The Black-Scholes method states a formula for certain strict assumptions that describes how to build a "risk-free" portfolio. Since this formula has been adopted by the Financial Accounting Services Board and widely used throughout the industry for predictions of options pricing it has garnered a fair amount of negative attention with the recent world economic crisis in the news. But is this unfair attention?

This news woke me up. I feel as though I must be missing something, like I've slept through the past decade of hedge funds, etc. never really understanding the mechanism... but I understand enough math now to at least comment on the apparently disasterous use of these ideas. Maybe I'm missing something, and a specialist can clarify why it's not really as bad as it sounds...

So, as I understand it, the basic mechanism is a clever way of observing the derivative of the valuation such that a hedge is perfectly placed so that either way the stock goes, the option will compensate and result in a magically "risk-free" portfolio. Obviously, this is huge news for anyone in Wall Street, since there is always substantial risk associated with every transaction.

Now, there are some known shortcomings with this model that economists were well aware of. For one, it makes certain strict assumptions about drift and volatility, namely that they are constant. Volatility is the more important concept here, because while it couldn't be known directly, there was a clever way of flipping the Black-Scholes around so that you could solve for the actual volatility given historical data. So quickly economists found that this value was anything but constant in the real world. In fact, they talk of volatility-spaces as a case of solution spaces that can have many shapes depending on the type of investment. But these are simple statistical generalizations -- there is no underlying theory of the shape of these spaces!! (at least not yet)

So, what we have is a system that allows us to concretely know the shape of the space AFTER it has happened, but has zero-predictive power to tell us what will happen. Now, economists apparently desparately wanted this space to have at least local-linearity, but keep in mind the essence of the space is unknown and is at least indirectly related to a geometric brownian factor (i.e. random!) -- so it's extremely unlikely that any linearity exists in the first place... but it's safer to say that we simply don't know what it is.

Now what mathematician in their right mind would suggest that this was anything more than simply shuffling one unknown into another variable and then claim that at least it is "somewhat useful as a first order approximation which can be adjusted"?

Are you kidding me? Based on what? Based on local linearity assumptions and cooperative historical data of the first few years?!? The space is unknown man! What made them think they could "adjust" it in any way predictively that made sense?

But it didn't even work that well when it was first introduced. Look at Merton's first board position at a hedge company that failed so spectacularly in 1998 that again the Fed was concerned about bailing out the industry. How did I miss that towards the end of the dot com bubble?! 4.6 billion dollars in four months!! (of course back then I'd never even heard of a hedge fund).

Yet the Financial Accounting Services Board adopted this method of calculating options value and now it's widespread use is implicated in the worldwide economic crisis you see before you.

Can someone please explain how this made any sense whatsoever?

[update 10/18]: Paul Wilmott's blog article in defense of Black-Scholes is an interesting read -- he's someone who does this for a living and so he has a much deeper view on it than I do certainly -- (it's a reference backing the wikipedia article and also provides additional references to check out)

He points out that Black-Scholes is very robust on the averages in spite of the obviously poor assumptions of the model, and therefore is a relatively simple model to use in practice (i.e. an industry workhorse even though it's not perfect). There are more complicated models but not a lot of evidence that they work any better. He asks why use more complicated models when the fat-tail isn't that important all the time?

Well, what about when it is?

Such as when the risk suddenly becomes very high market wide and the correction deltas are huge? Now you have a function that you assume is locally-linear (because of it's physics origins) -- but you don't. I think Black-Scholes might only be a proper "industry workhorse" when volatility is relatively constant.

It failed massively in 1998, and it failed massively now. Is it coincidence that the market was experiencing wide economic volatility at both those points in time?

I still have a big question about how worst-case scenarios were investigated on this model? If a handful of investments with high-volatility were considered in the context of stable market data, Black-Scholes might behave acceptably well on the averages. But was a "total market meltdown" considered closely?