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The measurement of option mispricing
Journal of Banking and Finance 12 (1988) 537-550. North-Holland
THE MEASUREMENT OF OPTION MISPRgCiLNG Dan W. FRENCH* Texas Christian University, Fort Worth, TX 76129, USA lnstituto de Postgrado en Administracion de Negocios, Guayaquil, Ecuador
Linda J. MARTIN* Arizona State University, Tempe, AZ 8.5287, USA Received June 1986, final version received November 1987 Different studies have examined the ability of the Black-Scholes option pricing model to e~'timate accurately market prices of publicly traded options and reached conflicting results. This study examines commonly used ex ante measures of option mispricing, finds that they can produce differing conclusions about option prices, and develops an aJ+.ernative measure for gauging option mispricing. Empirical analysis of returns to options selected using the various mispricing measures indicates that this new measure is more i/kely to detect mispricing and identify options that yield excess returns before commissions.
I. Introduction
The ability of the Black-Scholes option pricing model to estimate accurately the market price of publicly traded stock options has been the topic of a number of studies. [See, for example, Black (1975), Black and Scholes (1972), Galai (1977), Gultekin, Rogalski and Tinic (1982), Macbeth and Merville (1979) and Whaley (1982).:] These studies attempt to identify which types of call options the B~ack-Scholes model tends to price correctly and those it tends to misprice. The dift~rent studivs do not, however, always reach the same conclusion and often disagree on how the market prices call options relative to the Black-Scholes model price. There are a number of reasons which could explain conflicting results of option pricing mt...el tests. A list of these reasons includes, but is not necessarily limited to, the following: (1) different sample time periods, (2) different forms of the option pricing model used (e.g., Whaley adjusted for dividends while the others mentioned above did not), *The authors acknowledge helpful comments and suggestions by Georges Courtedon+ Stcven Manaster, Thomas O'Brien, and the participants of the finance research workshops of Southern Methodist University and Texas A & M University 0378-4266/88/$3,50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
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D.W. French and LoJ. Martin, Measurement of option mispricing
(31 different techniques for estimating the stock variance parameter of the model l'Rubinstein (1985) avoids this problem by using implied variances as a measure for determining the direction of mispricing], and (4) different methods of measuring the degree option mispricing. While any or all of these four areas may be sources of disagreement regarding the mispricing of options, this paper concentrates on the final item in the list. The use of different methods of measuring the amount of mispricing among options may be responsible for at leas;t some of the disagreement about the extent of option mispricing. This paper begins with a discussion of previously used measures of option mispricing and identifies the problems inherent in each approach. An alternative method for measuring the amount of option mispricing is then presented, and finally, the paper ends with empirical tests of the various measures. 2. A rev|ew of previous mispricing measures Them are basically three different measures that the majori~ty of previous studies have used to gauge option mispricing: the absolute prediction error, the relative prediction error, and the realized excess returns. Table 1 def'mes these m~asures and summarizes the results of some of the mispricing studies. Although the absolute raispricing error (APE) and the relative prediction error (RPE) may adequately indicate whether options were misprieed and the direction of mispricing (i.e., underprieed or overpriced) for a given sample, they may not adequately measure how much an option was mispriced, especially it" the potential profitability of an option is considered to be an indication of mispricing. To gauge the extent of option mispricing, a measure should not compare options of differ~:~t duration by simply identifying how much the option's price differs from its model price. As an example of this reasoning, consider two hypothetical options, a three-month and a six-month call, each overpriced in the market by 100 percent (i.e., RPE-1.0). As the RPE is positive, an option investor would attempt to reap excess return by writing these options. The RPE of 1.0 for each would seem to imply that the investor would be indifferent as to which one to write. Actually, the investor would prefer to write the three-month option because the excess returns would acrue to its writer at a faster rate than to the six-month option. In fact, the writer of the three-month call could establish his position, wait three months, collect excess returns on expiration, and repeat the procedure with another overpriced three-month option (assuming one is available) during the length of time it would take the six-month option to reach expiration. Therefore, wi~.h respect to potential trading profits, the three-month option must be mispriced to a greater extent than the six-month option, but the RP~ and the APE fail to recognize this. Similar reasoning would indicate the
D.W. French and L.J. Martin, Measurement of option mispricing
539
Table 1 Summary of some studies of option mispricing? Author(s)
Measure of mispricing
Summary of results
Black
Not identified
Galai
Ex post returns to riskless spread
Guletkin, Rogalski, Tinic
Prediction error Ca-Cm
Macbeth and Mercille
Relative prediction error
Model underprices short expiration options No significant evidence of degree of mispricing and time to expiration Model overprices in-the-money options and underprices out-ofthe-money options; degree of mispricing increases the further from the money Degree of mispricing increases as time to expiration increases Model overprices out-ofomoney options and underprices in-themoney options; degree of mispricing increa~cesthe further from the money Degree of mispr/cing increases as time to expiration increases No sj~ificant relationship between option mispricing and the extent by which the option is in or out of the money Degree of mispricin~ decreases as time to expiration increases
( Co - C~)/C,~
Whaley
Relative prediction error (Ca-C~)/C~
aC.-- actual market price; C m ~ Black-Scholes model price.
inadequacy of these measures when comparing out-of-the-money to in-themoney options. Measures similar to the RPE and APE have been used in studies comparing mispricing across other securities such as stocks. In such cases, the RPE and APE are good indicators of potential profit because the maturity parameter is not important. However, with options, every parameter plays an important role in translating mispricing into profit. The Black and Scholes (1972) and Galai (1977) technique, which measures the ex post risk-adjusted return to option positions is a good approach. If the positions yielded positive excess returns, Black-Scholes and Galai would conclude that the model underpriced options relative to the market, although Galai's study was principally concerned with tests of market efficiency rather than with determining pricing biases. In another study, Galai (1983) decomposed the total return to a hedged option position into its three components: the riskless return on the investment, the return from discrete adjustment of the hedge, and the return attributable to the change in the deviation of the
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D.W. French and L.J. Martin, Measurement of option mispricing
observed option price from its model price. As Galai found tke second component to be relatively small, he attributed any return greater than the risk-free rate to the mispricing represented by the third component. Galai and Geske (1984) show how the decomposition approach can be used to measure the performance of option traders or an option trading strategy.
