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Chicago board call options as predictors of common stock price changes
Journal of Econometrics
4 (1976) 101-l 13. @INorth-Holland
Publishing Company
CHICAGO BOARD CALL OPTIONS AS PREDICTORS OF COMMON STOCK PRICE CHANGES D. PANTON Uniwrsity
of Kansas, Lawrence,
Received November
Kansas 66045,
U.S.A.
1974. revised version received June 1975
The intent in this study is to determine, using response surface methodology, whether call option prices can be relied upon to predict future rises in common stock prices. If call option prices are bid up by insiders prior to the time that new information becomes available to stock traders, recognition of this price action could form the basis of a stock trading strategy yielding returns superior to buy and hold. Evidence of such an advantage would be inconsistent with the efficient market hypothesis.
1. Introduction If an insider were to acquire reliable information indicating an impending rise in the price of an identified common stock, he could use that information for personal benefit by purchasing any of several financial instruments whose performance is related to that of the common stock: warrants, convertible bonds, convertible preferred, and call options. Unless some advantage were to derive from purchasing one of the above assets in preference to the others, we would expect new information to be reflected simultaneously in all instruments having equity claim. However, the greater leverage possible through purchase of call options or warrants may constitute an important advantage to the insider having limited, liquid funds available for investment. Although, normally, leverage includes both the disadvantage of increased loss potential as well as the advantage of increased gain potential, the insider in possession of new information may reasonably discount the possibility of loss. Recognizing that insiders may be influenced by the leverage consideration enough to trade initially in the options’ market, we pose the following question: Do call option prices tend to increase prior to rises in the price of the underlying stock? Shulman (1967, p. 66) cites an example suggesting that this leading relation has held at least once in the past. ‘Although much of our analysis is also applicable to warrants and rights, we restrict our study to CBOE calls.
102
D. Panton,
Chicago Board call options
Five percent of option buying comes from employees, officers, or directors of firms who may have information of a favorable nature concerning major progress or increases in profits before such information is generally public. . . . In 1962 Shell Oil decided to expand in Canada by taking over the Canadian Oil Company. Canadian Oil was then selling for $35 per share, and the Shell board of directors had voted to offer the equivalent of $55 for every share of Canadian oil. Suddenly we were swamped with requests to buy thirty-day calls on Canadian Oil for which we were offered $3.50 per share, a huge sum considering that a ninety-five-day call normally would sell for that figure. These orders all emanated from New York and were unlimited; that is, they offered to buy any size. This is what actually happened. Options were purchased on many thousands of shares. The profits (to the buyers) were immense. The intent in this paper is to determine, using a well specified procedure, whether call option prices can be relied upon to predict future rises in common stock prices. If, in fact, call prices are bid up by insiders prior to the time that new information becomes available to stock traders, recognition of this price action of calls could form the basis of a stock trading strategy yielding returns superior to buy and hold. Evidence of such a strategy would be inconsistent with the efficient market hypothesis. The remainder of the paper is divided as follows: section 2 consists of a discussion of some relevant literature, section 3 is a description of the data used in the current study, section 4 is a presentation of the methodology, and section 5 is a summary of results. 2. Some relevant literature 2.1. The theory of call pricing Three basic approaches to call and warrant pricing models have been prevalent in the finance literature: (1) discounted expected-value models, (2) recursive optimization models, and (3) general equilibrium models. 2.1.1.
Discounted
expected-value
models
Most call pricing models are based upon calculation of the expected exercise value of the call immediately prior to expiration of the contract. For each possible price level, P, of the underlying stock, the authors derive the exercise value of the call, C(P), posit the form of the probability distribution of stock prices, F(P), and finally calculate the expected value. The expected (future) value is then discounted at a rate (p) which seems appropriate to the author. Present value price = eWPtf,” C(P) dF(P).
103
D. Panton, Chicago Board call oprions
Examples of call, or warrant, models fitting this general description by Sprenkle (1961), Boness (1964), and Baumol et al. (1966). 2.1.2. Recursive
are given
optimization models
The above studies implicitly assumed that the call would not be exercised prior to that point in time immediately preceding the call’s expiration. Samuelson (1965) and Chen (1970) took into consideration the privilege of exercising the call at any time before the expiration date. Essentially the method consists of a multiperiod decision problem in which the call owner: (1) decides to exercise if expected returns during the next period do not satisfy minimum requirements, or (2) to continue to hold the call for one more period, at the end of which he faces the same choices. Call life is divided into a series of periods - each period having a critical exercise value which, if attained by the call, results in immediate exercise of the option. Having decided upon a course of action for each possible value of the underlying stock at each period end, Samuelson used recursive optimization to find the value of a call with n periods of life remaining. Specifically, the valuation model states that calls (and warrants) are priced at the maximum of: (1) zero, (2) the present exercise value, or (3) expected exercise value one period in the future, discounted at a rate related to the risk class of the underlying stock. For a call having one discrete unit of time left before maturity, $1 striking price, the valuation equation may be expressed as FI(X)
= max [O, X-
1, e-O j,” F,(XZ) dP(Z;
I)].
continuously
and having a
(1)
Having determined the locus of points F,_,(X) for all values of stock prices (X) at tn-l, the locus of call prices for n time units before maturity may be derived from the recursive equation F”(X) = max [0, X-
1, emPs,” F”_,_,(XZ) dP(Z;
I)].
