The Black-Scholes option pricing model is one of the most famous equations in finance. With it mathematics replaced intuition as the means of pricing options. The knowable inputs in the equation are the stock or index price, the exercise price, the time to expiration and the risk-free rate (which is currently so low that its omission from the formula will not materially alter the results except for very long dated options).

The fifth and final input, the volatility of the underlying asset’s price, is where this model proves once again that there is *no* perfect forward-looking model. Black-Scholes uses historical volatility, which is the standard deviation of historical price moves times the square root of some given number of trading days, always results in an option price cheaper than the market. Sellers of options make a profit when the volatility implied by the price of their contracts is higher than the volatility realized in the past. To the extent volatility does not deviate too much from the recent norm, sellers make money. As you’ll read below, however, the difference between future volatility and historical, can be extreme.

Enter the Chicago Board of Options Exchange (CBOE) Volatility Index, known by its ticker symbol VIX, as a means of commercializing Black-Scholes. The VIX is a newbie in finance, with real-time data first available in 1993. It attempts to formulate a theoretical expectation of stock market volatility in the near future derived from short-term at-the-money S&P index contracts. It is quoted in percentage points and represents the expected the range of movement in the S&P 500 over time.

As recently as January 11, the VIX was under 10%. Less than a month later, on February 2, the VIX had crept up to 17.3%. By the close on the following Monday, February 5, it more than doubled to 37.5% as the S&P 500 plunged 4.2%, the biggest one-day decline since June 24, 2016. For the story on those who placed too much stock in the predictive power of the VIX, see the February 9 post, “There’s Never Just One Cockroach in the Kitchen.”

Among the VIX critics, Robert J. Shiller has argued that it would be circular reasoning to use VIX as predictive of future price moves because it relies on an input that requires unprovable assumptions about the future. Calculating VIX retrospectively back to 1929, Schiller concluded that the metric offered no warning of the tail risks that were to plague the market in the years to come.

The “unproveable assumptions about the future” is called implied volatility. As an essential component of options pricing, it attempts to provide the seller with a margin of safety over historical volatility. It is, working backwards through Black-Scholes, an imputed value derived from the price. Only after an option is traded can we determine the volatility implied by the transaction. If a one-year at-the-money contract has an implied volatility of 15%, it indicates a 68% chance the market will move within 15% of its current level (given a normal distribution of outcomes) over the year. That simple probability of future price moves is based solely upon a consensus arrived at by the market participants in the trade.

Implied volatility is inversely correlated with market price movements; i.e., it rises as markets fall and vice a versa. Further, the relationship is nonlinear, and volaility can rise dramatically in declining markets. During the January 26 to February 15 sell-off, the S&P 500 declined 10% but the implied volatility of a one-year 40% out-of-the-money put option on the index increased from 27.9% to 34.1%, a 22.2% increase. Consequentially, the estimated option price rocketed from $5.64 to $16.60.

This phenomenon, as you’ve likely observed, is logically counterintuitive. If the implied volatility of the put option contract specified above was a good predictor of future outcomes (if it’s the one variable, after all, that attempts to reconcile today’s price with that at the time of expiration, adjusted for the cost of money), it would be high at market peaks and low at market bottoms. Instead, implied volatility typically peaks near the end of a selloff. Obviously, a goodly portion of the implied volatility percentage at any given point in time is determined by looking in the rearview mirror.

Of course, this paradigm of primarily viewing the future through the lens of the present and the immediate past—of simple extrapolation—is how stocks themselves, or markets as stocks in the aggregate, are priced. Paraphrasing Warren Buffett’s lament, “investors are greedy when they should be fearful and fearful when they should be greedy.” Since this is true of both the underlying asset and its derivative—the option—the asymmetry of outcomes for the latter can indeed be extreme. See the paragraph above that begins with the phrase “Implied volatility.”

In his 2008 letter to the shareholders of Berkshire Hathaway, Warren Buffett wrote: “I believe the Black–Scholes formula, even though it is the standard for establishing the dollar liability for options, produces strange results when the long-term variety are being valued… The Black–Scholes formula has approached the status of holy writ in finance … If the formula is applied to extended time periods, however, it can produce absurd results.”

Pricing anomalies in long-term options result from models that simply cannot begin to adequately compensate for tail risks and are the source of potential windfalls for the informed and patient investor.