3. Measuring ex ante option misptieing Galai's method measures the observed performance of the hedge. To be of use to option traders, an ex ante approach is needed to measure option mispricing before the performance of the hedge is known. The basis for security analysis is that the more underpriced a security is, the greater its excess, risk-adjusted, expected returns. Securities that are mispriced to a greater extent will have greater expected returns than correctly priced securities of similar rsk. The excess expected return to the security is, therefore, a measure of mispricing, and an estimate of an option's expected excess return would be a vafid ex ante measure of how much the option is mispriced. Because all of the parameters affecting option prices play an important part in the translation of mispricing into return, the option mispricing measure should account for all of the option pricing parameters. A logical common-risk level for comparing options is that of zero risk because the Black-Scholes model provides the framework for the definition of a riskless option hedge constructed by purchasing one share of the underlying stock and writing a quantity of the call option equal to the Black-Scholes natural hedge ratio. The hedge is riskless only for a very short time interval, and its expected return, if the option is correctly priced, is equal to the riskless interest rate used in the Black-Scholes model (assuming validity of the Black-Scholes model). Accepting the standard Black-Scholes assumptions and using a reasonably short one-day holding period, the expected return on a riskless call option hedge is:
1 E(R)=
[E(¢)- C'_l
N{d,} SC N{d,}
where
(1)
E(R) =the expected one-day holding period return to a neutral option hedge, S --the price of the common stock, C = the price of the call option, N{-} = the cumulative normal distribution function,
dt
=(ln(SlK)+[r+(a212)]T)I~ET]~/2,
D.W. French and L.J. Martin, Measurement of option mispricing
541
(N{dt} is the first derivative of the Black-Scholes model, C/S. l/(N{dl}) is the Black-Scholes neutral hedge ratio, E(~) = the expected future market price of the stock at the end of the oneday holding period, E(g) = S(1 + m), E(¢~) = the expected future market price of the call at the end of the one-day holding period computed by solving the Black-Scholes call value equation evaluated at S(1 +m),K(1 +r), T,r, and [ ( l i T ) a 2 + ( 1 - 1/T)02] 1/2, K r m T 0-2
02
=the exercise price of the option, =the per day risk-free rate, = the per day expected stock return, =the time to expiration of the option, = the investor's estimate of the variance of the expected stock returns, and --the market's estimate of the variance of the expected stock returns (the implied variance of the market price of the option).
Eq. (1) is based on Rubinstein's (1984) formula for the expected future price of a call option. This formula assumes that option mispricing results from a difference between the investor's estimate of the stock variance and the market's estimate of the variance. It further asscnles that the mispricing falls as the difference between the two variance, estimates deteriorates linearly through time. The unknown parameter in eq. (1) is m, the expected stock return. Under Black-Scholes conditions of continuous trading and hedge adjustment, the neutral hedge return would be equal to the riskless rate regardless of the expected return on the stock if the option is correctly priced. Therefore, we could use any expected stock return in the formula, the simplest being a return of zero. In practice,, however, the hedge ratio changes as the stock price changes, and the hedge loses its neutral character. The overriding question, therefore, is whether daily adjustment of the hedge is sufficient to maintain an expected return for eq. (1) equal to the riskless rate for various expected stock returns. In support of the hypothesis that the discrete hedge would not lose its riskless characteristic over a reasonably short time span (e.g., one day), Galai (1983) finds that discrete adjustment of the hedge ratio is responsible for only a minor portion of the hedge return and that adjustments at one-day trading intervals do not significantly affect retdrns. Galai and Geske (1984) state that daily adjustment of the hedge ratio should be adequate, and that empirically it may be sufficient to adjust a hedged position only once per week. In addition to these arguments, we used Monte Carlo simulation to estimat¢ the
D.W. French and L.J. Martin, Measurement of option mispricing
542
expected return on a one-day risk-neutral opiton hedge. Table 2 presents the results of this simulation. The rows labelled 'Computed E(R)' represent the expected return on the position using eq. (1) less the riskless rate. To simulate expected daily returns, we generated a random vafiate from the lognormal distribution for a one-day stock return, computed the associated stock price, and used ~he Black-Scholes model to compute the call option price. We then used these option and stock prices to compute the simulated hedge return and repeated the procedure 2,000 times to find the expected hedge return. The rows labelled 'Simulated E(R)' report these returns (less the riskless rate). Note that the correctly valued options all heve a computed E(R) of zero as we would anticipate. Their counterparts in the simulated E(R) rows are all very close to zero, the small deviations resulting from random variation in the simulation process. This supports the contention that daily hedge adjustment justifies an expected riskless return for correctly valued options. Note in table 2 that all undervalued options (i.e., those whose market price is less than the true value) have negative excess expected returns, and all overvalued options' excess expected hedge returns are positive. In addition, their simulated excess returns are equal to or very close to the computed value. Table 2 also supports the idea that the expected hedge return is stable with respect to changes in the expected stock return. For example, the computed E(R) for the 30-day undervalued option is -0.091 percent for m=0, 0.10, and 0.20. An appropriate measure of option mispricing, then, is the exacted one° day excess return on a neutral option hedge. Because this return is independent of the expected stock return, any value for m may be used in the formula. Substituting a value of zero for m in eq. (1) and subtracting the riskless rate, the formula simplifies to: 1
N{d,} [ C - E ( ~ ] a= S,
C N{dl}
r, where
(2)
0t= the expected excess return to a risk-neutral option hedge (the measure of option mispricing). Correctly priced options should yield c~ values of zero. Overpriced options • (options whose market prices are greater than their model values) will yield positive values for 0t, and undervalued options will give negative ~s. Note that the mispricing measure explicitly considers all of the option pricing parameters.
