Thursday, March 31, 2011

forward volatility -- from basic principle

See also http://www.farmdoc.illinois.edu/nccc134/conf_2003/pdf/confp03-03.pdf -- variance is additive. More specifically, for n multiple independent experiments with each outcome having var1, var2, ...var_n, the sum of the outcomes has variance var1+var2+...var_n. Incidentally, the sum of the outcomes has mean of mean1+mean...mean_n. A random walker makes one "experiment" at each step, and the log(PR) is cumulative.
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One basic assumption in http://en.wikipedia.org/wiki/Forward_volatility is "Given that the underlying random variables for non overlapping time intervals are independent, the variance is additive."

Q: What's that random variable? I believe it's the "r" i.e. log(PR) described in in http://bigblog.tanbin.com/2011/12/h-vol-calc-using-price-relatives-right.html. This variable "r" is additive in itself. Intuitively, if over 4 days gold price moves by a ratio of 120% -> 101% -> 97% -> 99%, then we can ADD r2,r3,r4,r5... to get the overall Price-Relative of Day 5 over Day 1 closing.

[1]
where r1-2 represents log($Day2ClosingPrice / $Day1ClosingPrice) = log(PR over Day2), which is another label for the earlier r2.

It turns out this sum is the variance of log(PR1-5), or variance of r1-5 , i.e. the volatility over the 5 consecutive observations. Since the random variables r1-2 r2-3 r3-4 r4-5 follow the same pdf, each variance should be numerically identical.
==> variance over 96 hours and 5 observations (Price1 to Price5) is exactly 4 times the daily variance
==> If we assume 256 trading days in a year, then annual variance is 256 times daily variance
==> annualized vol is 16 times daily vol. If annualized vol is 80%, then log(PRdaily) has stdev = 5%.....

The sum in [1] is variance over Day 1-5. Forward variance over 2-5 can be derived from the 4 individual variance numbers, or from....
However, it's unfair to compare 2-5 fwd variance against "spot" variance of 1-5 when the holding periods are unequal. Therefore, all the variance numbers must be annualized for a fair comparison.

Basic statistics rule -- if Y = X_a + X_b, i.e. 2 independent normal variables, then the variance of Y is sum of the 2 variances.

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