Wednesday, August 23, 2017

Stock Market Prices Do Not Follow Random Walks

From Turing Finance, February 2016:

Because volatility seems to cluster in real life as well as the markets, it has been a while since my last article. Sorry about that. Today we will be taking our first giant leap along A Non-Random Walk down Wall Street.

The Non-Random Walk Series

A Non-Random Walk Down Wall Street is the cheeky title of an academically challenging textbook written by Lo and MacKinlay in response to the best-selling Wall Street classic, A Random Walk Down Wall Street, written by Professor Burton Malkiel. A Non-Random Walk Down Wall Street is a collection of papers which challenge the prevailing random walk hypothesis. Despite containing only outdated results and being mathematically unforgiving, it's an impressive textbook which has inspired me to write a series of articles about it.

This series of articles has the following goals: Bolster or invalidate my original findings using the NIST test suite; Translate Lo and MacKinlay's papers and tests into more intuitive terms; Extend the results to the present day to determine if they are still relevant; Extend the results to emerging markets with a strong focus on South Africa; And bridge the theoretical and practical gap between machine learning and market randomness. Whew!

This series is also inspired by many of the thoughtful comments I received after I published my post about the random walk hypothesis, Hacking The Random Walk Hypothesis. So please keep the comments coming.

P.S. Simply because market randomness tests don't exactly make for great WordPress featured images, the featured images for this series of articles will be screenshots from all of my favourite Wall Street inspired films. This article's featured image is from the most recent Wall Street inspired film to hit the box office: The Big Short. If you haven't seen it yet, do yourself a big favour and go watch it. Afterwards, if you want more information you can always suffer through an earlier, non technical article of mine: A Recipe for the 2008 Financial Crisis.

Article Outline

This series will début with Lo and MacKinlay's first paper: Stock Markets Do Not Follow Random Walks: Evidence from a Simple Specification Test. In this paper Lo and MacKinlay exploited the fact that under a Geometric Brownian Motion model with Stochastic Volatility variance estimates are linear in the sampling interval, to devise a statistical test for the random walk hypothesis. This post covers the theory and application of this test.
This post is broken up into the following sections:
  1. Efficiency, The Markov Property, and Random Walks
  2. Variants of the Random Walk Hypothesis
  3. Stochastic Model Specification
  4. Stochastic Model Calibration (NB!)
  5. Variance Ratio Properties and Statistics
  6. A Heteroskedasticity-Consistent Variance Ratio Test
  7. Results obtained on Simulated Asset Prices
  8. Results obtained on Real Asset Prices
  9. Remarks and Conclusions
  10. Appendix A: Why R?
Should you have any criticisms or comments about this post or the random walk hypothesis in general, please let me know via the comments section at the end of this article. I always appreciate the input.

Efficiency, The Markov Property, and Random Walks

The random walk hypothesis is a popular theory which purports that stock market prices cannot be predicted and evolve according to a random walk. This hypothesis is a logical consequent of the weak form of the efficient market hypothesis which states that: future prices cannot be predicted by analyzing prices from the past ...

To a statistician the assertion that future prices cannot be predicted by analyzing prices from the past goes by a different name: the Markov property or, more intuitively, memorylessness. Any time series which satisfies the Markov property is called a Markov process and Random Walks are just a type of Markov process.

The idea that stock market prices may evolve according to a Markov process or, rather, random walk was proposed in 1900 by Louis Bachelier, a young scholar, in his seminal thesis entitled: The Theory of Speculation. In his paper he proposed using Brownian motion, a Markov (and Martingale) process, to model stock options. That said, it wasn't really until 1973, when the Black Scholes formula for derivatives pricing was published, that the idea gained traction.
Physical Brownian Motion. Sourced from:
Since then the use of stochastic processes for derivatives pricing has become industry standard. That having been said, the philosophical question regarding whether or not stock market prices really evolve according to a random walk or, at the very least, according to the popular stochastic processes used in industry today, remains. To paraphrase Queen, we are left wondering: is this [The Random Walk Hypothesis] real life? Is this just fantasy?

Personally my mind rebels against the theory because it is too elegant; too simple. I like complexity; I like chaos. So I choose to spend my spare time learning more about the theory of randomness and I enjoy trying to find ways to test the conventional wisdom ... and hopefully someday learn to beat the market consistently ;-). 

Variants of the Random Walk Hypothesis

A typical test of the random walk hypothesis involves three steps. First off you assume that asset prices do evolve according to a random walk and you select an appropriate stochastic model. Secondly, you define which statistical properties you would therefore expect to see in asset prices. And lastly, you test whether or not the asset prices exhibit the expected properties. If the asset prices don't exhibit the expected properties, then the assets don't evolve according to the model of the random walk hypothesis you assumed they did to begin with.

It's not good enough to simply state that market returns aren't random, you need to also specify what type of random they aren't.

Admittedly the fact that I didn't follow this process exactly in my previous article on Hacking The Random Walk Hypothesis was its biggest shortcoming. Luckily a supportive statistician on Reddit helped me see the light: it is not good enough to simply state that market returns aren't random, you need to also specify what type of random they aren't. In light of this below I have defined three popular forms of the random walk hypothesis.

RW1: The first and strongest form of the random walk hypothesis assumes that the random disturbance, ϵt, is independent and identically distributed (IID). This corresponds to the Geometric Brownian Motion Model wherein volatility of the random disturbance, ϵt, allows only for homoskedastic increments (constant σ). Under this hypothesis, variance is a linear function of time (discussed in more detail in the next section).

RW2: The second, weaker form of the random walk hypothesis relaxes the identically distributed assumption and assumes that the random disturbance, ϵt, is independent and not identically distributed (INID). This corresponds to the Heston Model wherein the volatility of ϵtalso allows for unconditional heteroskedastic increments. Under this hypothesis, variance is a non-linear function of time (discussed in the next section).

Arch model
RW3: The third and weakest form of the random walk hypothesis relaxes the independence assumption meaning that it allows for conditional heteroskedastic increments in ϵt. This corresponds to some random walk process wherein the volatility either has some sort of non-linear structure (it is conditional on itself) or it is conditional on another random variable. Stochastic processes which employ ARCH (Autoregressive Conditional Heteroscedasticity) and GARCH (Generalized AutoRegressive Conditional Heteroscedasticity) models of volatility belong to this category.

In other words, any successful refutation of the random walk hypothesis must, ultimately, be model dependent. Furthermore, that model must clearly fall within the spectrum above. It just so happens that the weaker the form of the random walk hypothesis, the harder it is to disprove and the more powerful your statistical tests need to be.

To illustrate this point consider how easy it would be to show that some asset's prices don't evolve according to Brownian Motion but, on the other hand, how difficult it would be to show that the same asset's prices don't evolve according to some stochastic process without independent increments and with conditional heterskedasticity!
Dilbert random number generator
My favourite comic strip, Dilbert, gets it. Sourced from: