Friday, August 2, 2013

GBM random walk - again

Mostly this write-up will cover the discrete-time process. In continuous, it's no longer a walk [1]. Binomial tree, Monte Carlo and matlab are discrete.

Let's divide the total timespan T -- from last-observed to Expiry -- into n equal intervals. At each step, look at ln(S_new/S_old), denoted r. (Notation is important in this field. It's extremely useful to develop ascii-friendly symbols...) It's good to denote the current step as Step "i", so first step has i=1 i.e. r_1=ln(S_1/S_0). Let's denote interval length as h=T/n.

To keep things simple let's ignore the up/down and talk about the step size only. Here's the key point --

Each step size such as our r_i is ~norm(0, h). r_i is non-deterministic, as if controlled by a computer. If we generate 1000 "realizations" of this one-step stoch process, we get 1000 r_i values. We would see a bell-shaped histogram.

What's the "h" in the norm()? Well, this bell has a stdev, whose value depends on h. Given this is a Wiener process, sigma = sqrt(h). In other words, at each step the change is an independent random sample from a normal bell "generator" whose stdev = sqrt(step interval)

[1] more like a victim of incessant disturbance/jolt/bombardment. The magnitude of each movement would be smaller if the observation interval shortens so the path is continuous (-- an invariant result independent of which realization we pick). However, the same path isn't smooth or differentiable. On the surface, if we take one particular "realization" with interval=1microsec, we see many knee joints, but still a section (a sub-interval) may appear smooth. However, that's the end-to-end aggregate movement over that interval. Zooming into one such smooth-looking section of the path, now with a new interval=1nanosec, we are likely to see knees, virtually guaranteed given the Wiener definition. If not in every interval then in most intervals. If not in this realization then in other realizations. Note a knee joint is not always zigzag . If 2 consecutive intervals see identical increments then the path is smooth, otherwise the 2-interval section may look like a reversal or a broken stick.

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