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The option hedge's • measures excess return only as a measure of mispricing, which should not imply that such returns were actually available to traders. Several studies of market efficiency, including those of Phillips and Smith (1980), Wha!ey (!982) and French and Henderson ~198!), have shown that apparent excess returns on options disappear after considering commissions and transactions costs. One note of caution is in order. Boyle and Emanuel (1980) found that when investors appreciably misestimate the expected variance term, even daily hedge returns can become quite risky. Using eq. (2) to measure the extent of option mispricing is similar t ° the approach outlined by Cox and Rubinstein (1985). They develop a strategy for screening neutral option positions based on the potential profit for the hedged position. Their process yields an expected profit to a neutral option hedge, but it would yield different results from our eq. (2) and produce a different ranking of mispriced optivns for two fundamental reasons. First is that the Cox-Rubinstein formula measures potential profit as the absolute difference between the current option value and its market price; it does not use the expected future option value as does eq. (2). Second, the CoxRubenstein technique weights the expected profit by the sensitivity of the option hedge ratio to changes in the stock price. In other words, it will favor options whose hedge ratios tend to 0e more stable when t .I,,, _ . . . o,,,,.t. . . . p n ~• change~, minimizing the risk associated with shifting hedge ratios and the need to rebalance the hedge. The Cox-Rubenstein procedure should be more effective in identifying profitable option hedges when the indended holding period is of a relatively long time span. Their technique, therefore, might help identify neutral option positions which could be favorable to a trader who is subject to the risks and costs of frequent hedge rebalancing~ but it would not necessarily produce the same ranking of mispriced options as would eq. (2). Whether the Cox-Rubenstein procedure or eq. (2) would produce greater profits to the investor is grounds for fertile research, but beyond the scope and objectives of this paper. Eq. (2) is dependent upon the assumptions underlying the Black-Scholes model. Two particularly important assumptions are that the option is European and is on a stock which pays no cash dividend. As Roll (1977) notes, American options on dividend-paying stocks may appear to be misvalued just before the ex-dividend date. To adjust the option mispricing measure for American options on dividend-paying stocks, the Roll (1977), Geske (1~79a), Whaley (1982) option pricing formula can be used in eq. 12) to compute the hedge ratio and expected option prices. In addition, if th,, exdividend date falls ~thin the holding period, the probability of early exercise, the receipt of the cash dividend, and the expected ex-dividend decline in the stock price must be considered. There are four practical differences between the excess returns approach
D.W. French and L.J. Martin, Measurement of option mispricin£
545
and the prediction error approaches that could cause the them to show different amounts of mispricing. First, the hedge ratio is always greater for the short-term option (other factors equal), resulting in more of them being sold short to establish the hedge and hence more profit per share of stock purchase. Second, different options have different prices, which causes the cash investment to establish the hedge to differ. Third, when the options are of different duration, the pricing error deteriorates much more rapidly for a short-term option than for one of longer duration. When one is comparing options with different times to expiration, this final factor probably explains most of the difference between mispricing gauged by the ~ and mispricing gauged by the APE or RPE. The fourth reason is that when comparing outof-the-money options to in-the-money options, the hedge ratio is appreciably larger for out-of-the-money options. This factor may affect some of the results of the comparison of the in- and out-of-the-money options in the Gultekin, Rogaiski and Tinic study. For out-of-the-money options, Gultekin, Rogalski and Tinic correctly concluded that the positive mean prediction errors indicated that the model underpriced those options, and, similarly, the negative errors for the in-the-money group showed them to be overpriced. They also, however, compared the mean prediction errors of the in-themoney and out-of-the-money options, concluding that the larger errors for the in-the-money options indicate that these options were more mispriced by the model. Such a procedure is of questionable value, because of the fact that the hedge ratios of the out-of-the-money options would be significantly larger than their in-the-money options indicate that these options were more mispriced by the model. Whaley (1982) found n o significant relationships between the option mispricing and the amount by which the option was away from the money, although his measure of mispricing (relative prediction error) would not necessarily fully disclose option mispridng for the reasons mentioned above. This discussion suggests that the • would be preferred over the RPE and APE for computing option mispricing on an ex ante basis. Some might criticize the 0~by noting that its value is arbitrary because of the selection of the stock as the security for the denominator. That is, for a given mispriced option, an investor could increase his excess return by hedging with a leve: ~d stock position or by establishing an option spread, therefore investing less money while earning about the same profit. While this is true, this paper proposes that the option's • as defined by eq. (2) is acceptable as a standard measure for examining option mispricing because of its advantages over the commonly used RPE and APE and because it is consistent with the theory underlying the development of the Black-Scholes model. The real test, however, is how well the methods compare empirically.
546
D.W. French and LJ. Martin, Measurement of option mispricing
4. Empirical tests The sample of option market data is from the Berkeley Options Data Base for August and September of 1979. This data base has the advantage of offering every option trade for the day, along with concurrent stock prices. For each day in the data base, we extracted every call optiot~ trade occurring between ll:00a.m, and l:00p.m, that was not an exact duplicate of a trade previously extracted. ~ This plocedure yielded a sample of 26,556 observations. This first step in computing the mispricing measure was to estimate the investor's estimates of the expected standard deviation of stock returns. This study used the weighted implied standard deviation calculated by the ChirasManaster (1978) procedure. 2 The three ex ante measures of option mispricing (the APE, RPE, and 0c) were computed using this weighted implied standard deviation and an estimate of the riskless rate equal to the rate on three-month Treasury bills. Option and stock prices for the following day were used to calculate the actual return on a riskless option hedge (buying the stock and writing the option according to the hedge ratio), which serves as the ex post measure of option mispricing. The best before-the-fact indicator of how m~lch an option is mispriced should be the ex ante measure that best predicts actual excess return, that is, the measure with the highest correlation to the ex post measure of mispricing. The best measure would be of considerable interest to option traders, because only the ex ante measures are available to them while making tradl.ng decisions. Table 3 gives the correlation coefficients of the ex post measure with the RPE, APE, and 0c-measures. Although the three values appear relatively similar, the u's correlation coefficient is significantly greater than the other two at the 98 percent confidence level. The table also gives the 98 percent confidence limits for each of the correlation coefficients) Note also that the IUsing prices from ll:00a.m, to l:00p.m, served two purposes. First, it minimized any problems that might be a ~ ' i a t e d with transactions occurring during the market opening or closing times. Second, the da:..,~o~ ~ ~,J ~.~.:..,,st-,c,c~.~.~.:~ .,1:^.~ ~e .,,~.~.-~..~ r-~ *he time period, that a subset was needed in order to bring the sample down to a manageable size. Additionally, we omitted option quotations of 1/16 and 1/8, because the chance of such an option changing in price over a one day interval is remote (the chance of the 1/16 decr.-asing in price is nil). 2The ase of implied variance does introduce bias into Black-Scholes option price estimation. Butler and Schacter (1986) have shown that, in general, expected implied variances will not provide variance estimates which equal the true variance. They develop a variance estimator which yields unbiased estimates of option prices but which remains subject to sample error in estimating the moments of the sample variance. They note that the question remains empirical whether their method or an estimate based on implied variances is subject to greater sampling error. The effects of using implied variances in this study should be negligible because Butler and Schacter's tables show that differences between usual and unbiased Black-Scholes option price estimates are very small• 3See Kleinbaum and Kupper (1978) for a discussion of Fisher's Z transformation for the computation of the confidence limits of correlation coefficients. •
8
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D.W. French and L.J. Martin, Measurement of option r~Jspricing
547
Table 3 Correlation coeificients of ex ante option mispricing measures vs. ex post actual realized returns.
Technique
Correlation coelficients
98~o confidence intervals
RPE APE 0c
0.1402 0.1552 0.1854
0.1262--0.1542 0.1412--0.1691 0.1716-0.1991
Table 4 Returns on option portfolios selected using option mispricing measures.
