The impact of equity option expirations on the prices of non-expiring options.
The stock market crash of 1987 generated considerable debate over the impact of derivative security trading on the prices and volatilities of the underlying securities. The debate specifically focused on stock price behavior around the expiration date of index derivatives. In this paper we approach the issue of the impact of derivative security expiration from a different perspective. Rather than evaluate the effect of expiration on the underlying security, we analyze the impact of expiration on non-expiring options. This is an interesting issue because, just as options can be used in conjunction with the underlying asset for hedging, speculation and arbitrage purposes, option combination strategies can be employed for these purposes as well. In addition, just as there are linkages between the prices of options and the underlying security, there are linkages among the prices of options on the same underlying security.
Research on the effect of option expiration on non-expiring options is relatively scarce. Day and Lewis (1988) examine the implied volatility of call options on the Major Market Index, the New York Stock Exchange Composite Index and the Standard and Poor's 100 Index in the periods around both quarterly expirations of stock index futures contracts and monthly expirations of stock index options. They find that implied volatilities increase at both types of expirations, reflecting the market's expectation that these are periods of elevated volatility in the underlying stock indexes.
This study examines equity options listed on the Chicago Board Options Exchange (CBOE) over the period 1983 through 1985. Stocks with options listed on the CBOE are classified into one of three expiration cycles. These cycles are (1) January, April, July, and October; (2) February, May, August, and November; and (3) March, June, September, and December. The cycles are typically referred to by the first month in the cycle (e.g., the January cycle). Prior to late 1984, option expirations on a particular underlying stock occurred only in the months of the cycle to which the stock was assigned. As an example, because IBM is assigned to the January cycle, options on IBM expired only in January, April, July, and October. This system is referred to as a quarterly expiration regime.
In response to market demand for short maturity options, exchanges began modification of expiration cycles in late 1984. Under the new system, option expirations on a given underlying stock are available in the current month, the following month and the next two months of the expiration cycle to which the stock is assigned. With this change, there are expiring options on every underlying stock every month.
Our primary objective is to analyze the impact that option expiration has on non-expiring options. In order to do this, it is desirable to obtain contemporaneous observations of non-expiring options for which there is a nearby expiration event and non-expiring options for which there is no nearby expiration event. Our study covers 1983 through 1985, the most recent period during which quarterly expirations were the dominant regime. This period was specifically chosen so that we would have contemporaneous observations on groups of non-expiring options that differed only by the incidence of expiration of other options on the same underlying stock. It would not be possible to do this under the current system because some of the options on every underlying stock expire every month. As an example, consider the underlying stocks IBM and GM. IBM is in the January cycle and GM is in the March cycle. In January 1984, some of the IBM options expired, while no options on GM expired. We examine non-expiring options on both of these underlying stocks in the period around the January 1984 expiration date and conduct tests to determine whether there is any differential price behavior. The IBM options are referred to as 'non-expiring options in the current expiration cycle,' while the GM options are referred to as 'non-expiring options not in the current expiration cycle.' Note that under the new system of option expirations, some options on both IBM and GM expire every month. Our primary tests examine the behavior of non-expiring equity call option prices in the two week period centered around option expiration dates. Unexpected percentage returns are compared for two sets of non-expiring options: those that are in the current expiration cycle and those that are not. The absolute value of these percentage returns is used as a measure of the pricing efficiency of the options. We then conduct tests to determine whether the expiration event itself affects the efficiency of call option prices. The results indicate that the pricing efficiency of non-expiring options is greater for those in the current expiration cycle.
We can identify several possible impacts of option expiration. First, there is the potential for increased demand for non-expiring options on the underlying stock for which some options are expiring as both hedgers and speculators roll over their positions. Such behavior might he expected to increase the prices of the non-expiring options. Alternatively, call option expiration could result in a decrease in option prices if writers act to replace expired positions. The net impact of these effects will depend on the heterogeneity of the expectations of the holders and writers of the expiring options, as well as the extent to which the holding period of buyers as Opposed to writers terminates on the expiration days. Any impacts associated with the rolling over of positions would likely be observed in the period immediately prior to and including the expiration day.
Another potential impact is for heightened pricing efficiency for non-expiring options in the current expiration cycle throughout the period surrounding option expiration. Because short term options tend to have the largest open interest, we expect that these options reflect relatively greater amounts of information. As trading activity shifts to non-expiring options, there may he a transmission or 'spillover' of information into non-expiring option prices.
