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## Monday, May 28, 2012

### how to make volatility values ANNUALIZED

(Let’s assume a flat forward curve i.e. 0 drift, 0 dividends, 0 interests.) Suppose an implied vol for a 1-year option is 20%. If we record ln(PR) i.e. log of daily price relatives until expiry, we expect 68% of the 200+ daily readings to fall between -0.2 and 0.2. That’s because ln(PR) is supposed to follow a normal distribution.

Note we aren’t 68% sure about the expiration underlier price i.e. S(t=T) or S(T) for short. This S(T) has a lognormal distribution, so no 68% rule. However, we do know something about the S(t=T) because the end-to-end ln(PR) is the sum of ln(daily PR), and due to central limit theorem, the overall ln(PR) has a normal distribution with a variance = sum(variance of ln(daily PR)). We always assume the individual items in the sum() are independent and "identical", variance of ln(daily PR) is therefore 0.04/252days.

Also, Since ln(overal PR) = ln[S(T)/S(0)] has normal distribution, S(T) has a lognormal distribution. That's the reason for .

To answer any option pricing question, we invariably need to convert quoted, annualized vol to what I call raw-sigma or stdev.

Rule #1 -- we assume one-period variance will persist to 2 periods, 3 periods, 4 periods... (eg: a year consists of 12 one-month periods.)

Example 1: If one-year variance is 0.04, then a four-year raw-variance would be .04 * 48/12 = .16. The corresponding stdev i.e. raw-sigma would be 40%. This value is what goes into BS equation to price options with this maturity.

Example 2: If one-year variance is 0.04, then a three-month raw-variance would be .04 *3/12 = .01. The stdev i.e. raw-sigma would be sqrt(.01) = 10%. This value is what goes into BS equation to price options with this maturity. By Rule #1, we assume the same 3-month variance would persist in 2nd 3-month, the 3rd 3-month and 4th 3-month periods.