8: OptionsThe payoff diagram for the bond requires a little more explanation. The type of bond in this example
is very special. It is a risk-free, zero-coupon bond with a face value of $75. The payoff for an investor
who purchases this bond is simply $75, no matter what the price of the shares underlying the put and
call options turns out to be. That’s why the diagram shows a horizontal line at $75 for the long bond’s
payoff.^10
Next, consider a portfolio consisting of one share and one put option on that share, with a
strike price of $40. If, on the expiration date of the option, the share price is $40, or more, the
put option will be worthless. Therefore, the portfolio’s total value will equal the value of one
share. What happens if the share price is less than $40 on the option’s expiration date? In that
case, the put option has a positive payoff, which ensures that the portfolio’s value cannot drop
below $40, even if the share price does. Imagine that the share price falls to $30. At that point,
the put option’s payoff is $10, leaving the combined portfolio value at $40 ($30 from the share +
$10 from the put). Simply stated, the put option provides a kind of portfolio insurance, for it
guarantees that the share can be sold for at least $40. However, if the price of the shares rises, the
portfolio value will rise right along with it. Though the put option will be worthless, any increase
in the share price beyond $40 increases the portfolio’s value as well, as shown in Figure 8.5. This
strategy is known as a protective put.
Investors can construct portfolios containing options, shares and bonds in ways that generate
a wide range of interesting payoffs. We have illustrated how investors could construct a portfolio
not only to profit from a share’s volatility but also to protect themselves from that volatility, using
put options. As we see in the next section, no matter what kind of payoff structure an investor
wants to create, there is always more than one way to form a portfolio that generates the desired
payoffs.
8-2d PuT–Call ParITY
In the payoff diagrams we have studied thus far, the vertical axis shows the value of an option at a
particular point in time – the expiration date. Knowing what an option is worth when it expires is
important, but option traders need to know the value of options at any time, not just on the expiration
date. We explore option pricing in greater depth in the next two sections, but we can gain some basic
insights into the process of valuing options by examining payoff diagrams.
Suppose an investor forms a portfolio containing one risk-free, zero-coupon bond with a face value
of $75, and one call option with a strike price of $75. The bond matures in one year, which is also when
the call option expires. Figure 8.6 shows that in one year, this portfolio’s payoff will be at least $75.
Even if the call option expires out of the money, the bond will pay $75. In addition, if the price of the
underlying shares is high enough, the call option will have a positive payoff, too, and the portfolio’s payoff
will exceed $75.
Does this diagram look familiar? Notice that it has the same basic shape as the protective put shown
in Figure 8.5. In fact, we could create a new portfolio with exactly the same payoff as the one shown
10 Is it really possible to buy a risk-free bond with a face value of $75, given the usual face value of (risk-free) government debt is either $100 or
$1,000? Perhaps not, but an investor could buy 75 Treasury notes, each with a face value of $1,000, resulting in a risk-free bond portfolio with
a face value of $75,000. The assumption that investors can buy risk-free bonds with any face value is just a simplification to keep the numbers
in our examples manageable.
protective put
A portfolio containing a share
and a put option on that share