The dragon’s revenge

In the second article on the pitfalls of hedging, Neil Palmer considers one of the risks of managing options: dynamic hedging. He shows there is an awful lot that can go wrong in the quest for perfect risk elimination

Last time in Quant Talk we looked at basis risk. We found that if your hedging asset isn’t perfectly correlated to your exposure, the risk can be a lot more severe than might be expected. The message was that you can’t expect to slay the dragon of risk. You might just try to give it some sleeping pills and keep a fire extinguisher on hand for when it wakes up.

This time we look at a case where we do have perfect correlation between our potential liability and our hedging asset – because they’re one and the same thing. But the dragon of risk remains alive and well.

The problem we consider is option hedging. One famous solution published by Black and Scholes ultimately won a Nobel prize1 but don’t let that blind you to the practical problems that can crop up.

Options are funny things. They have a ‘now-you-see-me, now-you-don’t’ quality about them. No one knows for sure whether or not they’ll be exercised. So how do you manage them?

Well, it’s not really rocket science. Say you’ve sold to Mega Energy Inc the right (but not the obligation) to buy power from you at $50/MWh. It’s fair to suppose that the higher the market price for power rises, the greater the chance that Mega will ultimately exercise its option. So if the market does rise you should buy a little power forward to keep up your sleeve to offset the expected increase in your future liability. And when the market falls, you sell a little. This is the essence of dynamic hedging.

Here we can see that you’re hedging with the same asset you might ultimately have to deliver. In this sense there’s no basis risk at all between your exposure and your hedge. But perfect correlation doesn’t save us because we’ve introduced a new kind of risk.

The Black-Scholes formula has been described as the nearest thing in finance to pure physics. Let’s take a look at another point of contact between these two worlds. I’ll admit it’s fairly loose but I hope it makes the point.

This year is the centenary of Albert Einstein’s annus mirabilis in which he published several ground-breaking papers in physics. As far as I know, Einstein didn’t go in much for hedging options, but there’s a parallel between one of his famous results and the work of Black, Scholes and Merton.

Einstein’s theory of relativity says that as you go faster, time slows down. What Black, Scholes and Merton effectively said is that as you hedge faster, risk slows down.

According to Einstein – and the scientists who’ve not yet managed to prove him wrong – if you could travel at the speed of light then the passage of time would cease completely. And Black, Scholes and Merton pretty much said that if you can hedge at the speed of light then risk should vanish completely.

Black, Scholes and Merton didn’t use the term ‘hedging at the speed of light’ – they might have raised one or two eyebrows if they had. Their assumptions are commonly described using rather more innocuous-sounding terms like ‘continuous trading with zero transaction costs’ and so on. It doesn’t sound too unreasonable, does it? But these assumptions are really very strong.

Of course no one can really hedge at the speed of light. The real question to ask is: what are the consequences of our failure to do so? And it’s not pretty.

If you keep in place all the other assumptions Black, Scholes and Merton introduced and simply adjust your hedge at regular intervals, you immediately bring in a new source of risk. It’s a bit like basis risk: your ‘hedged portfolio’ is not quite hedged all the time.

If you want to reduce this risk it’s simple. Just dynamically re-hedge more often. But not so fast! If you want to halve your residual risk you need to hedge four times as often.

This is the essence of the ‘dragon’s revenge’: getting rid of risk via delta hedging can be very thirsty work and crucially dependent on market liquidity and model assumptions. And of course trading isn’t free. The more frequently you hedge the smaller your average volumes are – but your transaction costs overall are greater. Not many energy markets lend themselves to continuous trading.

One of the problems with the Black-Scholes approach is that it’s been hyped and elevated to an absurd degree. When Black, Scholes and Merton’s work was first made public, all of a sudden traders and quants started to believe that they could totally eliminate risk by trading the underlying asset. It’s rather like someone reading Einstein’s theory of relativity and concluding they can significantly slow down time by always driving in the fast lane of the motorway.

Last year I was lucky enough to attend Quant Congress 2004, an event run by the publishers of Energy Risk and Risk magazines. One of the most high-profile talks was given by Emanuel Derman. This gentleman is a widely respected elder statesman of the quant world, a former colleague of the late Fischer Black and a member of Risk magazine’s own Top 50 Hall of Fame.4

Professor Derman spoke about his own journey from the absolute purity of physics to the rather less pure social science that is finance. And then he said something shocking – well, shocking enough for me to write it down. Referring to the usual process of option hedging, he said: “Actually, dynamic replication is a bit of an illusion”.

What did he mean? Was he saying the emperor is really naked? Have the quants got it all wrong? Should Scholes and Merton be giving back their Nobel?

Of course he wasn’t saying any of these things. What I believe Derman meant was not to deny the ability of traders to hedge options, but to deny their practical ability to slay the dragon of risk finally and absolutely. Perfect risk-elimination is a cornerstone of the original option pricing theory. But there’s an awful lot that can go wrong, and often it does.

There’s absolutely no question that the Black, Scholes and Merton approach has been a wonderful contribution to the quant world: as well as the benchmark pricing tool it’s also a superb laboratory for asking ‘what ifs?’ about the various assumptions and different kinds of hedging strategies you might dream up.

But we’ve become a little blinkered by the central brilliance of the Black, Scholes and Merton approach. It’s just one more reminder of what we already knew: if you really want to make any money, you have to grit your teeth and make friends with the dragon.

Neil Palmer is a London-based quantitative structurer for a major energy trading company