UnRisk 7

The VIX Is A Volatility Ghost

Professor Robert Engle´s article on market volatility (“The threat that won´t go away”, Financial Times May 25th) toes a traditional line by equating the VIX index with, well, volatility. But the VIX, contrary to what many have believed and have been taught for years, is not quite equal to volatility. In fact, it may be miles away from being even a mild reflection of turbulence. I have a chapter devoted to this in my new Lecturing Birds On Flying.

The VIX has essentially been two things throughout its life, and none of them can be reasonably labeled as volatility. Initially, the CBOE calculated the VIX from the famous Black-Scholes option pricing model, by reverse-engineering the formula´s volatility parameter from quoted equity option market prices. That is, during its early years the VIX was what´s called “implied volatility”, which is supposed to represent the volatility expectation that traders input into the (supposedly employed) model; but implied volatility can´t be seen as a window into expected mayhem, for two basic reasons: traders may not be using the model when pricing options (who can guarantee that that´s always the case?), and even if they are the volatility input would contain lots of other stuff besides pure expectations (something called the volatility smile, glowing out there since October 1987, definitely tells us that implied volatility can´t equal true volatility expectations).

When the CBOE changed the calculation method in 2004, the VIX became the theoretical price for a type of derivative contract known as “variance swap”, obtained through very sophisticated mathematical ruminations (borrowing from prior work by Goldman Sachs quants). The new VIX does not assume the use of Black-Scholes in real life, and does not reverse engineer from a model. While the old VIX could somehow claim to represent implied volatility (if traders are using Black-Scholes), and thus be a window into traders' opinions, even if not an exact mirror into their volatility expectations, the modern VIX is not even implied volatility. It is not directly implied from option prices. Rather, the VIX is now linked to those prices through a mathematical sleight-of-hand, almost accidentally. When market option prices change, the VIX will change, but we can't imply that the VIX would reflect any precise information (such as volatility expectations) that those prices may or may not contain. There simply is no reverse engineering going on. No direct, incontrovertible peeking into traders’ minds here. The VIX is a wholly opaque and very dubious guide to the opinions and feelings of financial pros. If we see the VIX explode (tumble) all we can say with honest certainty is that the prices of some S&P 500 options are climbing (declining) quite steeply. We can't honestly imply anything about fear or exuberance levels along Wall Street.

Sadly for those yearning for precision-promising certainties, the VIX has never measured what we were told it could measure. By endowing it with fake powers, we are letting a deceitful ghost influence our markets.

Why Should Volatility Increase Option Prices?

It is conventional wisdom that if vol goes up, the price of (most) options would go up.

The typical explanation lies in the fact that options present assymetric payoffs: lots to gain, little to lose (known premium) from turbulence. Bring chaos on!

Another rationale is that enhanced vol makes the competition look worse. For many (hedgers particularly) the key decision is whether to go with options or forwards. Competition can be intense. Heated. Any small change can tilt the balance in favour of one or the other.

People who love forwards do so mostly because they don´t cost premium (extra leverage). This is of course due to their payoff linearity. That is, in this case, chaos can equally bring outsized gains and outsized losses. Options prevent the latter, and that is why they cost cold hard cash.

But I will not pay if I believe that mayhem will be very tame, because in that case the nightmares from the forward would be so limited. Too small to make me worry. Too small to make me reach for my wallet and enter the option shop.

If I expect vol to be small, the option loses competitive edge, and thus value. If I expect vol to be high, options would become comparatively more attractive.

ptriana@profesor.ie.edu

Volatile Puzzlement

As a result of my latest article on VIX Derivatives I have received some feedback from top players. I am certainly glad to see that leading pros take an interest in my humble musings (after all, they are my target audience). I continue to be a bit puzzled though by the fact that some practitioners continue to religiously believe that implied volatility (and thus the VIX) is a pretty good mirror into expected future real volatility. I mean, I can understand why academics (always in search of mechanistical relationships that can be econometricized) would be eager to defend such link, but real-world pros?