(2)
Given this valuation framework, Samuelson demonstrated the possibility of a situation arising such that present exercise value exceeds discounted future expected value. For exercise values above this ‘critical value’, C, = (Xa rational
investor
1) > e-pj,”
F,_,(XZ)
should exercise early.
dP(Z;
I),
(3)
104
2.1.3.
D. Panron, Chicago Board call options General equilibrium models
A more theoretically-satisfying, general equilibrium approach has been offered by Black and Scholes (1971). They demonstrate that, in equilibrium, the expected return on a hedged position consisting of calls and common stock must be equal to the return on a riskless asset. The only call price which would prevent the establishment of a profitable arbitrage operation was shown to be w = xN(d,) - ce-“*N(d,),
(4)
where w = the price of an option for a single share of stock, x = the current price of the stock, c = the striking price of the option, r = the short-term
rate of interest,
t* = the duration d
of the option, l/2 a2)t*
= In (x/c)+@+ 1
04/t*
,
d, = d,-o2/t*, N(d) = the value of the cumulative a2 = the variance
normal
density function,
rate of the return on the stock.
The Black and Scholes model is often cited in the current familiar to those interested in the theory of call pricing. 2.2. ESJient
literature
and is
market hypothesis
A concept which is fundamental to the issue raised in this paper is that of efficient capital markets. In such markets, information is widely, quickly and cheaply available to investors. This information includes all that is knowable and relevant for judging securities, and is very rapidly reflected in security prices. The hypothesis that capital markets are efficient is often made tenable with the assumption that the conditions of market equilibrium can be expressed in terms of expected returns. Fama (1970) has stated that most expected return theories can be expressed in the following manner:
(5) where E is the expected value operator; pjr is the price of security j at time t; pi, t+l is its price at t+ 1 (with reinvestment of any intermediate cash income from the security); rj, t+l is the one-period percentage return (pi, t+ 1 -pj,)/ pjt; Qj, is a general symbol for whatever set of information is assumed to be
D. Panton, Chicago Board call options
105
‘fully reflected’ in the price at t; and the tildes indicate that pi, 1+1 and rj, t+l are random variables at t. This model has the important implication that trading systems based on information contained in @ cannot generate expected returns in excess of the market equilibrium level of expected returns. Therefore, in an efficient market one would not expect readily available call price quotes to serve as the basis of a trading system which outperforms equivalent risk buy-and-hold strategies. Using the model developed in their 1971 paper, Black and &holes (1972) investigated the possibility of excess profit potential deriving from the identification of underpriced or overpriced call options. They found that, although the model tended to overestimate the value of a call on a high variance security and underestimate the value of a call on a low variance security, these deviations were not enough to offset transaction costs which would be incurred in a hedging operation. 3. Data Essentially, the primary data consist of three sets of observations: (1) call prices, (2) the values of recognized influencing variables (call life and underlying stock price at the time of the call transaction),’ and (3) percentage changes in the prices of underlying stocks over post-transaction time periods. These data will be used to determine whether those calls which are priced higher than explained by fitted response surfaces, are actually leading indicators of future stock price increases. The Chicago Board Options Exchange has furnished computer records of call option transactions on twenty-two stocks, covering the period April 26, 1973 to July 31, 1973. Each record contains the symbol of the underlying stock, the call expiration date, the exercise (striking) price, the stock price at the time of the call transaction, and the price of the call. From these data were selected the six call issues indicated below: Company
Striking3 price
1. 2. 3. 4. 5. 6.
$ 25 $100 $ 30 $130 $ 25 $160
Loews (LTR) Merck (MRK) Northwest Airlines Polaroid (PRD) Pennzoil (PZL) Xerox (XRX)
(NWA)
2Each fitted response surface was generated using empirical data from only those calls (1) related to the particular stock, and (2) having identical striking price. %a!1 options traded on the Chicago Board Options Exchange have fixed striking prices. These striking prices may differ greatly from the price of the underlying stock at the time of the call transaction.
106
D. Panton, Chicago Board call options
Only six issues were chosen because to have chosen more would have resulted in an unmanageable amount of computation in the analysis. These particular issues were chosen because: (I) they represented a diverse group of industries, (2) a large number of data points were available,4 and (3) appreciable variation in call prices and stock prices existed over the period of observation.
4. Methodology 4.1. The response surface Response surface methodology, sometimes referred to as RSM, is a body of techniques initially developed by Box and Wilson (1951) for the study of unknown functions through empirically-determined representations of those functions. The approximating representations usually take the form of a truncated power-series expansion (Maclaurin or Taylor) or, less frequently, a Fourier series expansion. Often the system being examined contains one particular feature of control or prediction interest; this feature is termed the response variable. Also in the system are control or a&ctor variables which are of interest primarily because of their effects on the response variable. The theory of call option pricing has been studied extensively in the academic literature. However, the pricing models proposed lack agreement not only on the list of variables which influence call price, but also on the nature of the structural relationship between response and influencing variables. In order to develop an underlying basis of theory for the structural relation, authors have been obliged to make assumptions regarding the distribution of stock price changes, feasible exercise timing, and/or risk attitudes of investors. Although the various models differ in structure considerably, each invariably includes the common stock price and option life as influencing variables. 5 In the absence of general agreement concerning the specific structure of the call option response surface, an approximation may be obtained by fitting a polynomial in the control variables (xi, . . ., x,,J, in which all terms including cross-product terms, through a given order d are included. This is equivalent to assuming that the true, but unknown, function may be represented to a sufficient approximation by a truncated power-series expansion, ignoring terms of order higher than d. For example, consider a response variable 2 %ixty-nine trading days were represented in the sample data. On most of those days, call issues having three different expiration dates were traded. The maximum number of observations, assuming three expiration dates for each call having specified exercise price, is therefore (69.3 = 207). As is often the case with empirical studies, additional observations would probably strengthen the analysis. 5Black and Scholes (1973, p. 641) state that, given the assumptions of their model, ‘the value of the option will depend only on the price of the stock and time and on variables that are taken to be known constants’.
