The Daiquiri Smile

Few people seem to understand that when you fudge the vol parameter in BSM (assuming that you are using BSM to price options) it no longer is BSM. It´s simple: the instructions did not say anything about vol fudging, vol is not supposed to be fudged with, vol is supposed to be independent of strike, if you fudge vol is not the same construct anymore it´s something else.

Let me provide what could be seen as a clarifying analogy (or, perhaps, as a complete waste of time). The Daiquiri cocktail was apparently invented by American engineer Jennings Cox in 1905. What could be deemed Cox´s drink is supposed to contain 4,5 cl of white rum, 2 cl of lime juice, and 0,5 cl of gomme syrup. That´s what Cox had in mind when coming up with the invention. Those were the instructions. Very precise.

Now, imagine that in real life you begin to observe that barmen all over the country are mixing it up differently. Depending on the room temperature, they would fudge the lime juice ingredient so as to obtain a more realistic output. If we plotted temperature and lime juice amounts we would get a smiling shape: the lime juice variable would be assigned higher values by barmen the more extreme the room temperature. That is, implied lime juice would smile at us.

Could we still call such real-world daiquiries Cox´s daiquiri? Of course not. Cox said nothing (I hope!) about fudging the lime juice figure. It was supposed to be constant, unaltered. If we do alter it, the final output can´t be called Cox´s daiquiri. It´s something else (Hemingway´s daiquiri?)

If you are given a toy with very precise instructions and you choose to violate such instructions through shameless manipulation, you are betraying the original spirit of the inventors, so much that it no longer is the invention that those original inventors devised.

It´s The Smile, Stupid!

In my humble opinion, the most insightful insight of Taleb&Haug´s analysis of the originality-validity-popularity of Black-Scholes-Merton is the following: when you fudge BSM through the vol parameter, the model you are using no longer is BSM. What had previously been described (including yours truly) as BSM´s number one advantage, becomes its exterminator. It was there for all of us to see, but sometimes the obvious refuses to be found.

So the vol smile, rather than representing BSM´s number one asset (the flexibility to allow traders to obtain desirable prices) represents in fact its demise.

Through two conduits: 1) if the smile is the result of vol fudging, then the model is no BSM, 2) if the smile is the result of supply-demand then prices are supply-demand based and no model, including BSM, is used

Smiling at BSM? More like frowning!

Good Point But Wrong Conclusions

A diligent Financial Times reader replies to Taleb´s recent finance theory-bashing op-ed piece in the following terms:

"The genius of the Black-Scholes formula lies in its elegant distillation of all the unknowns involved in the valuation of a future contingent claim into a single parameter: volatility. This has allowed practitioners, all well aware of the initial model's shortcomings, to enhance the formula in simple and intuitive ways, thus preserving its tractability under less idealised assumptions. As a consequence, it has survived the myriad stress tests it has been subject to since its 1973 inception, thus remaining the pricing foundation upon which billions of dollars change hands every day in the derivatives markets"

Having made the same point myself several times, I can´t deny the wisdom of the above words (even if they are meant to contradict Taleb´s excellent truth-filled article). However, the reader forgets a couple of things: 1) Practitioners might not have been using BS at all (see Taleb&Haug), 2) Assuming that the model is in fact used, the volatility fudging is done PRECISELY because traders know that BS is very very wrong; thus, highlighting such fudging (which, yes, is a wonderful thing that has allowed the model to remain alive in spite of its unworldliness) as a justification for the model receiving the Nobel misses the point completely. In fact, the need for volatility fudging sends a very powerful message to the Nobel commitee: you rewarded a very unrealistic (though still useful) construct.

What this reader doesn´t seem to understand is that resorting to the existence of the vol smile in order to criticize Taleb´s take-no-prisioners all-out assault of finance theory is the most effective way to recognize the accuracy of Taleb´s musings.

As Haug might say: good points but wrong conclusion.

The Day When Black Scholes Made Black Scholes

The 1987 stock market crash highlighted the structural flaws of the model but it also unveiled its usefulness.

