Anomalies in Option Pricing

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 â€Å"Managing Risk in the '90s: What Should You Be Asking about Derivatives? â€Å", an educational forum sponsored by the Boston Fed. Related Results Trust, E-innovation and Leadership in Change Foreign Banks in United States Since World War II: A Useful Fringe Building Your Brand With Brand Line Extensions The Impact of the Structure of Debt on Target Gains Project Management Standard Program. 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 odel 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 â€Å"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 â€Å"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. Suppose that [C. sub. t] + X[e. sup. -r(T-t)] [greater than] [P. sub. t] + [e. sup. -q(T-t)][S. sub. t], for example. In this case, the cost of the first position exceeds the cost of the second, but both must be worth the same at the option's expiration. The first position is overpriced relative to the second, and shrewd investors will go short the first and long the second; that is, they will write calls and sell bonds (borrow), while simultaneously buying both puts and the underlying stock. The result will be that, in equilibrium, equality will prevail and [C. sub. t] + X[e. sup. r(T-t)] = [P. sub. t] + [e. sup. -q(T-t)][S. sub. t]. Thus, arbitrage will force a parity between premiums of put and call options. Using this put-call parity, it can be shown that the premium for a European put option paying a continuous dividend at q percent of the stock price is: [P. sub. t] = -[e. sup. -q(T-t)][S. sub. t]N(-[d. sub. 1]) + X[e. sup. -r(T-t)]N(-[d. sub. 2]) where [d. sub. 1] and [d. sub. 2] are defined as in Box 2. The importance of arbitrage in the pricing of options is clear. However, many option pricing models can be derived from the assumption of complete arbitrage. Each would differ according to the probability distribution of the price of the underlying asset. What makes the Black-Scholes model unique is that it assumes that stock prices are log-normally distributed, that is, that the logarithm of the stock price is normally distributed. This is often expressed in a â€Å"diffusion model† (see Box 2) in which the (instantaneous) rate of change in the stock price is the sum of two parts, a â€Å"drift,† defined as the difference between the expected rate of change in the stock price and the dividend yield, and â€Å"noise,† defined as a random variable with zero mean and constant variance. The variance of the noise is called the â€Å"volatility† of the stock's rate of price change. Thus, the rate of change in a stock price vibrates randomly around its expected value in a fashion sometimes called â€Å"white noise. † The Black-Scholes models of put and call option pricing apply directly to European options as long as a continuous dividend is paid at a constant rate. If no dividends are paid, the models also apply to American call options, which can be exercised at any time. In this case, it can be shown that there is no incentive for early exercise, hence the American call option must trade like its European counterpart. However, the Black-Scholes model does not hold for American put options, because these might be exercised early, nor does it apply to any American option (put or call) when a dividend is paid. (2) Our empirical analysis will sidestep those problems by focusing on European-style options, which cannot be exercised early. A call option's intrinsic value is defined as max(S – X,0), that is, the largest of S – X or zero; a put option's intrinsic value is max(X – S,0). When the stock price (S) exceeds a call option's strike price (X), or falls short of a put option's strike price, the option has a positive intrinsic value because if it could be immediately exercised, the holder would receive a gain of S – X for a call, or X – S for a put. However, if S [less than] X, the holder of a call will not exercise the option and it has no intrinsic value; if X [greater than] S this will be true for a put. The intrinsic value of a call is the kinked line in Figure 1 (a put's intrinsic value, not shown, would have the opposite kink). When the stock price exceeds the strike price, the call option is said to be in-the-money. It is out-of-the-money when the stock price is below the strike price. Thus, the kinked line, or intrinsic value, is the income from immediately exercising the option: When the option is out-of-the-money, its intrinsic value is zero, and when it is in the money, the intrinsic value is the amount by which S exceeds X. Convexity, the Call Premium, and the Greek Chorus The premium, or price paid for the option, is shown by the curved line in Figure 1. This curvature, or â€Å"convexity,† is a key characteristic of the premium on a call option. Figure 1 shows the relationship between a call option's premium and the underlying stock price for a hypothetical option having a 60-day term, a strike price of $50, and a volatility of 20 percent. A 5 percent riskless interest rate is assumed. The call premium has an upward-sloping relationship with the stock price, and the slope rises as the stock p rice rises. This means that the sensitivity of the call premium to changes in the stock price is not constant and that the option-replicating portfolio changes with the stock price. The convexity of option premiums gives rise to a number of technical concepts which describe the response of the premium to changes in the variables and parameters of the model. For example, the