D. Punton, Chicago Board call options
107
which is believed to be related to two observable, control variables x1 and x2. A third-degree Taylor expansion of Z(x,, x2) around x1 = x2 = 0 may be written N
z =Neo$ Xl &+x2& -wl,
_(
The associated
least-squares
1
estimation
2
>
x2)-
(6)
of Z is
(7) or, in matrix notation,
2 = x$,, where
xj =
(1)
x1,
x2,
xl, . . ,, x3.
In general, there exist (“+j-’ ) terms of any given order u in a polynomial expansion having k control variables. As demonstrated by Feller (1957), the power series containing all terms up to and including the dth order, therefore, consists of
j.
(““6-l) =(d;k)
coded variables. An equation containing all coefficients up to the dth order is said to be of dth degree. Since the number of coefficients to be estimated increases rapidly with higherdegree expansions, the experimenter may be constrained to consider only those power-series polynomials of relatively low degree. Table 1 shows the number of terms contained in equations of degree 1, 2, 3 and 4 when the number of influencing variables is 2, 3, 4 or 5. The number of experiments (simultaneous observation of response and affector variables) must be at least as great as the number of coefficients fitted. For example, to fit a third-degree polynomial having five affector variables requires an absolute minimum of (3 < “) = 56 simultaneous observations on all variables. In our study the fitted response surface approximations are of little interest in themselves, but merely a mechanism to eliminate that portion of call price
108
D. Panton, Chicago Board call options
variability caused by two recognized influencing factors. By comparing observed option prices with those derived from the response surface, we may identify calls which cost more than would have been predicted in the basis of option life and underlying stock price alone. Although we have chosen to include only option life and stock price as major control variables (keeping other recognized control variables constant), the model ifself is completely general. The number of control variables that may be included is limited only by the expansion order desired and the number of available observations.
Number
k = number of influencing variables 2 3 4 5
Table 1 to be fitted for polynomial varying degree.
of coefficients
expansions
of
d = degree of fitted polynomial 1st
2nd
3rd
4th
5th
3 4
6 10 15 21
10 20 35 56
15 35 70 126
21 56 126 252
2
4.2. Test design First, second, and third degree response surfaces were approximated each of the six calls previously cited. These surfaces took the form First degree :
2 = b,+b,x,+b2x2,
Second degree:
2 = b, + b,x, + b,x,
Third degree :
2 = b,+b,xl+b,x,+b,x;+b4x1xZ+b5x;+bsx:
+ b,xf
+ b,x,x,
for
+ b,x&
+b,x:x,+b,x,x;+b,x;, where 2 = predicted (least squares) call price, x1 = call life (in days), and x2 = price of underlying stock. Calculated iso-price contours generated by first,- second,- and third-degree expansions for the Polaroid call are shown in fig. 1. Fig. 2 is a perspective representation of the third-degree expansion. Differences between calculated call prices and observed call prices were then expressed as percentages of the calculated call price. This resulted in residual vectors having elements rijk as defined below
109
D. Panton, Chicago Board call options
where i is the call issue, j is the expansion order, k is the observation number, Zijk is the observed call price, and giik is the calculated option price. Since we compared the actual call prices of six issues with estimated prices given by first-, second-, and third-degree fitted approximations to the unknown
20
50
80
110 Option
Option
140
170
200
life A. first degree
life C. third
230
260
20
50
80
110 Option
140
170
life 6. second
200
230
260
degree
degree
Fig. 1. Call option iso-price contours generated by first-, second-, and third-degree surface approximations (Polaroid/l30).
response
response surfaces, eighteen residual, or deviation, vectors were obtained. A positive residual indicated that a call had sold for an amount higher than that explained by the fitted response surface against which it was compared. Similarly, a negative residual indicated that the call option sold for less than would have been expected. Coefficients of determination (R’) for progressively higherorder expansions are shown in table 2 below.
110
D. Panton,
Fig. 2. Representation
Chicago Board call options
of third-degree
trend surface for Polaroid/$130
call option.
Table 2 Coefficients of determination for first-, second-, and third-degree response surface approximations. Degree of expansion Call issue
1
2
3
LTR/25 MRK/lOO NWA/30 PRD/130 PZL/25 XRX/160
0.851 0.850 0.869 0.862 0.858 0.844
0.855 0.861 0.886 0.901 0.862 0.898
0.870 0.873 0.900 0.907 0.874 0.907
From observations of stock prices subsequent to the call option transactions, we recorded the following three measures of stock price performance: (1) one-month percentage changep, = (MI - M,)/M,, where MI = the price of the underlying stock one month after the date of the call transaction, M, = the price of the underlying stock on the day of the call transaction; (2) two-month percentage changep, = (M, -M&M,, where M, = the price of the underlying stock two months after the date of the call transaction, M, = as previously defined ;
D. Panton, Chicago Board call options
111
(3) maximum percentage increase over two months pH = (MH-M,J/M,,, where MH = the highest price attained by the underlying stock over the twomonth period subsequent to the call transaction, M, = as previously defined. Sample (pi,
p2, pH)
correlations between the residuals (rijk) and were then calculated for all six call issues.