Imagine that you turn on the tv to check the latest news. Instantly, you feel the fear creep up inside. You turn sweaty and white-faced, still not quite comprehending what the little box at the bottom-right of the screen is displaying. Such anguish, though, would be fully understandable. WWIII it may not be, but a drop in the Dow Jones of 3000 points would surely qualify as a terrifying sight.

To the untrained eye, this fictional story seems way too fictional. After all, the market simply can´t tumble by 25% on a single day, right? Well, yes it can. In fact, a meltdown of such gigantic proportions already happened not so long ago. Only twenty years ago, to be exact. By the close of business on October 19th 1987, the Dow Jones had fallen by almost 23%. “Black Monday” was even graver in other parts of the world, with downfalls of close to 50% in some cases.

The October 87 crash is now part of financial markets legend. It was particularly important for the options markets. Bluntly stated, the crash showed that the Black-Scholes pricing model is wrong but it also motivated traders into showing why the model can be vastly useful. In effect, a Black-Scholes-demonizing event showed how reliable Black-Scholes can be in real-life.

As it is well known, so-called portfolio insurance strategies (which were heavily employed at the time of the crash) have been widely blamed for at least accelerating the market´s meltdown on that fateful Monday. Portfolio insurance was an attempt to synthetically replicate a short equity put position via Black-Scholes-inspired dynamic hedging techniques (which are, of course, at the very heart of the model´s machinery). Thus, as Black-Scholes dictates, insurers would sell the underlying when the market fell and would have to buy when it rose. Assuming that the underlying assumptions of perfect liquidity and continuous trading held, one could build a synthetic put in this manner, and in principle be protected from a market downturn.

As the stock market embarked on a bull run at the beginning of the 80s, portfolio insurers would have been forced to follow the herd upward. As the size of the portfolio pool being dynamically “protected” grew significantly, the required buying would have become larger and larger. Portfolio insurance, thus, quite likely provided a non-irrelevant push to the bullish market.

When the severe correction began to take place in mid-October 87, portfolio insurance-motivated selling helped drag the market to unknown depths (in terms of daily negative returns). Trading became illiquid and discontinuous and dynamic hedging inevitably broke down.

Many “insured” parties ended up with no protection from the ensuing mayhem. The crash (which could well be seen as Black-Scholes-inspired) thus unequivocably showed that the model is built on shaky foundations. In the real world, perfectly-replicating dynamic hedging is merely an illusion.

But at the same time, something funny happened as a direct result of the crash. The currently ubiquitous volatility smile was born, a reflection of freshly-developed crashophobia on the part of traders. After witnessing the massacre, it became clear to options pros that markets cannot be assumed to behave “normally” (as the mathematics behind Black-Scholes assume) and that rare events do happen and can be truly criminal. Traders realized that they had been hopelessly undervaluing crash-protecting out-of-the-money puts.

To correct for the mathematical insanity of Black-Scholes, now only too obvious, implied volatility was manipulated upwards so that the values of options with strikes at the extremes could be significantly pumped up, giving birth to the smile. Prior to the 87 crash, dealers had been content to charge the same volatility independent of the strike level, and the chart plotting implied volatility and strikes was more or less horizontally flat, exactly as the “pure” version of Black-Scholes would dictate. After a 20-sigma event in Wall Street option pros decided to change tact and take protective measures. The smile became such protection, and very graphically illustrated Black-Scholes’ number one comparative advantage, the real reason why it is embraced: a built-in self-correcting mechanism that very easily lets users correct for any theoretical nonsense and deliver reliable outputs. Conveniently allowing traders to adapt the model to real-world realities, Black-Scholes proved its practical worth.

In sum, the same event that highlighted the untrustworthiness of the model (which internal mechanics, arguably, contributed to the event taking place in the first place) helped underscore the reasons for its wild popularity. By producing the volatility smile, traders effectively rescued Black-Scholes from its self-dug graveyard.