Finally, for non-expiring options in the current expiration cycle, there are likely to be lower potential costs and risks of arbitraging discrepancies in relative option prices. As an example, consider the case in which a near-to-expiration option is deemed to be mispriced relative to an otherwise identical call option with a longer time to expiration. To profit from the apparent mispricing, the arbitrageur may construct a ratio spread, buying the underpriced call and writing the overpriced call in quantities that are proportional to the options' deltas. At a maximum, the position can be held until expiration of the shorter maturity option, and the spread is profitable only if the options are correctly priced by the closest expiration date. The arbitrageur bears the cost of financing the position and the risk that by expiration the perceived relative mispricing has increased. In the case where one of the options is near expiration, the trading cost represented by the bid-ask spread is likely to be low, due to the relatively high trading volume in short maturity options. In addition, the potential carrying cost and risk of this position are less than those for a ratio spread involving two long-term call options because the greater amount of time until expiration in the latter case creates the potential for a longer period over which the position must be financed and a greater likelihood that the mispricing will become exaggerated. Thus, an additional argument for greater pricing efficiency in the prices of non-expiring options in the current expiration cycle is based on the possibility of lower costs and risks of arbitraging relative option mispricing when one of the options is approaching expiration.
Sample and Methodology
We analyze call options traded on the CBOE over the period January 1983 to December 1985. Option and stock price data are obtained from the consolidated format of the Berkeley Options Database. The database consists of high and low option prices for each period during the day that the stock price remained constant, as well as option trading volume at the low and high prices and total trading volume during the constant stock price interval. The event window is defined as the week of option expiration and the following week. For every non-expiring common stock call option traded during this period, we obtain option and associated stock prices for the last constant stock price interval of each day. We compute a weighted average call price according to a procedure proposed by Rubinstein (1985):
C = [P.sub.L] ([V.sub.L]/[V.sub.T]) + [P.sub.H] ([V.sub.H]/[V.sub.T]) + [P.sub.L] + [P.sub.H]/2 ([V.sub.T] - [V.sub.L] - [V.sub.H]/[V.sub.T]), (1)
where: C = weighted average call option price
P = option price
V = option volume
L = option low
H = option high
T = option total
The call price computed according to Equation 1 is used as the observed call price in our tests. For each observation, the Black and Scholes (1973) formula with dividend correction is used to calculate an implied standard deviation. Dividend information is obtained from Moody's Dividend Record, and risk-free rates are obtained from the Wall Street Journal using treasury bills with maturity dates closest to the option expiration dates.
Observations are excluded from the sample if the present value of the dividends paid over the remaining life of the option exceeds the interest foregone from early exercise.(1) We also exclude observations for which there is violation of the no arbitrage condition:
C [greater than] [S.sup.*] - E[e.sup.-r[Tau]] (2)
where C = call price from Equation 1
[S.sup.*] = stock price less the present value of dividends paid over the life of the option
E = exercise price
r = risk free rate
[Tau] = time remaining until option expiration
For each option on day t, we first employ the dividend-adjusted Black-Scholes model to calculate an implied standard deviation for the same option from the previous trading day. We combine this volatility estimate with [Mathematical Expression Omitted], the risk-free rate, and time to expiration to calculate an expected call price, [Mathematical Expression Omitted], using the dividend-adjusted Black-Scholes model.(2) The deviation of this price from the observed option price on day t, [C.sub.t], is then divided by the observed call price from the previous trading day, [C.sub.t-1]. The resulting percentage call pricing error is denoted [Mathematical Expression Omitted]
[Mathematical Expression Omitted]
The statistic above is employed to test the joint hypothesis that the dividend corrected Black and Scholes model provides accurate estimates of call option prices and that call options are correctly priced around option expiration.(3) Mean values of [Mathematical Expression Omitted] that are significantly different from zero support rejection of the joint hypothesis.
In order to make comparisons regarding the efficiency of prices for current versus non-current expiration cycle options, we compute the mean of the absolute values of the percentage call pricing errors for each sub-sample, for each day t:
[Mathematical Expression Omitted],
where [Mathematical Expression Omitted] is the absolute value of the percentage call pricing error for options j = 1 ...... n on day t. A Wilcoxon test of the difference between samples for each day during the event window is performed.
