Category Archives: Finance

Working an example of von Neumann and Morgenstern utility

von Neumann and Morgenstern’s “Theory of Games and Economic Behavior” is the famous basis for game theory. One of the central accomplishments is the rigorous proof that comparative “preference methods” over fairly complicated “event spaces” are no more expressive than numeric (real number valued) utilities. That is: for a very wide class of event spaces and comparison functions “>” there is a utility function u() such that:

a > b (“>” representing the arbitrary comparison or preference for the event space) if and only if u(a) > u(b) (this time “>” representing the standard order on the reals).

However, an active reading of sections 1 through 3 and even the 2nd edition’s axiomatic appendix shows that the concept of “events” (what preferences and utilities are defined over) is deliberately left undefined. There is math and objects and spaces, but not all of them are explicitly defined in term of known structures (are they points in R^n, sets, multi-sets, sums over sets or what?). The word “event” is used early in the book and not in the index. Axiomatic treatments often rely on intentionally leaving ground-concepts undefined, but we are going to work a concrete example through von Neumann and Morgenstern to try and illustrate a bit more of the required intuition and deep nature of their formal notions of events and utility. I also will illustrate how, at least in discussion, von Neuman and Morgenstern may have held on to a naive “single outcome” intuition of events and a naive “direct dollars” intuition of utility despite erecting a theory carefully designed to support much more structure. This is possible because they never have to calculate in the general event space: they prove access to the preference allows them to construct the utility funciton u() and then work over the real numbers. Sections 1 through 3 are designed to eliminate the need for a theory of preference or utility and allow von Neuman and Morgenstern to work with real numbers (while achieving full generality). They never need to make the translations explicit, because soon after showing the translations are possible they assume they have already been applied. Continue reading

“Easy” Portfolio Allocation

This is an elementary mathematical finance article. This means if you know some math (linear algebra, differential calculus) you can find a quick solution to a simple finance question. The topic was inspired by a recent article in The American Mathematical Monthly (Volume 117, Number 1 January 2010, pp. 3-26): “Find Good Bets in the Lottery, and Why You Shouldn’t Take Them” by Aaron Abrams and Skip Garibaldi which said optimal asset allocation is now an undergraduate exercise. That may well be, but there are a lot of people with very deep mathematical backgrounds that have yet to have seen this. We will fill in the details here. The style is terse, but the content should be about what you would expect from one day of lecture in a mathematical finance course.

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A Discrete Model Gauging Market Efficiency

New paper: A Discrete Model Gauging Market Efficiency PDF

We highly recommend reading the PDF version, but please find below a HTML translation of the paper.

We follow up on some interesting work from the literature and explore some conditions that allow large predatory traders to dominate markets.

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It is not all the quants’ fault.

There is plenty of blame to go around from the current global financial crisis. But, I would like to point out that it is not “all the quants’ fault.” We are all now, unfortunately, sitting in the middle of a high quality (and extremely expensive) lesson in financial mathematics. I would hate for some of the truly important points to be lost to paying too much attention to some of the shiny details.

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A Quick Appreciation of the Sharpe Ratio

The current state of the global financial markets has gotten more people than usual worrying about the technical aspects of finance. One method for reasoning about investment returns and risk is a tool called the Sharpe Ratio. It is well worth reviewing this measure and seeing how, if used properly, it doesn’t favor any of the mistakes that underly our current financial crisis. Continue reading

Betting Best-Of Series

Betting Best of Series is a new expository paper describing the mathematics involved in betting on something like the United States’ Major League Baseball World Series. It isn’t so much about baseball as about demonstrating some of the really great ideas from mathematical finance in a simplified setting. This sort analysis is the “secret sauce” in a lot of financial models and I trying to share the thrilling feeling of working with these techniques in an elementary essay (with diagrams). Continue reading

Is Search Advertising a Market for Lemons?

author: John Mount, 5-13-2008

Anand Rajaraman recently wrote a very thought-provoking entry on his Datawocky blog. He asks “Is Search Advertising a Giffen Good?” As he explains a Giffen Good is a sort of economic doomsday machine that some segment of consumers are forced to buy more of an inferior good as the price of the inferior good goes up. His article is well written are really invites one to think about the issue. Anand’s question made me thing about a number of issues (which I will outline here) and I will leave off with a question of my own.

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