Lesson 6: Reliance on Continuous Hedging (Arguments)

One of the most important concepts in quantitative finance is that of delta or dynamic hedging. This is the idea that you can hedge risk in an option by buying and selling the underlying asset. This is called delta hedging since 'delta' is the Greek letter used to represent the amount of the asset you should sell. Classical theories require you to rebalance this hedge continuously. In some of these theories, and certainly in all the most popular, this hedging will perfectly eliminate all risk. Once you've got rid of risk from your portfolio it is easy to value since it should then get the same rate of return as putting money in the bank.

This is a beautiful, elegant, compact theory, with lots of important consequences. Two of the most important consequences (as well as the most important which is... no risk!) are that, first, only volatility matters in option valuation, the direction of the asset doesn't, and, second, if two people agree on the level of volatility they will agree on the value of an option, personal preferences are not relevant.

The assumption of continuous hedging seems to be crucial to this theory. But is this assumption valid?

Continuous hedging is not one of those model assumptions that may or may not be correct, requiring further study. It is blatantly obvious that hedging more frequently than every nanosecond is impossible. And even if it were possible, people are clearly not doing it. Therefore of all the assumptions in classical Black-Scholes theory this is one of the easiest to dismiss.

So why are there so few papers on hedging discretely when there are tens of thousands of papers on volatility modelling?

Perhaps because of these nice 'results,' most quants simply adore working within this sort of framework, in which continuous hedging is possible. Only a very tiny number are asking whether the framework is valid, and if it's not (which it isn't) then what are the consequences? This is unfortunate since it turns out that Black-Scholes is very robust to assumptions about volatility (see Ahmad & Wilmott, 2005) whereas robustness to discrete hedging is less well understood.

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.

Discrete hedging is the perfect example of this. 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 Wilmott (2006a). 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

Figure 5.9: Risk reduction when hedging discretely.

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 (Ahn & Wilmott, 2009).

Figure 5.9 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.

From this we can conclude that as long as people know the best way to dynamically hedge then we may be able to get away with using risk neutrality even though hedging is not continuous. But do they know this? Everyone is brought up on the results of continuous hedging, and they rely on them all the time, but they almost certainly do not have the necessary ability to make those results valid! The risk neutrals, even the cleverest and most well-read, almost certainly do not know the details of the mathematics of discrete hedging.

I think the risk neutrals need to focus their attention more on hedging than on making their volatility models even more complicated.

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