8.2Option Expiration

Exhibit 8.1, first seen in Chapter 7, graphs the value of a call and put option at the

expiration date against the value of the underlying asset at expiration. In this chapter,

we attach some algebra to the graphs of call and put values. For expositional simplic-

ity, we will often refer to the underlying asset as a share of common stock, but our

results also apply to options on virtually any financial instrument.

The uncertain future stock price at the expiration date, T,is denoted by S. The

T

strike price is denoted by K. The expiration value for the call option is the larger

ofzero and the difference between the stock price at the expiration date and the

strike price, denoted as max[0, S K]. For the put option, the expiration value is

T

max[0, K S].

T

³Typically, the underlying asset is common stock, a portfolio of stocks, foreign currencies, or futures

contracts, but there are many other assets or portfolios of assets on which options can be written. We use

the term “asset” loosely here to mean anything that has an uncertain value over time, be it an asset, a

liability, a contract, or a commodity. There is a vast over-the-counter market between financial institutions

in which options of almost any variety on virtually any underlying asset or portfolio of assets are traded.

⁴Deferred American options, not discussed here, have issue dates that precede their commencement

dates.

Grinblatt536Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw536Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

260	Part IIValuing Financial Assets
EXHIBIT8.1	The Value of a Call Option and a Put Option at Expiration

Call	Panel A: Call Option
value

Call: max [0,S T – K]

No
	Exercise region
exercise region

	45
		S T
K

Put	Panel B: Put Option
value

Put: max [0,K –S ] T

	No
Exercise region
Exercise region
	exercise region

45
		S T
	K

Note that the two graphs in Exhibit 8.1 never lie below the horizontal axis. They

either coincide with the axis or lie above it on the 45°line. In algebraic terms, the expres-

sionsfor the future call value, max[0, S K], and the future put value, max[0, K S],

TT

are never negative. Recall from Chapter 7 that options can never have a negative value

because options expire unexercised if option exercise hurts the option holder. The

absence of a negative future value for the option and the possibility of a positive future

value makes paying for an option worthwhile.

Future cash flows are never positive when writing an option. Exhibit 8.2 illustrates

the value at expiration of the short position generated by writing an option. When the

call’s strike price, K,exceeds the future stock price S(or Sexceeds Kfor the put), the

TT

option expires unexercised. On the other hand, if Sexceeds K,the call writer has to

T

sell a share of stock for less than its fair value. Similarly, if Kexceeds S,the put writer

T

has to buy a share of stock for more than it is worth. In all cases, there is no positive

future cash flow to the option writer. To compensate the option writer for these future

adverse consequences, the option buyer pays money to the writer to acquire the option.

Grinblatt538Titman: Financial	II. Valuing Financial Assets	8. Options	© The McGraw538Hill
Markets and Corporate			Companies, 2002
Strategy, Second Edition

	Chapter 8Options	261
EXHIBIT8.2	The Value of Short Positions in Call and Put Options at Expiration

Call	Panel A: Short Call Option
value

	K
			S T
No		Exercise region
exercise region

Call: max [0,ST – K]

Put	Panel B: Short Put Option
value

K
		S T
	No
Exercise region
	exercise region

Put: max [0,K –ST ]

Finally, observe that the nonrandom number S, which denotes the current stock

0

price, does not appear in Exhibits 8.1 and 8.2 because the focus is only on what hap-

pens at option expiration. One of the goals of this chapter is to translate the future rela-

tion between the stock value and the option value into a relation between the current

value of the stock and the current value of the option. The next section illustrates the

type of reasoning used to derive such a relation.