Mispricing measure
Mean continuously compounded return
Standard deviation
Frequency of highest returns
RPE APE ~,
0.019 0.024 0.036
0.021 0.017 0.022
4 7 ~0
confidence intervals for the APE and RPF, overlap, which indicates that the differences are not significant. The correlation coefficients were disappointingly low for all three of the measures. Several factors contribute to these low correlations, including the absence of a perfect estimate of the stock variance and discrete adjustment of the hedge. Another important likely explanation is that market option prices do not fully adjust over one-day intervals to mispricing to the extent suggested by Rubinstein (1984). If the 0o's do measure mispricing more accurately, then they Should be better at identifying positions that would lead to greater returns. One fmal test uses a trading strategy to see whether option portfolios selected using 0~ would yield higher returns than those chosen using the APE or RPE. We began by ranking the sample options on each day by the 0e mispricing measure from most undervalued to most overvalued. We then used the substitute neoge tecnntquc of /~-ren~h anti H ~ u , ~ o . l (i98i) an~, ,:,ii~.~,,~ (1982) to construct systematically riskless portfolios assuming purchase of the ten undervalued call options and short sale of the ten overvalued calls times the substitute ratio. (The mechanics of the hedge are in the appendix.) The profit to the portfolio was then computed assuming all positions were closed out on the option's first trade after ll:00a.m, on the following day. We then repeated this entire procedure ranking the options by APE and then by RPE.
Table 4 gives the results. The last column in table 4 presents the number
5,48
D.W. French and LJ. Martin, Measurement of option mispricing
of times out of 41 sample dates that the measure yielded a return higher than the other two measures. The 0h which resulted in the greatest return on 30 of the 41 dates, dominated the statistic. The 0c also yielded the highest mean return. To test hypotheses about the mean returns, we used the paired t-test. The results were as follows: Ho:/;(00
- F(APE),
Ho: ~a) ---~RPE), Ho: ~(APE)-~(RPE),
t ffi4.68 reject t = 5.62 reject t-- 1.52 do not reject.
These tests show that the portfolios identified by a yielded significantly higher returns at the 0.01 level than those chosen by APE and RPE. APE returns were not significantly different from RPE returns. The returns reported in table 4 should not be misrepresented. This is not a study of market efficiency; our purpose is only to show that a identifies options that are most likely to yield excess returns when trades are available at quoted prices. Upon recomputing the table 4 returns asseming transactions costs of even less than the average bid-ask spread reported by Phillips and Smith (1980) and only $1.00 per contract, apparent excess returns were virtually eliminated, a finding which is consistent with market efficiency in the options markets. 5. Coadmioa Our study identified one source of measurement error in option pricing. Prior researchers have identified additional factors, such as dividends, nonstationary variances, transaction costs and firm leverage, that may help to explain option mispricing? It should be noted that one ad hoc adjustment that could be made to the RPE is to divide the mtasure by the number of days to expiration in order to get a measure of the expected return per day. Such a measure could be used in the correlation analysis in order to enmpare the various technioues of option misnricing. However, theoretically such a measure is inappropriate because the decay function of the option pricing model is not linear. The a performs as a better measure of option mispricing than either the APE or RPE because it is more highly correlated with ex post hedge returns and it identified sample portfolios that consistently yielded higher returns that the APE or RPE. 4For a discussion of the effects of dividends on call option prices see Roll (1977), Geske (1979a) and Whaley (1981). The effects of non-stationary variance rates are discl~ssed in Merton (1976), Patell and Wolfson (1979), Whaley and Cheung (1982), Cox and Ross (197~# and F~nc!~ (!984). The effects of transaction costs are discussed in Philli~ a ~ S~J~.ith~9~,0). M i s ~ i n g effects caused by firm leverage are reported in Geske (1979b) and Christie (19~2).
D.W. French and L.J. Martin, Measurement of option mispricing
549
Appendix: Construction of the hedged portfofio Rather than equal-weighted portfolios, which result in an extremely large numbers of cheap options in the portfolio and a very small number of the more expensive options, we weighted each individual option in the po.qfolio by an amount equal to the, reciprocal of the option's beta. The theoretical result of this would be that a given change in the market would be accompanied by the same dollar change in each option position within the portfolio. This is consistent with the reason for using equal weighting in stock portfolios. The attendant French-Henderson (1981) substitute hedge ratio is then
H--C~=ICI. ~I/C~=I Cs,il,
where
H =the quantity of each option to short, the hedge ratio, CL.i=the price of the ith long option in the portfolio, Cs.~ =the price of the jth short option in the portfolio, m =the number of different long options in the portfolio, and n =the number of different short options in the portfolio. The beta from the appropriate issue of the Value Line served as an estimate of the stock's beta coefficient. References Black, F., 1975, Fact and fantasy! in the use of options, Financial Analysts Journal 31, 36-42. Black, F. and M. Scholes, 1972, The valuation of option contracts and test of market efficiency, Journal of Finance 27, 399-417. Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-654. Boyle, P.P. and D. Emanuel, 1980, Discretely adjus~,.d option hedges, Journal of Financial Economics 8, 259-282. Butler, £S. and B. Schacter, 1986, Unbiased estimation of the Biack/Scholes formula, Journal of Fi-~n~! ~eonomics 15, 341-357. Chiras, D.P. and S. Manaster, 1978, The information content of option prices and a test of market efficiency.Journal of Financial Economics 6, 213-234. Christie, A.A., 1982, The stochastic behavior of common stock prices: Value, leverage, and interest rates, Journal of Financial Economics 10, 407-432. Cox, J.C. and S.A. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, 145-166. Cox, J.C. and M. Rubinstein, 1985, Option markets (Prentice Hall, Englewood Cliffs, NJ). French, D.W., 1984, The weekend effect on the distribution of stock prices: Implications for option pricing, Journal of Financial Economics 13, 547-559. French, D.W. and G.V. Henderson, Jr., 1981, Substitute hedge option portfolios: Theory and evidence, Journal of Financial Research 4, 21-31. Galai, D., 1977, Tests of market efficiency of the Chicago Board Options Exchange, Journal of Business 50, 167-197. Galai, D., 1983, The component of the return from hedging options against stocks, Journal of Business 56, 45-54.
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D.W. French and L,J. Martin, Measurement of option mispricing
Galal, D. and R. Geske, 1984, Option performance measurement, Journal of Portfolio Management 10, 42-46. Geske, R., 1979a, A note on an analytical formula for unprotected American call options on stocks with known dividends, Journal cf Financial Economics 7, 375-380. Geske, R., 1979b, The valuation of compound options, JorJrnai of Financial Economics 7, 63-81. Gultekin, N.B., RJ. Rogalski and S.M. Tini¢, 1982, Option pricing model estimate~ Some empirical results, Financial Management 11, 58-69. Kleinbaum, D.G. and L.L. Kupper, 1978, Applied regression analysis and other multivariate methods (Duxbury Press, North Scituate, MA). Macbeth, J.D. and LJ. Merville, 1979, An empirical examination of the Black-Scholes call option pricin8 model, Journal of Finance 34, 1173-1186. Melton, R.C., 1976, Option pricin8 when underlying stock returns are discontinuous, Journal of Financial Economics 3, 125-144. Patell, J.M. and M.A. Wolfson, 1979, Anticipated information releases reflected in call option prices, Journal of Accounting and Economics 1, 117-140. Phillips, S.M. and CW. Smith, 1980, Trading costs for listed options, Journal of Financial Economics 8, 179-201. Roll, R., 1977, An analytic valuation formula for unprotected American call options on stocks with known dividends, Journal of Financial Economics 5, 251-258. Rubiustein, M~ 1983, Displaced diffusion option pricing, Journal of Finance 38, 213-217. Rubinstein, M., 1984, A simple formula for the expected rate of return of an option over a finite holding period, Journal of Finance 39, 1503-1509. Rubinstein, M., 1985, Nonparametri¢ tests of alternative option pricing models using all reported trades and quotes on the 30 most active CBOE option classes from August 23, 1976 through August 31, 1978, Journal of Finance 40, 455-480. Whale% R.E., 1981, On the valuation of American call options on stocks with known dividends, Journal o1"Financial Economics 9, 207-21L Whaley, R.E., 1982, Valuationof American call options on dividend-pricing stocks: Empirical test, Journal of Financial Economics 10, 29-58. Whaley, R.E. and J.K. Cheung, 1982, Anticipations of quarterly earnings announcements: A test of option market efficiency,Journal of Accounting and Economics 4, 57-83.
