Calculating the Convexity Effect...

This is another follow-up on the QIHU short trade that was generated through the optimization model I recently built.

In yesterday's post on the QIHU follow up, I noted the leverage gained by using options as a return driver.  I've talked about option leverage before but now I want to view it in the context of a multi-asset position - like the short QIHU trade which has 4 components... 1 stock and 3 options.

I started my career in traditional fixed income asset management doing portfolio analytics and risk reporting.  One of the key risk concepts of fixed income is convexity.  Convexity is the second derivative of the bond pricing model that accounts for the non-linearity of returns for changes in - mostly - interest rates (duration and convexity on non-investment grade debt tends to be more empirical).  Convexity is a major benefit to the holder of an asset... being long convexity means that you're making more as the value of your investment increases than you lose as the value drops.

Options are also non-linear assets and the second derivative of the options pricing model is gamma.  Gamma is a sensitivity measure that attributes the change in the delta of an option as the underlying asset value changes.  Gamma is highest for short term at-the-money options... this sounds great, but like leverage it's very much a double edged sword.

To better understand the impact that options have on a position, I'm going to use the QIHU short as an example and discuss it in terms of the convexity... mostly because it's easy to calculate and understand.

The formula for convexity is:

Where:

• P+ = Price of investment in a positive price shock
• P- = Price of investment in a negative price shock 
• P0 = Price of investment with no change in price
• i   = Price shock rate 

I re-entered the QIHU short into the model today to generate these figures so the numbers have naturally changed a little bit over the course of the past couple of days but the gist is still very much the same.

The selection of the shock rate is a little arbitrary but I'm going to use the projected standard deviation as my baseline.  The projected standard deviation is 14.28%

•  i = 14.28%, convexity = 3.89
•  i = 10%, convexity = -3.84
•  i = 15%, convexity = 4.27
•  i = 20%, convexity = 5.76

Again, these numbers are a measure of the benefit gained from using options - especially as the return drivers of the trade - relative to having a strictly linear position.   You can plainly notice that the more the prices are shocked - or the higher the volatility - the greater the  benefit.  Also, we see the impact of holding options through periods of low volatility as the position loses relative to a linear one as options expire out-of-the money.