Anomalies in option pricing: the Black-Scholes model revisited

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

The leverage assumption allows the riskless interest rate to be used in options pricing without reference to a trader's financial position, that is, to whether and how much he is borrowing or lending. Clearly this is an assumption adopted for convenience and is not strictly true. However, it is not clear how one would proceed if the rate on loans was related to traders' financial choices. This assumption is common to finance theory: For example, it is one of the assumptions of the Capital Asset Pricing Model. Furthermore, while private traders have credit risk, important players in the option markets, such as nonfinancial corporations and major financial institutions, have very low credit risk over the lifetime of most options (a year or less), suggesting that departures from this assumption might not be very important.

The homogeneity assumption, that traders share the same probability beliefs and opportunities, flies in the face of common sense. Clearly, traders differ in their judgments of such important things as the volatility of an asset's future returns, and they also differ in their time horizons, some thinking in hours, others in days, and still others in weeks, months, or years. Indeed, much of the actual trading that occurs must be due to differences in these judgments, for otherwise there would be no disagreements with "the market" and financial markets would be pretty dull and uninteresting.

The distribution assumption is that stock prices are generated by a specific statistical process, called a diffusion process, which leads to a normal distribution of the logarithm of the stock's price. Furthermore, the continuous price assumption means that any changes in prices that are observed reflect only different draws from the same underlying log-normal distribution, not a change in the underlying probability distribution itself.

II. Tests of the Black-Scholes Model

Assessments of a model's validity can be done in two ways. First, the model's predictions can be confronted with historical data to determine whether the predictions are accurate, at least within some statistical standard of confidence. Second, the assumptions made in developing the model can be assessed to determine if they are consistent with observed behavior or historical data.

A long tradition in economics focuses on the first type of tests, arguing that "the proof is in the pudding." It is argued that any theory requires assumptions that might be judged "unrealistic," and that if we focus on the assumptions, we can end up with no foundations for deriving the generalizations that make theories useful. The only proper test of a theory lies in its predictive ability: The theory that consistently predicts best is the best theory, regardless of the assumptions required to generate the theory.

Tests based on assumptions are justified by the principle of "garbage in-garbage out." This approach argues that no theory derived from invalid assumptions can be valid. Even if it appears to have predictive abilities, those can slip away quickly when changes in the environment make the invalid assumptions more pivotal.