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From an intuitive standpoint, the adjustments have two effects. First, the value of the asset is discounted back to the present at the dividend yield to take into account the expected drop in asset value resulting from dividend payments. Second, the interest rate is offset by the dividend yield to reflect the lower carrying cost from holding the asset (in the replicating portfolio). The net effect will be a reduction in the value of calls estimated using this model.

stopt.xls: This spreadsheet allows you to estimate the value of a short term option, when the expected dividends during the option life can be estimated.

ltopt.xls: This spreadsheet allows you to estimate the value of an option, when the underlying asset has a constant dividend yield.

Illustration 5.3: Valuing a short-term option with dividend adjustments - The Black Scholes Correction

Assume that it is March 6, 2001 and that AT&T is trading at $20.50 a share. Consider a call option on the stock with a strike price of $20, expiring on July 20, 2001. Using past stock prices, the variance in the log of stock prices for AT&T is estimated at 60%. There is one dividend, amounting to $0.15, and it will be paid in 23 days. The riskless rate is 4.63%.

Present value of expected dividend = —0 1523 = 0.15

1.0463365

Dividend-adjusted stock price = $20.50 - $0.15 = $20.35

Time to expiration = 103/365 = 0.2822

Variance in ln(stock prices) = 0.62=0.36

Riskless rate = 4.63%

The value from the Black-Scholes is:

Value of Call = (20.35)(0.6006)- (20>t(0 0463)(0-2822)(0.4745)= 2.85

The call option was trading at $2.60 on that day.

Illustration 5.4: Valuing a long term option with dividend adjustments - Primes and Scores

In recent years, the CBOT has introduced longer term call and put options on stocks. On AT&T, for instance, you could have purchased a call expiring on January 17, 2003, on March 6, 2001. The stock price for AT&T is $20.50 (as in the previous example). The following is the valuation of a call option with a strike price of $20. Instead of estimating the present value of dividends over the next two years, we will assume that AT&T's dividend yield will remain 2.51% over this period and that the riskfree rate for a two-year treasury bond is 4.85%. The inputs to the Black-Scholes model are: S = Current asset value = $20.50 K = Strike Price = $20.00 Time to expiration = 1.8333 years

Standard Deviation in ln(stock prices) = 60%

Riskless rate = 4.85%

Dividend Yield = 2.51%

The value from the Black Scholes is:

20.00

Value of Call= (20.50>e-(a0251)(1-8333)(0.6877)- (20>-(0 0485X1 8333) = 6 . 63 The call was trading at $5.80 on March 8, 2001.

2. Early Exercise

The Black-Scholes model was designed to value options that can be exercised only at expiration. Options with this characteristic are called European options. In contrast, most options that we encounter in practice can be exercised any time until expiration. These options are called American options. The possibility of early exercise makes American options more valuable than otherwise similar European options; it also makes them more difficult to value. In general, though, with traded options, it is almost always better to sell the option to someone else rather than exercise early, since options have a time premium, i.e., they sell for more than their exercise value. There are two exceptions. One occurs when the underlying asset pays large dividends, thus reducing the expected value of the asset. In this case, call options may be exercised just before an ex-dividend date, if the time premium on the options is less than the expected decline in asset value as a consequence of the dividend payment. The other exception arises when an investor holds both the underlying asset and deep in-the-money puts, i.e., puts with strike prices well above the current price of the underlying asset, on that asset and at a time when interest rates are high. In this case, the time premium on the put may be less than the potential gain from exercising the put early and earning interest on the exercise price.

There are two basic ways of dealing with the possibility of early exercise. One is to continue to use the unadjusted Black-Scholes model and regard the resulting value as a floor or conservative estimate of the true value. The other is to try to adjust the value of the option for the possibility of early exercise. There are two approaches for doing so. One uses the Black-Scholes to value the option to each potential exercise date. With options on stocks, this basically requires that we value options to each ex-dividend day and choose the maximum of the estimated call values. The second approach is to use a modified version of the binomial model to consider the possibility of early exercise. In this version, the up and down movements for asset prices in each period can be estimated from the variance and the length of each period2.

Approach 1: Pseudo-American Valuation

Step 1: Define when dividends will be paid and how much the dividends will be. Step 2: Value the call option to each ex-dividend date using the dividend-adjusted approach described above, where the stock price is reduced by the present value of expected dividends.

Step 3: Choose the maximum of the call values estimated for each ex-dividend day.

Illustration 5.5: Using Pseudo-American option valuation to adjust for early exercise

Consider an option, with a strike price of $35 on a stock trading at $40. The variance in the ln(stock prices) is 0.05, and the riskless rate is 4%. The option has a remaining life of eight months, and there are three dividends expected during this period: Expected Dividend Ex-Dividend Day $ 0.80 in 1 month

The call option is first valued to just before the first ex-dividend date:

2 To illustrate, if ct2 is the variance in ln(stock prices), the up and down movements in the binomial can be estimated as follows:

where u and d are the up and down movements per unit time for the binomial, T is the life of the option and m is the number of periods within that lifetime.

S = $40 K = $35 t= 1/12 a2 = 0.05 r = 0.04 The value from the Black-Scholes model is: Value of Call = $5.1312

The call option is then valued to before the second ex-dividend date: Adjusted Stock Price = $40 - $0.80/1.041/12 = $39.20 K = $35 t=4/12 a2 = 0.05 r = 0.04

The value of the call based upon these parameters is: Value of call = $5.0732

The call option is then valued to before the third ex-dividend date:

Adjusted Stock Price = $40 - $0.80/1.041/12 - $0.80/1.044/12 = $38.41

The value of the call based upon these parameters is:

The call option is then valued to expiration:

Adjusted Stock Price = $40 - $0.80/1.041/12 - $0.80/1.044/12 - $0.80/1.047/12 = $37.63 K = $35 t=8/12 a2 = 0.05 r = 0.04

The value of the call based upon these parameters is: Value of call = $4.7571

Pseudo-American value of the call = Maximum ($5.1312, $5.0732, $5.1285, $4.7571) = $5.1312

Approach 2: Using the binomial

The binomial model is much more capable of handling early exercise because it considers the cash flows at each time period rather than just the cash flows at expiration. The biggest limitation of the binomial is determining what stock prices will be at the end of each period, but this can be overcome by using a variant that allows us to estimate the up and the down movements in stock prices from the estimated variance. There are four steps involved.

Step 1: If the variance in ln(stock prices) has been estimated for the Black-Scholes, convert these into inputs for the Binomial

where u and d are the up and down movements per unit time for the binomial, T is the life of the option and m is the number of periods within that lifetime.

Step 2: Specify the period in which the dividends will be paid and make the assumption that the price will drop by the amount of the dividend in that period. Step 3: Value the call at each node of the tree, allowing for the possibility of early exercise just before ex-dividend dates. There will be early exercise if the remaining time premium on the option is less than the expected drop in option value as a consequence of the dividend payment.

Step 4: Value the call at time 0, using the standard binomial approach.

bstobin.xls: This spreadsheet allows you to estimate the parameters for a binomial model from the inputs to a Black-Scholes model.

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