Option Valuation and Dividend Payments F-1523

8/10/2019 Option Valuation and Dividend Payments F-1523

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This note was prepared by Professors Robert Conroy and Robert Harris. It was written as a basis for class discussion

rather than to illustrate effective or ineffective handling of an administrative situation. Copyright 2007 by theUniversity of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved. To order copies, send ane-mail to [email protected] part of this publication may be reproduced, stored in a retrieval system,used in a spreadsheet, or transmitted in any form or by any meanselectronic, mechanical, photocopying,

recording, or otherwisewithout the permission of the Darden School Foundation.

OPTION VALUATION AND DIVIDEND PAYMENTS

When a company pays dividends, option valuation requires careful attention to theparticulars of those payments. This note discusses how dividend payments affect option valuesand some approaches to handling those effects in valuation models. Valuation effects flowdirectly from the effects of dividend payments on share price and resulting investor behavior.First, lets consider some key features of dividend payments themselves.

Dividend Payments

Cash dividends are payments from the firm to shareholders. When a dividend payment isannounced, the firm also provides information on two important events. The first is the holder-of-record date. The list of shareholders on that date receives the dividend. The other date is thepayment date. The dividend will actually be paid on that date. The exchange on which a stock istraded also sets something known as the ex-dateor ex-dividend date. The ex-date is the date onwhich purchasers of the stock do not receive the upcoming dividend. For the New York StockExchange (NYSE), the ex-date is set by Rule 235:

NYSE Rule 235. Ex-Dividend, Ex-Rights: Transactions in stocks (except thosemade for cash) shall be ex-dividend or ex-rights on the second business day

preceding the record date fixed by the corporation or the date of the closing oftransfer books. Should such record date or such closing of transfer books occur

upon a day other than a business day, this Rule shall apply for the third preceding

business day.

On the NYSE, individuals who buy the stock two business days before the holder-of-record datedo not receive the dividend. For example, suppose ABC, an NYSE-listed company, announces adividend of $1.00 a share and the holder-of-record date is September 28, 2006 (a Thursday) andthe payment date is October 19, 2006. Here, the ex-date would be Tuesday, September 26, 2006.Individuals who purchase the stock on or after that date would not be entitled to the $1.00dividend to be paid on October 19. Because a purchaser on Monday, September 25, would get

the dividend and a purchaser on September 26 would not get the dividend, all things being equal,


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we would expect the stock price to drop by $1.00 between September 25 and September 26. Thisstock-price drop1is what complicates option valuation.

Dividend Payments and Options

Option prices depend on the current market value of the underlying asset. In the case ofoptions on stocks, the underlying asset is the stock price, which is affected by dividendpayments. In turn, the option value is also affected by dividend payments. We first consider howdividends affect the valuation of call options.

European call options: Known-dividend approach

The simplest case is a European call option where there are specific ex-dividend dates

prior to the options maturity. Consider a European Call option on one share of XYZ stock with amaturity time of five months and a strike price of $25. The stock is trading at $26 a share, andthe company has announced a quarterlydividend of $.60 with an ex-date in three months. Afterthe announcement, we know the upcoming dividend is to be paid. But because the option isEuropean and can only be exercised after the ex-date, we have an option on the stock without the$0.60 dividend. The underlying asset is the stock but without the right to receive the dividend.Note that we focus on the ex-date because this is when the stock price drops. If the optionsmaturity goes beyond the ex-date, we need to adjust for the dividend payment.

To adapt the Black-Scholes model for this known dividend payment, we redefine theunderlying asset value (UAV) to be the value of the stock without the dividend, or UAV =Current stock price Present value of the dividend payment, which will not be received by theoption holder.

We take the present value of the dividend payment, discounting it back from the paymentdate to the present, where tD is the dividend-payment date.

2Because it is usually easier to findinformation on ex-dates than on actual payment dates, practitioners often use the ex-date as anapproximation of the payment date, given that the two dates are typically so close together.

1

Technically speaking, we would expect the price decline on the ex-date to be equal to the present value of thedividend payment. For instance, in the text example, the ex-date is about three weeks prior to the actual paymentdate (September 26 vs. October 19), so the expected price drop would be less than $1.00 (by the time value ofmoney for the three weeks). Because this time period is so short, we approximate the ex-date price decline as justthe value of the dividend.

