The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 435 Wednesday, February 4, 2004 1:08 PM

Arbitrage Pricing: Finite-State Models 435

APT MODELS

In the previous sections we presented the general theory of arbitrage

pricing. The most fundamental principle of finance theory, absence of

arbitrage, applies to all price processes. In this section we present a spe-

cial case of the theory which applies to equity prices. In 1976 Stephen

Ross published a seminal paper^2 where he argued that equity returns

can be represented as a linear regression over a small set of factors and

that expected returns are determined by principle of absence of arbi-

trage. This pricing theory is called the Arbitrage Pricing Theory (APT).

APT is formulated in a one period setting. Suppose that equity

returns can be written as follows:

r = a + Bf + εεεε

where r is the n-vector of returns to be modeled, f is a k-vector of com-

mon factors with k << n, a is an n-vector of constants, B is a n×k matrix

and εεεε is an n-vector of random disturbances such that:

E[εεεεf] = 0

E[εεεεεεεε′ f] = ΣΣΣΣ

In the above relationships, the factors are stochastic variables. APT

states that, if there is no arbitrage, the constants a in the above relation-

ship must all be equal to the risk-free rate.

In a one period setting, if there are only a finite number of securities

traded at discrete dates and if the price of each security can take any

value regardless of the prices of other securities, clearly no arbitrage

opportunity is possible. In fact, given any portfolio, infinite price paths

can assume negative values. In a probabilistic context it might happen

that the probability of making a loss starting from zero investment

might be small but not zero.

APT holds in the limit of a large economy. Ross assumed that well-

diversified portfolios exist; this implies that stochastic fluctuations go to

zero in the limit of very large portfolios. This is not to say that portfolio

behavior becomes deterministic in the limit of large portfolios as factors are

assumed to be stochastic; it does however mean that uncertainty is com-

pletely captured by the dynamics of factors. Under this assumption, Ross

demonstrated that the following relationship holds for large economies:

(^2) Stephen Ross, “The Arbitrage Theory of Capital Asset Pricing,” Journal of Eco-
nomic Theory 13, no. 3 (December 1976), pp. 341–360.