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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 naive 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. 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. 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. 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 analysing hedge-fund-destroying risks. 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.