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Firstly, I'm sorry, if my question will come across as simple or even naive, but I have no formal background in statistics and I'm trying my best to learn it as much as I can, among other areas.

My question is two-fold and I will start with its simpler part:

What phenomenon or principle is the foundation that allows modeling a purely random (stochastic) process, described as Brownian motion (not drift-containing and trend-exhibiting Geometric Brownian motion)?

Based on the question above, I'd like to state another, more general question, as stated in the title (please note that, despite including econometrics tag, my questions are concerned with imaginary purely stochastic processes component of real-world complex hybrid economic, social, biological and other natural and artificial processes, containing "modellable" non-stochastic component):

Why "modeling volatility" is not an oxymoron?

I expect an answer to this question to expand a bit on the topic. Recommendations of and references to (preferably not math-heavy) introductory materials are welcome and are appreciated.

Aleksandr Blekh
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  • It is hard to determine what you are asking. Could you elaborate on (1) what would constitute a "phenomenon or principle" in this context (especially since the two words refer to such different things) and (2) exactly why you believe "modeling volatility" has an oxymoronic character to it? In particular, I cannot understand the sense in which any "phenomenon" would "allow modeling": what do you mean by that? – whuber Jan 05 '15 at 18:05
  • @whuber: Thank you for your comment. (1) I intentionally used both words to allow more freedom for answers to contain either A) mathematical, statistical, physical _principle(s)_, underlying some **systematic laws** of volatility's nature that allow its modeling, or B) **foundational** natural _phenomena_ (**building blocks**) of volatility. (2) Since _pure volatility_ implies complete **randomness** and **unpredictability**, but _modeling_ implies some **systematic underlying reason**, it seems to me that these concepts contradict each other, when applied together, hence, the oxymoron label. – Aleksandr Blekh Jan 05 '15 at 18:41
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    Think of a spore in Brownian motion. It is *impossible* to predict how exactly it would move, where exactly it'll end up. However, Brownian motion theory allows to predict how far the spores will move away from the starting *in average*. That's why in Einstein's paper he talks about a group of particles. – Aksakal Jan 05 '15 at 18:49
  • @Aksakal: I understand. I probably still need to review the paper you're referring to (and maybe other materials) in order to refresh my memory. However, as you've mentioned _averaging_ (group behavior), what is the underlying statistical principle/concept for that? – Aleksandr Blekh Jan 05 '15 at 19:00
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    Einstein's paper is interesting in historical context. I would rather recommend reading Zeldovich's reference, which I gave in my answer. I'm sure you can easily find it online. The principle is independence: each shock is independent both across time and across the particles. – Aksakal Jan 05 '15 at 19:03
  • @Aksakal: Thanks! I will definitely read both papers. I figured that, if I want to understand statistics better, I should (again) return to underlying principles and (again) work my way up to more complex concepts. – Aleksandr Blekh Jan 05 '15 at 19:17
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    One key thing to keep in mind is that you're describing the behavior of a *distribution* over time (in particular, with a diffusion model and no drift, a key part of the model describes how the variance changes with time). This is typical of statistical models (that they describe distributions, not that it's over time). With stochastic volatility, you again have a model for how variance changes with time (though it's not like a diffusion model). – Glen_b Jan 06 '15 at 01:01
  • @Glen_b: Couldn't we consider time as just another dimension in a statistical model (regardless of its different nature)? Or there is something special about the time dimension that "standard" models cannot be applied to model stochastic processes? – Aleksandr Blekh Jan 06 '15 at 01:08
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    I wasn't making any special point about time, just trying to be clear that modelling how distributions change (over something) was a typical situation. While it's common to look at how means change (and for people to forget that the model is of a distribution, of which the mean is only one component), some models are for things other than the mean. – Glen_b Jan 06 '15 at 01:16
  • @Glen_b: I see. I'm still feeling somewhat fuzzy on making an ordered structure of statistical concepts in my head. But that's normal, I guess, considering that I don't have formal background and basically jumped into that scary "ocean of knowledge", trying to swim. At least, I'm staying afloat (knocking on wood). – Aleksandr Blekh Jan 06 '15 at 01:21

3 Answers3

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The name Brownian motion comes from the guy named Brown, and his microscope observation of spores moving in the liquid. Einstein got Nobel prize for the theory. The physics are simple, as Einstein put it:

Soon after the appearance of my paper (*) on the movements of particles suspended in liquids demanded by the molecular theory of heat, Siedentopf (of Jena) informed me that he and other physicists-in the first instance, Prof. Gouy (of Lyons)-had been convinced by direct observation that the so-called Brownian motion is caused by the irregular thermal movements of the molecules of the liquid.

This is how Zeldovich and Myshkis set up the simple theoretical framework as follows:

Along the avenue on which the x-axis, there is a lot of stores equally spaced at $h$ distance from one another. From one of them serving as the reference point, the lady runs out: she only just heard that somewhere there's a sale on French umbrellas. She randomly chooses the direction along the alley, in a time $\tau$ she runs to a nearby store. Hearing that there are no French umbrellas, she forgets where she came from, and runs again at random to one of two sides. After the time $\tau$ she again gets a negative response and, once again losing the sense of direction, randomly selects the direction and so on. Where will she end up at time $n\tau$? (For simplicity, we assume that the rumor was false, so the search for French umbrellas goes on infinitely.

