Ross et al.: Fundamentals
of Corporate Finance, Sixth
Edition, Alternate EditionVIII. Topics in Corporate
Finance(^846) 24. Option Valuation © The McGraw−Hill
Companies, 2002
Looking back at our graph in Figure 24.1, we now see why the lines get progres-
sively steeper as the stock price rises for calls and falls for puts. The delta for a deeply
in-the-money option is close to one, whereas the delta for a deeply out-of-the-money op-
tion is close to zero.
Varying the Time to Expiration
The impact of changing the time to maturity on American-style options is also fairly ob-
vious. Since an American-style option can be exercised any time, increasing the option’s
time to expiration can’t possibly hurt and, especially for out-of-the-money options,
might help. Thus, for both puts and calls, increasing the time to expiration has a positive
effect.
For a European-style call option, increasing the time to expiration also never hurts be-
cause, as we discussed earlier, the option is always worth more alive than dead and any
extra time to expiration only adds to its “alive” value. With a European-style put, how-
ever, increasing the time to expiration may or may not increase the value of the option.
As we have discussed, for a deep in-the-money put, immediate exercise is often desir-
able, so increasing the time to expiration only reduces the value of the option. If a put is
out-of-the-money, then increasing the time to expiration will probably increase its value.
Figure 24.2 shows the effect of increasing the time to expiration on a put and a call.
As in Figure 24.1, the options are exactly at the money. In the figure, notice that once
time to maturity reaches about six months, further increases have little impact on the
put’s value. The call’s value, in contrast, just keeps on rising.
The sensitivity of an option’s value to the passage of time is called its theta. There is
a formula for theta, but it is fairly complicated, so we will not present it. The important
thing to realize is that option values are sensitive to the passage of time (especially call
option values). To see why this is important, imagine you buy an option today and you
hold it for a month. During the month, the stock price never changes. What happens to
the value of your option?
The answer is that the value of your option declines because time to expiration has
gotten shorter even though the underlying asset has not changed in value. We sometimes
say that an option is a “wasting” asset, meaning that its value declines as time goes by,
all else held constant. The tendency of an option’s value to decline as time passes is also
called “time decay.” An option’s theta is thus a measure of the rate of time decay.
CHAPTER 24 Option Valuation 821
Delta
Suppose you are given the following:
S$40
E$30
R6% per year, continuously compounded
90% per year
t3 months
What’s the delta for a call option? A put option? Which one is more sensitive to a change in the
stock price. Why?
We need to calculate N(d 1 ). See if you agree that it’s .815, which is the delta for the call.
The delta for the put is .815 1 .185, which is much smaller (in absolute value). The
reason is that the call option is well in the money and the put is out of the money.
EXAMPLE 24.6
theta
Measures the sensitivity
of an option’s value to a
change in the time to
expiration.