No, Small Probabilities Are Not "Attractive to Sell": A Comment

Owing to the convexity of the payoff of out -of-the money options, an extremely small probability of a large deviation unseen in past data justifies rationally buying them, or at least justifies excessive caution in not being exposed to them, particularly those options that are extremely nonlinear in response to market movement. On needs, for example, a minimum of 2000 years of stock market data to assert that some tail options are "expensive". The paper presents errors in Ilmanen (2012), which provides an exhaustive list of all arguments in favor of selling insurance on small probability events. The paper goes beyond Ilmanen (2012) and suggests an approach to analyze the payoff and risks of options based on the nonlinearities in the tails. In answering the question posed by his recent article (September/October 2012), Antti Ilmanen concluded— seemingly backed by a great deal of “empirical” examination and citing a large number of studies—that investors should not merely be uninsured but should also consider selling such insurance. Selling volatility on the left tail “adds value in the long term.” He also included carry trades because they imply tail-selling risk insurance. Perhaps Ilmanen cited too many papers and arguments for comfort. Just as in a complicated detective novel in which the character with the most alibis often turns out to be the murderer, the enumeration of “backup” arguments fails to mask a central methodological error: a combination of (1) cherry picking and (2) missing nonlinear effects and asymmetries in errors (deviations from the model result in considerably more harm when one is wrong than when one is right). Merely adding these nonlinear responses to tail events does more than reverse the result. Further, because Ilmanen included a review of all supporting arguments against the purchase of small-probability events, refuting his article allows the refutation of the prevailing arguments that posit the overpricing of small odds in finance. So, it turns out that there is not a single study that convincingly demonstrates the overpricing of small probabilities in finance or economics. 1 Ilmanen, Antti, 2012, "Do Financial Markets Reward Buying or Selling Insurance and Lottery Tickets?" Financial Analysts Journal, September/October, Vol. 68, No. 5 : 26 36. Abstract of Ilmanen's article: "Selling financial investments with insurance or lottery characteristics should earn positive long-run premiums if investors like positive skewness enough to overpay for these characteristics. The empirical evidence is unambiguous: Selling insurance and selling lottery tickets have delivered positive long-run rewards in a wide range of investment contexts. Conversely, buying financial catastrophe insurance and holding speculative lottery-like investments have delivered poor long-run rewards. Thus, bearing small risks is often well rewarded, bearing large risks not". There are two elephants in the room in the form of exclusion of central (i.e., nonlinear) evidence: First elephant. Ilmanen excluded the stock market crash of 1987 from his analysis. But because of the convexity of option payoffs, the return from such crashes is convex to distance from moneyness. So, to use a very extreme (but illustrative) case, an option located 20 standard deviations from the money would return 230,000 times its daily premium erosion in the event of a 20-standard-deviation move (i.e., standard deviations from the implied volatility at which the option was purchased). Hence, one would need more than 2,000 years of data showing an absence of 1987-style crashes— generously assuming that the environment is stable—to pronounce the sale of these options “safe.” Even those options that are closer to the money (and commonly traded) deliver large enough a payoff to forbid us from making claims from few decades worth of data; for instance an option 12 standard deviations away from the money return 5,000 the daily erosion. Another misunderstanding concerns the path dependence of these payoffs, which compounds the payoff asymmetry. When the implied volatility quadruples, a 15standard-deviation out-of-the-money option becomes a 4standard-deviation option and its value is multiplied by 3,300. Implied volatility (as represented by various volatility indices, such as the Chicago Board Options Exchange Volatility Index, or VIX) quadrupled at least six times over the past quarter century. Table 1 shows the convexity of options to changes in implied volatility. These changes in implied volatility induce a second layer of optionality that is missing from Ilmanen’s analysis—with opportunities for the option owner and an squeeze for the seller. (In a well-publicized debacle, the speculator Victor Niederhoffer went bust because of explosive changes in implied volatility in his option portfolio, not because of market movement; moreover, the options that bankrupted his fund ended up expiring worthless weeks