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Painted Black

This is an unconsidered piece.

I’m a little tired of reading about what a travesty Black-Scholes is. First of all, the real trouble isn’t Black-Scholes, it’s geometric Brownian motion. That’s the underlying error.

Black-Scholes is an engineering construction that would work if stocks really did evolve under GBM. They don’t. So, using Black-Scholes has plenty of problems. Stocks can jump, volatility isn’t constant, you can’t always short, there are transactions costs, and so on.

So, what can you do? Black-Scholes is a zeroth order approximation with (perhaps) a series of first- and second- and higher-order corrections. I say “perhaps” because claiming there are higher order corrections implies that someone knows the correct answer, and that’s not true. You have to think of Black-Scholes as being the right answer is a Platonic world that doesn’t match the one we live in.

It’s true that many devotees of Black-Scholes are naive. They assume that if you correct it to accommodate the things it neglects you can get there. Instead, if you’re a trader or a quant, you ought to think of Black-Scholes as a way of thinking about things, an ideal formula that doesn’t hold in the real world, and now it’s up to you to decide how to correct for its omissions. Live with it — you can’t do much better, at least for options. Even static hedging of the weak form (when there is no exact payoff matching) requires a model to construct the static hedge.

As someone I know once said: You can’t give someone a Black-Scholes calculator and turn him into a trader.

People put too much faith in the model in the past. Now there’s an over-reaction to its difficulties. What you have to do is look at the problems and then decide how to work your way around them, with a few calculations and a big dose of common sense.

The enthusiastic use of replication a la Black-Scholes is no doubt responsible for many disasters and market bubbles. But so is the naive reliance on anything: low future default rates, low P/E, The Nifty Fifty, you name it. Part of the trouble is the model itself — P/E is a model of sorts. But another part of the trouble, perhaps an equal or greater part, is human enthusiasm, in particular desperate enthusiasm for some metric. Metrics in the social sciences (someone I know told me his father said that any field that has the word ‘science’ in its name isn’t science) are always approximations.

If you get rid of Black-Scholes there will still be bubbles. Nevertheless, it’s compulsory to understand its limiations. Wild horses couldn’t drag me away??

Published in Models