Nassim Nicholas Taleb, author of the widely discussed The Black Swan and Fooled By Randomness, is out with a new paper. “The Fourth Quadrant: A Map of the Limits of Statistics” pursues a thesis very familiar to his readers, namely that economists and finance professionals put society at risk by offering false comfort in the form of statistical models.
Risk Does Not Equal Volatility
The novel effort here is Taleb’s attempt to map out which kinds of risks and events are more-or-less adequately captured by statistics, and which ones fall into the unquantifiable “black swan” category. Events that fall into this unanalyzable fourth quadrant (see image below) are characterized first by what logicians would call multivalent truth-conditions: they are not all-or-nothing, true or false occurrences, but rather may have varying degrees of realization. Second, the “payoffs” or impacts of such events may be nonlinear.
After categorizing some examples, Taleb concludes with nine practical rules for dealing with the existence of this fourth quadrant. We won’t list them all here, but we found one rule particularly notable:
Do not confuse absence of volatility with absence of risks. Recall how conventional metrics of using volatility as an indicator of stability has fooled Bernanke—as well as the banking system.
Figure 7 Random Walk—Characterized by volatility. You only find these in textbooks and in essays on probability by people who have never really taken decisions under uncertainty.
Figure 8 Random Jump process—It is not characterized by its volatility. Its exits the 80-120 range much less often, but its extremes are far more severe. Please tell Bernanke if you have the chance to meet him.
The difference between random walk distributions and what Taleb is calling the “random jump” process is precisely why using hedged positions is so important. A position with an absolute stop loss (appropriately sized, of course) can limit the impact of those fat-tail distributions. For a vanilla options spread, this is tantamount from moving from the third to the first quadrant – a much nicer place to be.
Cutting Off the Fat Tail
At the risk of seriously oversimplifying Taleb’s point, let’s look at a concrete example. Right now, implied volatility in S&P 500 options is quite high, and an aggressive trader might look to sell strangles to capture profits should implied volatility move back down toward its relative average. You could sell the SPY Oct 109/127 strangle (selling the October 109 puts and 127 calls) for a net credit of about $2.67 (or $267 on a 1-contract trade). But as the risk profile graph below demonstrates, you’re taking on an infinite risk for a rather small gain.
Let’s assume that normal probability distributions apply: you stand about a 68% chance of profiting from this trade at expiration, which is to say that this position can tolerate one standard deviation, or 68%, of possible outcomes. Even if we widen our expectations to include 99% of outcomes, we still wouldn’t expect to see SPY below 91 or above 147 at October expiration.
But that’s precisly the point of Taleb’s claim: the set of outcomes that are collectively less than 1% likely (and are individually less than 0.01% likely) have a disproportionately large impact, and are not reflected given our ordinary assumptions about probability. In this case, we are clearly not in the “fourth quadrant”: the payoff to this trade is linear and not complex. But it is vulnerable to fat tail outcomes or “black swan” events. From a trading standpoint, the risk/reward profile is bad enough under normal conditions; adding in the possibility of some 3+ standard deviation outcome makes the position unbearable.
So instead of just selling a short strangle, let’s also purchase a long strangle with slightly wider strike prices. We’ll buy the SPY Oct 107/129 strangle (buying the October 107 puts and 129 calls) for a net debit of about $2.03. The resulting 4-legged position will net us about $0.64, or $64 on a 1-contract trade. We’re obviously giving up some premium, but the resulting risk profile explains why.
By adding a long strangle to our short strangle, we have defined how much risk this trade will carry: we can lose no more than $136 a 1-contract trade. That means that, no matter where the underlying security is at expiration, we already know what our potential profits and losses will be. We are of course vulnerable to the ordinary range of adverse outcomes, but the important thing here is that, once we have hedged the position, fat tail outcomes and black swan events pose no additional risk.
If the risk profile of this pair of strangles looks just like that of an iron condor, that’s because it is. People are more prone to thinking of condors as pairs of vertical spreads, but you can also think of them as pairs of strangles, as we’ve shown.
But the terminology really doesn’t matter: all that really matters is risk and the management of it. By adding hedges to positions with linear payouts, we can “cut off” the fat tail and restrict our risk to the domain of normal probabilities.