relationship between the premium and the stock price is captured by the option's Delta ([Delta]) and its Gamma ([Gamma]). Defined as the slope of the premium at each stock price, the Delta tells the trader how sensitive the option price is to a change in the stock price. (3) It also tells the trader the value of the hedging ratio. (4) For each share of stock held, a perfect hedge requires writing 1/[[Delta]. ub. c] call options or buying 1/[[Delta]. sub. p] puts. Figure 2 shows the Delta for our hypothetical call option as a function of the stock price. As S increases, the value of Delta rises until it reaches its maximum at a stock price of about $60, or $10 in-the-money. After that point, the option premium and the stock price have a 1:1 relationship. The increasing Delta also means that th e hedging ratio falls as the stock price rises. At higher stock prices, fewer call options need to be written to insulate the investor from changes in the stock price. The Gamma is the change in the Delta when the stock price changes. (5) Gamma is positive for calls and negative for puts. The Gamma tells the trader how much the hedging ratio changes if the stock price changes. If Gamma is zero, Delta would be independent of S and changes in S would not require adjustment of the number of calls required to hedge against further changes in S. The greater is Gamma, the more â€Å"out-of-line† a hedge becomes when the stock price changes, and the more frequently the trader must adjust the hedge. Figure 2 shows the value of Gamma as a function of the amount by which our hypothetical call option is in-the-money. (6) Gamma is almost zero for deep-in-the-money and deep-out-of-the-money options, but it reaches a peak for near-the-money options. In short, traders holding near-the-money options will have to adjust their hedges frequently and sizably as the stock price vibrates. If traders want to go on long vacations without changing their hedges, they should focus on far-away-from-the-money options, which have near-zero Gammas. A third member of the Greek chorus is the option's Lambda, denoted by [Lambda], also called Vega. (7) Vega measures the sensitivity of the call premium to changes in volatility. The Vega is the same for calls and puts having the same strike price and expiration date. As Figure 2 shows, a call option's Vega conforms closely to the pattern of its Gamma, peaking for near-the-money options and falling to zero for deep-out or deep-in options. Thus, near-the-money options appear to be most sensitive to changes in volatility. Because an option's premium is directly related to its volatility – the higher the volatility, the greater the chance of it being deep-in-the-money at expiration – any propositions about an option's price can be translated into statements about the option's volatility, and vice versa. For example, other things equal, a high volatility is synonymous with a high option premium for both puts and calls. Thus, in many contexts we can use volatility and premium interchangeably. We will use this result below when we address an option's implied volatility. Other Greeks are present in the Black-Scholes pantheon, though they are lesser gods. The option's Rho ([Rho]) is the sensitivity of the call premium to changes in the riskless interest rate. (8) Rho is always positive for a call (negative for a put) because a rise in the interest rate reduces the present value of the strike price paid (or received) at expiration if the option is exercised. The option's Theta ([Theta]) measures the change in the premium as the term shortens by one time unit. (9) Theta is always negative because an option is less valuable the shorter the time remaining. The Black-Scholes Assumptions The assumptions underlying the Black-Scholes model are few, but strong. They are: * Arbitrage: Traders can, and will, eliminate any arbitrage profits by simultaneously buying (or writing) options and writing (or buying) the option-replicating portfolio whenever profitable opportunities appear. * Continuous Trading: Trading in both the option and the underlying security is continuous in time, that is, transactions can occur simultaneously in related markets at any instant. * Leverage: Traders can borrow or lend in unlimited amounts at the riskless rate of interest. Homogeneity: Traders agree on the values of the relevant parameters, for example, on the riskless rate of interest and on the volatility of the returns on the underlying security. * Distribution: The price of the underlying security is log-normally distributed with statistically independent price changes, and with constant mean and constant variance. * Continuous Prices: No discontinuous jumps occur in the price of the underlying security. * Transactions Costs: The cost of engaging in arbitrage is negligibly small. The arbitrage assumption, a fundamental proposition in economics, has been discussed above. The continuous trading assumption ensures that at all times traders can establish hedges by simultaneously trading in options and in the underlying portfolio. This is important because the Black-Scholes model derives its power from the assumption that at any instant, arbitrage will force an option's premium to be equal to the value of the replicating portfolio. This cannot be done if trading occurs in one market while trading in related markets is barred or delayed. For example, during a halt in trading of the underlying security one would not expect option premiums to conform to the Black-Scholes model. This would also be true if the underlying security were inactively traded, so that the trader had â€Å"stale† information on its price when contemplating an options transaction. 