Dynamic Suicide

I just read Nassim Taleb´s new post on his Wilmott blog. As usual, his musings are quite enlightening and right to the point (Taleb never ceases to offer us truth). While it is difficult to doubt the academic prominence achieved by both Harry Markowitz and Marc Rubinstein, it is at the same time difficult to deny the doubts that surround the real-life validity of both portfolio theory and portfolio insurance. Few people have explained the shortcomings of both approaches as clearly as Taleb.

Inspired by his words, I began to think a little bit about portfolio insurance (2007, after all, marks the 20th anniversary of its crash-motivated downfall) and I reached what may seem like a naive (maybe even flat-out wrong!) conclusion: portfolio insurance was assisted suicide for those who contracted the service.

Portfolio insurance was an attempt to synthetically replicate a short put position via Black-Scholes-inspired dynamic hedging techniques. Thus, insurers (such as Rubinstein and his colleagues) would sell the underlying when the market fell and would have to buy when it rose. Assuming perfect liquidity and continuous trading, one could build a synthetic OTC put in this manner.

As the stock market embarked on a bull run at the beginning of the 80s (those “it´s morning in America” days), portfolio insurers would have been forced to follow the herd upward (Leland-Obrien-Rubisntein was established in 1981). As the size of the portfolio pool being dynamically “protected” grew significantly, the required buying would have become larger and larger. For instance, just prior to the October 87 meltdown, LOR alone was insuring in excess of $50 billion. In essence, it wouldn´t be far-fetched to argue that portfolio insurance provided a non-irrelevant push to the bullish market. Portfolio insurance could help cause a bubble. In fact, the bubbles supported by dynamic insurers would be of the worst kind: those where much of the buying is done for no fundamental reason at all, thus extremely crash-prone.

When the severe correction began to take place in mid-October 87, portfolio insurance helped drag the market to unknown depths (in terms of daily negative returns). Trading became illiquid and discontinuous and dynamic hedging inevitably broke down. Many “insured” parties ended up with no protection from the ensuing mayhem, victims of a nightmarishly unimaginable crash that was exacerbated by the very techniques that were supposed to help and which would never had taken place had the prior bubble not existed first, possibly impulsed by those same replicating techniques.

So, portfolio insurance may help create the bubble that causes the crash that prevents the insurance from working. Were customers committing inevitable suicide by joining the portfolio insurance bandwagon? Were they invitably doomed after signing on the dotted line? Did they provoke the actions that would ultimately sink them? Quite likely, yes.

What We Know And What We Don´t Know

The price of an option should depend on several factors: Spot asset price, Strike level, Time to maturity, Interest rates, Volatility, Expected asset return, Liquidity concerns, Crash-o-phobia, Supply-demand disequilibriums, Upcoming elections, Artificial mark-ups by dealers trying to sink a counterpart, Etc, etc (anything deemed relevant by buyers and sellers).

In this list there are items that we know and items that we don´t know. Obviously, we only know a few. But the others are also important and should count. Of course, we can´t develop a model that includes a parameter for every single relevant factor (if only because these tend to change with time). Ideally, we would want a model that provides parameters for each of the known factors, plus an additional single parameter that acts as catch-all where dealers can dump at once all the other unknown factors. Black-Scholes wonderfully offers such service. The catch-all parameter is dubbed “implied volatility”, but it should be clear that the unknown volatility factor does not monopolize the catch-all. So many considerations go into that number that it would be unfair to define it by a single one of them. Thus, the price of an option is made up of stuff that we know and stuff that we don´t know. Of stuff that we know and something called implied volatility, not stuff that we know and volatility.

People say that implied volatility is the “market expected volatility”. It´s much more than that. Do not let volatility elbow out all the other unknown parameters and unfairly monopolize the ever-important figure of implied volatility.

Smiling At Black-Scholes

(this is a brief abstract of a piece that I published last month)

The existence of the volatility smile is a tremendous validation of the real-life usefulness of the Black-Scholes model. It very clearly explains why practitioners have embraced and continue to embrace the model, in spite of its wide-off-the-mark mathematical foundations (in particular, the assumption that asset prices are lognormally distributed). Derivatives players don´t trust “pure” results from Black-Scholes but they will continue using the model because they can so easily fix it into delivering the results that they, and not some unrealistic theoretical device, deem appropriate. The smile very graphically shows how easily, in fact.