2Note that the dividend-payment date, tD, and the maturity date, T, of the call option are different. Also notethat whether we need to do the dividend adjustment at all depends on whether the ex-date falls prior to the optionsmaturity because the ex-date is when the dividend affects the stock price. For instance, if a European call optionmatured in 30 days, the ex-date was in 20 days, and the payment date was in 43 days, we would still have to adjusttodays option valuation for the dividend payment.


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Assuming a risk-free rate of 5% (continuously compounded), the UAV is

407.25$60.026$ 12305.tD

0 ===

eeDividendSUAV fR

Assuming a volatility of .25, the Black-Scholes value of this European call can be calculated asfollows:

UAV = 407.25$60.026$ 12305.tD

0 ==

eeDividendS fR

X = $25T = .4167 years (5 months)Rf = 5% = .25 (assumed)

Black-Scholes call value = $2.107, adjusted for known dividend

As a comparison, suppose we had ignored the dividend and used an underlying asset value of$26.00. The resulting Black-Scholes value of the call would have been $2.492. The drop in thecalls value from $2.492 to $2.107 (adjusted for the dividend payment) is because the call ownerwill not capture the upcoming dividend payment.

In summary, when there are known dividend payments, we value European call optionsby calculating a new underlying asset value. This is done by taking the stock price and thensubtracting the present value of the dividend that we will not receive while holding the option. Ifthere is more than one dividend ex-date prior to the call options maturity, we would subtract the

present values of all those dividend payments. We then use the adjusted UAV in the Black-Scholes formula to value the call.

European call options: Constant-dividend-yield approach

Another way to account for dividends is to assume that dividends are paid outcontinuously at a certain dividend yield rate. This is an abstraction, but a useful one, if we arelooking at a relatively long time period that may include a whole set of dividend payments by afirm. This assumption allows us to effectively subtract the present value of a flow of dividendsfrom the share price to get at the true underlying asset value for the option holder.

Dividend yield is typically expressed as the annual dividend as a percentage of the stockprice. Hence, in the example used above, the dividend yield3for XYZ stock would be

Dividend yield = %23.900.26

40.2

Pr==

=

iceStock

DividendAnnualdy .

3A quarterly dividend of $0.60 translates to an annual dividend of $2.40.


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To take out the dividend flow that the option holder will not receive, we calculate theunderlying asset value as follows:

UAV = Current stock price discounted at the dividend yield rate, orTdyeSUAV = 0 ,

where S0 is the current stock price, dy is the appropriate dividend yield, and T is the time tomaturity of the option. The calculation reduces the stock price by the dividend rate. 4Using theexample from above, lets look at an option with a maturity of five years and a strike price of$25. Assuming that the dividend yield is 9.23%, the UAV would be

389.16$26$ 50923.0 === eeSUAV Tdy .

This UAV of $16.389 means that expected dividends over the next five years account for about$10 of the current share price of $26. To be precise, the five years of dividends are worth $9.611(i.e., 26 16.389 = 9.611). Because the option owner of a European call will not capture thesedividends, the owner effectively has an option on a non-dividend-paying stock that is worth$16.389. The value of the option would be as follows:

UAV = 389.16$26$ 50923.0 == eeS Tdy

X = $25T = 5.0 yearsRf = 5% = .25 (assumed)

Black-Scholes call value = $2.587

Typically, we use the known-dividend approach for shorter maturities and the constant-dividend-yield approach for longer maturities.

American call options: No dividend payments

Unlike European call options, American calls can be exercised at any time up to andincluding the maturity date. The possibility of early exercise complicates valuation because, aswe will discuss shortly, it sometimes makes sense to exercise early in order to capture a dividend

payment. As it turns out, the only time we can directly value an American options value using

4At first glance, a calculation that discounts at the dividend yield rate (i.e., e-dyT) may not appear logical. In fact,the calculation works because it is a shorthand way of accomplishing another calculation. The return on a stock (r) isthe sum of dividend yield (at rate dy) and capital gains (say, at rateg). But the European call owner doesnt get thedypart of the return because of dividends; the return on his underlying asset is onlyg. When the underlying stockprice grows atg and we discount this atrusing continuous compounding, we can use the rules of exponents to dothe calculations by taking the current share price to the power of (gr). But because r = (dy + g), (gr) is just equaltody.That is the result we have above: we discount the current share price at dy (taking it to thedypower).


