560 Part Six Options
bre44380_ch21_547-572.indd 560 10/05/15 12:53 PM
Step 3 Plug these numbers into the Black–Scholes formula. You can now calculate the
value of the Google call:[Delta × price] − [bank loan]
= [N(d 1 ) × P] − [N(d 2 ) × PV(EX)]
= [.5621 × 530] − [.4733 × (530/1.01)] = 297.89 − 248.36 = $49.52In other words, you can replicate the Google call option by investing $297.89 in the compa-
ny’s stock and borrowing $248.36. Subsequently, as time passes and the stock price changes,
you may need to borrow a little more to invest in the stock or you may need to sell some of
your stock to reduce your borrowing.Some More Practice Suppose you repeat the calculations for the Google call for a wide range
of stock prices. The result is shown in Figure 21.5. You can see that the option values lie along an
upward-sloping curve that starts its travels in the bottom left-hand corner of the diagram. As the
stock price increases, the option value rises and gradually becomes parallel to the lower bound
for the option value. This is exactly the shape we deduced in Chapter 20 (see Figure 20.10).
The height of this curve of course depends on risk and time to maturity. For example, if
the risk of Google stock had suddenly increased, the curve shown in Figure 21.5 would rise at
every possible stock price. For example, Figure 20.12 shows what would happen to the curve
if the risk of Google stock doubled.The Risk of an Option
How risky is the Google call option? We have seen that you can exactly replicate a call by a
combination of risk-free borrowing and an investment in the stock. So the risk of the option
must be the same as the risk of this replicating portfolio. We know that the beta of any portfo-
lio is simply a weighted average of the betas of the separate holdings. So the risk of the option
is just a weighted average of the betas of the investments in the loan and the stock.
On past evidence the beta of Google stock is βstock = 1.15; the beta of a risk-free loan is
βloan = 0. You are investing $297.89 in the stock and –$248.36 in the loan. (Notice that the
investment in the loan is negative—you are borrowing money.) Therefore the beta of the option
is βoption = (–248.36 × 0 + 297.89 × 1.15)/(–248.36 + 297.89) = 6.92. Notice that, because
a call option is equivalent to a levered position in the stock, it is always riskier than the stock
itself. In Google’s case the option is nearly seven times as risky as the stock. As time passes
and the price of Google stock changes, the risk of the option will also change.◗ FIGURE 21.5
The curved line shows how the value
of the Google call option changes as
the price of Google stock changes.0100200300400500600700800900Value of call, $Stock price, $
Exercise price = $5300(^53106159212265318371424477530583636689742795)
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Try It! The Black-
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