information is rep-

resented by a finite filtration ℑt, t = 0,1,...,T. In this case, the filtration ℑt

is not equivalent to an information structure It.

Each security i is characterized by a payoff process dti and by a

price process S i i

t

i. In this continuous-state setting, d

t and St are formed

by a finite number of continuous variables. As before, di t() ω and Si t() ω

are, respectively, the payoff and the price of the i-th asset at time t, 0 ≤ t

≤ T and in state ω ∈ Ω. Following Chapter 6, all payoffs and prices are

stochastic processes adapted to the filtration ℑ.

To develop an intuition for continuous-state arbitrage pricing, con-

sider the previous multiperiod, finite-state case with a very large number

M of states, M>>N where N is the number of securities. Recall from our

earlier discussion in this chapter that risk-neutral probabilities can be

computed solving the following system of linear equations:

qω

ω ∈ Akt,

∑

j

dj i () ω

Rtj,

• – SAi kt
  = t + 1

T

∑ =^0

M

∑ qω =^1

ω = 1

Recall also that at each date t the information structure It partitions the

set of states into Mt subsets. Each partition therefore yields N × Mt

equations and the system is formed by a total of