What I would ask those defenders of implied vol as expected vol is, how do you know? How do you know that the market is even thinking about future real vol at all when pricing options? Have you asked all traders around the world? Taken to the extreme, who is to guarantee that traders develop any view on volatility when pricing options? Yes, if you plot the VIX with some measure of real volatility it looks as if the graphs move somewhat in sync, but the gaps are anyway still quite sizable (that is, even if we assume that the market is forecasting, it appears to be a lousy forecaster). But of course such graphical representations could be the result of coincidence, luck, or external forces.

On the other side of the debate we have the strong evidence provided by the vol smile, which clearly tells us that implied vol is not expected real vol. Why? Because you can´t have twenty different consensual average forecasts for vol. Either vol for an underlying asset is consensually expected to be 20% or 30% or 40%, but it can´t be consensually expected to be 20% and 30% and 40%. Obviously, other factors are at play when determining implied vol. A vol forecast might (just might) be one of them, but certainly not the only one.

I know that I have already spoken about all this in the past, but my latest e-conversations with top players has re-awakened my impulse to defend implied vol's non-forecasting identity.

Of course, as I have humbly admitted to my readers, I could be wrong (perhaps suffering from "armchairism"), but my argument feels quite right.

ptriana@profesor.ie.edu

V & V, A Dangerous Couple

As it is well known, the problem with VaR is that it may become too popular, with lots of people buying into it and enslaving their position-taking tactics to the dictates of the statistical device. If VaR is too high, dump stuff. Wait, everybody is dumping stuff. The same stuff. All kinds of stuff. Everything has become correlated, everything is going down. Where did liquidity go??? It´s the end of the world!

A similarly twisted logic applies to the VIX index. Those in love with finding (or making up) mechanistical relationships in the financial markets have been telling us for years that if the VIX is so the markets will do this, and if the VIX is not so the markets will do that. It used to be that a low VIX meant that markets were going to be stable (ie, you can buy safely), while a high VIX meant upcoming trouble (you better sell mister). These days, so-call contrarians send, well, a contrarian message: low VIX equals dangerous complacency (sell!), high VIX means unduly worries and excessive fear-driven conservatism (buy!).

Now, imagine that a substantial number of players bought into one or both of these schools of thought and thus assume a fully-fledged, unquestionable mechanistical relationship between the markets and the level of the VIX. This would endow the VIX with tremendous power. Slave master, no less. If the VIX goes up or down, a whole bunch of supposedly sophisticated investors would buy or sell (or sell or buy) simply because the sacred, undoubted, gospel-like assumed mechanistical relationship says so.

Clearly, in such scenario the VIX (just like VaR) would have the power to move markets, simply because people chose to voluntarily enslave themselves to a certain theory. It might easily become one of the most-followed indicators, both by punters and policy makers.

Now, it is bad enough to assign such practical relevance to something simply because a few researchers claim to have found some kind of data relationship. But in the case of the VIX things get much worse. Because, you see, the VIX is not what people (including those researchers) assume it is. The whole idea behind the VIX-markets relationship (whether standard or contrarian) is that the index stands for expected future real volatility. It is taken for granted that the VIX reflects the market´s state of mind regarding upcoming turbulence, the only issue is how to interpret the message. But the VIX is not about real volatility, but implied volatility. Not the same thing (ask the vol smile if you don´t believe me).

So it turns out that in endowing VaR and VIX with imperial powers (a sin already commited in the first case, not so much yet in the latter's) we are commiting serious fraud. Just like the markets are not Normal (key assumption behind VaR calculations), implied volatility is not about real volatility (at least not entirely). If we keep this pace, soon markets may be moved entirely by absurd mechanistical rules to which apparently savvy investors have voluntarily enslaved themselves.

ptriana@profesor.ie.edu

Implied Asset

Some time ago there was a heated exchange over at the Financial Times between a US-based analyst and some honcho at the CBOE. The former had penned a piece criticizing the validity of the VIX as a hedge against stock market turmoil, saying some pretty nasty things along the way(I believe that "rigged game" was mentioned). In response, the CBOE wrote a complaint letter arguing its case. One of the things that the VIX-attacking fellow scribbled was (I seem to recall) that volatility is not an asset. In this, I think he is wrong.

