The Mathematics Behind the 1997 Nobel Prize in Economics

Guillermo Ferreyra [note]


On October 14 the Royal Swedish Academy of Sciences announced the winners of the 1997 Nobel Prize in Economics. The winners were Professor Robert C. Merton, of Harvard University, Cambridge, USA and Professor Myron S. Scholes, of Stanford University, Stanford, USA, for the discovery of "a new method to determine the value of derivatives".

The news media had ample coverage of this announcement and of the reason for the award to Merton and Scholes. In the words of the Swedish Academy

Robert C. Merton and Myron S. Scholes have, in collaboration with the late Fischer Black, developed a pioneering formula for the valuation of stock options. Their methodology has paved the way for economic valuations in many areas. It has also generated new types of financial instruments and facilitated more efficient risk management in society.
Financial analysts have reached the point where they are able to calculate, with high accuracy, the value of a stock option or derivative. The models and techniques employeed by today's analysts are rooted in a model developed by Black and Scholes in 1973, which today is known as the Black-Scholes formula.

What was little discussed in the media at the time the award was announced is the fact that the methodology employed by Merton, Black and Scholes is heavily indebted to the modern mathematical theory of probability. The work of the three economists in the 70's was a novel and extremely useful application to finance of the deep mathematical theory of stochastic processes that had culminated with the theory of stochastic differential equations (SDE's) less than thirty years before. The theory of SDE's is thriving today due to the many applications it has found in science and engineering, from physics, genetics, and the enviromental sciences to electrical engineering and computer science, specially in the de-noising of transmitted data. As examples of the lively current activity in this field, we point out two events that are occurring this month: A special session on Stochastic Systems at the American Mathematical Society-Sociedad Matematica Mexicana meeting in Oaxaca, and a Symposium on Stochastic Control and Nonlinear Filtering at the University of Southern California.

The simplest (from the mathematical point of view) financial derivative is the call option of European type. After explaining what this type of option is and the important effect it has for investors, we will describe what the theory of SDE's is and how it provides the mathematical tools on which the options model rests. At this point it is important to remark that students in the quantitative disciplines having a basic course in probability as a background, and willing to accept some facts from Lebesgue's theory of integration, can be taught the mathematical theory of SDE's rigorously in a month, and then the model for call options of European type can be done as a routine application in two or three lectures.

A call option of European type is a contract involving two parties, called the holder and the writer, containing three fixed clauses. These three fixed clauses are: an asset A to be purchased, a maturity date T and a purchase price P. The European call option gives the holder the right, but not the obligation, to purchase the asset A for the price P on the date T; thereby, the term call option. It is clear then that the holder will decide to purchase the asset A at time T if the market price S of A at time T is larger than P. If that is the case and if transaction costs are ignored, the holder will realize a profit of S - P at maturity. If on the other hand, the market price S of A at time T is smaller than P, then the holder will decide to do nothing. In this case, the holder's profit will be zero. The call option has the effect of an insurance for the holder. The holder of the option can purchase the asset A on date T at the price P, from the writer of the option, or at a lower price from the marketplace.

The first question that arises is, what is the price of such a contract? In other words, how much should the holder pay the writer for entering into this contract which, after all, is a form of insurance for the holder ? But that's not all. At a later date t after the contract for the option is established, and before the date T, the holder may wish to sell the option to a third party. Thus, the holder will need to establish the price of the option at that later date t. That value of the option, let's call it V, then depends on the date t and also depends on the price s of the asset A. So V is a function of two independent variables, V = V(s,t).

Pausing briefly, we have introduced the following parameters and variables:
A, the tangible asset
T, the maturity time of the call option
P, the price to be paid for A on date T
t, the time variable with t < T
s, the market value of the asset A
S = s(T), the market value of the asset A at maturity T
V(s,t), the value of the call option for a given asset price s on a given date t

The value of the asset A fluctuates according to its nature and to the forces of the market. We know that a certificate of deposit is a pretty secure investment, meaning that its payoff at maturity can be predicted with precision. On the other hand, the future value of an investment in a commodity such as oil or coffee is much more unpredictable. In this case, the future value of the asset is a random function of time. Such functions are called stochastic processes and are the subject-matter of probability theory.

The original work of Black and Scholes makes certain assumptions on the fluctuation of the asset price s so that the time evolution of s can be modeled as the solution of a stochastic differential equation.

A stochastic differential equation (SDE) is an equation where the unknown is a stochastic process. The SDE sets the time rate of change of the unknown stochastic process as the superposition of two effects. The first effect is the result of the evolution of time. The second effect results from the random time evolution of a stochastic process called Brownian motion. The evolution of time that determines the first effect is always in the positive direction. The evolution of Brownian motion can be positive or negative, with certain probabilities, thus accounting for the possibility that asset prices may go up or down. SDE's generalize in some sense the better known ordinary differential equations. Ordinary differential equations contain only the first effect described above.

SDE's were introduced in two brilliant papers by Kiyosi Itô (Proceedings of the Imperial Academy of Tokyo, 1944 and 1946). Using the theory of SDE's, Black and Scholes determined explicitly a stochastic process model for the evolution of the price s of certain assets, and, further, developed a formula for the value V = V(s,t) of the European call option. This is a function of the random variable s and of the time variable t. Once again, the work of K. Itô supplies the needed chain rule to differentiate V as a function of these two variables. Differentiation of composition of real valued functions is studied in freshman calculus. Itô's rule is similar in spirit, but it is the differentiation rule for functions of random processes when the random processes are solutions of SDE's. Moreover, the rule looks like the ordinary chain rule studied in calculus, but with an additional summand called Itô's correction term. Using this rule Black and Scholes were able to deduce their formula for V. As a byproduct of their deduction, they obtained a ratio for the mix between options and assets so that the resulting mix is hedged against fluctuations in the market price of the asset.

Given the fundamental role played by Itô's theory in the development of the financial models it would have been fitting that Kiyosi Itô shared the 1997 Nobel Prize in Economics.


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