Financial Mathematics

Although the IceQuants site is supposed to be about Mathematical Finance (Math-Fin), there are those among us who from time to time will indulge in Financial Mathematics (Fin-Math). For those of you who aren’t sure, the two words are not commutative! On the one hand we have such books as Hull’s (Math-Fin) and on the other hand that of Brigo (Fin-Math).

To make it completely clear, I’m sure there are many different opinions on the use of these words, but here is how I’m going to use them in this site. The major results of Fin-Math, are the so called First and Second Fundamental Theorems, discussed quite well, without any horrible depth, in the new version of Björk. The theorems clarify the conditions for a market to be:

  1. Arbitrage-free
  2. Complete

And this is done within a very axiomatic (and beautiful) framework. The result is essentially that a market is arbitrage-free if it admits a risk-neutral measure and it is also complete, is such a risk-neutral measure is unique. Notice that the risk-neutral measure is such that, with respect to it, the ratio of the traded securities to the numeraire, are martingales. To say it simpler, the risky assets are all fair game. Quite nice, and in addition, Fin-Math is very much occupied with studying so-called stochastic differential equations (SDE’s), equations of the form:

\displaystyle \textrm{d} X_t = \mu_t \, \textrm{d} t + \sigma_t \, \textrm{d} W_t

where \mu_t and \sigma_t are stochastic processes and hence, so is the solution X_t. The term \textrm{d} W_t is Gaussian noise.

So, an economist (and a physicist!) must wonder: “Thats like an empty assumption… can anything at all be done with nothing but random processes?” The answer is yes, and I was amazed myself at how much can actually be achieved by this general framework. The no-arbitrage conditions turns out to be quite strong. I’ve written some notes that I took (and constantly update) as I’m teaching myself this very interesting subject, the notes are here.

Well, if this sounds abstract, then you don’t want to go any further into Fin-Math. Math-Fin, on the other hand, is a much more pragmatic subject. Math-Fin focuses on modeling useful things, things which actually matter at work.