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the Black-Scholes model is when there are no dividend payments. This is because there will beno reason for the American call to be exercised early. So if there are no ex-dividend dates priorto the maturity of the American call, we can just pretend that we have a European option and use

Black-Scholes to value an American call.

The intuition for this result is easiest to see for a non-dividend-paying stock. So long aswe can buy and sell the American call in the market, we would never choose to exercise early. Ifwe exercise early, we collect the intrinsic value (SX), but if we sell the option, we get theintrinsic value plus the time value. Therefore, even if one thought the stock price was going todrop, one would never exercise early; rather, one would sell the option in the market. In thiscase, the values of the American call and the European call would be the same.

American call options: Dividends and early exercise

Valuation of American call options is different when we have dividend-paying stocksbecause early exercise may occur. On the dividend ex-date, the price of the stock drops by theamount of the dividend. Knowing this, the holder of a call option must decide on one of twostrategies just before the ex-date. The holder can exercise now (early exercise) and capture thecurrent intrinsic value of the call or wait and take the reduced intrinsic value (stock price dropsby the amount of the dividend) but keep the time value.

To illustrate the decision, consider an American call option with a maturity of sevenmonths and a strike price of $40. The current share price is $41, the volatility is .20, and the risk-free rate is 5%. There is a dividend of $.75 with an ex-date in four months. If the day before theex-datethe stock price is $43, the option holder must decide whether to exercise the option earlyand collect $3.00 (the intrinsic value of SX: $43.00 $40.00 = $3.00) or not exercise early andhave a call option on a stock with a price of $42.25 (S dividend5) and a remaining time tomaturity of three months. The value of this call would be as follows:

UAV = $42.25X = $40T = .25 years (3 months)Rf = 5% = .20 (assumed)


Call value = Intrinsic value + Time value$3.358 = $2.25 + $1.108

The value of the call is $3.358. Because it is greater than the $3.00 from early exercising, theholder will notexercise early. The driver of the decision not to exercise is the fact that the time

5Note that because the ex-date occurs essentially immediately, the dividend is already a present value and weare applying the European known-dividend-payment approach discussed earlier.


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value of the option ($1.108) more than offsets the reduction in the intrinsic value (from $3.00 to$2.25) when the stock goes ex-dividend.

In other circumstances, however, early exercise will make sense. If just before the ex-datethe stock price is $50, the value to exercise early is $10.00 (capturing the intrinsic value of $10 =$50 $40). The value of the option with the remaining three months to maturity would only be$9.768.6Because the value of early exercise is greater, the holder would choose to exercise early.

By trying different stock prices, we can determine that if just before the ex-date the shareprice is $45.13, then the value of early exercising would be $5.13 and the value of the optionwould be $5.13. Because the two values are the same, the holder would be indifferent 7betweenexercising early or not. Thus, we can see that if just before the ex-date the stock price is above$45.13, then the holder will exercise early. If the price is below $45.13, then the holder will notexercise early.

American call options: Valuing American call options on dividend-paying stocks

Fortunately, we can still get an approximation of the American calls value if we use theBlack-Scholes model in a more complicated way. The basic approach described below usesBlack-Scholes to figure out the value of two European options. One of the options has the samematurity as the American call. The other European calls maturity expires just prior to the ex-date. These two options capture two strategies available to the American calls owner. Thelonger-maturity option is what would happen if there were never any early exercise. The shorter-maturity option captures what happens if there is always early exercise. The valuation insight isthat the American option is worth at least as much as the more valuable of the two Europeanoptions. This is because the American calls owner can pick either strategy. We calculate thevalues of these never-exercise and always-exercise European calls. The American call isworth at least as much as the larger of these two values.

To implement this approach, lets return to the situation where we have a stock with acurrent price of $41, a dividend of $0.75 with an ex-date in four months, a volatility of .20, and arisk-free rate of 5%. We want to value an American call option with an exercise price of $40 andseven months to maturity.