Correlations
MRK/$lOO NWA/$30 PRD/$130 PZL/$25 XRX/$160
‘RejectH,:p
price
relatives
Table 3 between call option residuals and subsequent stock price relatives.
Call issue Stock/striking price LTR/$25
stock
Degree of expapsion 1 PI PZ PH p1 P2 PH Pl PZ PH Pl PZ PH Pl P2 PH PI P2 PH
0.1603 0.0007 -0.0285 -0.0397 0.1387 0.0250 -0.0123 -0.1588 -0.0232 0.2874a 0.2556” -0.0108 -0.0375 0.0608 0.0105 - 0.0202 0.2809” 0.0667
2 -0.1262 -0.0139 0.0623 - 0.0086 0.0196 -0.1115 - 0.0184 -0.1309 0.0063 0.3205” 0.3726” -0.1359 - 0.0414 -0.0035 -0.0412 0.0874 0.0043 -0.1003
3 -0.1096 -0.1403 0.1842a 0.0733 -0.1354 -0.1160 0.0153 -0.1468 - 0.0069 0.3491” 0.5327” -0.1442 - 0.0024 -0.1272 -0.1445 0.0702 -0.0812 - 0.0306
= 0; HA:p # Oat a = 0.01.
4.3. Results To support the research hypothesis of predictive information being contained in call prices, we would expect significant positive correlation between call price residuals and post-call transaction-stock price relatives.6 However, as shown in table 3, positive correlations are the exception rather than the rule. Summarizing the results, we find 6Testing hypotheses by counting positive correlation coefficients has its shortcomings, however. The residuals generated by a third-degree response surface are not independent of residuals generated by first- and second-degree response surfaces.
112
D. Panton, Chicago Board call options
(1) Of the fifty-four
Pearson product-moment correlations calculated, twentythree are positive and thirty-one are negative. At the 0.01 level, eight correlations are significantly different from zero; all eight are positive. are related to Polaroid Corporation. (2) Six of the eight significant correlations Residuals from first-, second-, and third-degree response surfaces are positively correlated with one- and two-month subsequent stock price relatives. Only the Polaroid call has significant positive correlation in more than one(3) degree expansion. 5. Summary and conclusions We have addressed several matters related to informational content in call option prices. First, we proposed a justification for insiders’ use of the options market in preference to the stock market. On a per dollar basis, calls provide greater claim to stock price gains than do most other financial instruments. This characteristic is an advantage to the insider wishing to reap maximum profits subject to the constraint of having limited, liquid investment capital. Second, we suggested a systematic procedure for identifying those call options which are underpriced or overpriced when compared to a fitted powerseries approximation of the unknown response surface. Third, deviations from the fitted surfaces were correlated with three sets of stock price relatives: (1) percentage change in stock price one month subsequent to the call transaction, (2) percentage change in stock price two months subsequent to the call transaction, and (3) maximum percentage increase in the stock price during the two-month period subsequent to the call transaction A high degree of positive correlation between the residual series and the stock price relatives would have been interpreted as indicating the presence of predictive informational content in call prices. However, since only in the case of Polaroid do we have significant positive correlation in a majority of the cases examined, we must admit that our results are not strong enough to state that call options, in general, are valid predictors of stock price movements. We must conclude that, if predictive information is present in call options, it has not been conclusively demonstrated here. References Baumol, W.J., B.G. Malkiel and R.E. Quandt, 1966, The valuation of convertible securities, Quarterly Journal of Economics 80,48859. Black, F. and M. Scholes, 1971, The pricing of options and corporate liabilities, mimeo. (M.I.T., Cambridge); revised version, 1973, Journal of Political Economy 81,637-654. Black, F. and M. Scholes, 1972, The valuation of option contracts and a test of market efficiency, Journal of Finance 27,399-417. Boness, A.J., 1964, Elements of a theory of stock-option value, Journal of Political Economy 72,163-175. Box, G.E.P. and N.R. Draper, 1959, A basis for the selection of a response surface design, Journal of the American Statistical Association 54,622-654.
D. Panton, Chicago Board call options
113
Box, G.E.P. and K.B. Wilson, 1951, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society (Series B) 13, l-45. Chayes, F., 1970, On deciding whether trend surfaces of progressively higher order are meaningful, Geological Society of America Bulletin 81,1273-1278. Chen, A.H.Y., 1970, A model of warrant pricing in a dynamic market, Journal of Finance 15, 1041-1059. Fama, E.F., 1970, Efficient capital markets: A review of theory and empirical work, Journal of Finance 15,383-417. Feller, W., 1957, An introduction to probability theory and its applications (Wiley, New York) 63-65. Hill, W.J. and W.G. Hunter, 1966, A review of response surface methodology: A literature survey, Technometrics 8570-590. Merton, R.C., 1973, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 141-183. Myers, R.H., 1971, Response surface methodology (Allyn and Bacon, Boston). Samuelson, P.A., 1965, Rational theory of warrant pricing, The Industrial Management Review (now Sloan Management Review) 6,13-31. Shulman, M., 1967, Anyone can make a million (McGraw-Hill, Toronto). Sprenkle, C.M., 1961, Warrant prices as indicators of expectations and preferences, Yale Economic Essays 1,178-231. Watson, G.S., 1971, Trend surface analysis, Journal of the International Association of Mathematical Geology 3,215-226.
