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Science in Finance IX: In defence of Black, Scholes and Merton

There’s been a lot of criticism of the Black-Scholes model of late, on our Forum, in our blogs, in the magazine (see Haug & Taleb, Wilmott magazine, January 2008) and in other media. Most is warranted, but perhaps not all. I would now like to speak in its defence! This may seem perverse since I have been highly critical of this model for the last 15 years. But as I will explain, Black-Scholes is a remarkably robust model that copes very well even when its underlying assumptions are violated, as they inevitably are in practice. Before detailing my views on this matter, I’d like to explain how my personal relationship with the Black-Scholes model has evolved.

I was introduced to options in around 1987, well before the October crash, while I was a postdoc researching in various problems of industrial/applicable maths. For a while I researched in several areas of finance simultaneously: technical analysis; chaos theory; stochastic calculus. (Thanks to the technical analysis I was short the market coming into the crash of ’87 but sadly only on paper!). I quickly dropped the TA and chaos theory, the latter seemed like a dead end, it was too easy to construct ‘toy models’ that looked plausible but were useless in practice. And so I began to focus on classical quant finance. Being in a maths department before most maths departments had heard of quant finance I had to rely on reading the literature in order to learn the subject. There were no courses for me to attend and no one more experienced to speak to. In those days whenever I read a paper I tended to believe everything in it. If the paper referred to volatility as a constant then I would believe that it was a constant. Black-Scholes was to me a good model, which just needed a minor bit of tweaking. My research from that era was on making small improvements to Black-Scholes to allow for transaction costs, and on the pricing of exotic derivatives in a constant-volatility world. This was the first phase in my relationship with Black and Scholes.

The second phase was as a consultant working for various investment banks, hedge funds and software companies. I was still in academia but moonlighting on the side. In this new capacity I finally got access to real data and was now speaking to practitioners rather than academics. (Fischer Black himself contacted me about the possibility of working for Goldman Sachs, and at this time I got to know Emanuel Derman. For a while I was sorely tempted to join them, but ultimately such a position would not have suited my personality.) It didn’t take long for me to realise how unrealistic were the assumptions in the Black-Scholes model. For example, volatility was certainly not constant, and the errors due to discrete hedging were enormous. My research during the mid and late ’90s was on making more dramatic improvements to the models for the underlyings and this was also the era when my interest in worst-case scenarios began. I worked with some very talented students and postdocs. Some great ideas and new models came out of this period. This was the height of my anti-Black-Scholes views.

A couple of years after leaving academia I became a partner in a volatility arbitrage hedge fund, and this was the start of phase three. In this fund we had to price and risk manage many hundreds of options series in real time. As much as I would have liked to, we just weren’t able to use the ‘better’ models that I’d been working on in phase two. There just wasn’t the time. So we ended up streamlining the complex models, reducing them to their simplest and most practical form. And this meant using good ol’ constant volatility Black-Scholes, but with a few innovations since we were actively looking for arbitrage opportunities. From a pragmatic point of view I developed an approach that used Gaussian models for pricing but worst-case scenarios for risk management of tail risk. And guess what? It worked. Sometimes you really need to work with something that while not perfect is just good enough and is understandable enough that you don’t do more harm than good. And that’s Black-Scholes.

I had gone from a naïve belief in Black-Scholes with all its simplifying assumptions at the start of my quant career, via some very sophisticated modelling, full circle back to basic Black-Scholes. But by making that journey I learned a lot about the robustness of Black-Scholes, when it works and when it doesn’t, and have learned to appreciate the model despite its flaws. This is a journey that to me seems, in retrospect, an obvious one to take. However, most people I know working as quants rarely get even half way along. (As discussed elsewhere, I believe this to be because most people rather like being blinded by science.)

My research now continues to be aimed at questioning commonly held beliefs, about the nature of ‘value,’ about how to use stochastic calculus to make money rather than in a no-arbitrage world, about the validity of calibration (it’s not valid!), and how people price risk (inconsistently is how!). All the time I strive to keep things understandable and meaningful, in the maths sweet spot that I’ve mentioned before.

That’s my journey. But what about the criticisms of Black-Scholes? There are several main ones: Black-Scholes was known well before Black, Scholes and Merton; traders don’t actually use Black-Scholes; Black-Scholes doesn’t work.

