13.2  Applications:  Stock Options

In this topic we apply the tools of option pricing theory to value options trading on IBM at the close of trading on March 24, 1994.  These are American options trading on (dividend-paying) IBM stock, traded on the Chicago Board of Trade (CBOT).

An American option on a dividend-paying stock immediately creates some practical problems.  The most formidable is that there is no closed-form solution to this type of valuation problem.  As a result, we have to resort to numerical methods to compute the option's value.

Online, you can apply the Option Calculator in Option Tutor to compute the values of (American) options traded on IBM.  Table 13.1 shows how option prices are quoted in the financial press (e.g. in The Wall Street Journal).   Column information is interpreted as follows.

Column 1:  The company name, IBM, and the stock’s closing price (from the NYSE).

Column 2:  The strike or exercise price of the option.

Column 3:  The expiration date of the option.  Recall that stock options expire on the Saturday after the third Friday of the month listed.

Column 4:  An estimate of open interest for the call option.  That is, the number of option contracts outstanding for all exchanges for the previous trading day (one contract is for 100 stocks).   Note that this information is one-day old, but the price information is current.

Column 5:  The last traded price for the call option.

Column 6:  Same as column 4, except for the put option; an estimate of open interest for the put option.

Column 7:  The last traded price for the put option.

A blank entry in the table means that the option did not trade.

Table 13.1

Option Prices

The timing of dividend-related complications are summarized in the time lines shown in Figure 13.1.   This figure shows you the  critical event dates for the four expiration months listed in Table 13.1.

Figure 13.1

IBM Option Event Dates

Critical Events

Markets have just completed trading on March 24, 1994.  This fixes the starting date for determining the remaining life of an option.

The options are settled on the Saturday after the third Friday of the expiration month. The expiration dates are shown on the time lines.

We have to convert the remaining life of each option into a proportion of a year.  For example, you can calculate that there are 58 days remaining in the life of any May expiration IBM option.  The option pricing model requires us to express the remaining life in days as an "annualized time," which here is 58/365 = 0.1589 of a year. 

Online, you can enter the May expiration date, 5/21/94, as well as “today's date” (which here is 3/24/94) into the date calculator.  If you click on OK, the date calculator will automatically place the value 0.1589 into the Maturity field of the calculator.

The remaining two events in the time line relate to dividends.  Observe that IBM stock goes ex-dividend twice during the lives of these options.  The first is in May, and this affects all but the April option.  The second is in September, which only affects the October option.  Dividends affect option values, because if you buy a stock after ex-dividend date, you do not get the dividends just declared (although you would be entitled to future dividends).

The effect of the May ex-dividend date is that IBM's stock price will adjust for the dividend on this date, (even though the actual payment date is June 6, 1994).   Since the value of the option depends on the stock price, the ex-dividend price is what is relevant to optionholders.  Similarly, the price will change when the stock goes ex-dividend in September.  The effects of these dividend payments need to be taken into account when valuing the option on March 24.

In order to apply the Black-Scholes option pricing model, we adjust the current IBM stock price by the present value of the projected dividend payment.  To do so we need to know five pieces of information:

1.  The contractual details of the option, such as maturity date, strike or exercise price, type (American or European), and whether it is a put or a call.

2.  The term structure of default-free interest rates.

3.  Estimated dividends over the life of the option.

4.  The volatility of IBM stock.

5.  The appropriate discount rate for critical cash flow events (such as the payment of dividends).

IBM options are American and the current strike prices being traded are provided in Table 13.1. The time to maturity is computed from the current date (now) to the Saturday after the third Friday of the option's expiration month.

For interest rates, we use the yields on Treasury bills that mature at dates very close to the option maturity date and also the ex-dividend date.  For the latter we will assume that the default-free interest rate is accurate to the cent.  These yields (on March 24) are shown in Table 13.2.

Table 13.2

Treasury Bill Yields 

This table gives you appropriate risk-free interest rates for the option valuation problem.  You may recall that in the Black-Scholes model, interest rates are assumed to be constant.  As the table reveals, in fact they vary with the time to maturity. 

To accommodate this problem, we simply make an approximation that is widely used in practice: we value the option using the yield to maturity of the Treasury bill closest to the maturity date.  For an appropriate discount rate for dividends, given the relatively short times involved, the term structure of Treasury bills provides a close enough estimate. 

IBM's last quarter dividend is our estimate for the current quarter's dividend.  Information on IBM's recent dividend history appears in Table 13.3.

Table 13.3

IBM’s Recent Dividend History