THE MEASUREMENT OF OPTION MISPRgCiLNG Dan W. FRENCH* Texas Christian University, Fort Worth, TX 76129, USA lnstituto de Postgrado en Administracion de Negocios, Guayaquil, Ecuador
Linda J. MARTIN* Arizona State University, Tempe, AZ 8.5287, USA Received June 1986, final version received November 1987 Different studies have examined the ability of the Black-Scholes option pricing model to e~'timate accurately market prices of publicly traded options and reached conflicting results. This study examines commonly used ex ante measures of option mispricing, finds that they can produce differing conclusions about option prices, and develops an aJ+.ernative measure for gauging option mispricing. Empirical analysis of returns to options selected using the various mispricing measures indicates that this new measure is more i/kely to detect mispricing and identify options that yield excess returns before commissions.
I. Introduction
The ability of the Black-Scholes option pricing model to estimate accurately the market price of publicly traded stock options has been the topic of a number of studies. [See, for example, Black (1975), Black and Scholes (1972), Galai (1977), Gultekin, Rogalski and Tinic (1982), Macbeth and Merville (1979) and Whaley (1982).:] These studies attempt to identify which types of call options the B~ack-Scholes model tends to price correctly and those it tends to misprice. The dift~rent studivs do not, however, always reach the same conclusion and often disagree on how the market prices call options relative to the Black-Scholes model price. There are a number of reasons which could explain conflicting results of option pricing mt...el tests. A list of these reasons includes, but is not necessarily limited to, the following: (1) different sample time periods, (2) different forms of the option pricing model used (e.g., Whaley adjusted for dividends while the others mentioned above did not), *The authors acknowledge helpful comments and suggestions by Georges Courtedon+ Stcven Manaster, Thomas O'Brien, and the participants of the finance research workshops of Southern Methodist University and Texas A & M University 0378-4266/88/$3,50 © 1988, Elsevier Science Publishers B.V. (North-Holland)
538
D.W. French and LoJ. Martin, Measurement of option mispricing
(31 different techniques for estimating the stock variance parameter of the model l'Rubinstein (1985) avoids this problem by using implied variances as a measure for determining the direction of mispricing], and (4) different methods of measuring the degree option mispricing. While any or all of these four areas may be sources of disagreement regarding the mispricing of options, this paper concentrates on the final item in the list. The use of different methods of measuring the amount of mispricing among options may be responsible for at leas;t some of the disagreement about the extent of option mispricing. This paper begins with a discussion of previously used measures of option mispricing and identifies the problems inherent in each approach. An alternative method for measuring the amount of option mispricing is then presented, and finally, the paper ends with empirical tests of the various measures. 2. A rev|ew of previous mispricing measures Them are basically three different measures that the majori~ty of previous studies have used to gauge option mispricing: the absolute prediction error, the relative prediction error, and the realized excess returns. Table 1 def'mes these m~asures and summarizes the results of some of the mispricing studies. Although the absolute raispricing error (APE) and the relative prediction error (RPE) may adequately indicate whether options were misprieed and the direction of mispricing (i.e., underprieed or overpriced) for a given sample, they may not adequately measure how much an option was mispriced, especially it" the potential profitability of an option is considered to be an indication of mispricing. To gauge the extent of option mispricing, a measure should not compare options of differ~:~t duration by simply identifying how much the option's price differs from its model price. As an example of this reasoning, consider two hypothetical options, a three-month and a six-month call, each overpriced in the market by 100 percent (i.e., RPE-1.0). As the RPE is positive, an option investor would attempt to reap excess return by writing these options. The RPE of 1.0 for each would seem to imply that the investor would be indifferent as to which one to write. Actually, the investor would prefer to write the three-month option because the excess returns would acrue to its writer at a faster rate than to the six-month option. In fact, the writer of the three-month call could establish his position, wait three months, collect excess returns on expiration, and repeat the procedure with another overpriced three-month option (assuming one is available) during the length of time it would take the six-month option to reach expiration. Therefore, wi~.h respect to potential trading profits, the three-month option must be mispriced to a greater extent than the six-month option, but the RP~ and the APE fail to recognize this. Similar reasoning would indicate the
D.W. French and L.J. Martin, Measurement of option mispricing
539
Table 1 Summary of some studies of option mispricing? Author(s)
Measure of mispricing
Summary of results
Black
Not identified
Galai
Ex post returns to riskless spread
Guletkin, Rogalski, Tinic
Prediction error Ca-Cm
Macbeth and Mercille
Relative prediction error
Model underprices short expiration options No significant evidence of degree of mispricing and time to expiration Model overprices in-the-money options and underprices out-ofthe-money options; degree of mispricing increases the further from the money Degree of mispricing increases as time to expiration increases Model overprices out-ofomoney options and underprices in-themoney options; degree of mispricing increa~cesthe further from the money Degree of mispr/cing increases as time to expiration increases No sj~ificant relationship between option mispricing and the extent by which the option is in or out of the money Degree of mispricin~ decreases as time to expiration increases
( Co - C~)/C,~
Whaley
Relative prediction error (Ca-C~)/C~
aC.-- actual market price; C m ~ Black-Scholes model price.
inadequacy of these measures when comparing out-of-the-money to in-themoney options. Measures similar to the RPE and APE have been used in studies comparing mispricing across other securities such as stocks. In such cases, the RPE and APE are good indicators of potential profit because the maturity parameter is not important. However, with options, every parameter plays an important role in translating mispricing into profit. The Black and Scholes (1972) and Galai (1977) technique, which measures the ex post risk-adjusted return to option positions is a good approach. If the positions yielded positive excess returns, Black-Scholes and Galai would conclude that the model underpriced options relative to the market, although Galai's study was principally concerned with tests of market efficiency rather than with determining pricing biases. In another study, Galai (1983) decomposed the total return to a hedged option position into its three components: the riskless return on the investment, the return from discrete adjustment of the hedge, and the return attributable to the change in the deviation of the
540
D.W. French and L.J. Martin, Measurement of option mispricing
observed option price from its model price. As Galai found tke second component to be relatively small, he attributed any return greater than the risk-free rate to the mispricing represented by the third component. Galai and Geske (1984) show how the decomposition approach can be used to measure the performance of option traders or an option trading strategy.