Now remember that we are talking about implied, not realized, vol here (as measured by the VIX). In one key respect, implied volatility behaves very much like an asset: its value depends entirely upon people´s opinions and actions. You will buy it and sell it depending on whether you think that people will buy it or sell it, and whether you eventually make money will depend on what other people have done. The value of implied volatility will depend on what people think that value should be.

Implied volatility is an opinion, a gut feeling, the result of supply-demand forces. Not very different from stock or bond prices. And just like those, quite possibly unforecastable.

ptriana@profesor.ie.edu

Volatility Is Not Volatility

(this is a reproduction of a piece that I recently published)

Few concepts in the financial markets are more talked-about than so-called implied volatility. Implied volatility essentially tells us what number (given a certain option price) traders are inputting under the volatility parameter that goes into the pricing model.

An implied volatility figure of, say, 30 per cent means the average consensus input for the volatility parameter that was entered by traders was 30 per cent. In other words, to reach the desired consensus price for the option, the volatility input has to be exactly 30 per cent.

Implied volatility is an important measure for several key reasons, but not for the reason that is most often cited. Most people take it as an article of faith that implied volatility represents the market's forecast of future realised volatility (ie, true expected market turbulence). If three-month implied volatility is 30 per cent then surely this must mean that traders are expecting the next quarter to be a wild ride, so would go the conventional wisdom.

Unfortunately, implied volatility is not a mirror into forthcoming turbulence. While the implied volatility number would contain traders' view on future real volatility, this is only one of its ingredients. Implied volatility is about much more.

Just like a sausage-making machine, the implied volatility number is fed with a bunch of different (and many times entirely unfamiliar) ingredients. To confuse implied volatility with the market volatility forecast would be akin to confusing the whole with one of its parts. Flatly stated, an implied volatility of 30 per cent does not mean an expectation of 30 per cent realised volatility. Implied volatility is, in fact, a liar.

Why the deceit? Blame it on Black and Scholes. When using this most popular of option pricing models (in fact, the place from where implied volatilities are quoted) traders input figures under volatility that do not stand (exclusively) for volatility. They do so to obtain conveniently the option price that they want. When deciding on the volatility number that would deliver the desired result traders obviously consider their expectations as to future turbulence, but would consider other things at the same time. Traders' opinions concerning any issue that can affect the price of an option would be included in the implied volatility number.

In effect, then, implied volatility is best redefined as the conduit through which traders obtain the prices that are deemed optimal, a catch-all where they correct for all those relevant aspects not rightly captured by the model's machinery. Real expected volatility will always be a key component of implied volatility, possibly the most important one, but it will never account for the whole picture. When it comes to option pricing, "volatility" does not always stand for volatility.

Thus, a higher implied volatility number does not automatically imply that traders are expecting higher market turbulence. A high level of implied volatility might imply the perception of high future real volatility, but it likely also reflects a host of other factors. Liquidity concerns, crash-o-phobia on the part of traders, and particular supply-demand disequilibriums all can result in bumped-up implied volatility numbers. Implied volatility is also many times inflated as a way to correct for the known deficiencies in the probabilistic assumptions behind Black-Scholes, which unrealistically assigns little chance to extreme market movements taking place.

Any unseemly factor could alter implied volatility. For instance, during the famed Long Term Capital Management episode of late 1998, equity options counterparts to the fund reportedly marked the volatility parameter up by a lot, as a way to protect their interests. A naive observer might have looked at the 40 per cent figure and concluded "the market is surely forecasting a wild ride".

Most academic studies on the subject have focused on trying to answer whether implied volatility is a reliable predictor of realised volatility. Those analyses are a bit like comparing apples with oranges. The truly interesting insight would be to explain what part of implied volatility is forecast volatility and what is "other stuff", and what exactly that stuff is. Only then could we discover the true market expectation for trouble on the horizon.

ptriana@profesor.ie.edu

Volatility Mumbo-Jumbo

I just read someone quote hedge-fund manager Stan Jonas as having recently said the following:

"Traditional "mumbo-jumbo" about skews, vol surfaces, have now become relegated to those that do not trade."

