Selecting an Index Option Expiration

A few weeks ago, I wrote about  what option strikes were best to sell when harvesting the variance premium. This is part of an ongoing project to find optimal option strategies for volatility trading, hedging and directional trading. As the next step, today I’m going to look at what expiration to trade when selling index volatility.

First, let’s look at the theoretical arguments in the Black-Scholes-Merton world. On average the total PL for an option sold at si when the realized volatility is (the lower) sr is

But it is the way that this comes about that is important to us here. This is actually the sum of gamma profits which occur continuously. In one time step

(Traders usually think in terms of the first equation but this second equation is probably more fundamental as it comes straight from the first term in the BSM differential equation).
So instantaneous PL is directly proportional to gamma. And short term (ATM) options have more gamma than long term (ATM) options. So you can expect the profits of mispriced options to be mainly realized as expiration approaches.

We can check this by looking at the example of a one year ATM straddle on a $100 stock, where we sold implied at 50% and realized volatility was 40%. This equates to selling the options for $39.5. Summing up the daily gamma profits we find that the first 11 months we made $5.60. In the last month we made $2.20. So 28% of our profits were realized in the last month.

This effect becomes even more pronounced as we get closer to expiration. One week options make profits more quickly. And one day options are even better.

This analysis has only considered average PL. Variance is also important. To look at the dispersion of results, I ran 10,000 simulations of the process. I looked at holding a one year straddle for 11 months and then liquidating at the fair value (i.e. volatility of 40%), and also selling the one month straddle and holding to expiration.

The average PLs were basically the same as our theory would indicate. But interestingly, shorter dated options also had lower profit variance. The standard deviation for the 11 month options was $24.70 (so return/standard deviation was 0.23). For the one month options the standard deviation was $6.85 (leading to a return/standard deviation ratio of 0.32). When short dated options “get away from you” they don’t have a chance to go as far as longer dated ones.

The simplified world of BSM pricing is very useful as a starting point, but real markets can be different. Fortunately, Adriano Tosi and Alexandre Ziegler made an empirical study of S&P 500 options in their paper “The Timing of Option Returns”. Specifically, they showed that the returns to short put options are concentrated in the few days preceding their expiration.

So, if you want to harvest the volatility premium you should short front-month options, preferably in the last week or so of the cycle, while investors wishing to go long volatility risk should buy back-month options. This is independent of the fact that the volatility premium will often be most significantly mispriced in shorter maturity options.

(If you are dynamically hedging I suspect the results will be similar but I will look at this in a future post).

It might be tempting to use these results to trade calendar spreads, selling front month options and buying longer dated options. This should allow collection of a lot of the volatility premium while also hedging against large volatility moves. To a degree this is true. But calendar spreads provide a new set of challenges. I will write about these another time.
Euan SinclairComment