3. Measuring ex ante option misptieing Galai's method measures the observed performance of the hedge. To be of use to option traders, an ex ante approach is needed to measure option mispricing before the performance of the hedge is known. The basis for security analysis is that the more underpriced a security is, the greater its excess, risk-adjusted, expected returns. Securities that are mispriced to a greater extent will have greater expected returns than correctly priced securities of similar rsk. The excess expected return to the security is, therefore, a measure of mispricing, and an estimate of an option's expected excess return would be a vafid ex ante measure of how much the option is mispriced. Because all of the parameters affecting option prices play an important part in the translation of mispricing into return, the option mispricing measure should account for all of the option pricing parameters. A logical common-risk level for comparing options is that of zero risk because the Black-Scholes model provides the framework for the definition of a riskless option hedge constructed by purchasing one share of the underlying stock and writing a quantity of the call option equal to the Black-Scholes natural hedge ratio. The hedge is riskless only for a very short time interval, and its expected return, if the option is correctly priced, is equal to the riskless interest rate used in the Black-Scholes model (assuming validity of the Black-Scholes model). Accepting the standard Black-Scholes assumptions and using a reasonably short one-day holding period, the expected return on a riskless call option hedge is:
1 E(R)=
[E(¢)- C'_l
N{d,} SC N{d,}
where
(1)
E(R) =the expected one-day holding period return to a neutral option hedge, S --the price of the common stock, C = the price of the call option, N{-} = the cumulative normal distribution function,
dt
=(ln(SlK)+[r+(a212)]T)I~ET]~/2,
D.W. French and L.J. Martin, Measurement of option mispricing
541
(N{dt} is the first derivative of the Black-Scholes model, C/S. l/(N{dl}) is the Black-Scholes neutral hedge ratio, E(~) = the expected future market price of the stock at the end of the oneday holding period, E(g) = S(1 + m), E(¢~) = the expected future market price of the call at the end of the one-day holding period computed by solving the Black-Scholes call value equation evaluated at S(1 +m),K(1 +r), T,r, and [ ( l i T ) a 2 + ( 1 - 1/T)02] 1/2, K r m T 0-2
02
=the exercise price of the option, =the per day risk-free rate, = the per day expected stock return, =the time to expiration of the option, = the investor's estimate of the variance of the expected stock returns, and --the market's estimate of the variance of the expected stock returns (the implied variance of the market price of the option).
Eq. (1) is based on Rubinstein's (1984) formula for the expected future price of a call option. This formula assumes that option mispricing results from a difference between the investor's estimate of the stock variance and the market's estimate of the variance. It further asscnles that the mispricing falls as the difference between the two variance, estimates deteriorates linearly through time. The unknown parameter in eq. (1) is m, the expected stock return. Under Black-Scholes conditions of continuous trading and hedge adjustment, the neutral hedge return would be equal to the riskless rate regardless of the expected return on the stock if the option is correctly priced. Therefore, we could use any expected stock return in the formula, the simplest being a return of zero. In practice,, however, the hedge ratio changes as the stock price changes, and the hedge loses its neutral character. The overriding question, therefore, is whether daily adjustment of the hedge is sufficient to maintain an expected return for eq. (1) equal to the riskless rate for various expected stock returns. In support of the hypothesis that the discrete hedge would not lose its riskless characteristic over a reasonably short time span (e.g., one day), Galai (1983) finds that discrete adjustment of the hedge ratio is responsible for only a minor portion of the hedge return and that adjustments at one-day trading intervals do not significantly affect retdrns. Galai and Geske (1984) state that daily adjustment of the hedge ratio should be adequate, and that empirically it may be sufficient to adjust a hedged position only once per week. In addition to these arguments, we used Monte Carlo simulation to estimat¢ the
D.W. French and L.J. Martin, Measurement of option mispricing
542
expected return on a one-day risk-neutral opiton hedge. Table 2 presents the results of this simulation. The rows labelled 'Computed E(R)' represent the expected return on the position using eq. (1) less the riskless rate. To simulate expected daily returns, we generated a random vafiate from the lognormal distribution for a one-day stock return, computed the associated stock price, and used ~he Black-Scholes model to compute the call option price. We then used these option and stock prices to compute the simulated hedge return and repeated the procedure 2,000 times to find the expected hedge return. The rows labelled 'Simulated E(R)' report these returns (less the riskless rate). Note that the correctly valued options all heve a computed E(R) of zero as we would anticipate. Their counterparts in the simulated E(R) rows are all very close to zero, the small deviations resulting from random variation in the simulation process. This supports the contention that daily hedge adjustment justifies an expected riskless return for correctly valued options. Note in table 2 that all undervalued options (i.e., those whose market price is less than the true value) have negative excess expected returns, and all overvalued options' excess expected hedge returns are positive. In addition, their simulated excess returns are equal to or very close to the computed value. Table 2 also supports the idea that the expected hedge return is stable with respect to changes in the expected stock return. For example, the computed E(R) for the 30-day undervalued option is -0.091 percent for m=0, 0.10, and 0.20. An appropriate measure of option mispricing, then, is the exacted one° day excess return on a neutral option hedge. Because this return is independent of the expected stock return, any value for m may be used in the formula. Substituting a value of zero for m in eq. (1) and subtracting the riskless rate, the formula simplifies to: 1
N{d,} [ C - E ( ~ ] a= S,
C N{dl}
r, where
(2)
0t= the expected excess return to a risk-neutral option hedge (the measure of option mispricing). Correctly priced options should yield c~ values of zero. Overpriced options • (options whose market prices are greater than their model values) will yield positive values for 0t, and undervalued options will give negative ~s. Note that the mispricing measure explicitly considers all of the option pricing parameters.