I will happily accept that the Black-Scholes formulae were around well before 1973. Espen Haug (“Collector”) has done an excellent job hunting down the real history of derivatives theory (see his Models on Models). Ed Thorp plays a large role in that history. In the first issue of our magazine (Wilmott magazine, September 2002) the cover story was about Ed Thorp and his discovery of the formulae and their use for making money (rather than for publication and a Nobel Prize!). Ed wrote a series of articles “What I Knew and When I Knew it” to clarify his role in the discovery, including his argument for what is now called risk-neutral pricing. I particularly like the story of how Fischer Black asked Ed out to dinner to ask him how to value American options. By the side of his chair Ed had his briefcase in which there was an algorithm for valuation and optimal exercise but he decided not to share the information with Black since it was not in the interests of Ed’s investors! Incorrect accreditation of discoveries is nothing new in mathematics, but usually there’s a quid pro quo that if you don’t get your name attached to your discovery then at some stage you’ll get your name attached to someone else’s!

They say traders don’t use Black-Scholes because traders use an implied volatility skew and smile that is inconsistent with the model. (Do these same people complain about the illegitimate use of the ‘bastard greek’ vega? This is a far worse sin.) I think this is a red herring. Yes, sometimes traders use the model in ways not originally intended but they are still using a model that is far simpler than modern-day ‘improvements.’ One of the most fascinating things about the Black-Scholes model is how well it performs compared with many of these improvements. For example, the deterministic volatility model is an attempt by quants to make Black-Scholes consistent with the volatility smile. But the complexity of the calibration of this model, its sensitivity to initial data and ultimately its lack of stability make this far more dangerous in practice than the inconsistent ‘trader approach’ it tries to ‘correct’!

The Black-Scholes assumptions are famously poor. Nevertheless my practical experience of seeking arbitrage opportunities, and my research on costs, hedging errors, volatility modelling and fat tails, for example, suggest that you won’t go far wrong using basic Black-Scholes, perhaps with the smallest of adjustments, either for pricing new instruments or for exploiting mispriced options. Let’s look at some of these model errors.

Transaction costs may be large or small, depending on which market you are in and who you are, but Black-Scholes doesn’t need much modification to accommodate them. The Black-Scholes equation can often be treated as the foundation to which you add new terms to incorporate corrections to allow for dropped assumptions. (See anything by Whalley & Wilmott from the 1990s.)

Discrete hedging is a good example of robustness. It’s easy to show that hedging errors can be very large. But even with hedging errors Black-Scholes is correct on average. (See PWOQF2.) If you only trade one option per year then, yes, worry about this. But if you are trading thousands then don’t. It also turns out that you can get many of the benefits of (impossible) continuous dynamic hedging by using static hedging with other options. (See Ahn & Wilmott, Wilmott magazine, May 2007 and January 2008.) Even continuous hedging is not as necessary as people think.

As for volatility modelling, the average profit you make from an option is very insensitive to what volatility you actually use for hedging (see Ahmad & Wilmott, Wilmott magazine, November 2005). That alone is enough of a reason to stick with the uncomplicated Black-Scholes model, it shows just how robust the model is to changes in volatility! You cannot say that a calibrated stochastic volatility model is similarly robust.

And when it comes to fat tails, sure it would be nice to have a theory to accommodate them but why use a far more complicated model that is harder to understand and that takes much longer to compute just to accommodate an event that probably won’t happen during the life of the option, or even during your trading career? No, keep it simple and price quickly and often, use a simpler model and focus more on diversification and risk management. I personally like worst-case scenarios for analyzing hedge-fund-destroying risks. (See anything from the 1990s by Hua & Wilmott.)

The many improvements on Black-Scholes are rarely improvements, the best that can be said for many of them is that they are just better at hiding their faults. Black-Scholes also has its faults, but at least you can see them.

As a financial model Black-Scholes is perfect in having just the right number of ‘free’ parameters. Had the model had many unobservable parameters it would have been useless, totally impractical. Had all its parameters been observable then it would have been equally useless since there would be no room for disagreement over value. No, having one unobservable parameter that sort of has meaning makes this model ideal for traders. I speak as a scientist who still seeks to improve Black-Scholes, yes it can be done and there are better models out there. It’s simply that more complexity is not the same as better, and the majority of models that people use in preference to Black-Scholes are not the great leaps forward that they claim, more often than not they are giant leaps backward.

P