4 (1976) 101-l 13. @INorth-Holland
Publishing Company
CHICAGO BOARD CALL OPTIONS AS PREDICTORS OF COMMON STOCK PRICE CHANGES D. PANTON Uniwrsity
of Kansas, Lawrence,
Received November
Kansas 66045,
U.S.A.
1974. revised version received June 1975
The intent in this study is to determine, using response surface methodology, whether call option prices can be relied upon to predict future rises in common stock prices. If call option prices are bid up by insiders prior to the time that new information becomes available to stock traders, recognition of this price action could form the basis of a stock trading strategy yielding returns superior to buy and hold. Evidence of such an advantage would be inconsistent with the efficient market hypothesis.
1. Introduction If an insider were to acquire reliable information indicating an impending rise in the price of an identified common stock, he could use that information for personal benefit by purchasing any of several financial instruments whose performance is related to that of the common stock: warrants, convertible bonds, convertible preferred, and call options. Unless some advantage were to derive from purchasing one of the above assets in preference to the others, we would expect new information to be reflected simultaneously in all instruments having equity claim. However, the greater leverage possible through purchase of call options or warrants may constitute an important advantage to the insider having limited, liquid funds available for investment. Although, normally, leverage includes both the disadvantage of increased loss potential as well as the advantage of increased gain potential, the insider in possession of new information may reasonably discount the possibility of loss. Recognizing that insiders may be influenced by the leverage consideration enough to trade initially in the options’ market, we pose the following question: Do call option prices tend to increase prior to rises in the price of the underlying stock? Shulman (1967, p. 66) cites an example suggesting that this leading relation has held at least once in the past. ‘Although much of our analysis is also applicable to warrants and rights, we restrict our study to CBOE calls.
102
D. Panton,
Chicago Board call options
Five percent of option buying comes from employees, officers, or directors of firms who may have information of a favorable nature concerning major progress or increases in profits before such information is generally public. . . . In 1962 Shell Oil decided to expand in Canada by taking over the Canadian Oil Company. Canadian Oil was then selling for $35 per share, and the Shell board of directors had voted to offer the equivalent of $55 for every share of Canadian oil. Suddenly we were swamped with requests to buy thirty-day calls on Canadian Oil for which we were offered $3.50 per share, a huge sum considering that a ninety-five-day call normally would sell for that figure. These orders all emanated from New York and were unlimited; that is, they offered to buy any size. This is what actually happened. Options were purchased on many thousands of shares. The profits (to the buyers) were immense. The intent in this paper is to determine, using a well specified procedure, whether call option prices can be relied upon to predict future rises in common stock prices. If, in fact, call prices are bid up by insiders prior to the time that new information becomes available to stock traders, recognition of this price action of calls could form the basis of a stock trading strategy yielding returns superior to buy and hold. Evidence of such a strategy would be inconsistent with the efficient market hypothesis. The remainder of the paper is divided as follows: section 2 consists of a discussion of some relevant literature, section 3 is a description of the data used in the current study, section 4 is a presentation of the methodology, and section 5 is a summary of results. 2. Some relevant literature 2.1. The theory of call pricing Three basic approaches to call and warrant pricing models have been prevalent in the finance literature: (1) discounted expected-value models, (2) recursive optimization models, and (3) general equilibrium models. 2.1.1.
Discounted
expected-value
models
Most call pricing models are based upon calculation of the expected exercise value of the call immediately prior to expiration of the contract. For each possible price level, P, of the underlying stock, the authors derive the exercise value of the call, C(P), posit the form of the probability distribution of stock prices, F(P), and finally calculate the expected value. The expected (future) value is then discounted at a rate (p) which seems appropriate to the author. Present value price = eWPtf,” C(P) dF(P).
103
D. Panton, Chicago Board call oprions
Examples of call, or warrant, models fitting this general description by Sprenkle (1961), Boness (1964), and Baumol et al. (1966). 2.1.2. Recursive
are given
optimization models
The above studies implicitly assumed that the call would not be exercised prior to that point in time immediately preceding the call’s expiration. Samuelson (1965) and Chen (1970) took into consideration the privilege of exercising the call at any time before the expiration date. Essentially the method consists of a multiperiod decision problem in which the call owner: (1) decides to exercise if expected returns during the next period do not satisfy minimum requirements, or (2) to continue to hold the call for one more period, at the end of which he faces the same choices. Call life is divided into a series of periods - each period having a critical exercise value which, if attained by the call, results in immediate exercise of the option. Having decided upon a course of action for each possible value of the underlying stock at each period end, Samuelson used recursive optimization to find the value of a call with n periods of life remaining. Specifically, the valuation model states that calls (and warrants) are priced at the maximum of: (1) zero, (2) the present exercise value, or (3) expected exercise value one period in the future, discounted at a rate related to the risk class of the underlying stock. For a call having one discrete unit of time left before maturity, $1 striking price, the valuation equation may be expressed as FI(X)
= max [O, X-
1, e-O j,” F,(XZ) dP(Z;
I)].
continuously
and having a
(1)
Having determined the locus of points F,_,(X) for all values of stock prices (X) at tn-l, the locus of call prices for n time units before maturity may be derived from the recursive equation F”(X) = max [0, X-
1, emPs,” F”_,_,(XZ) dP(Z;
I)].
(2)
Given this valuation framework, Samuelson demonstrated the possibility of a situation arising such that present exercise value exceeds discounted future expected value. For exercise values above this ‘critical value’, C, = (Xa rational
investor
1) > e-pj,”
F,_,(XZ)
should exercise early.
dP(Z;
I),
(3)
104
2.1.3.
