Café Mathematics II: Stefan Banach and Modern Finance

Stefan Banach (1892 – 1945), Polish mathematician, who is generally considered one of the world’s most important and influential 20th-century mathematicians.

A frequent patron of the Scottish Café (see my previous post), Stefan Banach (1892 – 1945) is not a name that is normally associated with finance. He was first and foremost a mathematician, and a very famous one at that. Banach is the best known as the father of Functional Analysis, the branch of math that generalizes real analysis (finite dimensional linear algebra) to infinite dimension vector spaces. Within this broad field, Banach proved many notable theorems including a fixed-point theorem named solely after him and the Hahn-Banach Theorem, after German mathematician Hans Hahn and Banach. It is this latter theorem that provides the link between Banach and modern finance which I will discuss below, skipping most of the technical details.

We start off with a very intuitive idea that underlies all of finance theory which is called the theory of no-arbitrage pricing. This theory says that in a competitive and “frictionless” market, there shouldn’t exist risk-free opportunities to make money by trading securities such as stocks, bonds or options. When this non-arbitrage condition is satisfied, the price of a security is exactly equal to its “fair price”. Roughly speaking, you get what you pay for, no more, no less.

Theoretical economists like to think that they are applied mathematicians, and as such, would not accept even something as “obvious” as the no-arbitrage theorem without a solid proof. It turns out that the Hahn-Banach separation theorem is instrumental in proving the no-arbitrage proposition. To be a bit more precise, the mathematical concept of no-arbitrage means that you cannot start with zero wealth, trade a security, and get a payoff which is on average positive. If you could, this would constitute a riskless profit opportunity. You and other people would buy this “cheap” asset and bid up its price until the easy profit is gone and equilibrium is restored. In the language of functional analysis, equilibrium in financial markets requires that there must a set of tradable outcomes which is disjoint from the set of positive wealth. This abstract idea is best explained graphically as shown in the figure below.

A separating hyperplane in two dimensional space.

The diagram shows two convex polygons, one in blue and the other, magenta. Think of the blue block as representing tradeable wealth and the magenta block as representing the set of strictly positive wealth. The Hahn-Banach separation theorem says there exists a separating line between these two sets. The power of the Hahn-Banach theorem is that It is not restricted to two-dimensional planes as depicted here, but also applies to infinite dimensional spaces, where the separating “lines” are now called separating hyperplanes. Whichever is the case, the theorem allows us to “draw a line” as it were, between the set of tradable wealth and the set of positive wealth. It is this separation, applied to the context of infinite dimension spaces that was used by mathematically skilled economists in the 1950s to rigorously proof the no-arbitrage theorem holds in equilibrium for any traded asset, be they stocks, bonds, options or commodities. It is this that links Banach to the world of modern finance. For more details, see note [1].


[1] As mentioned, the Hahn-Banach separation theorem plays a role in establishing general no-arbitrage pricing results which are classified as versions of the “Fundamental Theorem of Asset Pricing”. Here is an excerpt on this subject from Connor and Korajcyzk (1995):

“Ross (1976) and Kreps (1981) develop an exact non-arbitrage pricing theory. In the absence of exact arbitrage opportunities, there must exist a positive, linear pricing operator over state-contingent payoffs. Chamberlain and Rothschild (1983) show that in an infinite-asset model, the approximate-arbitrage Arbitrage Pricing Theory (APT) is an extension of the Ross-Kreps exact non-arbitrage pricing theory. In the absence of approximate arbitrage, the positive linear operator defined by Ross and Kreps must be continuous with respect to the second moment norm. Given an approximate factor model for asset returns, this continuity condition implies the same bound on APT pricing errors described above. Reisman (1988) extends the Chamberlain-Rothschild result to general normed linear spaces. He shows that the APT can be reduced to an application of the Hahn-Banach theorem using two assumptions: one, the non-existence of approximate arbitrage opportunities for limit portfolios, and two, the approximate factor model assumption on the countably infinite set of asset returns.”

Source: Gregory Connor and Robert Korajzyck, “The Arbitrage Pricing Theory and Multifactor Models of Asset Returns”, in Chapter 4 of Finance, Handbooks in Operations Research and Management Science, Volume 9, edited by R. Jarrow, V. Maksimovic, and W. Ziemba. Amsterdam: North Holland, 1995.

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