7.7. LIMITATIONS OF THE BLACK-SCHOLES MODEL 271
the problem. Still, changing volatility is an area of active research, both
practically and academically.
We also assumed that trading was continuous in time, and that security
prices moved continuously. Of course, continuous change is an idealizing
assumption. In fact, in October 1987, the markets dropped suddenly, almost
discontinuously, and market strategies based on continuous trading were not
able to keep with the selling panic that developed on Wall Street. Of course,
the October 1987 drop is yet another illustration that the markets do not
behave exactly as trading history would predict. [12]
Alternatives to Black-Scholes
Financial economists and mathematicians have suggested a number of alter-
natives to the Black-Scholes model. These alternatives include:
- stochastic volatilitymodels where the future volatility of a security
price is uncertain, - jump-diffusion modelswhere the security price experiences occa-
sional jumps rather than continuous change.
In spite of these flaws, the Black-Scholes model does a good job of gener-
ally predicting market prices. Generally, the empirical research is supportive
of the Black-Scholes model. Observed differences have been small compared
to transaction costs. Even more importantly, the Black-Scholes model shows
how to assign prices to risky assets by using the principle of no-arbitrage ap-
plied to a replicating portfolio and reducing the pricing to applying standard
mathematical tools.
Sources
This section is adapted from: “Financial Derivatives and Partial Differential
Equations” by Robert Almgren, in American Mathematical Monthly, Vol-
ume 109, January, 2002, pages 1–11 , and fromOptions, Futures, and other
Derivative Securities second edition, by John C. Hull, Prentice Hall, 1993,
pages 229–230, 448–449 andBlack-Scholes and Beyond: Option Pricing Mod-
els, by Neil A. Chriss, Irwin Professional Publishing, Chicago, 1997. Some
additional ideas are adapted fromWhen Genius Failedby Roger Lowenstein,
Random House, New York.