D. Panron, Chicago Board call options General equilibrium models
A more theoretically-satisfying, general equilibrium approach has been offered by Black and Scholes (1971). They demonstrate that, in equilibrium, the expected return on a hedged position consisting of calls and common stock must be equal to the return on a riskless asset. The only call price which would prevent the establishment of a profitable arbitrage operation was shown to be w = xN(d,) - ce-“*N(d,),
(4)
where w = the price of an option for a single share of stock, x = the current price of the stock, c = the striking price of the option, r = the short-term
rate of interest,
t* = the duration d
of the option, l/2 a2)t*
= In (x/c)+@+ 1
04/t*
,
d, = d,-o2/t*, N(d) = the value of the cumulative a2 = the variance
normal
density function,
rate of the return on the stock.
The Black and Scholes model is often cited in the current familiar to those interested in the theory of call pricing. 2.2. ESJient
literature
and is
market hypothesis
A concept which is fundamental to the issue raised in this paper is that of efficient capital markets. In such markets, information is widely, quickly and cheaply available to investors. This information includes all that is knowable and relevant for judging securities, and is very rapidly reflected in security prices. The hypothesis that capital markets are efficient is often made tenable with the assumption that the conditions of market equilibrium can be expressed in terms of expected returns. Fama (1970) has stated that most expected return theories can be expressed in the following manner:
(5) where E is the expected value operator; pjr is the price of security j at time t; pi, t+l is its price at t+ 1 (with reinvestment of any intermediate cash income from the security); rj, t+l is the one-period percentage return (pi, t+ 1 -pj,)/ pjt; Qj, is a general symbol for whatever set of information is assumed to be
D. Panton, Chicago Board call options
105
‘fully reflected’ in the price at t; and the tildes indicate that pi, 1+1 and rj, t+l are random variables at t. This model has the important implication that trading systems based on information contained in @ cannot generate expected returns in excess of the market equilibrium level of expected returns. Therefore, in an efficient market one would not expect readily available call price quotes to serve as the basis of a trading system which outperforms equivalent risk buy-and-hold strategies. Using the model developed in their 1971 paper, Black and &holes (1972) investigated the possibility of excess profit potential deriving from the identification of underpriced or overpriced call options. They found that, although the model tended to overestimate the value of a call on a high variance security and underestimate the value of a call on a low variance security, these deviations were not enough to offset transaction costs which would be incurred in a hedging operation. 3. Data Essentially, the primary data consist of three sets of observations: (1) call prices, (2) the values of recognized influencing variables (call life and underlying stock price at the time of the call transaction),’ and (3) percentage changes in the prices of underlying stocks over post-transaction time periods. These data will be used to determine whether those calls which are priced higher than explained by fitted response surfaces, are actually leading indicators of future stock price increases. The Chicago Board Options Exchange has furnished computer records of call option transactions on twenty-two stocks, covering the period April 26, 1973 to July 31, 1973. Each record contains the symbol of the underlying stock, the call expiration date, the exercise (striking) price, the stock price at the time of the call transaction, and the price of the call. From these data were selected the six call issues indicated below: Company
Striking3 price
1. 2. 3. 4. 5. 6.
$ 25 $100 $ 30 $130 $ 25 $160
Loews (LTR) Merck (MRK) Northwest Airlines Polaroid (PRD) Pennzoil (PZL) Xerox (XRX)
(NWA)
2Each fitted response surface was generated using empirical data from only those calls (1) related to the particular stock, and (2) having identical striking price. %a!1 options traded on the Chicago Board Options Exchange have fixed striking prices. These striking prices may differ greatly from the price of the underlying stock at the time of the call transaction.
106
D. Panton, Chicago Board call options
Only six issues were chosen because to have chosen more would have resulted in an unmanageable amount of computation in the analysis. These particular issues were chosen because: (I) they represented a diverse group of industries, (2) a large number of data points were available,4 and (3) appreciable variation in call prices and stock prices existed over the period of observation.
4. Methodology 4.1. The response surface Response surface methodology, sometimes referred to as RSM, is a body of techniques initially developed by Box and Wilson (1951) for the study of unknown functions through empirically-determined representations of those functions. The approximating representations usually take the form of a truncated power-series expansion (Maclaurin or Taylor) or, less frequently, a Fourier series expansion. Often the system being examined contains one particular feature of control or prediction interest; this feature is termed the response variable. Also in the system are control or a&ctor variables which are of interest primarily because of their effects on the response variable. The theory of call option pricing has been studied extensively in the academic literature. However, the pricing models proposed lack agreement not only on the list of variables which influence call price, but also on the nature of the structural relationship between response and influencing variables. In order to develop an underlying basis of theory for the structural relation, authors have been obliged to make assumptions regarding the distribution of stock price changes, feasible exercise timing, and/or risk attitudes of investors. Although the various models differ in structure considerably, each invariably includes the common stock price and option life as influencing variables. 5 In the absence of general agreement concerning the specific structure of the call option response surface, an approximation may be obtained by fitting a polynomial in the control variables (xi, . . ., x,,J, in which all terms including cross-product terms, through a given order d are included. This is equivalent to assuming that the true, but unknown, function may be represented to a sufficient approximation by a truncated power-series expansion, ignoring terms of order higher than d. For example, consider a response variable 2 %ixty-nine trading days were represented in the sample data. On most of those days, call issues having three different expiration dates were traded. The maximum number of observations, assuming three expiration dates for each call having specified exercise price, is therefore (69.3 = 207). As is often the case with empirical studies, additional observations would probably strengthen the analysis. 5Black and Scholes (1973, p. 641) state that, given the assumptions of their model, ‘the value of the option will depend only on the price of the stock and time and on variables that are taken to be known constants’.
