GeskeShastri EarlyExerAmPutsDiv JBF 9 1985

Journal of Banking and Finance 9 (1985) 207-219. North-Holland

THE EARLY EXERCISE OF AMERICAN PUTS*

Robert GESKE University of California, Los Angeles, CA 90024, USA

Kuldeep SHASTRI University of Pittsburgh, Pittsburgh, PA 15260, USA

Received April 1982, final version received May 1984

This paper shows that American puts on dividend paying stocks are most likely to be exercised either just after an ex-dividend date or just prior to expiration. At any other time the option to exercise an American put early may have less value. Thus, put writers and converters can predict when protection against premature exercise will be most valuable. The probability of early exercise is shown to be sensitive to managerial policy regarding the suspension of dividend payments, transaction costs, and interest rates. However, dividend payments are demonstrated to be the primary deterrent to early exercise.

1. Introduction

The American put is distinguished from its European counterpart by the allowance for early exercise. Merton (1973) has shown that this added flexibility implies that the American put will sell at a premium above the European put. Bounds on this premium are readily obtained when the underlying stock pays no dividends.

Geske and Johnson (1984) recently derived an analytic solution to the partial differential equation and boundary condition characterizing the American put. Before this solution most authors approximated the differential equation for the American put's value by numerical methods. Brennan and Schwartz (1977) used a finite difference approximation to value the American put, while Parkinson (1977) used a related form of numerical integration. Cox, Ross and Rubinstein (1979) have shown that a conceptually simple binomial process can also be employed for numerical approximation. All of these valuation schemes yield the same values for the American put [see Geske and Shastri (1982b)].

*The authors would like to thank John Cox, Richard Roll, Mark Rubinstein, participants at the American Stock Exchange Options Colloquium, and the referee for their comments. We thank the CIVITAS Foundation at UCLA and the Graduate School of Business, University of Pittsburgh for financial support.

0378-4266/85/$3.30 © 1985, Elsevier Science Publishers B.V. (North-Holland)

J.B.F.--B

208 R. Geske and K. Shastri, The early exercise of American puts

Empirically, the results reported to date using the approximation schemes are inconclusive. All of the papers confirm that the American put should be more valuable than the European put. In certain time periods, using market data, the approximation technique overvalues the American put [Brennan and Schwartz (1977)1 while in other periods the approximation undervalues the American put [Parkinson (1977)].

One observation about which there appears to be agreement is that few American puts are exercised early. Market makers trading put options and academics studying puts have both reported this fact. Brennan and Schwartz (1977) indicate that only about 2 percent of the puts in their over-the- counter (OTC) sample were exercised at dates earlier than just prior to expiration. However, the optimal policy of Brennan and Schwartz indicated that 24 percent should have been exercised earlier. They attribute this reluctance to exercise early to either a faulty model, gamblers' greed, or tax effects. The over-the-counter market adjusts the put's exercise price for dividend payments. This OTC adjustment actually increases the probability of early exercise. 1 The American puts currently traded on the Chicago and American exchanges are not dividend adjusted, and still purportedly few are exercised early.

Merton (1973) showed that the American put always has a positive probability of premature exercise. This always positive probability of exercise gives put writers reason to fear the unpredictability of 'being put'; covered writers and converters could be left unprotected. This paper proves that the most likely time to expect early exercise, other than just prior to expiration, is immediately after the last ex-dividend date. Thus, put writers could prepare for the greater likelihood of 'being put' at ex-dividend dates. Transaction costs, low interest rates, or taxes might be competing reasons for why puts are not exercised prematurely. However, by comparison, this paper demonstrates that dividends are the primary deterrent to early exercise. 2

In section 2, the critical stock price below which an in-the-money American put would be exercised prematurely is graphically presented for a variety of dividend policies. Here we also demonstrate the effects of varying the dividend suspension policy, transaction costs, and interest rates on the time path of critical stock prices.

Section 3 presents descriptive data regarding dividend payments. Table 1 shows that 93 percent of the puts currently traded on the Chicago and American exchanges pay quarterly dividends of sufficient magnitude to support the result that the most probable time to expect early exercise is immediately after an ex-dividend date. Section 4 summarizes the paper.

