The Mathematics of Financial Modelingand Investment Management

3-Milestones Page 89 Wednesday, February 4, 2004 12:47 PM

Milestones in Financial Modeling and Investment Management 89

this type are referred to as multifactor risk models. These models are

applied to portfolio management.

ARBITRAGE, HEDGING, AND OPTION THEORY: BLACK, SCHOLES,

AND MERTON

The idea of arbitrage pricing can be extended to any price process. A

general model of asset pricing will include a number of independent

price processes plus a number of price processes that depend on the first

process by arbitrage. The entire pricing structure may or may not be

cast in a general equilibrium framework.

Arbitrage pricing allowed derivative pricing. With the development

of derivatives trading, the requirement of a derivative valuation and

pricing model made itself felt. The first formal solution of the option

pricing model was developed independently by Fisher Black and Myron

Scholes in 1976,^22 working together, and in the same year by Robert

Merton.^23

The solution of the option pricing problem proposed by Black,

Scholes, and Merton was simple and elegant. Suppose that a market

contains a risk-free bond, a stock, and an option. Suppose also that the

market is arbitrage-free and that stock price processes follow a continu-

ous-time geometric Brownian motion (see Chapter 8). Black, Scholes,

and Merton demonstrated that it is possible to construct a portfolio

made up of the stock plus the bond that perfectly replicates the option.

The replicating portfolio can be exactly determined, without anticipa-

tion, solving a partial differential equation.

The idea of replicating portfolios has important consequences.

Whenever a financial instrument (security or derivative instrument) pro-

cess can be exactly replicated by a portfolio of other securities, absence

of arbitrage requires that the price of the original financial instrument

coincide with the price of the replicating portfolio. Most derivative pric-

ing algorithms are based on this principle: to price a derivative instru-

ment, one must identify a replicating portfolio whose price is known.

Pricing by portfolio replication received a powerful boost with the

discovery that calculations can be performed in a risk-neutral probabil-

ity space where processes assume a simplified form. The foundation was

thus laid for the notion of equivalent martingales, developed by Michael

(^22) Fischer Black and Myron Scholes, “The Pricing of Options and Corporate Liabil-
ities,” Journal of Political Economy (1973), pp. 637–654.
(^23) Robert C. Merton, “Theory of Rational Option Pricing,” Bell Journal of Econom-
ics and Management Science (1973), pp. 141–183.