D. Punton, Chicago Board call options
107
which is believed to be related to two observable, control variables x1 and x2. A third-degree Taylor expansion of Z(x,, x2) around x1 = x2 = 0 may be written N
z =Neo$ Xl &+x2& -wl,
_(
The associated
least-squares
1
estimation
2
>
x2)-
(6)
of Z is
(7) or, in matrix notation,
2 = x$,, where
xj =
(1)
x1,
x2,
xl, . . ,, x3.
In general, there exist (“+j-’ ) terms of any given order u in a polynomial expansion having k control variables. As demonstrated by Feller (1957), the power series containing all terms up to and including the dth order, therefore, consists of
j.
(““6-l) =(d;k)
coded variables. An equation containing all coefficients up to the dth order is said to be of dth degree. Since the number of coefficients to be estimated increases rapidly with higherdegree expansions, the experimenter may be constrained to consider only those power-series polynomials of relatively low degree. Table 1 shows the number of terms contained in equations of degree 1, 2, 3 and 4 when the number of influencing variables is 2, 3, 4 or 5. The number of experiments (simultaneous observation of response and affector variables) must be at least as great as the number of coefficients fitted. For example, to fit a third-degree polynomial having five affector variables requires an absolute minimum of (3 < “) = 56 simultaneous observations on all variables. In our study the fitted response surface approximations are of little interest in themselves, but merely a mechanism to eliminate that portion of call price
108
D. Panton, Chicago Board call options
variability caused by two recognized influencing factors. By comparing observed option prices with those derived from the response surface, we may identify calls which cost more than would have been predicted in the basis of option life and underlying stock price alone. Although we have chosen to include only option life and stock price as major control variables (keeping other recognized control variables constant), the model ifself is completely general. The number of control variables that may be included is limited only by the expansion order desired and the number of available observations.
Number
k = number of influencing variables 2 3 4 5
Table 1 to be fitted for polynomial varying degree.
of coefficients
expansions
of
d = degree of fitted polynomial 1st
2nd
3rd
4th
5th
3 4
6 10 15 21
10 20 35 56
15 35 70 126
21 56 126 252
2
4.2. Test design First, second, and third degree response surfaces were approximated each of the six calls previously cited. These surfaces took the form First degree :
2 = b,+b,x,+b2x2,
Second degree:
2 = b, + b,x, + b,x,
Third degree :
2 = b,+b,xl+b,x,+b,x;+b4x1xZ+b5x;+bsx:
+ b,xf
+ b,x,x,
for
+ b,x&
+b,x:x,+b,x,x;+b,x;, where 2 = predicted (least squares) call price, x1 = call life (in days), and x2 = price of underlying stock. Calculated iso-price contours generated by first,- second,- and third-degree expansions for the Polaroid call are shown in fig. 1. Fig. 2 is a perspective representation of the third-degree expansion. Differences between calculated call prices and observed call prices were then expressed as percentages of the calculated call price. This resulted in residual vectors having elements rijk as defined below
109
D. Panton, Chicago Board call options
where i is the call issue, j is the expansion order, k is the observation number, Zijk is the observed call price, and giik is the calculated option price. Since we compared the actual call prices of six issues with estimated prices given by first-, second-, and third-degree fitted approximations to the unknown
20
50
80
110 Option
Option
140
170
200
life A. first degree
life C. third
230
260
20
50
80
110 Option
140
170
life 6. second
200
230
260
degree
degree
Fig. 1. Call option iso-price contours generated by first-, second-, and third-degree surface approximations (Polaroid/l30).
response
response surfaces, eighteen residual, or deviation, vectors were obtained. A positive residual indicated that a call had sold for an amount higher than that explained by the fitted response surface against which it was compared. Similarly, a negative residual indicated that the call option sold for less than would have been expected. Coefficients of determination (R’) for progressively higherorder expansions are shown in table 2 below.
110
D. Panton,
Fig. 2. Representation
Chicago Board call options
of third-degree
trend surface for Polaroid/$130
call option.
Table 2 Coefficients of determination for first-, second-, and third-degree response surface approximations. Degree of expansion Call issue
1
2
3
LTR/25 MRK/lOO NWA/30 PRD/130 PZL/25 XRX/160
0.851 0.850 0.869 0.862 0.858 0.844
0.855 0.861 0.886 0.901 0.862 0.898
0.870 0.873 0.900 0.907 0.874 0.907
From observations of stock prices subsequent to the call option transactions, we recorded the following three measures of stock price performance: (1) one-month percentage changep, = (MI - M,)/M,, where MI = the price of the underlying stock one month after the date of the call transaction, M, = the price of the underlying stock on the day of the call transaction; (2) two-month percentage changep, = (M, -M&M,, where M, = the price of the underlying stock two months after the date of the call transaction, M, = as previously defined ;
D. Panton, Chicago Board call options
111
(3) maximum percentage increase over two months pH = (MH-M,J/M,,, where MH = the highest price attained by the underlying stock over the twomonth period subsequent to the call transaction, M, = as previously defined. Sample (pi,
p2, pH)
correlations between the residuals (rijk) and were then calculated for all six call issues.
