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(Notice that e T approximately equals 1 T, so the value of the dividend flow is approximately TS0.)
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

15 Option Valuation

F I G U R E 15.6
Value of a call (C )
Call option value
and hedge ratio


Slope = .6


while the loss on the 10 options written will be 10 $0.60, an equivalent $6. The stock price
movement leaves total wealth unaltered, which is what a hedged position is intended to do.
The investor holding both the stock and options in proportions dictated by their relative price
movements hedges the portfolio.
Black-Scholes hedge ratios are particularly easy to compute. The hedge ratio for a call is
N(d1), while the hedge ratio for a put is N(d1) 1. We defined N(d1) as part of the Black-
Scholes formula in Equation 15.1. Recall that N(d ) stands for the area under the standard nor-
mal curve up to d. Therefore, the call option hedge ratio must be positive and less than 1.0,
while the put option hedge ratio is negative and of smaller absolute value than 1.0.
Figure 15.6 verifies the insight that the slope of the call option valuation function is less
than 1.0, approaching 1.0 only as the stock price becomes extremely large. 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 certain it will be exercised.
In that case, every $1 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 moder-
ate 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 $1.
The fact that hedge ratios are less than 1.0 does not contradict our earlier observation that
options offer leverage and are sensitive to stock price movements. Although dollar move-
ments in option prices are slighter 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 0.6, an option with
exercise price $120 may sell for $5. If the stock price increases to $121, the call price would
option elasticity
be expected to increase by only $0.60, to $5.60. The percentage increase in the option value is
$0.60/$5.00 12%, however, while the stock price increase is only $1/$120 0.83%. The The percentage
ratio of the percent changes is 12%/0.83% 14.4. For every 1% increase in the stock price, increase in an
the option price increases by 14.4%. This ratio, the percent change in option price per percent option™s value
given a 1% increase
change in stock price, is called the option elasticity.
in the value of the
The hedge ratio is an essential tool in portfolio management and control. An example will
underlying security.
show why.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

552 Part FIVE Derivative Markets

Consider two portfolios, one holding 750 IBM calls and 200 shares of IBM and the other
holding 800 shares of IBM. Which portfolio has greater dollar exposure to IBM price move-
15.5 EXAMPLE ments? You can answer this question easily using the hedge ratio.
Each option changes in value by H dollars for each dollar change in stock price, where H
stands for the hedge ratio. Thus, if H equals 0.6, the 750 options are equivalent to 450
( 0.6 750) shares in terms of the response of their market value to IBM stock price
movements. The first portfolio has less dollar sensitivity to stock price change because the
450 share-equivalents of the options plus the 200 shares actually held are less than the 800
shares held in the second portfolio.
This is not to say, however, that the first portfolio is less sensitive to the stock™s rate of re-
turn. As we noted in discussing option elasticities, the first portfolio may be of lower total
value than the second, so despite its lower sensitivity in terms of total market value, it might
have greater rate of return sensitivity. Because a call option has a lower market value than
the stock, its price changes more than proportionally with stock price changes, even though
its hedge ratio is less than 1.0.

8. What is the elasticity of a put option currently selling for $4 with exercise price
$120, and hedge ratio 0.4 if the stock price is currently $122?

Portfolio Insurance
In Chapter 14, we showed that protective put strategies offer a sort of insurance policy on an
asset. The protective put has proven to be extremely popular with investors. Even if the asset
price falls, the put conveys the right to sell the asset for the exercise price, which is a way to
lock in a minimum portfolio value. With an at-the-money put (X S0), the maximum loss that
can be realized is the cost of the put. The asset can be sold for X, which equals its original
price, so even if the asset price falls, the investor™s net loss over the period is just the cost of
the put. If the asset value increases, however, upside potential is unlimited. Figure 15.7 graphs
the profit or loss on a protective put position as a function of the change in the value of the
underlying asset.
portfolio While the protective put is a simple and convenient way to achieve portfolio insurance,
insurance that is, to limit the worst-case portfolio rate of return, there are practical difficulties in trying
to insure a portfolio of stocks. First, unless the investor™s portfolio corresponds to a standard
market index for which puts are traded, a put option on the portfolio will not be available for
strategies that limit
purchase. And if index puts are used to protect a nonindexed portfolio, tracking error can re-
investment losses
while maintaining sult. For example, if the portfolio falls in value while the market index rises, the put will fail
upside potential.
to provide the intended protection. Tracking error limits the investor™s freedom to pursue ac-
tive stock selection because such error will be greater as the managed portfolio departs more
substantially from the market index.
Moreover, the desired horizon of the insurance program must match the maturity of a
traded put option in order to establish the appropriate protective put position. Today, long-term
index options called LEAPS (for Long-Term Equity AnticiPation Securities) trade on the
Chicago Board Options Exchange with maturities of several years. However, in the mid-
1980s, while most investors pursuing insurance programs had horizons of several years, ac-
tively traded puts were limited to maturities of less than a year. Rolling over a sequence of
short-term puts, which might be viewed as a response to this problem, introduces new risks
because the prices at which successive puts will be available in the future are not known today.
Providers of portfolio insurance with horizons of several years, therefore, cannot rely on
the simple expedient of purchasing protective puts for their clients™ portfolios. Instead, they
follow trading strategies that replicate the payoffs to the protective put position.
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

