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Updated: May 9, 2025
Black-Scholes is thought deficient on other issues as well. The implied volatilities of different options on the same stock tend to vary, defying the formula's postulate that a single stock can be associated with only one value of implied volatility. The model assumes a certain geometric Brownian distribution of stock prices that has been shown to not apply to US markets, among others.
Black-Scholes tackles stochastic volatility poorly. The formula also unrealistically assumes that the market dickers continuously, ignoring transaction costs and institutional constraints. No wonder that traders use Black-Scholes as a heuristic rather than a price-setting formula. Volatility also decreases in administered markets and over different spans of time.
The folly of this argument lies in the fact that stochastic volatility contradicts the assumption required by the B-S model if volatilities do change stochastically through time, the Black-Scholes formula is no longer the correct pricing formula and an implied volatility derived from the Black-Scholes formula provides no new information."
In the Risk annual congress in Boston two years ago, Myron Scholes of Black-Scholes fame and LTCM infamy, publicly recanted, admitting that, as quoted by Dwight Cass in the May 2002 issue of Risk Magazine: "It is impossible to fully account for risk in a fluid, chaotic world full of hidden feedback mechanisms." Jeff Skilling of Enron publicly begged to disagree with him.
Quantifying volatility has yielded an equal number of Nobel prizes and controversies. The vacillation of security prices is often measured by a coefficient of variation within the Black-Scholes formula published in 1973. Volatility is implicitly defined as the standard deviation of the yield of an asset. The value of an option increases with volatility.
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