Science in Finance X: Dynamic hedging and further defence of Black-Scholes

I continue to find myself in the middle of the argument over validity of Black-Scholes. On one side are those who we might call “the risk neutrals.” Those heavily invested in the concepts of complete markets, continuous hedging, no arbitrage, etc. Those with a relatively small comfort zone. On the other side there are those who tell us to throw away Black-Scholes because there are so many fallacious assumptions in the model that it is worthless. Let’s call them the “dumpers.” And then there are a tiny number of us saying yes, we agree, that there are many, many reasons why Black-Scholes should not work, but nevertheless the model is still incredibly robust to the model assumptions and to some extent you can pretend to be a risk neutral in practice.

A good example of this is the subject of discrete hedging. The theory says that to get the Black-Scholes model you need to hedge continuously. But this is impossible in practice. The risk neutrals bury their heads in the sand when this topic is discussed and carry on regardless, and the dumpers tell us to throw all the models away and start again. In the middle we say calm down, let’s look at the maths.

Yes, discrete hedging is the cause of large errors in practice. I’ve discussed this in depth in PWOQF2. Hedging error is large, of the order of the square root of the time between rehedges, it is path dependent, depending on the realized gamma. The distribution of errors on each rehedge is highly skewed (even worse in practice than in theory). But most analysis of hedging error assumes the simple model in which you rehedge at fixed time intervals. This is a very restrictive assumption. Can we do better than this? The answer is yes, if we are allowed a certain number of rehedges during the life of an option then rehedging at fixed intervals is not at all optimal. We can do much better.

The figure above shows a comparison between the values of an at-the-money call, strike 100, one year to expiration, 20% volatility, 5% interest rate, when hedged at fixed intervals (the red line) and hedged optimally (the green line). The lines are the mean value plus and minus one standard deviation. All curves converge to the Black-Scholes complete-market, risk-neutral, price of 10.45, but hedging optimally gets you there much faster. If you hedge optimally you will get as much risk reduction from just 10 rehedges as if you use 25 equally spaced rehedges.

You can see how this puts me firmly between the risk neutrals and the dumpers. The risk neutrals won’t go too far wrong as long as they know how to hedge well, and the dumpers are right if it turns out that the risk neutrals don’t know the maths of discrete hedging.

I suspect that the ratio of the number of papers on complete-market volatility models to the number of papers on discrete hedging is probably about a hundred to one, so I think the risk neutrals need to focus their attention elsewhere if they want to win this battle!