Black-Scholes comes with a built-in self-correcting mechanism that very conveniently lets traders manipulate the model so as to freely express their opinions and obtain the prices that are considered optimal from a practical point of view. This freedom-providing feature (which works by fudging the volatility parameter that goes into the formula) is the number one reason behind the spectacular real-life success of the model. The volatility smile, by reflecting those freely-expressed opinions and how Black-Scholes permits users to break free from its own mathematical straightjacket, symbolizes such success.

The smile is saying both things at the same time: “Black-Scholes is wrong, but it is right!”. The model is mathematically wrong, but it can be righted through its built-in self-correction mechanism. By easily making mathematics irrelevant, the model gained a prominent place in the hearts of traders (who are only too aware of the limitations of mathematical modelling when it comes to the financial markets).

The Message Of The Greeks

Let’s imagine a pre-Black-Scholes world, where people have no access to analytical option pricing tools and therefore have no way of methodically visualizing the price function and the associated risk parameters (what we now call the greeks). They can only think about them and imagine them. Or perhaps, draw them down with a pen. Let’s then put ourselves in the shoes of pre-1973 option traders who lack a formal pricing model but would nonetheless like to have an idea as to the characteristics of the options that they are selling to their customers. In particular, they would like to know the kind of risks that they are facing as option writers and how the value of the option will change as these parameters change.

After thinking for a few minutes, the traders realize that one obvious risk is so-called spot risk, as clearly the value must change when the underlying itself changes. This is of course what we nowadays call delta. Given the importance of spot risk, it would also be critical to know how such risk changes with the underlying, or in other words, how exposed the value of the option is to jumps in the underlying. This is what we know as gamma. Additionally, it should be clear that the option’s value is affected by how volatile the underlying is, as this will determine the probability that the option ends in or out of the money. This is our famous vega. Finally, time is also an important risk factor, given the fact that less time means less chances for movement in the underlying, and movement is always assumed to be good for option holders given the asymmetric payoffs. This is theta.

So, let’s picture these ancient traders sitting at their favourite restaurant (apparently, pre-exchanges option traders on Wall Street conducted their businesses from a restaurant, with the phone booths acting as offices). They have pen and paper and are going to try to draw graphs representing the change in value of the option with respect to each of the main risk parameters described above.

By putting ourselves into the shoes of pre-1973 option traders we try to get a no-nonsense view of the risk parameters of an option from the perspective of model-deprived, purely practical minds. What we would obtain are results that could be labelled as “street-smart reasoning”, the kind of results obtained from somebody who is unaware of the existence of any analytical option pricing model and who does not have any other tools at his disposal than his logic and his understanding for what makes sense. In a way, such analysis would detail how the greeks should behave in practice in real life. After all, we are reading seasoned traders’ minds here.

Interestingly, if one embarks on such exercise (if one tries to draw the greeks based on intuition, on what makes sense) the resulting figures are extraordinarily similar to what we would obtain from Black-Scholes. It is tempting to conclude that both the no-nonsense ancient traders and the model would agree on the greeks (on the shape of the graph, not the scale as this would depend on the volatility parameter).

Is this meaningful? Can we see this as an endorsement of Black-Scholes? After all, what the old traders lacked was a way of putting precise numbers to the greeks, a way of quantifying the risk parameters. What Black-Scholes did was provide a way, for the first time in history (yes?), for traders to assign a specific number to the many risks they face under different scenarios. Armed with the model, traders could tell if each landmine along the way could kill them or just inflict harmless injuries. Most important, once you know how big a risk you are facing you can equip yourself with the necessary matching hedge. Traders now had a key analytical tool at their disposal that would allow them to run trading books.

Could the message from the greeks be that the Black-Scholes framework is indeed the right one, because its outputs would be found agreeable by pre-modelling traders? Does Black-Scholes have a trader’s mind embedded into it?.