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Theory regarding B-S model in case of stock with dividends

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UVA-F-1523
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 the University of Virginia Darden School Foundation, Charlottesville, VA. All rights reserved.
To order copies, send an e-mail to
sales@dardenpublishing.com.
No part of this publication may be reproduced, stored in a retrieval system, used in a spreadsheet, or transmitted in any form or by any means—electronic, mechanical, photocopying, recording, or otherwise—without the permission of the Darden School Foundation.
◊
OPTION VALUATION AND DIVIDEND PAYMENTS
When a company pays dividends, option valuation requires careful attention to the particulars of those payments. This note discusses how dividend payments affect option values and some approaches to handling those effects in valuation models. Valuation effects flow directly from the effects of dividend payments on share price and resulting investor behavior. First, let’s consider some key features of dividend payments themselves.
Dividend Payments
Cash dividends are payments from the firm to shareholders. When a dividend payment is announced, 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 the
payment date
. The dividend will actually be paid on that date. The exchange on which a stock is traded also sets something known as the
ex-date
or
ex-dividend date
. The ex-date is the date on which purchasers of the stock do not receive the upcoming dividend. For the New York Stock Exchange (NYSE), the ex-date is set by Rule 235: NYSE Rule 235. Ex-Dividend, Ex-Rights:
Transactions in stocks (except those made 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 of transfer 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 date do not receive the dividend. For example, suppose ABC, an NYSE-listed company, announces a dividend of $1.00 a share and the holder-of-record date is September 28, 2006 (a Thursday) and the 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 $1.00 dividend 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,
UVA-F-1523 -2-we would expect the stock price to drop by $1.00 between September 25 and September 26. This stock-price drop
1
is what complicates option valuation.
Dividend Payments and Options
Option prices depend on the current market value of the underlying asset. In the case of options on stocks, the underlying asset is the stock price, which is affected by dividend payments. In turn, the option value is also affected by dividend payments. We first consider how dividends 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 option’s maturity. Consider a European Call option on one share of XYZ stock with a maturity time of five months and a strike price of $25. The stock is trading at $26 a share, and the company has announced a
quarterly
dividend of $.60 with an ex-date in three months. After the announcement, we know the upcoming dividend is to be paid. But because the option is European 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 option’s maturity 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 the
u
nderlying
a
sset
v
alue (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 the option holder. We take the present value of the dividend payment, discounting it back from the payment date to the present, where
t
D
is the dividend-payment date.
2
Because it is usually easier to find information on ex-dates than on actual payment dates, practitioners often use the ex-date as an approximation 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 the dividend payment. For instance, in the text example, the ex-date is about three weeks prior to the actual payment date (September 26 vs. October 19), so the expected price drop would be less than $1.00 (by the time value of money for the three weeks). Because this time period is so short, we approximate the ex-date price decline as just the value of the dividend.
2
Note that the dividend-payment date,
t
D
, and the maturity date,
T
, of the call option are different. Also note that whether we need to do the dividend adjustment at all depends on whether the ex-date falls prior to the option’s maturity because the ex-date is when the dividend affects the stock price. For instance, if a European call option matured in 30 days, the ex-date was in 20 days, and the payment date was in 43 days, we would still have to adjust today’s option valuation for the dividend payment.
UVA-F-1523 -3-Assuming a risk-free rate of 5% (continuously compounded), the UAV is 407.25$60.026$
12305.tD0
=⋅−=⋅−=
⋅−⋅−
ee Dividend S UAV
f
R
Assuming a volatility of .25, the Black-Scholes value of this European call can be calculated as follows: UAV = 407.25$60.026$
12305.tD0
=⋅−=⋅−
⋅−⋅−
ee Dividend S
f
R
X = $25 T = .4167 years (5 months) R
f
= 5%
σ
= .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 the call’s value from $2.492 to $2.107 (adjusted for the dividend payment) is because the call owner will not capture the upcoming dividend payment. In summary, when there are known dividend payments, we value European call options by calculating a new underlying asset value. This is done by taking the stock price and then subtracting the present value of the dividend that we will not receive while holding the option. If there is more than one dividend ex-date prior to the call option’s 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 out continuously at a certain dividend yield rate. This is an abstraction, but a useful one, if we are looking at a relatively long time period that may include a whole set of dividend payments by a firm. This assumption allows us to effectively subtract the present value of a flow of dividends from 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 stock price. Hence, in the example used above, the dividend yield
3
for XYZ stock would be Dividend yield = %23.9
00.2640.2Pr
==⋅⋅=
iceStock Dividend Annual dy
.
3
A quarterly dividend of $0.60 translates to an annual dividend of $2.40.
UVA-F-1523 -4-To take out the dividend flow that the option holder will not receive, we calculate the underlying asset value as follows: UAV = Current stock price discounted at the dividend yield rate, or
T dy
eS UAV
⋅−
⋅=
0
, where
S
0
is the current stock price,
dy
is the appropriate dividend yield, and
T
is the time to maturity of the option. The calculation reduces the stock price by the dividend rate.
4
Using the example from above, let’s 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
=⋅=⋅=
⋅−⋅−
eeS UAV
T dy
. 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 these dividends, 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
T dy
X = $25 T = 5.0 years R
f
= 5%
σ
= .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 and including the maturity date. The possibility of early exercise complicates valuation because, as we 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 option’s value using
4
At 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
) is the sum of dividend yield (at rate
dy
) and capital gains (say, at rate
g
). But the European call owner doesn’t get the
dy
part of the return because of dividends; the return on his underlying asset is only
g
. When the underlying stock price grows at
g
and we discount this at
r
using continuous compounding, we can use the rules of exponents to do the calculations by taking the current share price to the power of (
g
−
r
). But because
r = (dy + g)
, (
g
−
r
) is just equal to
–dy.
That is the result we have above: we discount the current share price at
dy
(taking it to the
–dy
power).

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