Black-Scholes-Merton as Beachhead

The above graph is a visual representation of the Black-Scholes model. Or Black-Scholes-Merton if you want credit shared equitably and "you're not into the whole brevity thing," Mr Lebowski.

As you can see, there are three axes. The x axis (the width of the box) is the price of the underlying asset, the stock price, treating the strike price in the center as 1. The y axis (the height of the box) is the volatility of that option. The z axis (the depth of the box) is the time to maturity [and either exercise or expiration].

As you can see, the plane of various shades of blue (the darker the blue, the higher) has a sharp crease at the front/center/bottom of the box, where the sharp difference between winners and losers on the lottery’s drawing date is indicated.

We should also mention that the volatility that goes into the calculations as we’ve described them above is historical volatility. One of the assumptions of the BSM model is constant volatility. This is a counter-factual assumption: volatility does in fact change.

Another phrase you might want to remember is: implied volatility. That comes up when a theorist is doing his calculations in the other direction. If we know the present price of both the underlying stock and the stock option, what degree of historical/constant volatility does their relationship imply?

As you might have grasped by now, the elegance of the BSM model, the elegance of that neatly sloping blue plane in the box, comes at some cost in heroic assumptions. We have to assume that the movement of stock values describes a bell curve rather than a non-normal sort of curve. We also have to assume that the standard deviation of that bell curve remains constant over time. We also have to assume the degree of liquidity and transparency in the markets necessary to make arbitrage easy, because that assumption is behind the various proofs of these equations.

Does all that make the model worthless? Not at all. As Emanuel Derman has written, all models sweep dirt under the rug. A good model is one that “makes explicit the dirt swept away,” and BSM is decidedly a good model in this sense.

BSM was a beachhead of an invasion. In showed the way, and other intellectual troops rushed in to widen the territory covered. Stock options, after all, are just one example of a broad category of instrument, known as derivatives. The value of a stock option depends upon that of the underlying stock, and in like manner the value of other derivatives depends upon the value of other sorts of underlying asset.

The brain power of the quantitative analysts (“quants”) of the world focused on this point: how far could the underlying logic of the BSM model be extended to other sorts of derivative? The answer turns out to be: quite far indeed. Yet extending its realm doesn’t change the nature of the assumptions built into it, nor the fact of their fallibility.

A Story About Coleridge

This is a quote from a memoir by Dorothy Wordsworth, reflecting on a trip she took with two famous poets, her brother, William Wordsworth, and their similarly gifted companion, Samuel Taylor Coleridge. We sat upon a bench, placed for the sake of one of these views, whence we looked down upon the waterfall, and over the open country ... A lady and gentleman, more expeditious tourists than ourselves, came to the spot; they left us at the seat, and we found them again at another station above the Falls. Coleridge, who is always good-natured enough to enter into conversation with anybody whom he meets in his way, began to talk with the gentleman, who observed that it was a majestic waterfall. Coleridge was delighted with the accuracy of the epithet, particularly as he had been settling in his own mind the precise meaning of the words grand, majestic, sublime, etc., and had discussed the subject with William at some length the day before. “Yes, sir,” says Coleridge, “it is a maj...

Searle: The Chinese Room

John Searle has become the object of accusations of improper conduct. These accusations even have some people in the world of academic philosophy saying that instructors in that world should try to avoid teaching Searle's views. That is an odd contention, and has given rise to heated exchanges in certain corners of the blogosphere. At Leiter Reports, I encountered a comment from someone describing himself as "grad student drop out." GSDO said: " This is a side question (and not at all an attempt to answer the question BL posed): How important is John Searle's work? Are people still working on speech act theory or is that just another dead end in the history of 20th century philosophy? My impression is that his reputation is somewhat inflated from all of his speaking engagements and NYRoB reviews. The Chinese room argument is a classic, but is there much more to his work than that?" I took it upon myself to answer that on LR. But here I'll tak...

Recent Controversies Involving Nassim Taleb, Part I

I've written about Nassim Taleb on earlier occasions in this blog. I'll let you do the search yourself, dear reader, for the full background. The short answer to the question "who is Taleb?" is this: he is a 57 year old man born in Lebanon, educated in France, who has been both a hedge fund manager and a derivatives trader. He retired from active participation from the financial world sometime between 2004 and 2006, and has been a full-time writer and provocateur ever since. Taleb's writings for the general public began where one might expect -- in the field where he had made his money -- and he explained certain financial issues to a broad audiences in a very dramatic non-technical way. Since then, he has widened has fields of study, writing about just about everything, applying the intellectual tools he honed in that earlier work. As you might have gather from the above, I respect Taleb, though I have sometimes been critical of him when my own writing ab...

Jamesian Philosophy Refreshed

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