Black-Scholes and Monte Carlo

Aside from the lattice method discussed in yesterday's entry, the two most important methods of valuing the expense of stock options as compensation are: the Black-Scholes-Merton (BSM) formula, and a Monte Carlo simulation.

BSM is here.

This amounts to valuing an option to buy a share of stock by making certain simplifying assumptions. The devisers of the formula were explicit about these assumptions. One of the more important of them is that extreme price changes are very rare, because the movements in the value of the underlying stock follow lognormal distribution. Many scholars have quarrelled with this, but BSM does still offer a useful approximation of the value of such options.

Still: the "Monte Carlo" method has by far the cooler name, paying homage as it does to the city whose image I've uploaded with this entry.

Through the miracle known as a digital computer, modelers can run thousands of simulation paths covering possible price movements in the underlying asset -- the stock - and can assign a value to the option for each of these paths. The present value of the average of all those values is the Monte Carlo value of the option.

More on Stock-based Compensation

It was only a few flicks of the calender away, it was as recently as 2004, that one could say that generally accepted accounting principles (GAAP) in the United States didn't require employers to recognize in the books they showed the investing public that by issuing stock options to their employees they were incurring an expense.

The move of the FASB that year toward standards explicitly requiring this was caught up in the election-year debates. The FASB held its ground, although if I remember correctly it had backed down under political pressure on the same score a decade before. But the second time around it held its ground and made the expensing of stock options mandatory for all annual and interim reports, effective beginning June 15, 2005.

Over the last few years the nature of the debate has shifted. Now that we know companies are supposed to expense their stock options, the question is: how?

There are three dominant methods thus far. There's the Black-Scholes-Merton (MSN)formula, the Monte Carlo os 'simulation' methods, and the lattice model.

The lattice model is the most intriguing of the three in my eyes so I'll expound upon that a bit.

The modelers start by dividing the time between the issuance of an option and its expiration into a definite number of discrete time periods. These periods might be, for example, months or quarters.

Given input variables and assumptions that the company and its modelers must makle explicit, they can then assign a probability to an "up" and to a "down" move in each period. This yields two (or more, depending on how the particular lattice is constructed) end points, which are then starting points (nodes) for the next jump.

The model continues to generate nodes in an ever-widening pattern, looking a bit like a branching tree turned on its side (because the horizontal axis by convention represents the passage of time) until time runs out; that is, until option expiration.

Modelers then employ the whole lattice to decide what would be the value of an option in the final time segment. It helps that the ending is in a sense binary. Either the exercise price of the option is less than the current stock price or it is not, and that simplicity makes calculation straightforward. From there, one can work backward, step by step, until one comes eventually to a value for the option at time 0.

BS v. BS: An update

Last December, the Financial Times ran an opinion column by Nassim Nicholas Taleb, author of Fooled by Randomness (2001) and The Black Swan (2007) with co-author Pablo Triana. They used the ongoing market chaos to reinforce an argument they had made before, that contemporary financial risk management in general is misguided and that the Black Scholes Merton model is part of the reason. I cannot help but think of this as the "Black Scholes versus Black Swans," or the BS v. BS controversy.

"Ask for the Nobel prize in economics to be withdrawn from the authors of these theories," they urged their readers. "Boycott professional associations that give certificates in financial analysis that promoted these methods. The fraud can be displaced only by shaming people, by boycotting the orthodox financial economics establishment and the institutions that allowed this to happen."

I've written of this here before and now seek only to update.

In May of this year, GQ ran a flattering article on Taleb called "The Thinker," in which the writer, Will Self, describes Taleb as "a genuinely significant philosopher ... someone who is able to change the way we view the struicture of the world through the strength, originality, and veracity of his ideas alone."

Taleb claims, naturally, to do a good deal more than theorize about finance -- he claims to have proven his own views in practice. In this regard, I note that the above passage in Self's essay appears soon after a quotation with a startling number in it, a number ($20 billion) that has attracted a lor of attention since the appearance of this issue of GQ, and has in fact become a new battleground in the war of BS v. BS.

"We didn't short the banks -- there's not much to be gained there, there were all these complex instruments, options and so forth. We'd been building out positions for a long while ... when they went to the wall we made $20 billion for our clients, half a billion for the Black Swan fund." So Taleb supposedly told Self.

So presumably, in addition to the profit he made for and through his own fund, he made another $19.5 billion for outside clients. Janet Tavakoli picked up on this right away, and she contacted Nassim about that $20 billion figure. He told her that the magazine made an error there. Did they just make the number up? Perhaps not. The $20 billion, Taleb said, "might correspond to the face value of positions."

The mistake is not a trivial one. It relates to the whole issue of the "scalability" of results. It is one thing to claim you've made some money picking up pennies in front of a steam roller because you've been nimble enough to dart in and out safely. It is another thing to say that the strategy can be increased indefinitely to any scale -- to even a $20 billion scale -- that there is that much money in front of aforesaid steamroller.