We proceed by considering two specific European calls. The first has an exercise price of$40 and a maturity of seven months. This is just a normal European call option, and it will have

6UAV = $49.25, X = $40, T = .25 years (3 months), Rf= 5%, = .20 (assumed)


Call value = Intrinsic value + Time value$9.768 = $9.25 + $0.518

7This indifference point is found through trial and error by finding the stock price just before the ex-date wherethe early-exercise value is equal to the value of the call if the option is not exercised early.


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the same ultimate payoff as if we held the American call option until maturity. The second is aEuropean call option with an exercise price of $40 and a maturity of four months (just before theex-date of the dividend). This call option provides the same payoff as if we always exercised the

American call option early prior to the ex-date of the dividend.

European call: Hold to maturity, never exercise early

X = $40 & T = 7 months

UAV = 262.40$75.041$ 12405.

0 ==

eeDividendSTRf



European call: Always exercise early

X = $40 & T = 4 months

UAV = S0= $41.00X = $40T = .333 years (4 months)Rf = 5% = .20 (assumed)


Because we can make either of these choices with the American call option, the American calloption must be worth at least as much as the more valuable of the two European calls. In thiscase, we can say that the American call option with a maturity of seven months and an exerciseprice of $40 is worth at least$3.178, the value of the seven-month European call, which, in turn,is worth more than the four-month European call.

But if the dividend payment is large enough, the value of the shorter-maturity Europeancall can exceed the value of the longer-maturity European call. This is because a larger dividendpayment would lead to a larger stock-price drop on the ex-date and make holding the option tomaturity less likely. For example, if the dividend were $1.50 instead of $0.75, the value would beas follows:


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European call: Never exercise early

X = $40 & T = 7 months

UAV = 525.39$50.141$ 12405.

0 ==

eeDividendSTRf



European call: Always exercise early

X = $40 & T = 4 months

UAV = S0= $41.00X = $40T = .333 years (4 months)Rf = 5% = .20 (assumed)


So when the dividend is $1.50, the value of the American call option with a maturity of sevenmonths and an exercise price of $40 is at least$2.797, the value of the four-month Europeanoption.

This approach of comparing two European calls to value an American call option is justan approximation, but it turns out to work fairly well. More-complicated approaches to pricingAmerican calls are available that get even better value estimates. Nonetheless, thisalways/never approach is a good approximation.

European put options

We can value European puts on dividend-paying stocks using a slightly modifiedequation for put-call parity. With dividends, the put-call-parity relationship needs to be restatedto acknowledge different dividend flows to stock ownership and call ownership. In essence, thestock owner captures some dividend payments that the call owner does not. For instance, if youhad a European call with a one-year maturity, you would not get any of the quarterly dividendpayments during the year. This would be reflected in a lower value for your call, as wevediscussed earlier. If you held the stock for the year, however, youd get the quarterly dividendpayments and still own the stock at the end of the year. To see how this affects put-call parity,recall that put-call parity is based on picking a strategy of owning stocks and puts that leads toexactly the same payoff at maturity as owning bonds and calls. In the presence of dividends, the


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extra wrinkle is that the strategy of owning stocks and puts also provides the dividend paymentsduring the life of the options (prior to maturity). Market forces will thus ensure that the stock-plus-put strategy should be worth more than the bond-plus-call strategy by exactly the value of

the interim dividend payments. In equation form, we have

Dff tRTR eDeXCallPutStock

++=+ ,

whereRfis the risk-free rate,Xis the exercise price of the put and call, Dis the dividend, and tDis the time to the dividend payment. Hence, the value of a European put option is

StockeDeXCallPut DfftRTR++=

.

Note that the calls value in the above equation is the value of a European put adjusted fordividends. For longer-term European puts, we can make a similar application of put-call parity if

we assume a flow of continuous dividend payments. We simply adjust put-call parity by addingback the value of the flow of dividends over the life of the option.

American put options

Valuing American puts is more complex because incentives for early exercise differbetween puts and calls. In the case of call options, we saw that it often does not make sense toexercise an American call early even in the presence of dividends. This is because the time valueof the call would be large enough to forestall early exercise. In the case of puts, however, earlyexercise can make sense even if there are no dividends, and dividends just add to the complexity.A good rule of thumb is that early exercise is typically not a large issue for out-of-the-money

puts. Their values can be approximated by assuming they are European. In contrast, deep-in-the-money puts tend to trade at or near their intrinsic value (early-exercise value,XS).

