## Jul 11, 2016

### PUT vs WPUT Analysis

Earlier this year Oleg Bondarenko, professor at U of Illinois published an excellent empirical analysis of CBOE's PUT index and more recent WPUT (same as PUT but using weekly options). I will review some of the points in the paper comparing theoretical values with the ones that were empirically observed.

Here is a summary, and complete research paper.

Professor Bondarenko notes:
Selling 1-month ATM puts 12 times a year can produce significant income. From 2006 to 2015, the average monthly premium is 2.01%.
From 2006 to 2015, the average weekly premium is 0.75%.
Although smaller, the premium is collected more frequently. Intuitively, the premium of the ATM put increases as the square root of maturity. This means that a one-week tenor option rolled over four times per month will approximately generate 2.0x the premium of a one-month tenor option rolled over once per month (i.e., 1/2 premium times 4).
Premium by itself, of course, does not guarantee profit. Let's take a look at some basic formulas (assuming ABM,  zero rates, and unit price for simplicity. All formulas were generated using Mathematica )

$\inline&space;\dpi{200}&space;\bg_black&space;\large&space;putprice&space;=&space;\frac{\sigma&space;\sqrt{T}}{\sqrt{2&space;\pi}}$

So, as stated, assuming 4 weeks per month, we would expect theoretically to collect 2x more premium. Since in reality we are collecting about 1.5x more premium, it means that weekly premiums are smaller than expected, which is pretty much what we would expect as vol term structure is in contango most of the time.

Similarly, expected vol arb (implied vs realized) PL suffers from contango.

$\inline&space;\dpi{200}&space;\bg_black&space;\large&space;volarb&space;=&space;\frac{(\sigma_{implied}&space;-\sigma_{realized})\sqrt{T}}{\sqrt{2&space;\pi}}$

In theory, for the same implied vol value we would expect 2x more profit for WPUT, while in practice WPUT has slightly underperformed PUT.
This implies an interesting relationship (not a statistical or theoretical, just merely an observation) if we simply re-arrange the terms, that value of weekly implied volatility is in between monthly implied volatility and realized volatility.

As an approximation I took VXST index (available since 2011), VIX, and SPX returns. Because of the period mismatch, the results are not exact - average value of VXST is 17.14, between  average of VIX of 17.47, and realized volatility of 15.54.

Now, let's consider another aspect of put selling - what is the risk? We can derive expected volatility of put selling strategy:

$\inline&space;\dpi{200}&space;\bg_black&space;\large&space;vol&space;\&space;of&space;\&space;volarb&space;=&space;\frac{\sqrt{\pi-1}&space;}{\sqrt{2&space;\pi}}&space;\sigma_{realized}\sqrt{T}$

Just like in the formulas above, the expression is "per trade". That means that individual weekly trade will have half the volatility of monthly trade, but since there are 4 of them, and vol scales with the square root of # of trades, the expected volatility of PUT and WPUT should be the same. In reality, WPUT has somewhat smaller standard deviation (2.84% vs 3.32% ). Since expected volatility of put selling strategy only (theoretically) depends on realized volatility, the difference is quite surprising.

Finally we can derive a formula for sharpe ratio for selling ATM options, it is

$\inline&space;\dpi{200}&space;\bg_black&space;\large&space;SR&space;=&space;\frac{\sigma_{implied}-\sigma_{realized}}{\sigma_{realized}} * \frac{1}{\sqrt{\pi-1}}$

We can see that sharpe ratio does not depend on frequency of put sales, but rather proportional to the percent difference in implied vs realized volatility. As expected, SR for WPUT is lower than that of the PUT.