"A thought from a former student" on quantitative finance

A real letter I received a few days ago.


Dear Professor Derman,

I graduated from Columbia last May. I took your seminar on FE important papers during my senior year and enjoyed it very much. I went to work at ***** doing things that have pretty much no relation to the things I learned in school. I am however still very interested in quantitative finance and it is something I like to read about and think about fairly often.

I am also very interested in economics in a more general sense. I am one of those people that still thinks people are rational and markets work. I know these are ideas that have fallen severely out of fashion, but it is the only thing that ever made any sense to me from a theoretical/analytical point of view. I don't mean to say that I don't believe in the experiments carried out by the behavioral economists, but I am yet to be convinced that these things can have any real effects in the aggregate. Having worked in at least one part of the financial industry for a little while, I can say that the approach to valuation and overall decision making seems pretty rational over here. I really don't think that we give higher valuations to a company on a given day simply because we just paid a high price for our lunch (That one was called anchoring, right?).

Lately I have been having certain doubts about some of the first principles on which so much of what I studied in school is built and I thought perhaps I would share them with you and you would have some interesting insight or comment.

Basically it boils down to the following: in most areas of economics, when I think of the first principles, it is a collection of assumptions that all make logical sense. Many of the variables are not directly observable, but we know the general form and it is something one would have a hard time refuting. For example, while I can't easily observe the demand curve for apples, i can safely say that it has to be "downward sloping"--If apples are more expensive I will buy less apples. That is simple and about as true as we can expect anything in the realm of human behavior to be.

My issue comes when I try to think of first principles, or just the "form of first principles" in the area of quantitative finance. We have the law of no arbitrage and the efficient market hypothesis (both of which I think make a lot of logical sense, which I like), but those two alone don't really get you anywhere. We need some assumption for how asset prices evolve. And that is where I think things get murky. There is no logical explanation that gets us to Brownian motion or anything of the like. I can see an explanation for assuming some sort of evolution that is a "martingale", which can logically follow from the efficient market hypothesis (in fact, its almost like saying the same thing), but any assumption we make as far as the form of the distribution seems to come out of thin air. Sure, we can inform the guess with past data, but that does not seem theoretically rigorous. So I look at the entire paradigm of the mean-variance (or risk-return) approach to investing and I am left thinking: risk is not even well defined. Markowitz (or Bachelier) woke up one morning and said "these returns look normally distributed" and we started defining risk in terms of standard deviations, but that's just one random (pun not intended) guess!

And it feels like from then on it just gets worse. Will Sharpe came up with his Capital Asset Pricing Model (which we use at **** all the time, in its most simplistic form) and now it is part of the dogma that asset returns are linearly related to market returns. What is the logical basis to assume a linear relationship here? He could have just as easily assumed a cubic relationship (which will almost by definition fit past data better) and done the same work. His math would have probably gotten messier, but it is hard to find any merits to the linear assumption other than the fact that it is simple. I am all for simplicity, and perhaps if I asked Sharpe he'd tell me that was just a reasonable first approximation, but that is certainly not the way in which people use it now (maybe they are not so rational after all?).

Well, I think this e-mail is well past the point of "long enough". Maybe this is just me longing for the more intellectually challenging days of school, but as I said earlier, I have been struggling with these questions. I would greatly appreciate a response.