Financial Mathematics Text

Thursday, January 31, 2013

Grassmann's Real versus Formal distinction

So I'm currently reading A History of Vector Analysis written by Michael J. Crowe (or to say it another way, I'm not normal). I came across the book after reading this article: Hermann Grassmann and the Creation of Linear Algebra which I read because I have an interest in the development of linear algebra (aka I'm not normal).

Crowe quotes at some length from a book written by Grassmann entitled: Die lineale Ausdehnungslehre ein neuer Zweig der Mathematik: dargestellt und durchAnwendungen auf die übrigen Zweige der Mathematik, wie auch auf die Statik,Mechanik, die Lehre vom Magnetismus und die Krystallonomie erläutert (which in German means tl;dr). But he makes the following distinction:
The primary division in all the sciences is into the real and the formal. The former represent in thought the existent as existing independently of thought, and their truth consists in their correspondence with the existent. The formal sciences on the other hand have as their object what has been produced by thought alone, and their truth consists in the correspondence between the thought processes themselves.
From what I'm gathering from what little bits Crowe has quoted, Grassmann thinks that his system (which from what I gather was an early form of modern day linear algebra) as being merely "formal". It's existence has been posited through pure thought but it doesn't correspond to anything in the world (outside of the correspondence to human thought). Contrast that with geometry which corresponds with space and is therefore an example of a "real" science. He even went so far as to suggest that geometry properly ought not be included within mathematics:
It had for a long time been evident to me that geometry can in no way be viewed, like arithmetic or combination theory, as a branch of mathematics; instead geometry relates to something already given in nature, namely, space.
What he calls "pure mathematics", by contrast is:
... the science of the particular existent which has come to be through thought. The particular existent, viewed in this sense, we name a thought-form or simply a form. Thus pure mathematics is the theory of forms.
I'm not sure how far these distinctions would go today. Consider space for example. In a post-Einstein world, space has different structure when in the presence of objects of mass. Mass actually bends or warps space. And it's not entirely restricted to 3 dimensions as it was in Grassman's day.

On the more "abstract" side of things, the lines are even more blurred. We now deal with infinitely dimensional complex vector spaces to explain such phenomenon as matter waves.

At this point there should be a cue for a joke about Hilbert spaces but I'm too lazy to write one. So I'll just get a room in his hotel and crash for the night.

Sunday, January 13, 2013

What is a Financial Bubble? Part I

The idea that drives this metaphor is blowing a bubble. One can blow it larger and larger but eventually it pops. Some economists have denied the existence of bubbles while others have expressed skepticism that the concept is well-defined enough to be scientifically meaningful. Eugene Fama, a prominent supporter of the Efficient Markets Hypothesis, stated as much in an interview in The New Yorker.

I share these reservations over defining a bubble but I still suspect it's a useful concept. As a result, I'd like to explore the issue a bit. I'll be doing so in a series looking at various examples and aspects of financial bubbles in order to get a better grip on whether or not we can give it a meaningful definition.

Sunday, January 6, 2013

Gettier Intuition

One common criterion used to determine whether or not something is knowledge is the "justified true belief" criterion (JTB). It has many forms but the basic idea is this:

S knows that P means:
  1. S believes that P.
  2. S has justification for believing P.
  3. S is true.
In 1963 Edmund Gettier proposed counterexamples to the above relationship. The idea is pretty simple. Construct a scenario in which (1), (2) and (3) are satisfied, but it fails to be knowledge. We'll look at an example from his original paper: Is Justified True Belief Knowledge? In particular, we'll look at Case I.

Wednesday, January 2, 2013

Downside Risk Investing

I'd like to give some consideration to a class of investment strategies which I call Downside Risk Investing. These strategies earn returns by taking on downside risk while at the same time having limited upside potential. There are a number of strategies that fit this bill which I'll outline below giving several examples.

There's a metaphor that goes around talking about "picking up pennies in front of a steamroller". It's in reference to investment strategies in which the investor risks getting run over by a steamroller (taking the risk of getting wiped out) for the benefit of receiving a couple of pennies. This entire reference is to Downside Risk Investing.

Now in some sense, many investment strategies have the potential for huge losses with limited upside. The question we should be asking is this: how many pennies do I need to be able to pick up for a particular Downside Risk Investing to be a good strategy? I don't know if I have an answer to that question but I'd like to present a discussion. So without further ado, here are some examples of strategies that I think fit the bill (some of which may be good strategies to employ).