15 Option Valuation

F I G U R E 15.7
Change in value
of protected position Profit on a
put strategy

0 Change in value
of underlying asset
Cost of put

Here is the general idea. Even if a put option on the desired portfolio with the desired ex-
piration date does not exist, a theoretical option-pricing model (such as the Black-Scholes
model) can be used to determine how that option™s price would respond to the portfolio™s value
if the option did trade. For example, if stock prices were to fall, the put option would increase
in value. The option model could quantify this relationship. The net exposure of the (hypo-
thetical) protective put portfolio to swings in stock prices is the sum of the exposures of the
two components of the portfolio: the stock and the put. The net exposure of the portfolio
equals the equity exposure less the (offsetting) put option exposure.
We can create “synthetic” protective put positions by holding a quantity of stocks with the
same net exposure to market swings as the hypothetical protective put position. The key to this
strategy is the option delta, or hedge ratio, that is, the change in the price of the protective put
option per change in the value of the underlying stock portfolio.

Suppose a portfolio is currently valued at $100 million. An at-the-money put option on the
portfolio might have a hedge ratio or delta of 0.6, meaning the option™s value swings $0.60
for every dollar change in portfolio value, but in an opposite direction. Suppose the stock port-
folio falls in value by 2%. The profit on a hypothetical protective put position (if the put existed) Synthetic
would be as follows (in millions of dollars): Protective
Loss on stocks 2% of $100 $2.00
Gain on put: 0.6 $2.00 1.20
Net loss $0.80

We create the synthetic option position by selling a proportion of shares equal to the put
option™s delta (i.e., selling 60% of the shares) and placing the proceeds in risk-free T-bills. The
rationale is that the hypothetical put option would have offset 60% of any change in the stock
portfolio™s value, so one must reduce portfolio risk directly by selling 60% of the equity and
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition

554 Part FIVE Derivative Markets

putting the proceeds into a risk-free asset. Total return on a synthetic protective put position
with $60 million in risk-free investments such as T-bills and $40 million in equity is

Loss on stocks: 2% of $40 $0.80
Loss on bills: 0
Net loss $0.80

The synthetic and actual protective put positions have equal returns. We conclude that if
you sell a proportion of shares equal to the put option™s delta and place the proceeds in cash
equivalents, your exposure to the stock market will equal that of the desired protective put

F I G U R E 15.8 Value of a put (P)
Hedge ratios change
as the stock price Higher slope =
High hedge ratio

Low slope =
Low hedge ratio

The difficulty with synthetic positions is that deltas constantly change. Figure 15.8 shows
that as the stock price falls, the absolute value of the appropriate hedge ratio increases. There-
fore, market declines require extra hedging, that is, additional conversion of equity into cash.
This constant updating of the hedge ratio is called dynamic hedging, as discussed in Section
dynamic hedging
15.2. Another term for such hedging is delta hedging, because the option delta is used to de-
Constant updating of
termine the number of shares that need to be bought or sold.
hedge positions as
Dynamic hedging is one reason portfolio insurance has been said to contribute to market
market conditions
change. volatility. Market declines trigger additional sales of stock as portfolio insurers strive to in-
crease their hedging. These additional sales are seen as reinforcing or exaggerating market
In practice, portfolio insurers do not actually buy or sell stocks directly when they update
their hedge positions. Instead, they minimize trading costs by buying or selling stock index fu-
tures as a substitute for sale of the stocks themselves. As you will see in the next chapter, stock
prices and index future prices usually are very tightly linked by cross-market arbitrageurs
so that futures transactions can be used as reliable proxies for stock transactions. Instead of
Bodie’Kane’Marcus: V. Derivative Markets 15. Option Valuation © The McGraw’Hill
Essentials of Investments, Companies, 2003
Fifth Edition


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