Part 2 of a series on the use, and misuse, of probability distributions in modelling financial data.
“All models are wrong, but some are useful.”George E. P. Box
Why are all models wrong? And if all of them are wrong, then why do we keep using them? All models are wrong because they are a simplification of reality based on assumptions. We keep using them because it is not possible for us to visualize the scale of a system in real life and / or it may be too dangerous for us to test a system in real life without, first, gaining some level of confidence by testing it out as a model. A map of New York City falls into the first category, while a model airplane in an aerospace wind tunnel falls into the second category.
A map of New York City may not scale perfectly to the actual city and it may have some missing natural landmarks, incorrect elevations and it may not even show some of the skyscrapers; but it can still help us drive from point A to point B. At the same time, we should not be using such a map to fly over the city in our Cessna aircraft; lest we crash into one of the unmarked skyscrapers. Thus, in some cases it helps to have a model, while in other cases a model can actually enhance our chances of disaster (by making us dangerously over-confident). In the latter case, it is better to not have any kind of model, thereby either not undertaking the endeavour, or if we must, then acting in an overly cautious manner relying on the inherent instinct (gut feeling) that has successfully ensured our survival as a species since the caveman days, for decision making and risk management.
Thus, the key is not only to understand how to model systems, but to know when to use which model; and most importantly, to know when to not use a model at all. The former relates to efficiency (profit and loss, in finance lingo); the latter relates to survival.
The Normal distribution was doing a great job after being fully crystallized by the German mathematician - Johann Carl Friedrich Gauss (1777 - 1855). Much like the honeycomb hexagon and the snowflake crystal, another example of nature organizing its components neatly to find the most efficient solution while spending the least amount of effort. However, within a century, the Normal distribution's own elegance was about to become its biggest vulnerability due to the emergence of two interesting phenomena in the area of social sciences: physics-envy and the Great Scientist effect.
Physics, primarily due to the efforts of Sir Isaac Newton's Principia Mathematica (1687), was able to (successfully) mathematicize itself beyond other areas of inquiry, disproportionately. In simple terms, physics research papers looked like maths papers, providing rigorous underlying mathematics equations to support empirical results in a manner that was envied by other fields of study. In an effort to compete with physics, researchers in the fields of social sciences, including finance, were also eager to follow this approach. What physics researchers had (via mathematics) achieved in the deterministic domain, researchers in finance wanted to achieve in the stochastic domain; specifically in the stochastic non-physical domain, that is, in the informational domain.
The first step in this direction was (rather inadvertently) taken in 1900, by a French mathematician—Louis Bachelier—, who (much like Einstein, at the same time) modeled Brownian motion in his PhD thesis, Théorie de la Spéculation. However, while Einstein used Brownian motion to define the existence of atoms, Bachelier discussed Brownian motion as the first statistical method for analysing the random behaviour of stocks, bonds, and options. Bachelier’s model is known today as the famous Random Walk of prices. Its two main postulates state that price movements are like coin tosses; they are statistically independent and Normally distributed.
Bachelier's research was destined to obscurity, until it was discovered six decades later in the USA. By 1960s, computing power was becoming more easily available and, simultaneously, the field of finance was trying to become more grounded in mathematics. This led to the research efforts of, arguably, the greatest name to be associated with the Normal distribution since Gauss himself - MIT economics professor and the first American to win the Nobel prize in economics, Paul Samuelson. Samuelson rediscovered and championed Bachelier’s work, which was published in English in 1964, helping to launch the field of quantitative finance; Samuelson himself became widely known as the father of modern economics. Today, key building blocks of quantitative finance - Efficient Market Hypothesis (EMH), Modern Portfolio Theory (MPT), Capital Asset Pricing Model (CAPM), Arbitrage Theory, to name a few, are all based on the two key assumptions of Random walk - statistical independence and Normal distribution. The field of economics (and finance) has mathematicized itself into econometrics - the branch of economics concerned with the use of mathematical methods (especially statistics) in describing economic systems -, overwhelmingly.
The question that needs to be answered is whether economics has mathematicized itself successfully, like physics. Or are its pilots now using incoherent maps based on inaccurate assumptions, putting their passengers at more risk than if they had no maps to begin with.
If we know the historical rewards (µ) and associated risks (σ²) for the price (x) of any category of financial products, we can predict the future rewards and associated risks using the ever-reliable Normal distribution. The above simplifies finance significantly, making it easy to build an underlying quantitative framework. One need not construct sophisticated statistical toolkits to manage the complex interactions of a large number of parameters / variables / constants to predict the future; two parameters suffice, rather conveniently. As per econometrics, nature has not only provided us, via the Normal distribution, with a simple solution for modeling the relatively docile stochastic physical domain; it also allows us to use the same distribution to model the significantly more turbulent informational domain.
While Paul Samuelson (along with his colleagues and students at MIT) was laying the foundations of quantitative finance, simultaneously, one of the greatest mathematicians of the past century - Benoit Mandelbrot - at IBM (and at Harvard) was finding major holes in the same models. In 1962, while studying the movements of cotton prices, Mandelbrot discovered they were neither statistically independent nor Normally distributed. In fact, he recognized this anomaly in all price movements he studied. Mandelbrot showed that empirical financial data accumulated since 1900 for price movements, displayed high values of kurtosis (the peak of the central hump in the probability curve), far beyond those for a Normal distribution. Specifically, he showed that price movements displayed three characteristics - jumps, volatility clustering and nonstationarity - due to which they cannot be modelled correctly by the Normal distribution. Based on this, Mandelbrot, in a landmark paper in 1963, redefined probability distributions for modelling price movements.