Correlations
MRK/$lOO NWA/$30 PRD/$130 PZL/$25 XRX/$160
‘RejectH,:p
price
relatives
Table 3 between call option residuals and subsequent stock price relatives.
Call issue Stock/striking price LTR/$25
stock
Degree of expapsion 1 PI PZ PH p1 P2 PH Pl PZ PH Pl PZ PH Pl P2 PH PI P2 PH
0.1603 0.0007 -0.0285 -0.0397 0.1387 0.0250 -0.0123 -0.1588 -0.0232 0.2874a 0.2556” -0.0108 -0.0375 0.0608 0.0105 - 0.0202 0.2809” 0.0667
2 -0.1262 -0.0139 0.0623 - 0.0086 0.0196 -0.1115 - 0.0184 -0.1309 0.0063 0.3205” 0.3726” -0.1359 - 0.0414 -0.0035 -0.0412 0.0874 0.0043 -0.1003
3 -0.1096 -0.1403 0.1842a 0.0733 -0.1354 -0.1160 0.0153 -0.1468 - 0.0069 0.3491” 0.5327” -0.1442 - 0.0024 -0.1272 -0.1445 0.0702 -0.0812 - 0.0306
= 0; HA:p # Oat a = 0.01.
4.3. Results To support the research hypothesis of predictive information being contained in call prices, we would expect significant positive correlation between call price residuals and post-call transaction-stock price relatives.6 However, as shown in table 3, positive correlations are the exception rather than the rule. Summarizing the results, we find 6Testing hypotheses by counting positive correlation coefficients has its shortcomings, however. The residuals generated by a third-degree response surface are not independent of residuals generated by first- and second-degree response surfaces.
112
D. Panton, Chicago Board call options
(1) Of the fifty-four
Pearson product-moment correlations calculated, twentythree are positive and thirty-one are negative. At the 0.01 level, eight correlations are significantly different from zero; all eight are positive. are related to Polaroid Corporation. (2) Six of the eight significant correlations Residuals from first-, second-, and third-degree response surfaces are positively correlated with one- and two-month subsequent stock price relatives. Only the Polaroid call has significant positive correlation in more than one(3) degree expansion. 5. Summary and conclusions We have addressed several matters related to informational content in call option prices. First, we proposed a justification for insiders’ use of the options market in preference to the stock market. On a per dollar basis, calls provide greater claim to stock price gains than do most other financial instruments. This characteristic is an advantage to the insider wishing to reap maximum profits subject to the constraint of having limited, liquid investment capital. Second, we suggested a systematic procedure for identifying those call options which are underpriced or overpriced when compared to a fitted powerseries approximation of the unknown response surface. Third, deviations from the fitted surfaces were correlated with three sets of stock price relatives: (1) percentage change in stock price one month subsequent to the call transaction, (2) percentage change in stock price two months subsequent to the call transaction, and (3) maximum percentage increase in the stock price during the two-month period subsequent to the call transaction A high degree of positive correlation between the residual series and the stock price relatives would have been interpreted as indicating the presence of predictive informational content in call prices. However, since only in the case of Polaroid do we have significant positive correlation in a majority of the cases examined, we must admit that our results are not strong enough to state that call options, in general, are valid predictors of stock price movements. We must conclude that, if predictive information is present in call options, it has not been conclusively demonstrated here. References Baumol, W.J., B.G. Malkiel and R.E. Quandt, 1966, The valuation of convertible securities, Quarterly Journal of Economics 80,48859. Black, F. and M. Scholes, 1971, The pricing of options and corporate liabilities, mimeo. (M.I.T., Cambridge); revised version, 1973, Journal of Political Economy 81,637-654. Black, F. and M. Scholes, 1972, The valuation of option contracts and a test of market efficiency, Journal of Finance 27,399-417. Boness, A.J., 1964, Elements of a theory of stock-option value, Journal of Political Economy 72,163-175. Box, G.E.P. and N.R. Draper, 1959, A basis for the selection of a response surface design, Journal of the American Statistical Association 54,622-654.
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Box, G.E.P. and K.B. Wilson, 1951, On the experimental attainment of optimum conditions, Journal of the Royal Statistical Society (Series B) 13, l-45. Chayes, F., 1970, On deciding whether trend surfaces of progressively higher order are meaningful, Geological Society of America Bulletin 81,1273-1278. Chen, A.H.Y., 1970, A model of warrant pricing in a dynamic market, Journal of Finance 15, 1041-1059. Fama, E.F., 1970, Efficient capital markets: A review of theory and empirical work, Journal of Finance 15,383-417. Feller, W., 1957, An introduction to probability theory and its applications (Wiley, New York) 63-65. Hill, W.J. and W.G. Hunter, 1966, A review of response surface methodology: A literature survey, Technometrics 8570-590. Merton, R.C., 1973, Theory of rational option pricing, The Bell Journal of Economics and Management Science, 141-183. Myers, R.H., 1971, Response surface methodology (Allyn and Bacon, Boston). Samuelson, P.A., 1965, Rational theory of warrant pricing, The Industrial Management Review (now Sloan Management Review) 6,13-31. Shulman, M., 1967, Anyone can make a million (McGraw-Hill, Toronto). Sprenkle, C.M., 1961, Warrant prices as indicators of expectations and preferences, Yale Economic Essays 1,178-231. Watson, G.S., 1971, Trend surface analysis, Journal of the International Association of Mathematical Geology 3,215-226.