In order to evaluate the robustness of our results, the procedures and tests described above are also performed over non-expiration time periods.(4) Non-expiration time periods are defined as the two week periods prior to the week of expiration. For these tests, event day zero is defined as the trading day falling two weeks prior to the expiration date.
For the expiration period sample, we run a regression for each day t during the event window to determine whether the expiration event itself impacts the pricing efficiency of non-expiring call options. The specific form of the model is:
[Mathematical Expression Omitted],
where [Mathematical Expression Omitted] = absolute percentage call pricing error for option j on day t,
[CYCLE.sub.jt] = 0, 1 variable with a value of 1 if options on the same underlying stock as option j expire on day t = 0,
[SHORTINT.sub.jt] = the number of shares of the underlying stock sold short as a percent of the total number of shares outstanding for option j on day t,(5)
[ABSMONEY.sub.jt] = the absolute value of the difference between the stock price and the exercise price divided by the exercise price for option j on day t,
[TTEXPIRY.sub.jt] = the time to expiration for option j on day t, expressed in years, and
[e.sub.jt] = disturbance term for option j on day t.
In the above regression equation, the sign and significance of the CYCLE coefficient will indicate whether expiration of options in the current cycle is associated with smaller discrepancies between actual and predicted option prices. Brent, Morse and Stice (1990) find a positive relationship between short interest in the underlying stock and option open interest. Because we do not have open interest data, we use the variable SHORTINT to proxy for option open interest. Large open interest in an option or option series is expected to be associated with a high degree of market scrutiny of option prices and a corresponding increase in the efficiency of option prices. Therefore, we predict that the coefficient on the SHORTINT variable will be negative. Inclusion of the ABSMONEY and TTEXPIRY variables is primarily intended to control for any biases associated with the choice of the option pricing model.(6) Because the dependent variable is a measure of trading day deviations arising from the model and not a measure of price levels, we expect that model related biases are largely mitigated. As such, these variables should provide reliable information on the impact of moneyness and time to expiration on the efficiency of option prices.
Table 1 provides descriptive statistics on the two expiration period sub-samples. The average moneyness (stock price divided by exercise price) for the two samples is approximately equal. Both the mean call price elasticity and implied standard deviation are somewhat lower for the options in the current expiration cycle, while the mean time to expiration is somewhat longer. Note that the mean implied volatility for the current cycle sample appears to shift downward the week after expiration relative to the week of expiration. This is consistent with the finding of Day and Lewis (1988) that option expiration dates are viewed by market participants as times of heightened volatility for the underlying stocks. For the non-current cycle sample, there appears to be an upward shift in the mean implied volatility after the expiration date. Note that after expiration this group includes some options that will expire on the next expiration date. This may trigger increased demand for these options resulting in higher prices and, consequently, higher measured volatility.
Panel 1 of Table 2 reports the mean percentage call pricing error for each day relative to the expiration date for the two samples drawn from the expiration periods. For non-expiring calls in the current expiration cycle, mean percentage call pricing errors are insignificantly different from zero over the period covering event day -3 though the expiration date (event day 0). Furthermore, the magnitude of the mean percentage call pricing error diminishes in monotonic fashion from -0.3 percent on day -3 to 0.0005 percent on the expiration date. These results indicate that, for non-expiring options in the current expiration cycle, observed call option prices conform to the prices predicted by the dividend-corrected Black-Scholes model in the period just prior to, and including, the expiration date. In contrast, significant, non-zero pricing errors are observed for nine of the ten days examined for calls not in the current expiration cycle.
Panel 2 of Table 2 reports the mean percentage call pricing errors for each day during the non-expiration time periods. For this sample period, non-zero mean [TABULAR DATA FOR TABLE 1 OMITTED] [TABULAR DATA FOR TABLE 2 OMITTED] pricing errors are observed for eight out of the ten days examined. Because of the similarity between the results over this non-expiration time period and the expiration period results for options not in the current expiration cycle, we find some support for the notion that the expiration event is not relevant for options that are not in the current expiration cycle.
