Espen Gaarder Haug & Nassim Nicholas Taleb have produced a highly entertaining – but, alas, somewhat less than informative – polemic: Why We Have Never Used the Black-Scholes-Merton Option Pricing Formula:
Options traders use a pricing formula which they adapt by fudging and changing the tails and skewness by varying one parameter, the standard deviation of a Gaussian. Such formula is popularly called “Black-Scholes-Merton” owing to an attributed eponymous discovery (though changing the standard deviation parameter is in contradiction with it). However we have historical evidence that 1) Black, Scholes and Merton did not invent any formula, just found an argument to make a well known (and used) formula compatible with the economics establishment, by removing the “risk” parameter through “dynamic hedging”, 2) Option traders use (and evidently have used since 1902) heuristics and tricks more compatible with the previous versions of the formula of Louis Bachelier and Edward O. Thorp (that allow a broad choice of probability distributions) and removed the risk parameter by using put-call parity. 3) Option traders did not use formulas after 1973 but continued their bottom-up heuristics. The Bachelier-Thorp approach is more robust (among other things) to the high impact rare event. The paper draws on historical trading methods and 19th and early 20th century references ignored by the finance literature. It is time to stop calling the formula by the wrong name.
The tone of the paper is evident in the first angry footnote:
For us, practitioners, theories should arise from practice.
Footnote: For us, in this discussion, a practitioner is deemed to be someone involved in repeated decisions about option hedging, not a support quant who writes pricing software or an academic who provides “consulting” advice.
The main thrust of the article is that the premise of the Black-Scholes model is incorrect:
Referring to Thorp and Kassouf (1967), Black, Scholes and Merton took the idea of delta hedging one step further, Black and Scholes (1973):
If the hedge is maintained continuously, then the approximations mentioned above become exact, and the return on the hedged position is completely independent of the change in the value of the stock. In fact, the return on the hedged position becomes certain. This was pointed out to us by Robert Merton.
This may be a brilliant mathematical idea, but option trading is not mathematical theory. It is not enough to have a theoretical idea so far removed from reality that is far from robust in practice.
The authors point out that
- Option trading has been around for a long time
- The only way to hedge options properly is with other options, due to pricing discontinuities
- Put-Call Parity is the basic theoretical foundation of proper hedging
The second main point of the article is that, consistent with the idea that only options are a proper hedge against options, the job of an options trader is not to value options based on some theory; it is to make money with a market-neutral book:
In that sense, traders do not perform “valuation” with some “pricing kernel” until the expiration of the security, but, rather, produce a price of an option compatible with other instruments in the markets, with a holding time that is stochastic. They do not need topdown “science”.
…
This raises a critical point: option traders do not “estimate” the odds of rare events by pricing out-ofthe-money options. They just respond to supply and demand. The notion of “implied probability distribution” is merely a Dutch-book compatibility type of proposition.
They conclude:
One could easily attribute the explosion in option volume to the computer age and the ease of processing transactions, added to the long stretch of peaceful economic growth and absence of hyperinflation. From the evidence (once one removes the propaganda), the development of scholastic finance appears to be an epiphenomenon rather than a cause of option trading. Once again, lecturing birds how to fly does not allow one to take subsequent credit.
This is why we call the equation Bachelier-Thorp. We were using it all along and gave it the wrong name, after the wrong method and with attribution to the wrong persons. It does not mean that dynamic hedging is out of the question; it is just not a central part of the pricing paradigm.
I must point out that Mr. Taleb’s rose-tinted vision of the good old days – while probably quite true in most respects – do not square completely with what I have read in other sources.
If I recall correctly, Morton Schulman recounted in his book “Anybody can still be a millionaire” his adventures as partner in a small Toronto brokerage in the … late ’60’s? early ’70’s?. He and his partners were willing to write puts and became, he says, amazingly popular with his New York counterparts because there was bottomless demand for them.