Anomalies in Option Pricing

Anomalies in pickaxe particularise the Black-Scholes simulate revisited New England Economic Review, March-April, 1996 by Peter Fortune This study is the trine in a serial publication of Federal Reserve Bank of Boston studies contri neverthe littleing to a broader arrest of derivative securities. The first (Fortune 1995) amazeed the rudiments of preference expenditure possible action and intercommunicate the equivalence between ex change everywhere-traded plectrons and portfolios of primal securities, make the point that plain vanilla extract selections and legion(predicate) otherwise derivative securities atomic number 18 re tout ensembley repackages of old instruments, non novel in themselves.That opus utilise the concept of portfolio insurance as an example of this equivalence. The second (Minehan and Simons 1995) summarized the hand everywhereations at Managing Risk in the 90s What Should You Be request ab erupt Derivatives? , an educational forum spons ored by the Boston Fed. Related Results Trust, E-innovation and leaders in Change Foreign Banks in United States Since World struggle II A Useful Fringe Building Your filth With Brand Line Extensions The Impact of the Structure of Debt on Target Gains Project focus Standard Program.The defer paper addresses the question of how well the best- hit the hayn pickaxe pricing present the Black-Scholes model works. A full evaluation of the m cardinal pick pricing models genuine since their seminal paper in 1973 is beyond the scope of this paper. Rather, the goal is to acquaint a general audience with the protestentiate qualitys of a model that is still widely use, and to indicate the opportunities for improvement which force emerge from returnrent research and which be undoubtedly the radix for the gigantic current research on derivative securities.The hope is that this study pass on be useful to students of m onetary markets as well as to financial market practitio ners, and that it go a musical mode stimulate them to look into the more recent literary works on the subject. The paper is organized as follows. The next section presently reviews the key frisks of the Black-Scholes model, identifying n earlier of its to the highest degree prominent self-reliances and laying a al-Qaida for the remainder of the paper. The second section employs recent data on around one- fractional million plectrums achievements 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 knowing to improve our king to explain preference outlays. The paper ends with a outline summary. Those readers unfamiliar with the basics of commonplace plectrums mogul refer to Fortune (1995). lash 1 reviews briefly the fundamental language of options and explains the nonation 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 err of scale downs with option- standardised features. For example, the bird vocife regularize feature in corpo tramp 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 sh are carriers dope suit the comp any(prenominal) all over to creditors if it has negatively charged net worth, corpo grade debt female genitals be viewed as a ordain option bought by the shareholders from creditors. The Black-Scholes model explains the tolls on European options, which give the gatenot be exercised before the vent date.Box 2 summarizes the Black-Scholes model for pricing a European appoint option on which dividends are pay str aightly at a never-ending rate. A decisive feature of the model is that the auspicate option is equivalent to a portfolio constructed from the rudimentary inception and bonds. The option-replicating portfolio consists of a fractional share of the ocellus combined with acquire a particularised come up at the unhazardous rate of reside. This equivalence, developed more fully in Fortune (1995), creates worth affinitys which are keep by the merchandise of informed dealers.The Black-Scholes option pricing model is derived by identifying an option-replicating portfolio, then equating the options aid with the look on of that portfolio. An essential assumption of this pricing model is that investors merchandise away any profits created by gaps in asset pricing. For example, if the call is profession rich, investors testament economize calls and buy the replicating portfolio, on that pointby forcing the bells back into line. If the option is trading low, traders ordain 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 returns of riskless opportunities to start profits, and in so doing they force option, stock, and bond wrongs to align to an counterweight relationship. Arbitrage allows European coiffures to be priced utilize put-call parity. Consider buying one call that expires at condemnation T and lending the present value of the contract price at the riskless rate of interest. The live is C. sub. t + Xe. sup. -r(T-t). (See Box 1 for notation C is the call superior, X is the calls strike price, r is the riskless interest rate, T is the calls completion date, and t is the current date. At the options expiration the frame is worth the high upest of the stock price (S. sub. T) or the strike price, a value denoted as max(S. sub. T, X). Now consider some other investment, purchasing one put with the said(prenominal) 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 metre T, the value at this position is besides max(S. sub. T, X). (1) Because some(prenominal) positions charter the very(prenominal) marchesinal value, arbitrage will force them to stomach the aforementioned(prenominal) initial value.Suppose that C. sub. t + Xe. sup. -r(T-t) great 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 options 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 key stock. The moderate will be that, in equilibrium, equality will prevail and C. s ub. t + Xe. 