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Hedge Ratios and the Black-Scholes Formula
In the last chapter, we considered two investments in FinCorp stock: 100 shares or 1,000
call options. We saw that the call option position was more sensitive to swings in the stock
price than was the all-stock position. To analyze the overall exposure to a stock price more
precisely, however, it is necessary to quantify these relative sensitivities. We can summa-
rize the overall exposure of portfolios of options with various exercise prices and times to
expiration using the hedge ratio , the change in option price for a $1 increase in the stock
price. A call option, therefore, has a positive hedge ratio and a put option a negative hedge
ratio. The hedge ratio is commonly called the option’s delta .
If you were to graph the option value as a function of the stock value, as we have done
for a call option in Figure 21.9 , the hedge ratio is simply the slope of the curve evaluated
at the current stock price. For example, suppose the slope of the curve at S
0
5 $120 equals .60.
As the stock increases in value by $1, the option increases by approximately $.60, as the
figure shows.
For every call option written, .60 share of stock would be needed to hedge the investor’s
portfolio. If one writes 10 options and holds six shares of stock, according to the hedge ratio
of .6, a $1 increase in stock price will result in a gain of $6 on the stock holdings, whereas
the loss on the 10 options written will be 10 3 $.60, an equivalent $6. The stock price
movement leaves total wealth unaltered, which is what a hedged position is intended to do.
Black-Scholes hedge ratios are particularly easy to compute. The hedge ratio for a call is
N ( d
1
), whereas the hedge ratio for a put is N ( d
1
) 2 1. We defined N ( d
1
) as part of the Black-
Scholes formula in Equation 21.1. Recall that N ( d ) stands for the area under the standard
normal curve up to d. Therefore, the call option hedge ratio must be positive and less than
1.0, whereas the put option hedge ratio is negative and of smaller absolute value than 1.0.
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C H A P T E R
2 1
Option
Valuation
747
Figure 21.9 verifies that the slope of the
call option valuation function is less than
1.0, approaching 1.0 only as the stock price
becomes much greater than the exercise price.
This tells us that option values change less
than one-for-one with changes in stock prices.
Why should this be? Suppose an option is so
far in the money that you are absolutely cer-
tain it will be exercised. In that case, every
dollar increase in the stock price would
increase the option value by $1. But if there is
a reasonable chance the call option will expire
out of the money, even after a moderate stock
price gain, a $1 increase in the stock price will
not necessarily increase the ultimate payoff
to the call; therefore, the call price will not
respond by a full dollar.
The fact that hedge ratios are less than 1.0
does not contradict our earlier observation
that options offer leverage and disproportion-
ate sensitivity to stock price movements. Although dollar movements in option prices
are less than dollar movements in the stock price, the rate of return volatility of options
remains greater than stock return volatility because options sell at lower prices. In our
example, with the stock selling at $120, and a hedge ratio of .6, an option with exercise
price $120 may sell for $5. If the stock price increases to $121, the call price would be
expected to increase by only $.60 to $5.60. The percentage increase in the option value is
$.60/$5.00 5 12%, however, whereas the stock price increase is only $1/$120 5 .83%. The
ratio of the percentage changes is 12%/.83% 5 14.4. For every 1% increase in the stock
price, the option price increases by 14.4%. This ratio, the percentage change in option
price per percentage change in stock price, is called the option elasticity.
The hedge ratio is an essential tool in portfolio management and control. An example
will show why.
Value of a Call ( C)
S
0
40
20
0
120
Slope
5 .6
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