Anomalies in option pricing: the Black-Scholes model revisited

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

Because each record shows the actual S&P 500 at almost the same time as the option transaction, the MDR provides an excellent basis for estimating the theoretically correct option premium and evaluating its relationship to actual option premiums. There are, however, some minor problems with interpreting the MDR data as providing a trader's-eye view of option pricing. The transaction data are not entered into the CBOE computer at the exact moment of the trade. Instead, a ticket is filled out and then entered into the computer, and it is only at that time that the actual level of the S&P 500 is recorded. In short, the S&P 500 entries necessarily lag behind the option premium entries, so if the S&P 500 is rising (falling) rapidly, the reported value of the SPX will be above (below) the true value known to traders at the time of the transaction.

Test 1: An Implied Volatility Test

A key variable in the Black-Scholes model is the volatility of returns on the underlying asset, the SPX in our case. Investors are assumed to know the true standard deviation of the rate of return over the term of the option, and this information is embedded in the option premium. While the true volatility is an unobservable variable, the market's estimate of it can be inferred from option premiums. The Black-Scholes model assumes that this "implied volatility" is an optimal forecast of the volatility in SPX returns observed over the term of the option.

The calculation of an option's implied volatility is reasonably straightforward. Six variables are needed to compute the predicted premium on a call or put option using the Black-Scholes model. Five of these can be objectively measured within reasonable tolerance levels: the stock price (S), the strike price (X), the remaining life of the option (T - t), the riskless rate of interest over the remaining life of the option (r), typically measured by the rate of interest on U.S. Treasury securities that mature on the option's expiration date, and the dividend yield (q). The sixth variable, the "volatility" of the return on the stock price, denoted by [Sigma], is unobservable and must be estimated using numerical methods. Using reasonable values of all the known variables, the implied volatility of an option can be computed as the value of [Sigma] that makes the predicted Black-Scholes premium exactly equal to the actual premium. An example of the computation of the implied volatility on an option is shown in Box 3.

The Black-Scholes model assumes that investors know the volatility of the rate of return on the underlying asset, and that this volatility is measured by the (population) standard deviation. If so, an option's implied volatility should differ from the true volatility only because of random events. While these discrepancies might occur, they should be very short-lived and random: Informed investors will observe the discrepancy and engage in arbitrage, which quickly returns things to their normal relationships.

Figure 3 reports two measures of the volatility in the rate of return on the S&P 500 index for each trading day in the 1992-94 period.(10) The "actual" volatility is the ex post standard deviation of the daily change in the logarithm of the S&P 500 over a 60-day horizon, converted to a percentage at an annual rate. For example, for January 5, 1993 the standard deviation of the daily change in lnS&P500 was computed for the next 60 calendar days; this became the actual volatility for that day. Note that the actual volatility is the realization of one outcome from the entire probability distribution of the standard deviation of the rate of return. While no single realization will be equal to the "true" volatility, the actual volatility should equal the true volatility, "on average."