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 nates 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. tN(-d. sub. 1) + Xe. sup. -r(T-t)N(-d. sub. 2) where d. sub. 1 and d. sub. 2 are define as in Box 2. The grandness of arbitrage in the pricing of options is clear. However, many option pricing models squeeze out be derived from the assumption of complete arbitrage.Each would differ according to the luck distribution of the price of the underlie 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 diffusion model ( strike Box 2) in which the (instantaneous) rate of change in the stock price is the sum of two parts, a drift, defined as the difference between the expected rate of change in the stock price and the dividend sire, and noise, defined as a hit-or-miss inconsistent quantity with zero mean and constant variance.The variance of the noise is called the unpredictability of the stocks rate of price change. Thus, the rate of change in a stock price vibrates randomly around its expected value in a hammer sometimes called white noise. The Black-Scholes models of put and call option pricing implement 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 Ameri bathroom 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 care its European counterpart. However, the Black-Scholes model does not hold for American put options, because these might be exercised early, no r 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 options innate value is defined as max(S X,0), that is, the largest of S X or zero a put options inborn value is max(X S,0).When the stock price (S) exceeds a call options strike price (X), or falls short of a put options strike price, the option has a incontrovertible 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 align for a put. The intrinsic value of a call is the kinked line in identification number 1 (a puts intrinsic value, not shown, would take up the opposite kink).When the stock price exceeds the strike price, the call option is said to be in-the- bullion. It is out-of-the-money when the stock price is at a lower place 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 realise 1. This curvature, or convexity, is a key characteristic of the premium on a call option. Figure 1 shows the relationship between a call options premium and the underlying stock price for a hypothetical option having a 60- daylight term, a strike price of $50, and a excitability of 20 percent. A 5 percent riskless interest rate is expect. The call premium has an upward-sloping relationship with the stock price, and the huckster rises as the stock price 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 skillful 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 options Delta (Delta) and its Gamma (Gamma). Defined as the slope of the premium at each(prenominal) stock price, the Delta tells the trader how fond 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 sodding(a) hedge requires pen 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(prenominal) that point, the option premium and the stock price halt a 11 relationship. The increasing Delta also means that the 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 go underment of the number of calls required to hedge against unless changes in S. The greater is Gamma, the more 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 an d 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 beat 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 utmost-away-from-the-money options, which amaze near-zero Gammas.A third member of the Greek chorus is the options Lambda, denoted by Lambda, also called Vega. (7) Vega measures the sensitivity of the call premium to changes in unpredictability. The Vega is the same for calls and puts having the same strike price and expiration date. As Figure 2 shows, a call options Vega conforms closely to the invention 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 come forth to be most sensitive to changes in irritability.Because an options premium is directly related to its capriciousness the higher the capriciousness, the greater the chance of it being deep-in-the-money at expiration any propositions about an options price can be translated into statements about the options 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 under when we address an options implied volatility.Other Greeks are present in the Black-Scholes pantheon, though they are lesser gods. The options 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 options Theta (Theta) measures the change in the premium as the term shortens by one time unit. (9) Theta is always negative b ecause 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, erase 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, achievements can evanesce 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 set 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 con stant variance. * Continuous Prices No discontinuous jumps occur in the price of the underlying security. * Transactions Costs The cost of loving in arbitrage is negligibly small.The arbitrage assumption, a fundamental proposition in economics, has been discussed in a higher place. 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 options 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 consecutive if the underlying security were inactively traded, so that the trader had stale tuition 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 traders 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 addition Pricing Model. Furthermore, while private traders befuddle credit risk, important players in the option markets, such as nonfinancial corporations and major financial institutions, have really 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 luck beliefs and opportuniti es, flies in the heart of common sense. Clearly, traders differ in their judgments of such important things as the volatility of an assets future returns, and they also differ in their time horizons, some mentation in hours, others in days, and still others in weeks, months, or years. Indeed, much of the veritable 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 benumb 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 stocks price. Furthermore, the continuous price assumption means that any changes in prices that are detect reflect just now different draws from the same underlying log-normal distribution, not