# Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

## BR: The Binomial Representation Theorem – I

The second half of the second chapter of BR’s book uses the binomial tree model discussed so far to introduce some of the basic probabilistic concepts in the theory of mathematical finance (in particular, the ones they need to build the theory in continuous time)

1. Process: The set of of possible values the underlying can take. The random variable $S_i$, then, denotes the value of the process at time $i$.

2. Measure: How probabilities evolve over the tree

3. Filtration: The history of the asset price $S_i$ up until the time $\displaystyle i$ on the tree: $F_i$. It corresponds to the history of $S_i$ on a particular node as of time $i$. The binomial structure ensures that there is only history corresponding to any node. Given a node and a point in time filtration fixes the history “so far”. It is a useful construct to talk about where we are at a point in time and how we reached there.

4. Claim: The derivative in question being priced, which gives a payoff at “maturity”: $X$ – it’s value is a function of the filtration $F_T$.

5. Conditional Expectation of a Claim: Given a claim $X$, we can talk about it’s expectation given the history “so far”, i.e. a filtration $F_i$ – or in other words, the expectation $E^Q[X|F_i]$. Note that this makes this conditional expectation also a random variable, as it depends on the filtration – i.e. where we are on the tree.

For each node/point in time, given the probability measure $Q$, $E^Q[X|F_i]$ denotes the expectation of the claim $X$ if we have observed a filtration $F_i$. It is clear that the unconditional expectation is the same as $\displaystyle E^Q[X|F_0]$

6. Previsible process: It’s a process $\phi_i$ whose value at any time is dependent only the history upto one time earlier, i.e. $F_{i - 1}$ (e.g. $\phi_i = S_{i - 1}$)

7. Martingale: A process $S$ is a martingale with respect to a measure $Q$ and a filtration $F$ if $\forall j\leq i$:

$E^Q[S_i | F_j] = S_j$

That is, the process $S$, if it is a martingale does not move up or down systematically – i.e. it has no drift.

8. Tower Property: $\forall k\leq j\leq i$

$E^Q[E^Q[S_i | F_j]|F_k] = E^Q[S_i|F_k]$

It’s a useful mathematical property of expectations not only in mathematical finance but also in time series analysis. Another way to interpret the Tower property is to say that the conditional expectation process of a claim is always a martingale (obviously, this is just a consequence of the mathematical property of conditional expectations).

Please note that previsibility is closely related to the notion of predictability. It essentially says that a process whose value is “known” at any time given the “history so far” is previsible. Technically, a process $X_t$is previsible, if given a filtration $F_t$ over a probability space $(\Omega, P, F)$, $X_t$ is adapted to $F_t$ or, alternatively, is $F_t$ – measurable. On the other hand, the martingale property is a probabilistic concept relating best forecast to conditional expectation of the random variable (in finance with a clear physical interpretation of “no drift”).

Written by Vineet

August 27, 2010 at 4:26 pm

## 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.

Written by Vineet

August 20, 2010 at 5:16 am