Thursday, August 15, 2013

Stoch lesson 05 - Level^Stepsize ..

A3: stepsize_i. The Level_i value may or may not be normally distributed if we plot the 9999 realizations, but that depends on some factors. For example, if the noisegen is identical and independent on every query then we can rely on Central Limit Theorem. For stock prices that's not the case.
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In this series of lessons, I will create a set of "local jargon" used in later blog posts. First, Imagine the "Level" [Note 1] of a time-varying random variable W is a random walker taking up or down steps at regular intervals. At step_i, the stepsize_i [Note 2] is generated from a (Gaussian or otherwise) noisegen such as a computer. Level_i is the sum of all previous steps, positive or negative, i.e.

    Level_i = stepsize_1 + stepsize_2 + ....stepsize_i

It's important to differentiate Level_i vs stepsize_i. Q3: which one of them has a normal distribution? Answer is hidden somewhere.

Notation is important here. It's extremely useful to develop ascii-friendly symbols, with optional font sizing. These notations will be used in subsequent "lessons". Here are a few more notations and jargon --

Let's divide the total timespan T -- from last-observation to Expiry -- into n equal intervals. Denote a particular step as Step "i", so first step has i=1. Let's denote interval length as h=T/n = t_i+1 - t_i

I will use norm(a,b) to denote a Gaussian noisegen with mean=a and variance=b, so stdev=sqrt(b).

[1] The word "value" is too vague compared to Level.
[2] a.k.a. increment_i but less precise.

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