Very interesting. So all those academics that make a living producing an endless stream of papers on the vol skew have basically been talking to themselves? Been involved in irrelevantly theoretical mumbo-jumbo? All I can say is, my goodness!

Volatile Irrelevance

It is sacred conventional wisdom that volatility is of paramount importance when it comes to options. The most important aspect, say some. Unavoidably relevant, imply others. You will be hard pressed to find an options book or attend an options course where volatility is not assigned a preeminent place. You would likely be reminded that volatility is the only input of the Black-Scholes pricing model that is not fixed, thus making it the most critical of the bunch and accentuating the need for accurate volatility forecasting methodologies.

And yet, the argument could be made that Black-Scholes makes volatility irrelevant. Heretical as this might sound, Black-Scholes may have produced volatility-independent option pricing. Thanks to the Nobel-caliber model, traders can price options without taking volatility into account, if they so wish. There is no need to develop a view as to future market turbulence. The desired option price can be obtained even if no volatility estimate is taken into consideration.

Of course, the Black-Scholes model does include a parameter named “volatility” that is supposed to stand for, well, volatility. That was the very intention of Fisher Black and Myron Scholes when they developed the model. You can´t price an option without inputting some number under the volatility parameter. However, that does not mean that that number stands for expected real future volatility. That number may simply be a conduit used by traders to arrive at the desired optimal option price, the ingredient necessary to churn out exactly the correct outcome from the pricing sausage machine. For all we know, no actual volatility forecast may be embedded into that number. Who is to guarantee that traders are even expressing a view on future turbulence when inputting a number under volatility?.

In theory, of course, volatility should affect the value of an option. Generally speaking, the more expected turbulence the merrier for the owner of options. But this is only theory. In practice, option prices are determined entirely by human action, not theoretical truth. We assume that humans take future volatility into account when pricing options, but we just don´t know for sure. Perhaps the biggest advantage of the Black-Scholes model is that one can get desired prices without having to develop a view on future realized volatility (just like the model does not force you to develop a view as to the expected return of the underlying asset).

Another reason to argue that expected volatility may be irrelevant to traders is that their concerns lie entirely with implied volatility (the number consensually inputted under “volatility” by the market), as they live or die by the mark-to-market of their books, and this is of course determined by changes in implied volatility (essentially, traders´ opinions). Unlike many on the “buy side”, which usually keep the option until maturity and thus care about future volatility (due to the need to develop a view as to whether the contract would end up in-the-money), dealers are judged (and compensated) exclusively by the market value of the position.

Implied volatility and expected realized volatility are simply not the same thing (the existence of the volatility smile, among other things, very clearly tells us so). While the latter may be embedded into the former, it is just one of the component parts. Blunty stated, implied volatility is a bunch of stuff. A catch-all that allows traders to make relevant all the crucial aspects that the pricing model does not take into account, things like liquidity concerns, crash-o-phobia on the part of traders, demand-supply disequilibriums and in general anything that is deemed practically relevant when it comes to pricing an option. Traders love Black-Scholes precisely because it allows them to easily and conveniently make such corrections. The known deficiencies in the Black-Scholes machinery thus result in the paradox that when it comes to option pricing, volatility is not volatility.

If implied volatility is all that matters to a trader, what´s left for poor neglected real volatility?. Even if we assume that expected turbulence is a part of implied volatility (ie, that traders do in fact develop a view as to future realized volatility when pricing the option), this probably wouldn´t help much. Unless one can develop a view on future implied volatiltiy as a whole, just being able to forecast one of its components won´t help traders develop a sense as to their future mark-to-market exposure. But getting implied volatility right is bound to be challenging. It is one thing to forecast real market turbulence or the expectation of it (parameters that could be reasonably calculated to a certain degree), and another thing to predict other traders´ opinions about a whole bunch of stuff (real volatility but also anything else that would affect their gut feelings).