~Geske, Roll and Shastri (1983) demonstrated (but did not report) that the OTC dividend adjustment actually increases the expected number of puts which would be exercised early by increasing the probability of exercise.

2The tax deterrent to early exercise was not considered here due to the relative heterogeneity among investors' tax brackets.

R. Geske and K. Shastri, The early exercise of American puts 209

2. The early exercise strategy

The 'critical' stock price is the key to analyzing the early exercise strategy. In order to solve for a time series of critical stock prices when the underlying stock pays dividends, numerical approximation is necessary. In this paper the numbers reported in the tables and figures are based on an explicit finite difference approximation. 3 Define:

St - t h e market value of one share of stock on which the put is written at current time t;

S ' - t h e critical stock price below which the put will be exercised early; S s - t h e suspension stock price below which the dividend payment will be

suspended; X - t h e exercise price of the put; P(S, X, T - t ) - the value of a put to sell one share of stock at the exercise

price X any time between current time t and expiration date T; z( = T - t ) - the time remaining to expiration; r - t h e continuously compounded risk free rate of interest; D - t h e amount of the discrete dividend payment to the stock.

Between dividend dates the stock price is assumed to follow the stochastic process

dS S = #dt + adz, (1)

where dz is a Gauss-Weiner process. Thus the value of the put must obey the following partial differential equation, as shown by Black and Scholes (1973):

r P - Pt-rSPs --½o2S2Pss =0, (2)

where the subscripts denote partial differentiation. Furthermore, the put must satisfy certain well-known dominance con-

ditions developed by Merton (1973). In particular the possibility of early exercise prevents the value of the American put from falling below the exercise value. Thus,

P(S,, X, T- t )>max(O,X-S , ) . (3)

aSee Brennan and Schwartz (1977) or Geske and Shastri (1982b) for an explanation of finite difference procedures.


Merton (1973) has shown that there will always exist a critical stock price, S c, below which the put should be rationally exercised. One explanation for this is if interest rates are positive, then for some stock price greater than zero, receiving X - S and reinvesting at the risk free rate dominates holding or selling the put. Thus, for stock prices less than the critical stock price the put will be exercised.

As long as the American put's market value is greater than its exercise value ( P > X - S ) the put should be held or sold in the secondary market rather than exercised. Only when the American put's market value is equal to its exercise value should it be exercised early. Thus, in this paper, the critical stock price is determined by checking condition (3) (for each stock price in the vector of stock prices considered by the finite difference approximation) until the particular stock price is found which forces condition (3) to hold as an equality (i.e., P =X-SC). By iterating current time and repeating this process a time series of critical stock prices is generated. It is obvious from this procedure that the critical stock price is independent of the current stock price.

If the stock pays known discrete dividends, and the dividend can be suspended, the condition for determining the critical stock price is slightly different at each ex-dividend date. Since little is known regarding managerial behavior with respect to dividend policy, this paper assumes that for low values of the firm, and correspondingly low stock prices, dividend payments will be suspended. Thus we assume management will decide to pay or suspend the dividend at the ex-dividend date whenever the stock price is less than or equal to the known suspension price, S < SS. 4

When the stock price is greater than the suspension price at the ex- dividend date, the dividend will be paid. In the case of a certain dividend payment, an American put will never be exercised just before the ex-dividend date. Notationally, if t - is the instant before the ex-dividend date and t + is the instant after, then whenever S t - > S ~ the put will not be exercised. The reason is the value of the put an instant before the ex-dividend date is the maximum of the exercise value an instant before the ex-dividend date and the put value an instant after. Thus, algebraically, the put value at t - satisfies

P(St-, X, T - t-) =max I X - St-, P(St +, X, T - t+)]

=P(St+,X, T-- t +) if St- >S ~. (4)

In condition (4) the second equality must hold to avoid riskless arbi- trage, and it reflects the fact that P(St+, X, T-t+)>_max[O, X - S , ÷ ] >

4Although it may seem less consistent to consider the ex-payout date instead of the declaration date as the decision point for paying or suspending the dividend, it is actually appropriate when using numerical analysis and working backwards.


m a x [ 0 , X - S t - ] . Since the put will never be exercised the instant before the ex-dividend date whenever the stock price is greater than the dividend suspension price, this implies that the critical stock price an instant before the exdividend date cannot be greater than the suspension price.