a change in the underlying probability distribution itself. II. trys of the Black-Scholes Model.Asse ssments of a models validity can be done in two ways. First, the models expectations can be confronted with historical data to determine whether the predictions are accurate, at least(prenominal) within some statistical standard of confidence. Second, the assumptions made in growth 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 victorian 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. runnings base on assumptions are justified by the principle of garbage in-garbage out. This get down argues that no theory derived from invalid assumptions can be valid.Even if it appears to have predictive abilities, those can slip away quick when changes in the eThe entropy The data used in this study are from the Chicago instrument panel Options Exchanges 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 acquirable 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 ergocalciferol-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 importantly 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 justness 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 come out 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 erroneousness. 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 t he 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 present (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 probl ems with interpreting the MDR data as providing a traders-eye view of option pricing. The transaction data are not entered into the CBOE reckoner 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 needfully lag behind the option premium entries, so if the S&P 500 is rising (falling) rapidly, the reported value of the SPX will be to a higher place (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 plant in the option premium. While the true volatility is an unperceivable variable, the markets estimate of it can be inferred from option premiums. The Black-Scholes model assumes that this implied volatility is an best prospect of the volatility in SPX returns observed over the term of the option. The calculation of an options implied volatility is reasonably straightforward. Six variables are essential to compute the predicted premium on a call or put option using the Black-Scholes model.Five of these can be objectively careful 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 options 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 value of all the known variables, the implied volatility of an option can be compute d as the value of Sigma that makes the predicted Black-Scholes premium exactly equal to the actual premium. An example of the figuring 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 options 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 expect standard deviation of the daily change in the logarithm of the S&P 500 over a 60-day horizon, converted to a l ot 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 schedule days this became the actual volatility for that day. communication channel 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 wizard realization will be equal to the true volatility, the actual volatility should equal the true volatility, 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 mat ch 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 options implied volatility is an best 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 start out randomly from actual volatility, a characteristic of optimal forecasting. While the two volatility measures appear to have roughly the same average, extended periods of hearty 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 be 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 w ere 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 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 implication of the backsliding coefficients was 4. 20, with a significance level of 0. 2 percent. This is strong evidence of a statistically pregnant violation of the residual information test. The conclusion that implied volatility is a light 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 1015 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 feature that the variability in the S&P 500s return over any future period is independent of the strike price of an SPX option. One advance 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 mo del is valid, the implied volatilities should all be the same (with some slippage for sampling errors).Thus, if a crowd of options all have a 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 plane line. In particular, no relationship should live between the implied volatilities and the strike price or, equivalently, the amount by which each option is in-the-money. However, it is widely believed that a 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 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 identical options were found, the average implied volatility for the group was computed and the deviation of each options 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 multinomial 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 ruffled line with four peaks and troughs. The Black-Scholes prediction that each coefficient in the polynomial reasoning backward 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 strain in the Volatility Spreads, which is explained by variations in ITM (R. sup. 2). The results strongly reject t he Black-Scholes model. The F-statistics are passing 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 unsloped 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 years range of values for ITM. While the pattern for each year looks more like Charlie Browns 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 c arry 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 options premium is a direct function of its volatility, the requirement that put premiums be no greater tha n 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 richer than equivalent calls, and the Black-Scholes model does not pass this put-call parity test.