While the volatility smile corrects for the non-Normality of the market, such adjustment may not be enough. Crash protection might still be underpriced unless a further, less straightforward, adjustment is made, something that here will be called “the path-dependence premium in option pricing”.
This concept builds upon the idea that non-Normality (eg, a market crash) is inevitable when people live under a Normality fantasy. People´s belief in the absence of rare events will eventually cause the rare event to take place. The assumption of Normality will make people take actions that will render the actual distribution non-Normal. The religion of thin tails will deliver the paganism of fat tails.
It is the belief in Normality on the part of investors and their advisers that can set-up the perfect conditions for Normality to be eventually erased. The more participants in the market (attracted by the prospects of surprise-free, tragedy-devoid developments), the more chances that someone, somewhere would react negatively to a new event (such as large corporate losses, an accounting scandal, or disappointing economic figures), would panic and would liquidate as a result, prompting other investors to panic, liquidate, and so on all the way to a crash. In essence, when faced with the unexpected presence of the unexpected, Normality-believers will tremble and exacerbate the downfall. They never believed in outliers until they experienced one, and their reaction gives strength to the outlier. The belief in the Normal curve will make the real distribution non-Normal. The belief in thin-tails will produce fat-tails. The belief in white swans will produce the black swan.
The key idea is that the probability of a deep out-of-the-money put delivering a payout at maturity could be higher for spot prices way above the strike than for spot prices closer to the strike (with everything else constant). This assertion, which of course goes against conventional wisdom in a big way, is equivalent to saying that the probability of a market crashing down could experience a dramatic increase once spot has reached a certain, high level.
In layman’s terms, a bubble won´t crash unless a certain peak has been reached first. If the market never reaches such milestone, then it may not accumulate enough steam to plummet to the ground once panic sets in. For instance, the probability of the market going down to 60 in the next month may be higher when it is trading at 120 than when it is trading at 90. In the latter case, the level of confidence and sense of invincibility among investors may not have reached the skyrocketing highs that make the market so vulnerable to the slightest piece of bad (or simply not-so-good) news. A mere correction would take care of any temporary setback. At 120, the market may have become saturated with Normality-long investors, those who cannot even imagine the notion of a black swan and as such are prone to panic and liquidate if faced with the unexpected. The chances of a quick meltdown become quite real.
What this implies is that, during a sustained market rally, option traders should value deep out-of-the-money puts dearer at a very high level of spot. Once the put is really deep out-of-the-money its price should experience a jump. The 60-strike put should be worth more when spot has reached 120 than when it was at 90, since the fallout from 120 to 60 (and beyond) may be more probable than the fallout from 90 to 60. In other words, the price of the put would depend on the level reached by the underlying during the life of the transaction and, crucially, on how it got there. That is, the option´s premium would become path-dependent.
In such scenario, traders would need to make an extra adjustment to their prices, on top of the volatility smile adjustment. Under spot levels deemed high enough to potentially trigger a rare event, the implied volatility assigned to deep-out-of-the-money puts should experience a jump as traders rush to protect themselves from the now-forseeable mayhem (that is, the implied volatility for a 60 strike should be sufficiently larger at a spot of 120 than at a spot of 90 to make the put costlier in the former case). Effectively, the smile would need to be more pronounced. Happier.
The volatility smile adjustment compensates for the fact that rare events do take place quite often in the real world. The path-dependence adjustment would compensate for the fact that the rare event may become more likely depending on the level of spot and how it got there. While the volatility smile adjustment will always be present (and necessary), the path-dependence adjustment may appear and disappear as the market swings and could spend long periods in hidding (for instance, during the presumably-lenghty healing period following a debacle). In other words, the regular smile will always be there as it symbolizes the structural correction to a failure in the pricing model. The path-dependence smile, on the other hand, would come and go with market developments.
Like the fans of a sports team that regularly displays a decent performance and that every few years is capable of winning the championship, an option´s implied volatility always smiles and has the potential to glow.