interpolation using variance vs. standard deviation.

[tina]

I was hoping someone could shed some light on a problem we are encountering with the interpolation methods for volatility. Should you use standard deviation or variance?

If you have a .5 year contract with an annualized volatility of 20% and a 1 year contract with an annualized volatility of 30%. You then want to solve for the .75 year contract's annualized volatility. which one of the following methods would make more sense?

Should you take the mid point of stand deviation of the .50 year contract, which is .14142 (sqrt(.5)*20%) and stand deviation of the 1 year contract, which is .30 (sqrt(1)*30%). So the standard deviation for the .75 day contract is .2207, implying a annualized volatility of 25.48%.

Or

Should you take the mid point of variance of the .50 year contract, which is .02 ((.5)*20%*20%) and variance of the 1 year contract, which is .09 ((1)*30%). So the variance for the .75 day contract is .055, implying a annualized volatility of 27.08%.

 

[David Harper CFA FRM]

Hi Tine,

I replied to your email. I think this understandably confuses the 'square root rule' with the 'volatility term structure.' I mean, if you asked me to interpolate here, I'd probably intepolate variances: average[4%,9%] = 6.5% variance = 25.5% volatility.

But, IMO, that is no more or less correct than EITHER of your approaches. There is no linear interpolation logic that is really justified here. If you said, instead, six month (periodic) variance is 4% and annual variance is 8%, then the square root rule would justify interpolating variances: 9-month variance = average [4%,8%] = 6%. But that's interpolation to scale from one 'periodicity' to another...

You instead are interpolating along the term structure. Books etc have been written on this, I don't think there is a linear approach. Most volatility surfaces (term structure + skew) are sort of exponential...implying, in your example, I am just making this up, something like 21% or even 20% at 0.75 is possible (much nearer to the 0.5 contract; some surfaces may be *flat* in the short run). So, IMO, this is one of those cases where you can be "precisely wrong" by leaning on the math too much. Unless you have more information on the term structure, both of your approaches can be viewed as non-linear weighted average interpolations that are no more/less informed than any other...David