C H A P T E R
2 1
Option
Valuation
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The Chicago Board Options Exchange
regularly computes the implied volatil-
ity of major stock indexes. Figure 21.8 is
a graph of the implied (30-day) volatility
of the S&P 500 since 1990. During peri-
ods of turmoil, implied volatility can spike
quickly. Notice the peaks in January 1991
(Gulf War), August 1998 (collapse of Long-
Term Capital Management), September 11,
2001, 2002 (build-up to invasion of Iraq),
and, most dramatically, during the credit cri-
sis of 2008. Because implied volatility cor-
relates with crisis, it is sometimes called an
“investor fear gauge.”
A futures contract on the 30-day implied
volatility of the S&P 500 has traded on the
CBOE Futures Exchange since 2004. The
payoff of the contract depends on market
implied volatility at the expiration of the contract. The ticker symbol of the contract is VIX.
As the nearby box makes clear, observers use it to infer the market’s assessment of possible
stock price swings in coming months. In this case, the article questioned the relatively low level
of the VIX in light of tense political negotiations at the end of 2012 over the so-called fiscal
cliff. The question was whether the price of the VIX contract indicated that investors were being
too complacent about the potential for market disruption if those negotiations were to fail.
Figure 21.8 reveals an awkward empirical fact. While the Black-Scholes formula is
derived assuming that stock volatility is constant, the time series of implied volatilities
derived from that formula is in fact far from constant. This contradiction reminds us that
the Black-Scholes model (like all models) is a simplification that does not capture all
aspects of real markets. In this particular context, extensions of the pricing model that
allow stock volatility to evolve randomly over time would be desirable, and, in fact, many
extensions of the model along these lines have been developed.
12
The fact that volatility changes unpredictably means that it can be difficult to choose
the proper volatility input to use in any option-pricing model. A considerable amount of
recent research has been devoted to techniques to predict changes in volatility. These tech-
niques, known as ARCH and stochastic volatility models, posit that changes in volatility
are partially predictable and that by analyzing recent levels and trends in volatility, one can
improve predictions of future volatility.
13
Gulf War
LTCM
9/11
Iraq
Subprime and
Credit Crises
U.S. Debt
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Im
p
lie
d
V
o
la
tilit
y
(
%
)
Figure 21.8
Implied volatility of the S&P 500 (VIX index)
Source: Chicago Board Options Exchange, www.cboe.com .
Suppose the call option in Spreadsheet 21.1 actually is selling for $8. Is its implied volatility more or less
than 27.83%? Use the spreadsheet (available at the Online Learning Center) and Goal Seek to find its
implied volatility at this price.
CONCEPT CHECK
21.7
12
Influential articles on this topic are J. Hull and A. White, “The Pricing of Options on Assets with Stochastic Vola-
tilities,”
Journal of Finance (June 1987), pp. 281–300; J. Wiggins, “Option Values under Stochastic Volatility,”
Jour-
nal of Financial Economics (December 1987), pp. 351–72; and S. Heston, “A Closed-Form Solution for Options
with Stochastic Volatility with Applications to Bonds and Currency Options,” Review of Financial Studies 6 (1993),
pp. 327–43. For a more recent review, see E. Ghysels, A. Harvey, and E. Renault, “Stochastic Volatility,” in Hand-
book of Statistics, Vol. 14: Statistical Methods in Finance, ed. G. S. Maddala (Amsterdam: North Holland, 1996).
13
For an introduction to these models see C. Alexander, Market Models (Chichester, England: Wiley, 2001).
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