D.W. French and L.J. Martin, Measurement of option mispricing
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The option hedge's • measures excess return only as a measure of mispricing, which should not imply that such returns were actually available to traders. Several studies of market efficiency, including those of Phillips and Smith (1980), Wha!ey (!982) and French and Henderson ~198!), have shown that apparent excess returns on options disappear after considering commissions and transactions costs. One note of caution is in order. Boyle and Emanuel (1980) found that when investors appreciably misestimate the expected variance term, even daily hedge returns can become quite risky. Using eq. (2) to measure the extent of option mispricing is similar t ° the approach outlined by Cox and Rubinstein (1985). They develop a strategy for screening neutral option positions based on the potential profit for the hedged position. Their process yields an expected profit to a neutral option hedge, but it would yield different results from our eq. (2) and produce a different ranking of mispriced optivns for two fundamental reasons. First is that the Cox-Rubinstein formula measures potential profit as the absolute difference between the current option value and its market price; it does not use the expected future option value as does eq. (2). Second, the CoxRubenstein technique weights the expected profit by the sensitivity of the option hedge ratio to changes in the stock price. In other words, it will favor options whose hedge ratios tend to 0e more stable when t .I,,, _ . . . o,,,,.t. . . . p n ~• change~, minimizing the risk associated with shifting hedge ratios and the need to rebalance the hedge. The Cox-Rubenstein procedure should be more effective in identifying profitable option hedges when the indended holding period is of a relatively long time span. Their technique, therefore, might help identify neutral option positions which could be favorable to a trader who is subject to the risks and costs of frequent hedge rebalancing~ but it would not necessarily produce the same ranking of mispriced options as would eq. (2). Whether the Cox-Rubenstein procedure or eq. (2) would produce greater profits to the investor is grounds for fertile research, but beyond the scope and objectives of this paper. Eq. (2) is dependent upon the assumptions underlying the Black-Scholes model. Two particularly important assumptions are that the option is European and is on a stock which pays no cash dividend. As Roll (1977) notes, American options on dividend-paying stocks may appear to be misvalued just before the ex-dividend date. To adjust the option mispricing measure for American options on dividend-paying stocks, the Roll (1977), Geske (1~79a), Whaley (1982) option pricing formula can be used in eq. 12) to compute the hedge ratio and expected option prices. In addition, if th,, exdividend date falls ~thin the holding period, the probability of early exercise, the receipt of the cash dividend, and the expected ex-dividend decline in the stock price must be considered. There are four practical differences between the excess returns approach
D.W. French and L.J. Martin, Measurement of option mispricin£
545
and the prediction error approaches that could cause the them to show different amounts of mispricing. First, the hedge ratio is always greater for the short-term option (other factors equal), resulting in more of them being sold short to establish the hedge and hence more profit per share of stock purchase. Second, different options have different prices, which causes the cash investment to establish the hedge to differ. Third, when the options are of different duration, the pricing error deteriorates much more rapidly for a short-term option than for one of longer duration. When one is comparing options with different times to expiration, this final factor probably explains most of the difference between mispricing gauged by the ~ and mispricing gauged by the APE or RPE. The fourth reason is that when comparing outof-the-money options to in-the-money options, the hedge ratio is appreciably larger for out-of-the-money options. This factor may affect some of the results of the comparison of the in- and out-of-the-money options in the Gultekin, Rogaiski and Tinic study. For out-of-the-money options, Gultekin, Rogalski and Tinic correctly concluded that the positive mean prediction errors indicated that the model underpriced those options, and, similarly, the negative errors for the in-the-money group showed them to be overpriced. They also, however, compared the mean prediction errors of the in-themoney and out-of-the-money options, concluding that the larger errors for the in-the-money options indicate that these options were more mispriced by the model. Such a procedure is of questionable value, because of the fact that the hedge ratios of the out-of-the-money options would be significantly larger than their in-the-money options indicate that these options were more mispriced by the model. Whaley (1982) found n o significant relationships between the option mispricing and the amount by which the option was away from the money, although his measure of mispricing (relative prediction error) would not necessarily fully disclose option mispridng for the reasons mentioned above. This discussion suggests that the • would be preferred over the RPE and APE for computing option mispricing on an ex ante basis. Some might criticize the 0~by noting that its value is arbitrary because of the selection of the stock as the security for the denominator. That is, for a given mispriced option, an investor could increase his excess return by hedging with a leve: ~d stock position or by establishing an option spread, therefore investing less money while earning about the same profit. While this is true, this paper proposes that the option's • as defined by eq. (2) is acceptable as a standard measure for examining option mispricing because of its advantages over the commonly used RPE and APE and because it is consistent with the theory underlying the development of the Black-Scholes model. The real test, however, is how well the methods compare empirically.
546
D.W. French and LJ. Martin, Measurement of option mispricing
4. Empirical tests The sample of option market data is from the Berkeley Options Data Base for August and September of 1979. This data base has the advantage of offering every option trade for the day, along with concurrent stock prices. For each day in the data base, we extracted every call optiot~ trade occurring between ll:00a.m, and l:00p.m, that was not an exact duplicate of a trade previously extracted. ~ This plocedure yielded a sample of 26,556 observations. This first step in computing the mispricing measure was to estimate the investor's estimates of the expected standard deviation of stock returns. This study used the weighted implied standard deviation calculated by the ChirasManaster (1978) procedure. 2 The three ex ante measures of option mispricing (the APE, RPE, and 0c) were computed using this weighted implied standard deviation and an estimate of the riskless rate equal to the rate on three-month Treasury bills. Option and stock prices for the following day were used to calculate the actual return on a riskless option hedge (buying the stock and writing the option according to the hedge ratio), which serves as the ex post measure of option mispricing. The best before-the-fact indicator of how m~lch an option is mispriced should be the ex ante measure that best predicts actual excess return, that is, the measure with the highest correlation to the ex post measure of mispricing. The best measure would be of considerable interest to option traders, because only the ex ante measures are available to them while making tradl.ng decisions. Table 3 gives the correlation coefficients of the ex post measure with the RPE, APE, and 0c-measures. Although the three values appear relatively similar, the u's correlation coefficient is significantly greater than the other two at the 98 percent confidence level. The table also gives the 98 percent confidence limits for each of the correlation coefficients) Note also that the IUsing prices from ll:00a.m, to l:00p.m, served two purposes. First, it minimized any problems that might be a ~ ' i a t e d with transactions occurring during the market opening or closing times. Second, the da:..,~o~ ~ ~,J ~.~.:..,,st-,c,c~.~.~.:~ .,1:^.~ ~e .,,~.~.-~..~ r-~ *he time period, that a subset was needed in order to bring the sample down to a manageable size. Additionally, we omitted option quotations of 1/16 and 1/8, because the chance of such an option changing in price over a one day interval is remote (the chance of the 1/16 decr.-asing in price is nil). 2The ase of implied variance does introduce bias into Black-Scholes option price estimation. Butler and Schacter (1986) have shown that, in general, expected implied variances will not provide variance estimates which equal the true variance. They develop a variance estimator which yields unbiased estimates of option prices but which remains subject to sample error in estimating the moments of the sample variance. They note that the question remains empirical whether their method or an estimate based on implied variances is subject to greater sampling error. The effects of using implied variances in this study should be negligible because Butler and Schacter's tables show that differences between usual and unbiased Black-Scholes option price estimates are very small• 3See Kleinbaum and Kupper (1978) for a discussion of Fisher's Z transformation for the computation of the confidence limits of correlation coefficients. •
8
°
*
D.W. French and L.J. Martin, Measurement of option r~Jspricing
547
Table 3 Correlation coeificients of ex ante option mispricing measures vs. ex post actual realized returns.
