Anomalies in option pricing: the Black-Scholes model revisited

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

Our analysis takes an agnostic position on this methodological debate, looking at both predictions and assumptions of the Black-Scholes model.

The Data

The data used in this study are from the Chicago Board Options Exchange's Market Data Retrieval System. The MDR reports the number of contracts traded, the time of the transaction, the premium paid, the characteristics of the option (put or call, expiration date, strike price), and the price of the underlying stock at its last trade. This information is available for each option listed on the CBOE, providing as close to a real-time record of transactions as can be found. While our analysis uses only records of actual transactions, the MDR also reports the same information for every request of a quote. Quote records differ from the transaction records only in that they show both the bid and asked premiums and have a zero number of contracts traded.

The data used are for the 1992-94 period. We selected the MDR data for the S&P 500-stock index (SPX) for several reasons. First, the SPX options contract is the only European-style stock index option traded on the CBOE. All options on individual stocks and on other indices (for example, the S&P 100 index, the Major Market Index, the NASDAQ 100 index) are American options for which the Black-Scholes model would not apply. The ability to focus on a European-style option has several advantages. By allowing us to ignore the potential influence of early exercise, a possibility that significantly affects the premiums on American options on dividend-paying stocks as well as the premiums on deep-in-the-money American put options, we can focus on options for which the Black-Scholes model was designed. In addition, our interest is not in individual stocks and their options, but in the predictive power of the Black-Scholes option pricing model. Thus, an index option allows us to make broader generalizations about model performance than would a select set of equity options. Finally, the S&P 500 index options trade in a very active market, while options on many individual stocks and on some other indices are thinly traded.

The full MDR data set for the SPX over the roughly 758 trading days in the 1992-94 period consisted of more than 100 million records. In order to bring this down to a manageable size, we eliminated all records that were requests for quotes, selecting only records reflecting actual transactions. Some of these transaction records were cancellations of previous trades, for example, trades made in error. If a trade was canceled, we included the records of the original transaction because they represented market conditions at the time of the trade, and because there is no way to determine precisely which transaction was being canceled. We eliminated cancellations because they record the S&P 500 at the time of the cancellation, not the time of the original trade. Thus, cancellation records will contain stale prices.

This screening created a data set with over 726,000 records. In order to complete the data required for each transaction, the bond-equivalent yield (average of bid and asked prices) on the Treasury bill with maturity closest to the expiration date of the option was used as a riskless interest rate. These data were available for 180-day terms or less, so we excluded options with a term longer than 180 days, leaving over 486,000 usable records having both CBOE and Treasury bill data. For each of these, we assigned a dividend yield based on the S&P 500 dividend yield in the month of the option trade.