Archive for the ‘Clarity’ Category

What do normal subgroups look like?

September 26, 2012

I am in the process of writing a longer post on Galois Theory (see here), and one of the central concepts is that of a normal subgroup.

We all know the definition (and their equivalents) from classes/books, but anyone who likes to ‘see their mathematics’ is left with the question:

… but what do they look like?

In this post I give a few different interpretations of the standard definitions, and go some way to explaining why it is difficult to answer this question in concrete terms.

Spoiling the punchline a bit, it seems to me that problem is actually the other way round: most of the groups that you have an intuition about are normal, so the question is really when is a group not normal? (more…)

Understanding the credit crisis

June 17, 2012

In an earlier post (see here) I put a link to an online book that explains all the main concepts of economics in terms of the classic ‘Island Economy’.

If you think about it in theoretical terms, this book is a ‘proof’ that the Island Economy is a sufficiently rich model into which many important elements of economic life can be incorporated.

And like all good models, it is light and understandable.

Very importantly, this means that everyone can get involved in a healthy debate about its assumptions & simplifications, and any policy implications coming from it.

This is something that undermines the quality of our mathematical models of financial or economic processes: they require a very large amount of technical ability before you can even start to understand the terms of the model (for derivatives pricing this means stochastic calculus & arbitrage-free pricing theory; for systematic trading this means advanced statistical analysis; for economics this means advanced calculus and plenty of background reading of fragmented economic theories).

Experience tells me that the more technically ‘perfect’ we make our models, the less able the users become of giving an informed critique of the theory’s assumptions and main features.

The lesson I draw from this is that it is the responsibility of the model builders to work even harder to make our theories and models accessible to an averagely intelligent user.

What does this hard work look like?

Well here are four examples:

  1. Irwin Schiff’s book on economics (see my post here).
  2. Ray Dalio’s paper on why economies rise and fall (see my post here).
  3. A series of articles by Isabella Kaminski from FTAlphaville that present the credit crisis in terms of a felt cartoon story about water gathering and storing (see here).
  4. My work on understanding the Black-Scholes formula for out-of-the-money options (see my post here).

The value of Good Explanations

January 12, 2012

I am a firm believer in the value of good explanations, and I’d say a lot of my time is spent on finding the right explanation or intuition for a process or event (this blog itself is an example).

Rather like a good user interface, a good explanation can be the difference between a success and failure in a project. Here are some qualities of good explanations:

  • they are intellectually satisfying when you find them,
  • they allow you to reach out to a larger audience than just your fellow specialists, and therefore are a better way to have an external audit of the work you are doing,
  • they can show the direction your research should go,
  • a good explanation of a trade/model/etc inspires confidence in your traders & clients that you know what you are doing and are taking care with their money.

The key to all this is that:

finding a good explanation is about asking the right questions

and that itself is the basis of good research.

The power of notation in problem solving

December 13, 2011

It’s trivial when you think about it:

good mathematical notation is one way of making a problem easier to solve.

In my introduction to advanced probability theory I put emphasis on how probability theory has developed a clever and natural way to describe the processes we deal with. If you think about it some more, the design of the notation is an important part of slicing the subject into manageable blocks.