Anomalies in option pricing: the Black-Scholes model revisited

New England Economic Review, March-April, 1996 by Peter Fortune

This study is the third in a series of Federal Reserve Bank of Boston studies contributing to a broader understanding of derivative securities. The first (Fortune 1995) presented the rudiments of option pricing theory and addressed the equivalence between exchange-traded options and portfolios of underlying securities, making the point that plain vanilla options - and many other derivative securities - are really repackages of old instruments, not novel in themselves. That paper used the concept of portfolio insurance as an example of this equivalence. The second (Minehan and Simons 1995) summarized the presentations at "Managing Risk in the '90s: What Should You Be Asking about Derivatives?", an educational forum sponsored by the Boston Fed.

The present paper addresses the question of how well the best-known option pricing model - the Black-Scholes model - works. A full evaluation of the many option pricing models developed since their seminal paper in 1973 is beyond the scope of this paper. Rather, the goal is to acquaint a general audience with the key characteristics of a model that is still widely used, and to indicate the opportunities for improvement which might emerge from current research and which are undoubtedly the basis for the considerable current research on derivative securities. The hope is that this study will be useful to students of financial markets as well as to financial market practitioners, and that it will stimulate them to look into the more recent literature on the subject.

The paper is organized as follows. The next section briefly reviews the key features of the Black-Scholes model, identifying some of its most prominent assumptions and laying a foundation for the remainder of the paper. The second section employs recent data on almost one-half million options transactions to evaluate the Black-Scholes model. The third section discusses some of the reasons why the Black-Scholes model falls short and assesses some recent research designed to improve our ability to explain option prices. The paper ends with a brief summary. Those readers unfamiliar with the basics of stock options might refer to Fortune (1995). Box 1 reviews briefly the fundamental language of options and explains the notation used in the paper.

I. The Black-Scholes Model

In 1973, Myron Scholes and the late Fischer Black published their seminal paper on option pricing (Black and Scholes 1973). The Black-Scholes model revolutionized financial economics in several ways. First, it contributed to our understanding of a wide range of contracts with option-like features. For example, the call feature in corporate and municipal bonds is clearly an option, as is the refinancing privilege in mortgages. Second, it allowed us to revise our understanding of traditional financial instruments. For example, because shareholders can turn the company over to creditors if it has negative net worth, corporate debt can be viewed as a put option bought by the shareholders from creditors.

The Black-Scholes model explains the prices on European options, which cannot be exercised before the expiration date. Box 2 summarizes the Black-Scholes model for pricing a European call option on which dividends are paid continuously at a constant rate. A crucial feature of the model is that the call option is equivalent to a portfolio constructed from the underlying stock and bonds. The "option-replicating portfolio" consists of a fractional share of the stock combined with borrowing a specific amount at the riskless rate of interest. This equivalence, developed more fully in Fortune (1995), creates price relationships which are maintained by the arbitrage of informed traders. The Black-Scholes option pricing model is derived by identifying an option-replicating portfolio, then equating the option's premium with the value of that portfolio.

An essential assumption of this pricing model is that investors arbitrage away any profits created by gaps in asset pricing. For example, if the call is trading "rich," investors will write calls and buy the replicating portfolio, thereby forcing the prices back into line. If the option is trading low, traders will buy the option and short the option-replicating portfolio (that is, sell stocks and buy bonds in the correct proportions). By doing so, traders take advantage of riskless opportunities to make profits, and in so doing they force option, stock, and bond prices to conform to an equilibrium relationship.

Arbitrage allows European puts to be priced using put-call parity. Consider purchasing one call that expires at time T and lending the present value of the strike price at the riskless rate of interest. The cost is [C.sub.t] X[e.sup.-r(T-t)]. (See Box 1 for notation: C is the call premium, X is the call's strike price, r is the riskless interest rate, T is the call's expiration date, and t is the current date.) At the option's expiration the position is worth the highest of the stock price ([S.sub.T]) or the strike price, a value denoted as max([S.sub.T], X). Now consider another investment, purchasing one put with the same strike price as the call, plus buying the fraction [e.sup.-q(T-t)] of one share of the stock. Denoting the put premium by P and the stock price by S, then the cost of this is [P.sub.t] [e.sup.-q(T-t)][S.sub.t], and, at time T, the value at this position is also max([S.sub.T], X).(1) Because both positions have the same terminal value, arbitrage will force them to have the same initial value.