The Mathematics of Financial Modelingand Investment Management

14-Arbitrage Page 394 Wednesday, February 4, 2004 1:08 PM

394 The Mathematics of Financial Modeling and Investment Management

principle of finance called the law of one price, which states that a given

asset must have the same price regardless of the location where the asset

is traded and the means by which one goes about creating that asset.

The law of one price implies that if the payoff of an asset can be syn-

thetically created by a package of assets, the price of the package and

the price of the asset whose payoff it replicates must be equal.

When a situation is discovered whereby the price of the package of

assets differs from that of an asset with the same payoff, rational inves-

tors will trade these assets in such a way so as to restore price equilib-

rium. This market mechanism is founded on the fact that an arbitrage

transaction does not expose the investor to any adverse movement in

the market price of the assets in the transaction.

For example, consider how we can produce an arbitrage opportu-

nity involving three assets A, B, and C. These assets can be purchased

today at the prices shown below, and can each produce only one of two

payoffs (referred to as State 1 and State 2) a year from now:

Asset Price Payoff in State 1 Payoff in State 2

A $70 $50 $100

B 60 30 120

C 80 38 112

While it is not obvious from the data presented above, an investor

can construct a portfolio of assets A and B that will have the identical

return as asset C in both State 1 and State 2. Let wA and wB be the pro-

portion of assets A and B, respectively, in the portfolio. Then the payoff

(i.e., the terminal value of the portfolio) under the two states can be

expressed mathematically as follows:

■ If State 1 occurs: $50 wA + $30 wB

■ If State 2 occurs: $100 wA + $120 wB

We create a portfolio consisting of A and B that will reproduce the

payoff of C regardless of the state that occurs one year from now. Here

is how: for either condition (State 1 and State 2) we set the payoff of the

portfolio equal to the payoff for C as follows:

■ State 1: $50 wA + $30 wB = $38

■ State 2: $100 wA + $120 wB = $112