Tuesday, 22 July 2014

What's gamma trading?

23:19 Posted by The Thalesians (@thalesians) 2 comments

Given this blog is supposed to be interactive, I asked my readers for ideas for this article. By popular demand (well at least 3 readers @krs, @JeremyWS and @jaredwoodard), it was suggested that I write an article explaining what gamma trading is.

Let's first consider the simplest option trade we can do to get exposure to short volatility. We could sell a straddle (so basically a call and a put) in this instance. If spot stays close to the strike, we would make money at expiry, as we would pick up the option premium (and the payoff would be close to flat). However, in the event of a large directional move, the trade would result in a large loss. We could of course delta hedge our exposure and run a delta neutral position. If we are short vol, this would involve buying spot as spot rises and conversely selling spot as it falls. The conundrum is obviously how frequently we delta hedge (see my paper for backtested results on this!) Clearly, continuous delta hedging is not possible (and would be too costly in terms of transaction costs).

If we think about how we can calculate the P&L from a delta hedged short vol position, we need to mark to market the option and also calculate the P&L on the various spot trades which have been undertaken for delta hedging. However, what we want is some more intuitive way to relate this P&L to implied and realised volatility.

It can be shown that our P&L can roughly be expressed as a gamma weighted difference between implied and realised variance (assuming we don't have large exposure to other greeks and that we have short vol exposure). This equation is given below. Sorry, the quant part of me loves formulae - skip it if you find mathematics offensive! Note, that the formula is different to a variance swap, which obviously doesn't have the gamma term.

Looking at the formula, it is not surprising that trading options in this way and delta hedging is known as gamma trading. Shorter dated options have more gamma and also when spot is trading closer to the strike, gamma will be higher. Hence, if we are short gamma and very close to expiry, with spot oscillating rapidly around the strike, we could be faced with large losses (if we take a look at the formula). Conversely, if realised vol picks up and we are short gamma, but spot is very far from the strike, P&L is not impacted as much.

On the plot at the start of the blog, we have plotted the returns for being short EUR/USD vol (short-dated straddles) splitting up the returns into option P&L and delta hedging P&L (but excluding transaction costs). We can see that in 2008, whilst our option had a drawdown, our spot delta hedges made money, illustrating our earlier point that delta hedges can offset the losses from large directional spot moves.

We note that short gamma is generally profitable as a strategy in the plot (and actually more broadly). This is because of the volatility risk premium. This simply describes the fact that implied is generally above realised volatility. So by selling volatility we are harvesting this risk premium. Obviously, there are drawdowns associated with capturing risk premium in this way, as we have mentioned.

In this very short summary, we have very briefly described gamma trading and related it to the differential between implied and realised volatility, using a simple formula. Please also see my interview at Global Derivatives by Robert Almgren where I describe systematic gamma trading in FX.

To read more about this topic please see the below work I've done, which goes through the subject in much more detail, including systematic strategies for trading gamma in FX (and discussing optimal ways for delta hedging). My book Trading Thalesians also has some colour on this topic (mixed in with a bit of ancient history).

Thalesians - Gamma, gamma, gamma - Explaining gamma trading in FX markets - 14 Apr 2014 (available for Thalesians clients only)


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