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.
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