# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: The Binomial Model

BR’s second chapter is devoted to introducing the main probabilistic concepts in mathematical finance using the binomial model (they call it the “branch model”).

They start off with the standard one-period “branch model”, and then take forward the discussion over a binomial tree. This is mostly standard stuff. The examples and the general description follows on familiar lines.

They conclude their discussion about the “branch model” by pointing out that arbitrage ensures that there is only one “fair” price of a derivative. Also, that one can think of the price as a discounted value of “local expectation” with probabilities measured differently.

There is very little one can do new when discussing binomial model – so not much to comment here. Still a few things stand out in their approach:

1. First thing that strikes as a little odd is that they go through the entire sub-section on the binomial model without ever talking about hedging or even defining arbitrage. One could argue that the warning in chapter 1 is sufficient enough, but they obviously expect more than a bit from their readers.
2. They again get themselves stuck in the “muddy waters” they created for themselves by repeatedly having to warn the readers that “expectation pricing” doesn’t work. And then they have to clarify that while “expectation pricing” doesn’t work, one could still think of pricing a derivative in the binomial world as an expectation but with different probabilities – totally unnecessary, but inevitable given the way they motivated no-arbitrage pricing.
3. At this stage, I think, they also miss out on utilizing the opportunity of introducing the idea of a risk-neutral world/measure. In fact, they have no discussion on risk at all till this point in the book (and am not sure they will get such an opportunity later)
4. It’s a good sign, however, that they clearly mention that the “construction portfolios $(\phi_i, \psi_i)$  are also random” – a fact which is often not made explicit in introductory mathematical finance texts.

As an aside – their choice of words is a little odd (non-standard use of “expectation pricing”, “construction portfolio” instead of hedging portfolio etc). Not sure a guy who studied finance at a business school would enjoy it too much. But given that the authors are ex-mathematicians and heading a quant-desk may have something to do with it. In fact, their terminology may even be preferred by quants – who themselves have a physics/math background – working in banks.

Their next section is devoted to “binomial representation theorem”. From the name it seems it’s going to be a discussion of the martingale representation theorem in the binomial setting. Not sure I have come across this elsewhere before, and I am quite looking forward to reading it.