When the stock price is less than the suspension price at the ex-dividend date, the critical stock price is calculated as in the no dividend case. If the dividend is likely to be suspended and the suspension price is less than the critical stock price, then the critical stock price will converge to the suspension price an instant before the ex-dividend date. This is necessary to avoid a positive probability that the American put will be exercised just before the dividend is paid. However, the put might be exercised at the scheduled ex-dividend date if the dividend is actually suspended. 5

Figs. 1, 2, and 3 portray critical stock prices as a function of time to expiration for both dividend and non-dividend cases. Of the first three figures, fig. 2 is most representative of the dividend payments for currently listed puts (See section 3, table 1). If this put is currently at-the-money (i.e., S = X = $45.00) a $0.50 quarterly dividend payment is approximately a 4 percent dividend yield. In each of figs. 1, 2, and 3, X=$45.00, r= 5 percent per year, and a=0.3 annually. Time to expiration ranges from 7 months to zero, and quarterly dividends are paid at 0.5, 3.5, and 6.5 months prior to expiration. The dividends are $0.01, $0.50, and $1.00 in figs. 1, 2, and 3, respectively.

In each of these figures the critical stock price, S c, for the no dividend case is a convex, monotonically decreasing function of time to expiration. When there is no time remaining to expiration (i.e., ~=0) the critical stock price equals the exercise price (i.e., S¢= X = $45.00). As time to expiration increases to 7 months, the critical stock price falls monotonically to about $34.00. This demonstrates that a put that remains in-the-money will always be exercised prior to expiration since the critical stock price approaches the exercise price as the expiration date approaches.

When the stock pays dividends, the critical stock price is a function of the dividend. In this paper it is initially assumed that managers suspend the dividend payment whenever the stock price is less than or equal to the dividend (i.e., S<D). More realistic dividend suspension policies are also examined.

For the multiple dividend cases, figs. 1, 2 and 3 each demonstrate that the critical stock price follows a cyclical pattern. When there are no dividends

SContrary to Brennan and Schwartz (1977), one cannot say that it will never pay the put holder to 'expect' to exercise immediately before the 'expected' dividend payment. This will depend upon whether or not it is the manager's policy to pay a liquidating dividend, as Brennan and Schwartz assumed, or instead to suspend dividends payments for low stock prices. Since relatively few corporations pay liquidating dividends, whenever the current stock price is much less than both the critical and suspension prices, there will be a positive probability of expected exercise at the scheduled ex-dividend date.

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remaining the critical stock price tracks the no dividend case. For this suspension policy, immediately before each dividend payment, S c falls to the level of the dividend. This shows that for this policy there is little probability of 'expecting' to exercise an American put option immediately before a dividend payment. As current time is moved backward, increasing time to expiration, the critical stock price rises and then falls to a low by the next ex- dividend date. This cycle is repeated at regular intervals when the dividend payments are quarterly. 6

As expected, except for the abrupt drop in the critical stock price at the ex-dividend dates, the $0.01 dividend case in fig. 1 is almost coincident with no dividend case. However, figs. 2 and 3 show a remarkable distinction as the dividend is raised to $0.50 and $1.00, respectively.

Unlike the ho dividend case, for reasonable dividends the critical stock price between dividend dates is an increasing function of time to expiration (see fig. 2). As time to the option's expiration diminishes, immediately before and after each ex-dividend date, the critical stock price, S c, jumps from the dividend suspension price just before the ex-dividend date to a peak. From this peak the critical stock price decreases to become equal to the suspension price at the next ex-dividend date. The path the critical stock price takes between these local maxima and minima at the ex-dividend dates can be both concave and convex from below.

