Interpolation using variance vs. standard deviation.

[tina]

I was hoping you could shed some light on a problem we are encountering with the interpolation methods for volatility. Our assumption is that the standard deviation moves at the square root of time. and variance moves at the rate of time. I am taking the standard deviation which is (sqrt(T)*annualized volatility). This is according to Natenberg who says one standard deviation over a period of time, T, is represented by V*sqrt(T), V being the annualized implied volatility determined by the market place and T beginning measure of years to expire. Variance is that number square.

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 do a linear interpolation of the sqrt(T) and Standard deviations? For example the stand deviation of the .50 year contract, which is .14142 (calculated by sqrt(.5)*20%) and stand deviation of the 1 year contract, which is .30 (calculated by sqrt(1)*30%). So the standard deviation for the .75 day contract is .2274( doing a linear interpolation of the sqare root of time and standard deviation) implying a annualized volatility of 26.25%.

Or

Should you do a linear interpolation of the Time and variance? For example variance of the .50 year contract, which is .02 (calculated by (.5)*20%*20%) and variance of the 1 year contract, which is .09 (calculated by (1)*30%). So the variance for the .75 day contract is .055 ( doing a linear interpolation of time and variance), implying a annualized volatility of 27.08%.


Should you use standard deviation or variance? Which one make sense and why? Any light you could shed on this issue would be much appreciated.

 

[pardasani]

I would go for linear interpolation in variances implying a vol of 27.08% you calculated...

the reason being when you are trading options you are trading variance -- in the sense that a delta hedged option gives a PnL of vol*vol* delT ; where we notice vol is the realized vol in the time delT....

 

[tina]

thanks

I was hoping you coukld expand a little on what you mean that the pnl of the option is caculated by vol*vol*delta?

 

[pardasani]

Okay let me give this a try (although I guess its difficult to do it without a picture):

So lets say today is day 0 and I go long a ATM call option... the concept of so called "risk neutrality" would say I should sell off some stock (approximately 0.5 because ATM) so that I am delta neutral. I do that, and so in our portfolio now I am long a call option and I am short some stock, resulting in approximately 0 delta wrt underlying...

now I come in tomorrow, and the stock moves... if it moves up, the delta of the call would increase, and conversely if the spot moves down the delta would decrease... therefore to rehedge the portfolio it so happens that we sell stock when the spot moves higher and buy it whenever the spot goes down.... and we do it not just on day 1, but throughout the life of the option...

the reason that it so happens that I need to buy low and sell high (therefore creating a "profit") is because of the structure of a call option: is convex (remember the saying convexity comes with a price?). This tiny bit profit I make up is already paid for upfront because I bought a call option, and now i (or a typical option market maker) am "recovering" its value doing this delta hedging stuff...

Lets go back to day 1. Remember on day 0 I had a delta 0.5... Say the spot went higher by dS. I could then approximate the delta on day 1 as
delta (day1) = delta (day 0) + Gamma(0) * dS

Waving some things under the carpet, for the sake of understanding, lets say dS could be positive or negative. Lets say a 50-50 chance. So you are buying stock at S-dS/2 and selling off at S+dS/2

Effectively your expected profit on day 1 is proportional to
( Gamma * dS) * dS = Gamma * (dS)^2
where the first term is the number of shares you need to adjust and dS is the profit because you buy low and sell high..

Basic stochastic calculus says that (dS)^2 is proportional to (vol*vol)*delT (this is where the lognormal assumption comes in) and hence the magic that in a small interval your expected PnL has a term related to (ds)^2

In general the exact formulation would be

V(T - 1) = V(T) + 0.5* Gamma*ds*ds + Thetha*dT (assuming you remain delta hedged), where V(T) is the value of the option

the last term is because the option loses time value when time moves from T to T-1

Hope this helps !