758
P A R T V I
Options, Futures, and Other Derivatives
position, so that both options are priced at implied volatilities of 30%. You expect to
profit from the increase in the value of the call purchased as well as from the decrease
in the value of the call written. The option prices at 30% volatility are given in panel
B of Table 21.4 and the values of your position for various stock prices are presented
in panel C. Although the profit or loss on each option is affected by the stock price,
the value of the delta-neutral option portfolio is positive and essentially independent
of the price of IBM. Moreover, we saw in panel A that the portfolio would have been
established without ever requiring a cash outlay. You would have cash inflows both
when you establish the portfolio and when you liquidate it after the implied volatilities
converge to 30%.
This unusual profit opportunity arises because you have identified prices out of align-
ment. Such opportunities could not arise if prices were at equilibrium levels. By exploiting
the pricing discrepancy using a delta-neutral strategy, you should earn profits regardless of
the price movement in IBM stock.
Delta-neutral hedging strategies are also subject to practical problems, the most impor-
tant of which is the difficulty in assessing the proper volatility for the coming period. If the
volatility estimate is incorrect, so will be the deltas, and the overall position will not truly
be hedged. Moreover, option or option-plus-stock positions generally will not be neutral
with respect to changes in volatility. For example, a put option hedged by a stock might be
delta neutral, but it is not volatility neutral. Changes in the market assessments of volatility
will affect the option price even if the stock price is unchanged.
These problems can be serious, because volatility estimates are never fully reliable.
First, volatility cannot be observed directly and must be estimated from past data which
imparts measurement error to the forecast. Second, we’ve seen that both historical and
implied volatilities fluctuate over time. Therefore, we are always shooting at a moving
target. Although delta-neutral positions are hedged against changes in the price of the
underlying asset, they still are subject to volatility risk, the risk incurred from unpre-
dictable changes in volatility. The sensitivity of an option price to changes in volatility
is called the option’s vega . Thus, although delta-neutral option hedges might eliminate
exposure to risk from fluctuations in the value of the underlying asset, they do not elimi-
nate volatility risk.
21.6
Empirical Evidence on Option Pricing
The Black-Scholes option-pricing model has been subject to an enormous number of
empirical tests. For the most part, the results of the studies have been positive in that the
Black-Scholes model generates option values fairly close to the actual prices at which
options trade. At the same time, some regular empirical failures of the model have been
noted.
The biggest problem concerns volatility. If the model were accurate, the implied volatil-
ity of all options on a particular stock with the same expiration date would be equal—after
all, the underlying asset and expiration date are the same for each option, so the volatility
inferred from each also ought to be the same. But in fact, when one actually plots implied
volatility as a function of exercise price, the typical results appear as in Figure 21.15 , which
treats S&P 500 index options as the underlying asset. Implied volatility steadily falls as the
exercise price rises. Clearly, the Black-Scholes model is missing something.
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C H A P T E R
2 1
Option
Valuation
759
Rubinstein
19
suggests that the prob-
lem with the model has to do with fears of
a market crash like that of October 1987.
The idea is that deep out-of-the-money puts
would be nearly worthless if stock prices
evolve smoothly, because the probabil-
ity of the stock falling by a large amount
(and the put option thereby moving into
the money) in a short time would be very
small. But a possibility of a sudden large
downward jump that could move the puts
into the money, as in a market crash, would
impart greater value to these options. Thus,
the market might price these options as
though there is a bigger chance of a large
drop in the stock price than would be sug-
gested by the Black-Scholes assumptions.
The result of the higher option price is a
greater implied volatility derived from the
Black-Scholes model.
Interestingly, Rubinstein points out
that prior to the 1987 market crash, plots
of implied volatility like the one in Figure 21.15 were relatively flat, consistent with the
notion that the market was then less attuned to fears of a crash. However, postcrash plots
have been consistently downward sloping, exhibiting a shape often called the option smirk.
When we use option-pricing models that allow for more general stock price distributions,
including crash risk and random changes in volatility, they generate downward-sloping
implied volatility curves similar to the one observed in Figure 21.15 .
20
0.84
0.89
0.94
0.99
1.04
1.09
Implied V
olatility (%)
Ratio of Exercise Price to Current Value of Index
25
20
15
10
5
0
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