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 â€Å"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 â€Å"the proof is in the pudding. It is argued that any theory requires assumptions that might be judged â€Å"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 â€Å"garbag e 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 eThe 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. nvironment make the invalid assumptions more pivotal. 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. 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 â€Å"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 â€Å"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 â€Å"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 â€Å"true† volatility, the actual volatility should equal the true volatility, â€Å"on average. † The second measure of volatility is the implied volatility. This was constructed as follows, using the data described above. For each trading day, the implied volatility on call options meeting two criteria was computed. The criteria were that the option had 45 to 75 calendar days to expiration (the average was 61 days) and that it be near the money (defined as a spread between S&P 500 and strike price no more than 2. 5 percent of the S&P 500). The first criterion was adopted to match the term of the implied volatility with the 60-day term of the actual volatility. The second criterion was chosen because, as we shall see later, near-the-money options are most likely to conform to Black-Scholes predictions. The Black-Scholes model assumes that an option's implied volatility is an optimal forecast of the volatility in SPX returns observed over the term of the option. Figure 3 does not provide visual support for the idea that implied volatilities deviate randomly from actual volatility, a characteristic of optimal forecasting. While the two volatility measures appear to have roughly the same average, extended periods of significant differences are seen. For example, in the last half of 1992 the implied volatility remained well above the actual volatility, and after the two came together in the first half of 1993, they once again diverged for an extended period. It is clear from this visual record that implied volatility does not track actual volatility well. However, this does not mean that implied volatility provides an inferior forecast of actual volatility: It could be that implied volatility satisfies all the scientific requirements of a good forecast in the sense that no other forecasts of actual volatility are better. In order to pursue the question of the informational content of implied volatility, several simple tests of the hypothesis that implied volatility is an optimal forecast of actual volatility can be applied. One characteristic of an optimal forecast is that the forecast should be unbiased, that is, the forecast error (actual volatility less implied volatility) should have a zero mean. The average forecast error for the data shown in Figure 3 is -0. 7283, with a t-statistic of -8. 22. This indicates that implied volatility is a biased forecast of actual volatility. A second characteristic of an optimal forecast is that the forecast error should not depend on any information available at the time the forecast is made. If information were available that would improve the forecast, the forecaster should have already included it in making his forecast. Any remaining forecasting errors should be random and uncorrelated with information available before the day of the forecast. To implement this â€Å"residual information test,† the forecast error was regressed on the lagged values of the S&P 500 in the three days prior to the forecast. 11) The F-statistic for the significance of the regression coefficients was 4. 20, with a significance level of 0. 2 percent. This is strong evidence of a statistically significant violation of the residual information test. The conclusion that implied volatility is a poor forecast of actual volatility has been reached in several other studies using different methods and data. For example, Canina and Figlewski (1993), using data for the S&P 100 in the years 1983 to 1987, found that implied volatility had almost no informational content as a prediction of actual volatility. However, a recent review of the literature on implied volatility (Mayhew 1995) mentions a number of papers that give more support for the forecasting ability of implied volatility. Test 2: The Smile Test One of the predictions of the Black-Scholes model is that at any moment all SPX options that differ only in the strike price (having the same term to expiration) should have the same implied volatility. For example, suppose that at 10:15 a. m. on November 3, transactions occur in several SPX call options that differ only in the strike price. Because each of the options is for the same interval of time, the value of volatility embedded in the option premiums should be the same. This is a natural consequence of the fact that the variability in the S&P 500's return over any future period is independent of the strike price of an SPX option. One approach to testing this is to calculate the implied volatilities on a set of options identical in all respects except the strike price. If the Black-Scholes model is valid, the implied volatilities should all be the same (with some slippage for sampling