Should traders then lose much sleep over future real volatility? Maybe not.

ptriana@profesor.ie.edu

A Happier Smile

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.

ptriana@profesor.ie.edu

The Central Bank Of Volatility

I have always found it puzzling that most of the discussion surrounding the LTCM story has focused on the “math stuff”. To many observers, the fund failed because its funky models went awry, because the extravagant quantitative techniques used by the geeky principals ceased to work in the face of unprecedented market turmoil. This line of thought fits the commonly-held stereotype of LTCM as a theory-driven laboratory where ultra-sophisticated mathematical alchemy was devised by PhD-holding rocket scientists who pocketed untold sums of money because of their superior brainpower.

Now, I don´t doubt for a second that LTCM was full of very intelligent people who had, by the way, proven their worth as traders for many years prior to the creation of the fund. I don´t pretend to play down the techniques employed in devising the trading strategies. But I have always been a bit suspect about the overriding role of the quant stuff when it comes to LTCM. I mean, DE Shaw they weren´t.

Again, exaggerating the quanty (non-human) aspect behind LTCM´s decision-making processes serves the purpose of satisfying those comfortable in the belief that hedge funds and other derivatives users are driving the world into chaos through the mischievous use of esoteric equations and computer programs. To learn that perhaps some of the strategies were less-than-complex, in fact quite simple, would be akin to heresy to such deeply-entrenched conventional wisdom.

Take the “short long-dated equity volatility” strategy, which was a late starter but ended up causing untold mayhem. Call me naive, but as hard as I try I can´t see anything out-of-this-world about nakedly selling options into the market. I am possibly missing something, but what LTCM basically decided to do (apparently in a quest to diversify away from its traditionally dominant fixed-income trades) was to sell equity index options to the structured products players that needed to buy long-dated volatility as a way to hedge their positions. Nakedly selling options is a simple strategy, isn´t it?. In fact, some people may see it as an easy way to generate income without having to think too much.

But while LTCM´s equity volatility trades may not deserve to belong under the “funkily complex” category, they deserve our attention for one crucial reason (besides the fact that it cost the fund hundreds of millions in losses), something that is as relevant today as it was back then: the tricks that implied volatility can play on you.

Apparently, LTCM believed that shorting (implied) volatility at 22% was a bargain because historical (realized) volatility had been around 15%. The thinking seems to have been that 22% was way too high compared to the usual 15%, and thus anyone selling at 22% had to make money eventually. The problem, of course, is that this analysis was akin to comparing apples with oranges. Implied volatility and realized volatility are simply not the same thing. The latter is true real market turbulence. The former is whatever number traders are consensually inputting under the pricing model´s volatility parameter. Part of that number may reflect the market´s expectation of future real turbulence, but it likely also contains other stuff. As we know, traders use the volatility input to manipulate the model so as to obtain the price that they want; in doing so, they may produce an implied volatility number that is useful for their pricing intentions, but that has very little to do with what they think real volatility is going to be like. Implied volatility is thus, by defition, a very distorted picture of expected volatility. An approximation at best.

LTCM looks as if they overlooked this basic fact. They seem to have believed that they were betting on real volatility (or at least real volatility estimates by traders) being below 22%, when in fact they were gambling that the number that traders would input as volatility in the model would be below 22%. Two widely different things of course. It is one thing to gamble on real market turbulence or the expectation of it (parameters that could be reasonably calculated to a certain degree), and another thing to gamble on traders´opinions about a whole bunch of stuff (real volatility but also liquidity concerns, crash-o-phobia, demand-supply disequilibriums, and in general anything that would affect their gut feelings).

Eventually, long-dated implied equity volatility did rise to staggering levels (30%, then 40%), thus killing LTCM, and for weird (ie, very difficult to forecast) reasons. After the Asian and Russian crisis, the number of players in the structured products market went down dramatically and “volatility supply” simply dried down. As supply goes down, up goes its price. Anyone who wanted to purchase long-dated volatility had to pay dearly. As LTCM began to hurt badly, counterparts apparently marked up the implied volatility against the fund as a way to protect their interests. Demand-supply travails and traders´ survival instincts are what truly drove implied volatility to crazy levels well above 22%, not a sustained belief that true market turbulence was going to be that high.

In the options markets oranges are oranges and apples are apples.

ptriana@profesor.ie.edu