# Why isn't the maximum entropy approach to modelling stock return distributions from options prices more well studied

–1 vote
edited ago
I recently came up with an idea of modeling the distribution of asset returns from the price of options using a maximum entropy approach.  (It should be noted that I'm not really in finance so my grasp of the literature is not good).  Turns out someone more than 20 years ago beat me to that idea.  It's here: https://www.jstor.org/stable/2331391?seq=1#metadata_info_tab_contents and apparently people have been adding on to it for years.

It doesn't bother me that much that it's been done way before I was thinking about econ, but that it doesn't seem like there is that much empirical work done using this approach.  Maybe I'm wrong, but most of the work seems theoretical.  I talked to a professor of finance (I'm a PhD student), and he wasn't even aware of this line of work, suggesting state of the art was deriving moments from a Black-Scholes equation.  At least if I am not mistaken, this is a 100000x better approach for modeling the distribution of future returns of assets given constraints (i.e. that the expected value of the option equals the price adjusted for time value of money).  The reasoning is that you don't have to assume a normal distribution, in fact, the method can, I think, approximate any distribution given enough constraints.   Second, it also approximates the expected return and doesn't just treat that as an exogenous variable which would be the case if you use a Black-Scholes approach.

All this being said, I'm curious if there are a reason I'm missing why the Black-Scholes approach is used, or empirical asset pricing researchers don't even bother using an approach like this, where you can get an estimate of the expected return of an asset without waiting for the realization of that return.   Maybe I'm missing a broad swath of empirical literature, but I'd be curious why this doesn't have more than 300 cites.  I'd love to hear from someone what I am missing that makes this work much less important than I think it is.  I am sorry I'm just beside myself right now.   I think Buchen and Kelly, whoever they are, got cheated out of a really good paper (but maybe I'm saying that because I got attached to an idea I came up with myself).

+1 vote
I don't know about the reasons why this particular approach was not followed. I do know, however, quite about about the criticisms one gets in the rational inattention literature, a part of which is also based on Shannon entropy.

One concern people have is that they do not understand the basis for using that particular measure of entropy. It's derived from information theoretic considerations, so why should it be relevant to particular economic settings, where considerations about channel capacity and optimal information transmission play no role.  Why should we use Shannon entropy, rather than any of the half-dozen other entropy measures? At the same time, everyone is intimately familiar with central limit theorems. And while they are, of course, aware that the following is false, they nonetheless often seem to make intuitive assessments as if the central limit theorems implied that all probability distributions are normal.

Another reason might be the fact that the authors are not very well known. If you decide to build on somebody else's research, you are taking a risk. The higher the chance the research you're building on will be accepted, the higher the chance your own work will be accepted. And that chance is higher for better-known authors.
commented ago by (300 points)
This is me still trying to boost the authors work for the empirical asset pricing people out there.  I appreciate the response and I think it is a good one and so I upvoted.

You only have a finite number of constraints and an infinite dimensional object to optimize so you have to choose something else to constrain it with.  But why choose the more restrictive constraint.  The maximum entropy distribution with a set mean and set standard deviation is a Gaussian, so if the constraints can imply that, you will get a Gaussian out of the model.   However at the same time to critique the more restrictive Gaussian assumption, we know stock returns don't often behave in a Gaussian manner and that they there is the option smile/ fat tails, so why make a distributional assumption that is more restrictive than you have to.  Meanwhile Shannon entropy is the natural entropy to use because AFAIK, it is the only entropy that basically doesn't when applied to a distribution doesn't allow any other "spurious" constraints to be embedded in it.  Given a set of constraints, it satisfies them and is as natural as possible to any other constraint that could be applied but is not applied.  There is a more full throated defense in the seminal papers by Jaynes (see this wikipedia: https://en.wikipedia.org/wiki/Principle_of_maximum_entropy), that I have not read.  But even if you don't like Shannon entropy, use another entropy if you can get an equation out of it.  Either way, there is no reason to restrict oneself to a Gaussian.

I do agree with the fact that the authors are not well known probably is a big reason.  However, there have been a decent number of theory papers building on this but I could hardly find one empirical paper.
+1 vote