Last week, Tavakoli -- the principal of Tavakoli Structured Finance -- revisited the matter of the disappearing $20 billion on her website, in a piece called "Where Were the Drama Pundits [Whitney, Taleb, and Gasparino] When It Mattered?"

She notes that Taleb has posted the GQ article, with its $20 billion figure, on his website, www.fooledbyrandomness.com, and he is silent there about the error. Indeed, Taleb praises Self's profile of him as one of the "most representative overall" yet done.

Silence on a little matter of $20 billion may be taken, Tavakoli submits, "as endorsement whatever the source of the original error," an endorsement of the suggestion that a strategy of running in front of the Black-Scholes steamroller applies to large investments -- a point for which there is "actually no empirical evidence."

It is a good point, and reinforces my suspicion that, pennies notwthstanding, there may be life in the old BSM steamroller yet. Enough life so that it is better to be in its driver's seat than to dart around in front of it.

P.S. Tavakoli has asked that I clarify two points in the above. First, she says, "I wrote that Taleb has 'corrected' the error [re: the phantom $20 billion], but he did so more than two months after his original posting, and only in the face of media pressure."

Taleb's more recent position is that the $20 billion figure stands for a "notional amount," and that the actual gains produced thereby were between $250 and $500 million. This is the subject of Tavakoli's second requested clarification. she questions "how Taleb made so little on bearish derivatives for the 2007-November 2008 timeframe in question...." The top of that range, $500 million, is only 2.5% of the $20 billion notional amount.

BS v. BS II

So: what do I think about the "Black Swans versus Black-Scholes" dispute I tried to chronicle in yesterday's entry?

By way of answering, let me re-introduce you to Emanuel Derman. I mentioned him yesterday -- he was a co-author, with Taleb, of "The illusions of dynamic replication," the 2005 article that seems to have started the whole anti-Black-Scholes campaign.

Derman dropped out of the story then, because Taleb started shifting his line of attack and acquired new co-authors for the new approach.

But Derman, a one-time subatomic physicist, and the 2006 recipient of the Wilmott Award for Contributions to Quantitative Finance, has re-appeared. Now he shows up more as a defender of Black-Scholes than as a critic. See his New Years' Day blog comment on the Wilmott website.

So far as I understand these things: I think Derman is right. the Black-Scholes-Merton model was a real advance in the understanding of financial economics, one that has the benefit of being very clear about its simplifying assumptions. To oppose simplifying assumptions, after all, is to oppose a good deal more than this particular model for the pricing of stock options. It is to oppose a crucial step in the achievement of every major scientific advance on record.

And yes, I think with the anonymous recent poster at FT, that BSM has been on the whole a force for good in this crazy mixed-up world.

One intruguing sidebar to this controversy is the whole "teaching birds to fly" meme. Taleb and others on his side of the debate sometimes speak as if the idea of teaching birds to fly is inherently ridiculous, and if Black, Merton, and Scholes were trying to teach options traders to trade, describing methods the options traders already (instinctively?) possessed.

The linguist Noam Chomsky has invoked the same meme, as it happens, in his evaluation of primate-language research. Chomsky believes that langauge is a distinctively human attribute, that humans are hard-wired for it, and that the efforts to teach human language to other primates have failed in a predictable way, making this point. In an interview he gave in 1983 link, he said: "That's exactly what we should expect, I think. Why should we expect it? Because, if it turned out, contrary to what has so far been shown, if it turned out that apes really did have something like a capacity for human language, we would be faced with a kind of biological paradox. We would be faced with something analogous to, say, the discovery on a previously unexplored island that there is a species of bird with all the mechanisms for flight that has never thought of flying, until somebody comes along and trains it and says, look, you can fly. That's not impossible, but it's so unlikely that nobody would take the possibility very seriously."

Chomsky and Taleb, then, share the teaching-birds-to-fly meme.

But it seems to me, from either source, a bit presumptuous. There is nothing inherently absurd in teaching birds to fly. In fact, when I googled that phrase just now, I came up with this.

Birds are, it appears, taught how to fly, just as Elsa was taught how to be a free-ranging lion.

Cue the Born Free theme music please.

BS v. BS: Black Swans against Black-Scholes

On December 30, the Economist's website posted an anonymous column about the Black-Scholes-Merton options pricing formula. Access it here if you please.

The columnist observes that BSM has come under attack of late, and it has even been used as a prime example of why the Nobel Prize in Economics ought to be discontinued. Scholes and Merton received that august prize in 1997 for their work on this model in articles published in 1973 [Fischer Black had died in 1995 and the award is never given posthumously.]

The critics of the BSM model to whom the Economist alludes include Nassim Nicholas Taleb, the author of Fooled by Randomness (2001) and The Black Swan (2007), who attributes much of the recent financial dislocation to what he sees as the fallacious view of financial risk and risk management of which the BSM model is an important part.

