Stochastic Modeling in Academia
1900 | Randomness (Bachelier)
Contribution
Bachelier was well ahead of his time. His observation of “random walks” predated Einstein’s work on Brownian motion by five years, and the work of Maurice Kendall and Eugene Fama by half a century.
Quotation
“The influences which determine the movements of the Stock Exchange are innumerable. Events past, present or even anticipated, often showing no apparent connection with its fluctuations, yet have repercussions on its course.”
Translation
“Price changes seem random.”
1953 | Independence (Kendall)
Contribution
Kendall is credited with the first systematic study of the statistical properties of equity returns.
Quotation
“[Market prices are] much less systematic than is originally believed…the series looks like a wandering one…”
Translation
“Price changes seem independent.”
1959 | Normality (Osborne)
Contribution
Observing that returns seemed random and independent, Osborne advanced the notion that they conform to the normal distribution based on the central limit theorem.
Quotation
“The normal distribution arises in many stochastic processes involving large numbers of independent variables, and certainly the market place should fulfill this condition, at least.”
Translation
"Random and independent? We’ve got a bell curve here, folks.”
1960-61 | Leptokurtosis (Larson & Alexander)
Contribution
Larson and Alexander raised serious doubts about the normality of market returns based on “outliers” observed in the empirical data.
Quotation
“The distribution ... is symmetrical and very nearly normally distributed…but there is an excessive number of extreme values.” (Larson)
“Osborne did not rigorously test the normality of the distribution. A rigorous test…would lead us strongly to dismiss the hypothesis of normality.” (Alexander)
Translation
“Wrong. Check out these fat tails.”
1963 | Infinite Variance (Mandelbrot)
Contribution
Mandelbrot observed that price changes could be described as having an infinite variance. Fat-tailed Paretian distributions appeared to fit the data better than the Normal (Gaussian) model. This observation carried extreme implications—it implied that traditional statistical tools like sample variance (or standard deviation) were useless when applied to financial data.
Quotation
“… the empirical distributions of price changes are usually too ‘peaked’ to be relative to samples from Gaussian populations…the tails…are in fact so extraordinarily long…it is my opinion that these facts warrant a radically new approach to the problem of price variation.”
Translation
“Beat it, nerds. The tails are fat because the variance is infinite. Your models are useless."
1964-65 | Infinite Variance (Fama & Cootner)
Contribution
Cootner and Fama acknowledged the merits of Mandelbrot’s findings, touching on the potentially devastating implications for the field’s body of work up to this point.
Quotations
“The Gaussian hypothesis was not seriously questioned until recently when the work of Benoit Mandelbrot first began to appear…Mandelbrot’s hypothesis does seem to be supported by the data…” (Fama)
“If [Mandelbrot] is right, almost all of our statistical tools are obsolete…almost without exception, past econometric work is meaningless…surely before consigning centuries of work to the ash pile, we should like to have some assurance that our work is truly useless.” (Cootner)
Translation
"Welp.”
1972-73 | Finite Variance (Praetz & Clark)
Observation
Trade activity and the availability of information affecting stocks vary from period to period. This phenomenon could potentially explain fat tails without an infinite variance assumption.
Quotations
“The information which affects prices does not come uniformly, but rather in bursts of activity.” (Praetz)
“The number of individual effects added together to give the price change during a day is variable and in fact random, making the central limit theorem inapplicable…on days when new information violates old expectations, trading is brisk, and the price process evolves much faster.” (Clark)
Translation
“There's still hope for our traditional models: fat tails can be reconciled with a finite variance."
1974 | Student’s t (Blattberg & Gonedes)
Observation
Blattberg and Gonedes observed that daily changes in stock prices conform materially to the Student’s t distribution.
Quotations
“…[we] consider another family of symmetric distributions that can also account for the observed ‘fat tails’…our interpretations of the empirical results were…for daily rates of return, the Student model has greater descriptive validity than the symmetric-stable model.”
Translation
"Student's t distribution, anyone?”
1976 | Gaussian Jump Diffusion (Merton)
Contribution
Merton modified the traditional Gaussian time series model to include unusually large and random “jumps” to account for “fat tails.”
Quotation
“In essence, such a process allows for a positive probability of a stock price change of extraordinary magnitude, no matter how small the time interval between successive observations.”
Translation
“Let's use a model that describes what we're seeing as opposed to trying to explain it."
1982-86 | ARCH & GARCH (Engle & Bollerslev)
Contribution
The Autoregressive Conditional Heteroskedasticity (ARCH) (Engel) and Generalized Autoregressive Conditional Heteroskedasticity (GARCH) (Bollerslev) models were advanced to describe “volatility clustering” initially observed by Mandelbrot in 1963.
Quotation
“Traditional economic models assume a constant one-period forecast variance…in conventional econometric models…variance does not depend [upon the past]…[I] propose a class of models where the variance does depend upon the past and will argue for their usefulness in economics.” (Engle)
Translation
“We shouldn't model volatility as a constant. It's always changing and its erratic behavior tends to cluster."
1984 | Mixture of Normals (Kon)
Contribution
Kon advanced evidence that stock returns can be described using a “discrete mixture of normals.” The reasoning is that returns are not normally distributed because of non-stationarity. That is, the parameters (mean and standard deviation) are constantly shifting due to exogenous factors.
Quotation
“…the true distribution of stock returns may be normal, [but] its parameters shift among a finite set of values…stationarity tests on the parameter estimates of the discrete mixture of normal distributions model revealed significant differences in the mean estimates that can explain the observed skewness in security returns. Significant differences in the variance estimates can explain the observed kurtosis.”
Translation
“Fat tails and other non-normal behavior could be caused by constant changes to expected return and standard deviation. We can account for this by "mixing" normal distributions."