This critical stock price pattern occurs because the put's value is affected by two opposing forces as time elapses. First, diminishing time to expiration decreases the put value, while the approaching dividend payment and subsequent stock price drop increases the put value, and both of these factors influence the critical stock price in opposite directions. 7 One way to see this is to recall that the American put, if 'alive', must be worth more than X - S c. Thus, if P declines with diminishing time to expiration, since X is constant, the critical stock price, S c, will rise. Alternatively, as P increases due to an impending dividend, the critical stock price will drop all the way to the dividend suspension price at the ex-dividend date. This must occur because as the ex-dividend date approaches, if the dividend is going to be paid, no investor would exercise early since the interest earned on the exercise proceeds invested at the risk-free rate could not exceed the dividend. Exercise would only occur before an impending dividend if it were likely that the dividend would be suspended. Thus, the tradeoff between the time to expiration, the dividend, and the suspension effects determines the shape of the critical stock price path between dividends. All the graphs demonstrate

6Merton, Scholes and Gladstein (1982, pp. 11-16), offer empirical support for this cyclical shape of the critical stock price path. Their result also supports this discrete payout and suspension policy rather than a policy of continuously adjusting the dividend as the firm value changes.

7This also affects the sensitivity of the American put's value with respect to time to expiration. See Geske and Shastri (1982a) for details.


the inverse relation between time to expiration and the critical stock price for the no dividend cases. Furthermore, with dividends, the graphs show that the critical stock price typically falls between dividend payments, and is directly related to time to expiration between ex-dividend dates.

Between dividends the cycle of the critical stock price path is repeated, although the width and amplitude of the peaks diminish as time to expiration increases. This pattern of critical stock prices demonstrates that the most likely time to exercise an American put is immediately after the dividend payment. It also shows that for reasonable dividend policies there is less probability of exercising the American put at any other time prior to payment of the last dividend scheduled before the put expires.

Fig. 3 shows that for a larger dividend payment of $1.00, the critical stock price never rises much above the dividend. Thus, American puts on stocks paying large dividends may have little likelihood of early exercise until after the last ex-dividend rate. If the exercise price is decreased, the critical stock pattern will be similar to that in fig. 2 except the probability of early exercise will be reduced. 8

More restrictive bond convenants would require suspension of the dividend payments long before the firm value fell enough to imply a stock price below the scheduled, known dividend. This should increase the probability of exercising an American put prematurely, as shown in figs. 4 and 5. All parameters are the same as in fig. 2 except it is assumed that dividend payments are suspended whenever the stock price drops below either $10.00 or $20.00, respectively. The higher suspension price increases the probability of exercising an American put early. (Note that for any current stock price the conditional probability of 'expected' exercise at a scheduled ex-dividend date is now considerably higher.) This more severe payout restriction would be more likely for levered firms. Figs. 4 and 5 still confirm that it is less likely to expect an American put to be prematurely exercised at any date other than just after a dividend payment.

For the purpose of illustration, transactions costs as an impediment to early exercise are approximated by altering the equality used to solve for critical stock price versus time to the following:

X-TS,=P(St, X,T-t), (5)

where 7 represents the percent of transaction costs. For example, a 2 percent round trip transaction cost (7 equal 1.02) would reduce the exercise proceeds, and thus the desire to exercise. As expected, the critical stock price path depicted in fig. 6 shows that transaction costs further reduce the inclination

SThe critical stock price is independent of the current stock price, and thus independent of whether the put is in, at, or out-of-the-money. However, the probability of early exercise is obviously conditional on the current stock price.


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to exercise early, although the effects on the critical stock price are small relative to the dividend effects.

It is well known that increases in interest rates imply a higher opportunity cost to holding an in-the-money put. Thus, raising the interest rate will increase the probability of early exercise. Fig. 7 demonstrates this while showing that the cyclical pattern of the critical stock price remains basically unchanged for higher interest rates.

3. Dividend data

Although dividends can significantly reduce the probability of early exercise, if few of the currently listed put options are on dividend paying stocks this result would have little empirical significance. A survey of the dividend policy of stocks with listed put options is necessary to determine the potential relevance of the dividend impact on early exercise.