In 2005, Taleb and co-author Emanuel Derman wrote "The illusions of dynamic replication," in the journal Quantitative Finance. This was a brief nerdy paper in quant jargon. In essence, "dynamic replication" refers to the replication of the value of a derivative by the continuous ("dynamic") trading of its underlyings.

Presumably, dynamic replication would render the derivative itself redundant, and this fact allows for the determination of its value by means of the no-arbitrage principle.

The Derman-Taleb paper made the case that dynamic replication is much too complicated—there are simpler ways to get the results. The most important of these simpler ways is static replication. Their idea was to drop the principle of continuous trading and construct a portfolio consisting of a long position in a call and a short position in a put. The traditionally discounted expected value of their payoffs must replicate a forward contract.

The Black-Scholes option pricing formula could have been discovered much more rapidly than it was, Taleb and Derman maintained, had this simpler route to that goal been adopted.

In 2006, the same journal published a responsive comment from two students of Robert Merton, Doriana Ruffino and Jonathan Treussard, who took issue with the Derman-Taleb reasoning. Ms. Ruffino and Mr. Treussard wrote in their paper that although it's conceivable the same result could have been reached by mathematicians making use of put-call parity without dynamic replication, it would have been a fluke, a "matter of pure chance."

In 2007, Taleb changed partners and somewhat changed his line of attack. He wrote a paper with Espen Haug entitled "Why We Have Never Used the Black-Scholes-Merton Options Pricing Formula." They proposed that the name "Black-Scholes" itself be relegated to the dustbins. They would call the options pricing formula Bachelier-Thorp, after the early stochastic-process theorist Louis Bachelier and blackjack-sharp hedge-fund pioneer Ed Thorp.

The Taleb-Haug paper attracted more attention than did its Derman-Taleb precursor. I wrote about the controversy myself at that point.

On December 7, 2008, the Financial Times ran an opinion column by Taleb and another co-author, Pablo Triana, using the intervening market chaos to reinforce their argument against contemporary financial risk management in general and the BSM model in particular.

"Ask for the Nobel prize in economics to be withdrawn from the authors of these theories," they urged their readers. "Boycott professional associations that give certificates in financial analysis that promoted these methods. The fraud can be displaced only by shaming people, by boycotting the orthodox financial economics establishment and the institutions that allowed this to happen."

So it was that the same Pablo Triana wrote an open letter to the Swedish central bank, which gives out the Nobel Prize in this field (it isn't one of the real Nobel Prizes, instituted by the old peace-loving dynamiter of that name.) Triana said: "Stop doing this! You only encourage theorists who come up with these terrible ideas that lead us all off a cliff!" -- that my paraphrase of his letter, but I submit it's a fair one.

And so we come back to the column in The Economist, which takes issue with Triana and with the whole campaign, contending that the BSM model has been "a force for good," i.e. that the world is much better off in terms of how risks are managed than it would have been had the three authors written nothing at all.

So much for the history of this dispute. I'll wade into it (unworthily, for I'm a humble scribe who has just named several people all of whom are much smarter than am I) tomorrow.

News Sense

Okay, my news sense isn't always infallible. Sometimes I think I'm on to something big, and it fizzles.

Such is the case with the Microsoft annual meeting held yesterday. There were two shareholder resolutions, and I discussed them in Monday's entry. There's really nothing to say about yesterday's meeting, though, except that all members of the board of directors were re-elected and, as the company management had recommended, both resolutions went down to defeat.

Sometimes I sense a story in the world of academic in-fighting, too. This can work out, but might not.

On Friday, I wrote a story for HedgeWorld (my dayjob) about such an academic dispute, in the world of quantitative finance. I over-state the degree to which I understand such things when I write of them, but hey -- I did take a course in calculus once.

The underlying conflict is between Nassim Taleb and the remaining authors of the famous/infamous Black-Scholes articles concerning the pricing of stock options. Fischer Black, alas, is deceased. The other namesake of the formula, Myron Scholes, is very much alive, as is Robert Merton.

Both Scholes and Merton won a Nobel Prize for their work on Black-Scholes, sometimes more generously called Black-Scholes-Merton. But Nassin Taleb, the author of a couple of widely-read books on risk and its management, says that the formula in the form they offered it, doesn't work very well. Options traders don't use it. Further, he says, it wasn't original enough with them to have their names on it, so it should be called Bachelier-Thorp if referenced any more at al.

I thought this was a big story. I did the usual consacientious reporterly work, wrote up various views of Black-Scholes on the one hand and Taleb's challenge on the other, and my editors posted the result at HedgeWorld.

One of the responses I've had since then has been to the effect that it isn't really newsworthy. Some bitter second-rate fellow envies the Nobel Prize winners and is trying to tear them down: why is that a story? one reader asked me.

All I could say is that arguing over what is newsworthy and what isn't is a mugs game, and I declined to get involved in it. She might be right, and Taleb might simply disappear.

Or, this might be the start of something big, and my readers would have heard of it early on. Damned if I know which is the case.