14-Arbitrage Page 393 Wednesday, February 4, 2004 1:08 PM

CHAPTER

14

Arbitrage Pricing:

Finite-State Models

T

he Principle of Absence of Arbitrage is perhaps the most fundamental

principle of finance theory. In the presence of arbitrage opportunities,

there is no trade-off between risk and returns because it is possible to

make unbounded risk-free gains. The principle of absence of arbitrage is

fundamental for understanding asset valuation in a competitive market.

This chapter discusses arbitrage pricing in a finite-state, discrete-time

setting. In the following chapter we extend the discussion to a continu-

ous-time, continuous-state setting.

THE ARBITRAGE PRINCIPLE

Let’s begin by defining what is meant by arbitrage. In its simple form,

arbitrage is the simultaneous buying and selling of an asset at two differ-

ent prices in two different markets. The arbitrageur profits without risk

by buying cheap in one market and simultaneously selling at the higher

price in the other market. Such opportunities for arbitrage are rare. In

fact, a single arbitrageur with unlimited ability to sell short could correct

a mispricing condition by financing purchases in the underpriced market

with proceeds of short sales in the overpriced market. (Short-selling

means selling an asset that is not owned in anticipation of a price

decline. The mechanism for doing this is described in Chapter 2.) This

means that riskless arbitrage opportunities are short-lived.

Less obvious arbitrage opportunities exist in situations where a

package of assets can produce a payoff (expected return) identical to an

asset that is priced differently. This arbitrage relies on a fundamental

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