All of the mean percentage call pricing errors reported in Table 2 are small enough to be completely consumed by transaction costs.(7) Thus, if we were to accept the accuracy of the Black-Scholes model corrected for dividends, we would still conclude that call option prices are, on average, economically efficient for each of the samples for each day in the event windows. It is, nonetheless, noteworthy that call prices for non-expiring options in the current expiration cycles are consistent with the pricing model for the four days leading up to and including the expiration date, while this is not true outside this window. Nor is it generally true that the prices of call options that are not subject to a nearby expiration event conform well to the pricing model. This evidence is interesting in that it suggests that there are certain factors, apart from moneyness and time to expiration, that affect the biasedness of the Black-Scholes model.
Table 3 presents evidence that, during the expiration period, the Black-Scholes model provides more efficient price estimates for the current cycle sample than for the non-current cycle sample. A graph of these results is presented in Figure 1. For each of the ten days in the expiration period event window, the mean absolute value of the percentage call pricing error is lower for the current cycle sample, with six of the ten Wilcoxon test statistics significant at .02. It is interesting to note, however, that pricing efficiency appears to deteriorate for the current cycle sample in the two days immediately following expiration. This is consistent with the downward shift in mean implied volatilities reported for event day 1 in Table 1 for the current expiration sample. Any measured shifts in implied volatility would be accompanied by increases in call pricing errors. If market makers and traders are rolling over written positions in the two days following expiration, this could induce the observed decline in mean implied volatility and increase in mean absolute percentage call pricing error.
Examination of the mean absolute call pricing errors for the non-expiration period sample generally confirms the greater efficiency of prices for the current expiration cycle sample. For nine out of ten days, the mean absolute value of the percentage call pricing error is lower for the current cycle sample than for the non-expiration period sample, with six of these differences significant at the .02 level. Five of these significant differences occur on consecutive days beginning with event day -4 and ending on event day 0. On event day 1, we observe that the mean absolute percentage call pricing error is significantly lower for the non-expiration sample. This reflects the significant increase in mean error for the current cycle sample that is described above.
Comparison of the mean absolute errors for the non-expiration period with those for non-current expiration cycle options during expiration periods does not indicate any strong pattern. Seven of the ten differences are not significantly different from zero. Two of the significant differences indicate a larger mean error for non-current [TABULAR DATA FOR TABLE 3 OMITTED] expiration cycle options during expiration periods (event days 1 and 5). One of the significant differences indicates a larger mean error for the non-expiration period (event day -2).
Taken together, the results reported in Table 3 support two conclusions. The first is that the period leading up to an option expiration date is one of increased pricing efficiency for non-expiring options on stocks for which there are other expiring options. The second is that increased efficiency during this period is not shared by non-expiring options that are not in the current expiration cycle.
Table 4 reports the results of the regression model specified in equation 5, and provides further support for the finding that the expiration period is one of increased pricing efficiency for non-expiring call options in the current expiration cycle. For seven of the ten days in the event window, the coefficient for the expiration dummy variable, [[Beta].sub.1], is negative and significant. Consistent with the results from Panel 2 of Table 2, there is no significant differential impact of expiration on either of the two days following expiration.
A further interesting result is the negative relationship between the mean absolute call pricing error and the degree of short interest in the underlying stock. The coefficients for SHORTINT are negative for each of the ten days in the event period, with six of the ten coefficients significantly different from zero. In addition, five of the significantly negative coefficients occur consecutively on event days-1 [TABULAR DATA FOR TABLE 4 OMITTED] through +3. This result supports the notion that short interest acts as a proxy for market scrutiny of call prices.
As expected, the moneyness variable, defined as the absolute value of the difference between the stock and exercise price divided by the exercise price, is found to have positive coefficients for each day during the event window. This indicates that pricing efficiency is greater the closer to the money is the call option. Because near-the-money options are generally preferred by traders, we expect their prices to reflect the most information, and thus, be most efficient.
A somewhat surprising result is the consistently negative relationship between time to expiration and the mean absolute percentage call pricing error. Just as near-the-money options are preferred by traders, there is ample evidence that traders also prefer shorter times to expiration.(8) By the logic employed above, we would expect that call pricing errors would be smaller, the shorter the time to expiration. A possible explanation for the result obtained involves the definition of the implied variance from the Black-Scholes model as the average variance per unit of time from the valuation date to the option expiration (see Patell and Wolfson, 1979, 1981). Thus, the impact of information events on the implied variance of longer term options will be less than for shorter term options, due to the larger amount of time over which the variance impact is averaged. This lower sensitivity translates into greater consistency in the day to day estimates of implied volatility which, in turn, results in smaller call pricing errors.
