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## Sunday, July 7, 2013

### variance is additive, explained briefly ] %% own words

In 2012, I told an option pricing interviewer that variance is additive along the time horizon. Important statement worth a closer look. Here are some necessary conditions

* 2 consecutive periods
* independent
* normal distribution

See P281 [[hull]]. In my own language, Suppose we denote the 1year-from-now value of a variable (say, temperature in New Oleans) as A1. (think of log return...) We aren't so interested in A1 as A1-A0, denoted X_01. We can assume X_01 has a normal distribution so X_01's histogram (simulated) is bell-shaped. Mean 0 variance assumed 13.

Consider X_02, defined as A2-A0. This is viewed as 2 consecutive PROCESSes, each over a year. Identical and independent. X_02 is therefore the sum of X_01 + X_12, two variables of normal distributions. This sum is therefore another normal distribution with mean = 0, variance = 2 * 13.

variance of 2 iid variables is twice the individual variance.

Stdev is therefore 13*√ 2 .