errors). Thus, if a group of options all have a â€Å"true† volatility of, say, 12 percent, we should find that the implied volatilities differ from the true level only because of random errors. Possible reasons for these errors are temporary deviations of premiums from equilibrium levels, or a lag in the reporting of the trade so that the value of the SPX at the time stamp is not the value at the time of the trade, or that two options might have the same time stamp but one was delayed more than the other in getting into the computer. This means that a graph of the implied volatilities against any economic variable should show a flat line. In particular, no relationship should exist between the implied volatilities and the strike price or, equivalently, the amount by which each option is â€Å"in-the-money. † However, it is widely believed that a â€Å"smile† is present in option prices, that is, options far out of the money or far in the money have higher implied volatilities than near-the-money options. Stated differently, deep-out and far-in options trade â€Å"rich† (overpriced) relative to near-the-money options. If true, this would make a graph of the implied volatilities against the value by which the option is in-the-money look like a smile: high implied volatilities at the extremes and lower volatilities in the middle. In order to test this hypothesis, our MDR data were screened for each day to identify any options that have the same characteristics but different strike [TABULAR DATA FOR TABLE 1 OMITTED] prices. If 10 or more of these â€Å"identical† options were found, the average implied volatility for the group was computed and the deviation of each option's implied volatility from its group average, the Volatility Spread, was computed. For each of these options, the amount by which it is in-the-money was computed, creating a variable called ITM (an acronym for in-the-money). ITM is the amount by which an option is in-the-money. It is negative when the option is out-of-the-money. ITM is measured relative to the S&P 500 index level, so it is expressed as a percentage of the S&P 500. The Volatility Spread was then regressed against a fifth-order polynomial equation in ITM. This allows for a variety of shapes of the relationship between the two variables, ranging from a flat line if Black-Scholes is valid (that is, if all coefficients are zero), through a wavy line with four peaks and troughs. The Black-Scholes prediction that each coefficient in the polynomial regression is zero, leading to a flat line, can be tested by the F-statistic for the regression. The results are reported in Table 1, which shows the F-statistic for the hypothesis that all coefficients of the fifth-degree polynomial are jointly zero. Also reported is the proportion of the variation in the Volatility Spreads, which is explained by variations in ITM ([R. sup. 2]). The results strongly reject the Black-Scholes model. The F-statistics are extremely high, indicating virtually no chance that the value of ITM is irrelevant to the explanation of implied volatilities. The values of [R. sup. 2] are also high, indicating that ITM explains about 40 to 60 percent of the variation in the Volatility Spread. Figure 4 shows, for call options only, the pattern of the relationship between the Volatility Spread and the amount by which an option is in-the-money. The vertical axis, labeled Volatility Spread, is the deviation of the implied volatility predicted by the polynomial regression from the group mean of implied volatilities for all options trading on the same day with the same expiration date. For each year the pattern is shown throughout that year's range of values for ITM. While the pattern for each year looks more like Charlie Brown's smile than the standard smile, it is clear that there is a smile in the implied volatilities: Options that are further in or out of the money appear to carry higher volatilities than slightly out-of-the-money options. The pattern for extreme values of ITM is more mixed. Test 3: A Put-Call Parity Test Another prediction of the Black-Scholes model is that put options and call options identical in all other respects should have the same implied volatilities and should trade at the same premium. This is a consequence of the arbitrage that enforces put-call parity. Recall that put-call parity implies [P. sub. t] + [e. sup. -q(T – t)][S. sub. t] = [C. sub. t] + [Xe. sup. -r(T – t)]. A put and a call, having identical strike prices and terms, should have equal premiums if they are just at-the-money in a present value sense. If, as this paper does, we interpret at-the-money in current dollars rather than present value (that is, as S = X rather than S = [Xe. sup. -r(t – q)(T – t)]), at-the-money puts should have a premium slightly below calls. Because an option's premium is a direct function of its volatility, the requirement that put premiums be no greater than call premiums for equivalent at-the-money options implies that implied volatilities for puts be no greater than for calls. For each trading day in the 1992-94 period, the difference between implied volatilities for at-the-money puts and calls having the same expiration dates was computed, using the [+ or -]2. 5 percent criterion used above. (12) Figure 5 shows this difference. While puts sometimes have implied volatility less than calls, the norm is for higher implied volatilities for puts. Thus, puts tend to trade â€Å"richer† than equivalent calls, and the Black-Scholes model does not pass this put-call parity test.