In simple terms, volatility clustering implies that financial data has memory, due to which large changes are followed by other large changes and small changes are followed by small changes; the changes being the volatility and their consecutive recurrence being the clustering. This runs counter to a Normal distribution model. A jump is a sudden non-continuous shift in price movements. This also runs counter to Normal distributions, which prefers continuous movements. Non-stationarity implies infinite (inconsistent) variance. Infinite variance means that the data is all over the place. This can happen for a Normal distribution also, however, the probability of this occurring is low in case of Normal distributions (stationarity), while in case of non-stationary distributions (like the Cauchy distribution) the probability is much higher. Stationary processes are time-independent, implying the average value of the measurements is a constant, as in case of a Normal distribution, where the variance (σ²) must be constant. While it is beyond the scope of this discussion, however, detecting whether (the distribution of) events actually shows constant variance can require an extremely long time scale over which the measurements are made (confusing the two-humped camel for a single-humped camel problem).
The logical action when these modelling inconsistencies were highlighted was to step back, accept the fact that the Normal distribution did not model price movements accurately, and attempt to find a distribution that does. However, the Normal distribution was far too seductive to be discarded so easily. This is where the Great Scientist Effect came into play; great scientists and their disciples have such a defining impact on their disciplines, to the point that even the defects in their research become part of the mainstream. Not only had Paul Samuelson and his close colleagues won Nobel prizes based on the Normal distribution assumption, but his students (and even their students) were in the Nobel prize league. In fact, the Random walk model has had such a huge impact on finance, that not merely have Nobel prizes been awarded for sophisticated models built on top of Random walk; they have also been awarded for even more sophisticated models that attempt to correct the inaccuracies - volatility clustering, jumps and stationarity - of the original models. New, difficult to pronounce (and even more difficult to understand) statistical techniques and terminologies exploded on the quantitative finance scene post-60s - , heteroskedasticity-consistent covariance-matrix estimator, autoregressive distributed lags (ARDL), and everyone's favorite, generalized autoregressive conditional heteroskedasticity (GARCH), to name a few. So much so, that six of the top ten most cited papers ever in economics do not build upon the Normal distribution; they actually define corrective modelling techniques to fit empirical financial data (which consistently does not follow the Normal distribution) onto a Normal distribution. They do not add to the existing accepted research; they try to fill in holes for why the existing research does not work in all cases. Much like the additions made to the heliocentric model of the universe, when the planets would not show up where the model predicted they should.
This brings up another question: what was the need for such complicated correction techniques, when the simple and elegant Normal distribution was already available? A Normal distribution, in all its elegance, is so simple that even a high school mathematics student can understand it without difficulty. While the correction techniques to fit empirical financial data onto a Normal distribution are so complex that most doctorate students cannot get their heads around them.
Why doesn't the curve of nature seamlessly transition into the curve of finance?
To answer this foundational question, perhaps, we need to understand how nature organizes itself. We keep discovering deep mathematical connections in nature, even though mathematics is supposed to be a disconnected and abstract field of study. In many cases, we cannot derive the formulas that cause such events in nature; yet we can describe the recurring patterns in the empirical data, mathematically. We discover equations, constants and / or a probability distribution in nature, but we cannot quite explain why they are there. It is as if nature is communicating with us through mathematics; Φ - the golden ratio of the DNA double helix, snail shells and elephant tusks; H: the Hurst exponent defining the long term memory of water flow leading to floods and droughts; the perfect hexagon of a honeycomb, the bilateral symmetry of a leaf, the radial symmetry of a snowflake and the Fibonacci sequence of the patterns of seeds in a sunflower, to name a few. The unexplainable existence of physical constants in the universe from the speed of light (c) to the constant of gravitation (G) is another example. And, of course, the recurring appearance of the Normal distribution in the stochastic physical domain - the domain with bounded randomness.
This may be why physics has been able to mathematicize itself so seamlessly. Because physical objects and events have been organized mathematically by nature, to begin with. Our increasing knowledge of mathematics is simply unveiling the already existent patterns in those objects and events. Why, then, is mathematics having so much difficulty in mathematicizing the informational domain? Why aren't simple ratios and constants and probability distributions unveiling themselves for the informational domain, forcing us to rely on increasingly complicated correction equations?
Perhaps, it is because nature evolved its stochastic physical processes into a Normal distribution over millions of years. Molecule by molecule, cell by cell, correcting its mistakes (via adjustments to tail events) till it was able to reach its ultimate goal of finding the most efficient solution while spending the least amount of effort. This evolution continues to fine-tune itself, at a microscopically slow pace, as we speak. The world of finance is simply too new. It has not been fine tuned into its own elegant probability distribution, and try as we might, our (rather inelegant) mathematical techniques have been unable to (forcefully) map misbehaving empirical financial data onto the recurring patterns of a Normal distribution (or for that matter, any distribution). Or perhaps, nature concluded, a long time ago that humanity has a greater chance of survival if it keeps all its options open, including the ability to act with unbounded (and thus unmodellable) randomness, when dealing with informational uncertainty (as an example, the fight-or-flight instinct when deciding between a stick and a snake); thereby giving us unbounded options to manage our risks when reacting to perceived threats (like a perceived crash of the stock market), while bounding our options when deciding on the preferable height of our next child.
Mandelbrot's research, in the 1960s, highlighted what was happening, that is, it clearly showed that price movements do not follow the Normal distribution. However, it did not explain the why of this question. That would not happen till another landmark paper in 1979, that led to the first Nobel prize in economics ever awarded to a psychologist, in 2002; explaining that the secret(s) of empirical financial data did not lie in the field of mathematics, but in the field of psychology. How did this happen? That is the subject of Part III.