To see the forces at work with put options, lets recall what happens with calls. Earlier,we discussed why a call holder would not exercise an American call prior to maturity. For such acall, it was always better to sell the option in the market and capture the calls time value. Onlywith dividend payments did we have to worry about early exercise of American calls, and eventhen early exercise often did not make sense as doing so would give up the time value. ForAmerican puts, incentives are different. Consider an extreme example: suppose the stock price ofa non-dividend-paying stock goes to zero and you own an American put. In this case, the put is atits maximum value because the price can go no lower. The optimal strategy is to exercise early,

collect the difference between the exercise price and zero, and then invest in the risk-free assetuntil the puts original maturity date. Waiting to exercise means you (and any other potentialbuyer of the put) would forgo the time value of money without any chance of an offsettingbenefit. The American put would be exercised early and be worth more than a European put. Ingeneral, for deep-in-the-money puts (Xmuch greater than S), it can be optimal to exercise earlyin order to collect the intrinsic value and invest at the risk-free rate for the remaining time tomaturity. In such cases, the American puts value can be approximated as the intrinsic value (XS) from early exercise.


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Another way to see the incentives surrounding early exercise is to look at put-call parity.From put-call parity (no dividends), we get the following:

StockeXCallPut

eXCallPutStockTR

TR

f

f

+=

+=+

It makes sense to exercise a put early only if the early-exercise value (XS) exceeds the value ofthe put from put-call parity. This is because the put owner has the alternative to hold the put untilmaturity, which would yield the same value as the European call (calculated from put-callparity). In symbols, early exercise will make sense only if

SeXCallPutSXTRf+=>

CalleX

TRf

>

1 .

The left-hand side of the inequality is essentially the present value of getting the exercise pricetoday (early exercise of the put) versus forgoing early exercise and waiting until maturity (T) toget the exercise price. The right-hand side of the inequality is the value of the call. Early exercisemakes sense only when the calls value is relatively small. When will this occur? Call values arevery small precisely when they are far out of the money (S is much lower than X). But that isexactly when puts are deep in the money. So early exercise most likely makes sense for deep-in-the-money puts.

Overall, valuing puts is more complicated than valuing options. In the case of European

puts, we can use put-call parity to value a put, taking advantage of having already valued the call.We can also extend this approach to European puts on dividend-paying stocks, with appropriateadjustments to put-call parity.

American puts, however, are even more complicated than American calls owing toincentives surrounding early exercise. Even for stocks that dont pay dividends, it may makesense to exercise an American put early. Prospective ex-dividend dates during the life of a putactually reduce incentives for early exercise because stock prices decline on ex-dates. Rememberthat put owners profit from lower prices. Understanding the issues surrounding early exerciseleads to some key valuation insights. If the probability of early exercise is very high, theAmerican put will tend to trade near its intrinsic value (the amount X S, which could be

realized upon immediate exercise). This situation best fits deep-in-the-money puts. If theprobability of early exercise is very low, the American puts value can be approximated by justassuming that it is European. This situation best fits American puts that are way out of themoney.


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Summary

Whenever a stock pays dividends prior to an options maturity, valuing the option is more

complex. This is because the option owners underlying asset is effectively the stock minus thevalue of the dividend payments. A similar insight applies to a host of options on other assets. Forinstance, if an oil well is currently pumping and selling oil, the owner of a long-term Europeancall on that well does not capture the value of current production.

By making appropriate adjustments to the underlying asset value, we can still deployoption-pricing models (e.g., the Black-Scholes model) to value European call options ondividend-paying stocks. Valuing American calls is more complicated because such options maybe exercised prior to maturity in order to capture a dividend.

Valuing puts is more complex than valuing calls. In the case of European puts, we can

use put-call parity to value a put, taking advantage of having already valued the call. We can alsoextend this approach to European puts on dividend-paying stocks, with appropriate adjustmentsto put-call parity. American puts, however, are even more complicated than American callsowing to incentives surrounding early exercise. By understanding the factors that influenceexercise decisions, we can glean practical insights into how to approximate the values ofAmerican puts.

Documents

Option Valuation and Dividend Payments F-1523