Then next passage sets the expectations of what we can get from the model:

Of course, it is impossible to predict exactly, one can only determine the probability of the fact that the lady at the time $n\tau$ will have a certain coordinate.

Note, how they emphasized the fact the Lady was a bit out of her mind and sense of direction: she doesn't remember where she came from. That's the crucial detail: each move is independent of each other. Then the central limit theorem (CLT) kicks in, so you have the $x(n\tau)=\sum_{i=1}^nx_i=\sum_{i=1}^n(-1)^\xi_ih$, where $\xi_i$ - bernulli process with $p=1/2$. According to CLT this must converge to some kind of normal distrobution.

What's the problem with "modeling volatility"? It is a strong assumption to state that the volatility is constant. In economics and finance it fails again and again. Think of volatility as a noise "level", it doesn't have to remain constant and it doesn't. What guys like Heston noticed, was that volatility seems to be stationary and persistent, hence the family of stochastic volatility models.

Aksakal
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  • +1 Thank you for your answer. I still don't understand how something completely random (I'm talking about that part only) can at all be modeled, when modeling implies some systematic underlying reason/law/concept, which complete randomness lacks by definition. Please see my comment to @whuber above. – Aleksandr Blekh Jan 05 '15 at 18:46
  • Let's start with the diffusion. It's a completely random process. Does it strike you that it can be modeled? – Aksakal Jan 05 '15 at 18:47
  • I might be wrong (parts of my physics knowledge are rusty), but I think that diffusion (as many other natural processes) can be modeled only because its model considers finite values of materials' quantity and space/volume (or other parameters for other processes). – Aleksandr Blekh Jan 05 '15 at 18:54
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    Diffusion is a stochastic process, where molecules of one material randomly get inside another material. It is impossible to predict how exactly each molecule will move and where it will end up, but when you consider enormous number of molecules you can easily and very precisely predict the average diffusion speed. It's the same thing with all this stochastic modeling. It's the same math. In fact when you think Brownian motion, it's safe to think *diffusion*. – Aksakal Jan 05 '15 at 18:59
  • Thank you, now it's clearer. But the question remains (repeating from my comment above): what is the underlying statistical principle/concept for averaging (group behavior)? – Aleksandr Blekh Jan 05 '15 at 19:03
  • Thank you for the updates - it is even more interesting now. – Aleksandr Blekh Jan 05 '15 at 19:33
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Many financial instruments are often modeled as being stochastic partial differential equations or difference equations if modeled in discrete time.

It is not so different as in time series analysis where series are modeled as some sort of low order process having trends, seasonal components and irregular noise superimposed in the same process.

In more classical setting volatility is an parameter, not an variable to be modeled but this has changed too. It might not be theoretically justifiable to incorporate deterministic parameters and their sample estimates in used model, but of course in reality many might do that. For example in Black & Scholes valuation formula has fixed parameters and users often supplant sample estimates into the formulae.

Users generally can fall into the trap of model risk if they believe too much that their formulas incorporate all relevant aspects of reality.

Analyst
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  • +1 Thank you for your answer. Actually, the origin of this question is my recent interest in time series analysis (for example, [this](http://stats.stackexchange.com/a/131133/31372) and [this](http://datascience.stackexchange.com/a/3764/2452)). In particular, I came across the term "modeling volatility", which I felt as an oxymoron, in the 2003 Nobel Prize in Economics [research paper](http://www.kva.se/globalassets/priser/nobel/2003/sciback_ek_en_03.pdf), which I've referred to in my first answer. – Aleksandr Blekh Jan 05 '15 at 19:29
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I think you are having problems with the notion that a random process can be described by rules - eg in a model.

Standard Brownian motion ${B_t}$ is defined as the process that

  1. starts at 0 at time zero,
  2. the time differences $B_t-B_s$ are normally distributed with mean 0 and variance $t-s$ independently of past ${B_u}_{0\le u<s}$ (for $t>s>0$)
  3. has continuous sample paths

So it should be clear that the Gaussian independent increment assumption is a clear modelling assumption - one could make other assumptions and have different sorts of randomness with different statistical properties wiki brownian motion

seanv507
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  • +1 Thank you for your answer. I believe that your first sentence nicely captures the essence of my fuzziness. Please see my comments above (2 threads). From your reference to normal distribution, it seems that an underlying statistical principle/concept for Brownian motion and, thus, randomness, is Gaussian function. But why? – Aleksandr Blekh Jan 05 '15 at 19:09
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    Brownian motion is a type of randomness. Eg I could say my position would be determined by throw of dice, or my change in position, these all give rise to different statistical distributions. All are random processes – seanv507 Jan 05 '15 at 19:35
  • Thank you - that I understand. I'm confused about what is the **core underlying reason/concept/principle/law** that allows us to describe _random_ processes **analytically** (PDF, etc.). In other words, what is the basis for _averaging_ (_group behavior_) that can be described analytically (modeled)? – Aleksandr Blekh Jan 05 '15 at 19:42