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Table. 1 summarizes the dividend record of all stocks with put options listed on the Chicago Board Options Exchange (CBOE) or the American Stock Exchange (AMEX) as recorded in the Wall Street Journal on October 21, 1980. A total of 106 different companies had listed put options trading on their stock. The dividend record was available in the Standard and Poor Dividend Record for 105 of these stocks. Of the 105, about 93 percent paid dividends, all quarterly. The average annual dividend yield was approximately 3.76 percent, and the yields ranged from 0.2 to 8.2 percent. By comparison, the $0.50 dividend considered in many of the examples in this paper on an at-the-money put (S=$45.00) represents about a 4 percent annual dividend yield, slightly above average. A slightly lower average dividend yield would increase the probability of early exercise, but this would not qualitatively change the above results.

If the most likely time to expect early exercise of in-the-money American puts is immediately after a dividend payment, the open interest on average

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Table 1 Dividend record of stock with listed puts?

Company Exchange Yield Company Exchange Yield

AMF Inc. AMEX 6.5 Marathon Oil AMEX 2.9 ASA Ltd. AMEX 5.8 Merrill Lynch & Co. CBOE/AMEX 3.1 Aetna Life & Casualty AMEX 5.7 Mesa Petroleum AMEX 0.2 AMAX, Inc. AMEX 5.0 Mobil Corp. CBOE 4.2 American Cyanamid AMEX 6.0 Motorola AMEX 2.0 American Home Products AMEX 6.1 NCR Corp. CBOE 2.8 Asarco Inc. AMEX 2.1 NLT Corp. AMEX 5.7 Atlantic Richfield CBOE 2.9 Natl.Distillers,&Chem. AMEX 6.3 AVNET Inc. AMEX 2.3 Natl. Semi Conductor CBOE/AMEX 0 Bally MFG. Corp. CBOE/AMEX 0.4 Northwest Industry CBOE 6.2 Bausch & Lomb AMEX 2.1 Notomas AMEX 2.4 Boeing Co. CBOE 3.3 Occidental Petr. CBOE 6.1 Bristol-Myers CBOE 3.8 Owens-Illinois Inc. CBOE 5.3 Browning-Ferris Ind. AMEX 2.9 Plrtzer Inc. AMEX 3.2 Bucyrus-Erie Co. AMEX 3.7 Phelps Dodge Corp. AMEX 4.2 Burroughs AMEX 4.6 Philip Morris AMEX 3.7 Chase Manhattan Corp. AMEX 6,8 Phillips Petroleum AMEX 3.3 Coastal Corp. AMEX 1.0 Polaroid CBOE 3.6 Control Data CBOE 0.8 Prime Computer Inc. AMEX 0 Corning Glass Wks. CBOE 2.9 Proctor & Gamble AMEX 5.3 Digital Equipment AMEX 0 Ralston Purina CBOE 5.5 Disney Productions AMEX 1.6 Revlon Inc. CBOE 3.2 Dow Chemical CBOE 5.0 Rockwell Intl. CBOE 4.1 Du Pont CBOE/AMEX 4.7 Sante Fe Industries AMEX 3.4 E1 Paso Co. AMEX 6.4 Schlumberger Ltd. CBOE 0.9 Eastman Kodak Co. CBOE 4.2 Searle & Co. AMEX 2.5 Esmark Inc. CBOE 3.4 Standard Oil Cal. AMEX 4.0 Exxon Corp. CBOE 7.0 Standard Oil Ohio AMEX 1.9 Federal Express CBOE 0 Sterling Drugs AMEX 3.9 Fluor Corp. CBOE 1.3 Storage Technology CBOE 0 Freeport Minerals CBOE 1.6 Superior Oil CBOE 0.3 General Dynamics CBOE 2.2 Syntex Corp. CBOE 2.3 General Electric CBOE 5.6 TRW Inc. AMEX 4.3 General Motors CBOE 7.5 Tandy Brands AMEX 0.5 Goodyear Tire & Rubber AMEX 7.9 Tektronix Inc. CBOE 1.4 Greyhound Corp. AMEX 8.2 Teledyne Inc. CBOE 0 Gulf Oil Corp. AMEX 5.3 Teleprompter Corp. AMEX 0 Halliburton Co. CBOE 1.4 Tenneco Inc. AMEX 5.6 Hewlett Packard Co. CBOE 0.5 Texaco Inc. AMEX 5.9 Holiday Inns Inc. CBOE 2.6 Tiger Intl. AMEX 3.4 Homestake Mining CBOE 2.6 Tosco Corp. AMEX - - Honeywell Inc. CBOE 3.4 Union Carbide AMEX 6.8 Houston Oil & Minerals CBOE 1.9 US Home Corp. AMEX 2.3 Hughes Tool CBOE 1.4 US Steel AMEX 7.2 Hutton Group AMEX 1.8 Valero Energy AMEX 0.5 IBM CBOE 5.0 Warner International Harvester CBOE 7.9 Communications CBOE 1.7 Kaneb Services Amex 2.7 Warner Lambert AMEX 6.5 Kennecott Corp. CBOE 4.3 Westinghouse Elec. AMEX 4.8 Kerr-McGee Corp. CBOE 2.0 Whittaker Corp. AMEX 3.1 Lilly & Co. AMEX 4.5 Williams Cos CBOE 2.6 Litton Industries CBOE 1.7 Xerox Corp. CBOE 4.3 MGIC Invest. Corp. AMEX 3.8 Zenith Radio Corp. AMEX 3.1