Technique
Correlation coelficients
98~o confidence intervals
RPE APE 0c
0.1402 0.1552 0.1854
0.1262--0.1542 0.1412--0.1691 0.1716-0.1991
Table 4 Returns on option portfolios selected using option mispricing measures.
Mispricing measure
Mean continuously compounded return
Standard deviation
Frequency of highest returns
RPE APE ~,
0.019 0.024 0.036
0.021 0.017 0.022
4 7 ~0
confidence intervals for the APE and RPF, overlap, which indicates that the differences are not significant. The correlation coefficients were disappointingly low for all three of the measures. Several factors contribute to these low correlations, including the absence of a perfect estimate of the stock variance and discrete adjustment of the hedge. Another important likely explanation is that market option prices do not fully adjust over one-day intervals to mispricing to the extent suggested by Rubinstein (1984). If the 0o's do measure mispricing more accurately, then they Should be better at identifying positions that would lead to greater returns. One fmal test uses a trading strategy to see whether option portfolios selected using 0~ would yield higher returns than those chosen using the APE or RPE. We began by ranking the sample options on each day by the 0e mispricing measure from most undervalued to most overvalued. We then used the substitute neoge tecnntquc of /~-ren~h anti H ~ u , ~ o . l (i98i) an~, ,:,ii~.~,,~ (1982) to construct systematically riskless portfolios assuming purchase of the ten undervalued call options and short sale of the ten overvalued calls times the substitute ratio. (The mechanics of the hedge are in the appendix.) The profit to the portfolio was then computed assuming all positions were closed out on the option's first trade after ll:00a.m, on the following day. We then repeated this entire procedure ranking the options by APE and then by RPE.
Table 4 gives the results. The last column in table 4 presents the number
5,48
D.W. French and LJ. Martin, Measurement of option mispricing
of times out of 41 sample dates that the measure yielded a return higher than the other two measures. The 0h which resulted in the greatest return on 30 of the 41 dates, dominated the statistic. The 0c also yielded the highest mean return. To test hypotheses about the mean returns, we used the paired t-test. The results were as follows: Ho:/;(00
- F(APE),
Ho: ~a) ---~RPE), Ho: ~(APE)-~(RPE),
t ffi4.68 reject t = 5.62 reject t-- 1.52 do not reject.
These tests show that the portfolios identified by a yielded significantly higher returns at the 0.01 level than those chosen by APE and RPE. APE returns were not significantly different from RPE returns. The returns reported in table 4 should not be misrepresented. This is not a study of market efficiency; our purpose is only to show that a identifies options that are most likely to yield excess returns when trades are available at quoted prices. Upon recomputing the table 4 returns asseming transactions costs of even less than the average bid-ask spread reported by Phillips and Smith (1980) and only $1.00 per contract, apparent excess returns were virtually eliminated, a finding which is consistent with market efficiency in the options markets. 5. Coadmioa Our study identified one source of measurement error in option pricing. Prior researchers have identified additional factors, such as dividends, nonstationary variances, transaction costs and firm leverage, that may help to explain option mispricing? It should be noted that one ad hoc adjustment that could be made to the RPE is to divide the mtasure by the number of days to expiration in order to get a measure of the expected return per day. Such a measure could be used in the correlation analysis in order to enmpare the various technioues of option misnricing. However, theoretically such a measure is inappropriate because the decay function of the option pricing model is not linear. The a performs as a better measure of option mispricing than either the APE or RPE because it is more highly correlated with ex post hedge returns and it identified sample portfolios that consistently yielded higher returns that the APE or RPE. 4For a discussion of the effects of dividends on call option prices see Roll (1977), Geske (1979a) and Whaley (1981). The effects of non-stationary variance rates are discl~ssed in Merton (1976), Patell and Wolfson (1979), Whaley and Cheung (1982), Cox and Ross (197~# and F~nc!~ (!984). The effects of transaction costs are discussed in Philli~ a ~ S~J~.ith~9~,0). M i s ~ i n g effects caused by firm leverage are reported in Geske (1979b) and Christie (19~2).
D.W. French and L.J. Martin, Measurement of option mispricing
549
Appendix: Construction of the hedged portfofio Rather than equal-weighted portfolios, which result in an extremely large numbers of cheap options in the portfolio and a very small number of the more expensive options, we weighted each individual option in the po.qfolio by an amount equal to the, reciprocal of the option's beta. The theoretical result of this would be that a given change in the market would be accompanied by the same dollar change in each option position within the portfolio. This is consistent with the reason for using equal weighting in stock portfolios. The attendant French-Henderson (1981) substitute hedge ratio is then
H--C~=ICI. ~I/C~=I Cs,il,
where
H =the quantity of each option to short, the hedge ratio, CL.i=the price of the ith long option in the portfolio, Cs.~ =the price of the jth short option in the portfolio, m =the number of different long options in the portfolio, and n =the number of different short options in the portfolio. The beta from the appropriate issue of the Value Line served as an estimate of the stock's beta coefficient. References Black, F., 1975, Fact and fantasy! in the use of options, Financial Analysts Journal 31, 36-42. Black, F. and M. Scholes, 1972, The valuation of option contracts and test of market efficiency, Journal of Finance 27, 399-417. Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Journal of Political Economy 81, 637-654. Boyle, P.P. and D. Emanuel, 1980, Discretely adjus~,.d option hedges, Journal of Financial Economics 8, 259-282. Butler, £S. and B. Schacter, 1986, Unbiased estimation of the Biack/Scholes formula, Journal of Fi-~n~! ~eonomics 15, 341-357. Chiras, D.P. and S. Manaster, 1978, The information content of option prices and a test of market efficiency.Journal of Financial Economics 6, 213-234. Christie, A.A., 1982, The stochastic behavior of common stock prices: Value, leverage, and interest rates, Journal of Financial Economics 10, 407-432. Cox, J.C. and S.A. Ross, 1976, The valuation of options for alternative stochastic processes, Journal of Financial Economics 3, 145-166. Cox, J.C. and M. Rubinstein, 1985, Option markets (Prentice Hall, Englewood Cliffs, NJ). French, D.W., 1984, The weekend effect on the distribution of stock prices: Implications for option pricing, Journal of Financial Economics 13, 547-559. French, D.W. and G.V. Henderson, Jr., 1981, Substitute hedge option portfolios: Theory and evidence, Journal of Financial Research 4, 21-31. Galai, D., 1977, Tests of market efficiency of the Chicago Board Options Exchange, Journal of Business 50, 167-197. Galai, D., 1983, The component of the return from hedging options against stocks, Journal of Business 56, 45-54.
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D.W. French and L,J. Martin, Measurement of option mispricing
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