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”).