aTotal no. of companies: 106; No. paying dividends: 98(92.5%); No. not paying dividends: 7(6.6%); No. for which dividend data not available: 1(0.9%); Average dividend yield for 105 companies=3.51% (range: 0 to 8.2%); Average dividend yield for 98 companies that pay dividends = 3.76% (range: 2 to 8.2%).


should decline at the ex-dividend date. However, open interest data are not best suited to checking for early exercise because open interest will also vary with opening and closing purchase or sale transactions. A better measure would be the actual exercise data collected by the Options Clearing Corporation, but this is not currently available for study. Casual empiricism by market makers supports the early exercise pattern discussed in this paper.

4. Conclusion

This paper demonstrates that American puts are most likely to be exercised just after the ex-dividend dates or just prior to expiration. Managerial policy regarding the suspension of dividend payments, transaction costs, and interest rates are shown to affect the probability of early exercise, but the dividend effect is dominant. Thus, put writers and converters can predict that ex-dividend dates are the times when protection against early exercise is most valuable.

References

Black, F. and M. Scholes, 1973, The pricing of options and corporate liabilities, Jourfial of Political Economy, May.

Brennan, M. and E.~ Schwartz, 1977, The valuation of American put options, Journal of Finance, May.

Cox, J., S. Ross and M. Rubinstein, 1979, Option pricing: A simplified approach, Journal of Financial Economics, Sept.

Geske, R. and H. Johnson, 1984, The American put option valued analytically, Journal of Finance, Dec.

Geske, R. and K. Shastri, 1982a, The effects of payouts on the rational pricing of American options, Working paper no. 12-82 (University of California, Los Angeles, CA).

Geske, R. and K. Shastri, 1982b, Valuation by approximation: A comparison of alternatives, Finance working paper no. 13-82 (University of California, Los Angeles, CA) forthcoming in: Journal of Financial and Quantitative Analysis.

Geske, R., R. Roll and K. Shastri, 1983, Over-the-counter options market dividend protection and 'biases' in the Black-Scholes model, Journal of Finance, Sept.

Merton, R., 1973, The theory of rational option pricing, Bell Journal of Economics and Management Science, Spring.

Merton, R., M. Scholes and M. Gladstein, 1982, The returns and risks of alternative put option portfolio investment strategies, The Journal of Business, Jan.

Parkinson, M., 1977, Option pricing:. The American put, The Journal of Business, Jan.

GeskeShastri EarlyExerAmPutsDiv JBF 9 1985

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Transcript of GeskeShastri EarlyExerAmPutsDiv JBF 9 1985