Summary and Conclusions
We examine the impact of option expiration on the prices of non-expiring common stock call option prices. We find that for non-expiring options in the current expiration cycle, observed prices conform to the prices predicted by the dividend-corrected Black-Scholes model in the period prior to, and including, the expiration date. This is generally not true for options that are not in the current expiration cycle or for options during non-expiration time periods. This evidence is interesting in that it suggests that the dividend-corrected Black-Scholes model is more useful as a pricing tool when there is a nearby expiration event affecting options on the same underlying stock.
We also find that, during the expiration period, mean absolute pricing errors, a measure of pricing efficiency, are smaller for non-expiring options in the current expiration cycle than for non-current expiration cycle options. Pricing efficiency during expiration periods for current expiration cycle options is also generally found to be higher than for options during non-expiration periods. A regression test is performed to evaluate the impact of the expiration event, the degree of short interest in the underlying stock, and the moneyness and time to expiration of the option on the mean absolute percentage pricing error. We find that pricing efficiency is positively related to impending expiration of other options on the same underlying stock, short interest in the underlying stock, the nearness of the option to-the-money, and time to expiration. The results reported here suggest that the ability to arbitrage security mispricing at relatively low cost and risk through the use of related or derivative instruments enhances pricing efficiency. The evidence is also consistent with a 'spillover' of information from expiring options into non-expiring option prices.
Acknowledgment: The authors thank David Dubofsky and participants of seminars at Florida State University and The University of Albany for helpful comments.
1. We recognize that by employing the Black-Scholes model with dividend correction and excluding options carrying an early exercise premium there is a possible loss of generality of the results. However, we know of no reason a priori why the results would be systematically different if this restriction were not imposed. On average, our sample consists of between 14,000 and 15,000 observations per event day. The choice of the dividend corrected Black-Scholes model was made to balance considerations of sample size and computational constraints against possible loss of generality.
2. For a similar application of this procedure, see Vijh (1990).
3. The problem of joint tests is one faced frequently in empirical finance. For example, event studies examining the reaction of stock prices to various events typically employ the market model to estimate an expected stock return. Consequently, all tests using the market model are tests of the joint hypothesis of the correctness of the market model and the existence of abnormal returns. We cannot develop a call pricing error unless we estimate an expected call price. The Black-Scholes model provides our expected call price. Whaley (1982) uses a related procedure in his tests of alternative option pricing models. He finds that implied volatility estimates obtained from the dividend-adjusted Black-Scholes model are virtually identical to those calculated using the exact model developed by Roll (1977), Geske (1979), and Whaley (1981). We use the implied volatility estimate from t-1 to estimate the call price on day t. Thus, in a sense, we are calibrating the model to produce an expected call price on day t that is consistent with the observed call price on day t-1. The expected call price produced by this procedure almost certainly contains less error than the expected stock return produced by applying market model parameters that have been calculated over an arbitrarily defined 'estimation period.'
4. We are grateful to the referee for suggesting this test.
5. Monthly short interest data were obtained from the Daily Stock Price Record. The data consist of the number of shares sold short as of the fifteenth of the month, or the closest preceding trading day if the fifteenth is not a trading day. To make the short interest variable comparable across firms, short interest is divided by the total number of shares outstanding. The percent of shares sold short is then multiplied by 100.
6. The biases of the Black-Scholes model that have been documented in the literature involve moneyness and time to expiration. There is not a large difference in the mean values of these two parameters between the current and non-current expiration cycle samples, especially in the period preceding the expiration date (see Table 1). Thus, even if these biases were evident, they would not have a significant differential impact between samples. Furthermore, the inclusion of ABSMONEY and TTEXPIRY in the regression equation should control for any differences between samples and provide a test of the impacts of the expiration event and the degree of short interest.
7. See Phillips and Smith (1980) for estimates of transactions costs incurred by arbitrageurs, option market makers and individual traders.
8. The CBOE's move to add monthly expirations is indicative of the demand for short expiration options.
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|Author:||Broughton, John B.; Chance, Don M.; Smith, David M.|
|Publication:||Review of Financial Economics|
|Date:||Mar 22, 1995|
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