Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

BR: The Binomial Representation Theorem – II

Having laid down the building blocks, now we are ready to define the Binomial Representation Theorem (BRP).

The Binomial Representation Theorem

Given a binomial price process $S$ which is a $Q$ martingale, if there exist another process $N$ which is also a $Q$ martingale, then there exists a previsible process $\phi$ such that:

$N_i = N_0 + \displaystyle\sum\limits_{k = 1}^n \phi_k \Delta S_k$

The basic idea is that if there are two martingale processes under the same measure, then we can find the value of one given the other in a ‘previsible’ (read deterministic) way. As long as we know the possible states of the world in both measures,  we just need to “match the widths” (how much the two processes vary – variance if you will) and “match the offsets” (how the mean of the two processes differ). Stated simply, it’s just like change of cordinates in geometry. In geometry we require that the change of co-ordinates be in the same X-Y plane, and here we require that both processes be martingale under the same measure $Q$.

Next step is to see what to exploit this useful change of co-ordinate trick in finance.

The first thing to note is that we have the claim $X$, which is a random variable and not a martingale. However, we know from the Tower Property of expectations discussed earlier that the conditional process of a claim is always a martingale under any arbitrary measure. This immediately tells us that we can, in principle, given the BRP, we should be able to find a previsible process $\phi_k$ that allows us to go from $S$ to $E[X|F_i]$

But before we do it systematically, we need to set up a little bit more machinery:

9. Bond process: Don’t think it needs any explanation. Process for bond price is a previsible process. We call $B_i$ (with $B_0 = 1$) the bond process

10. Discount process: Since $B_i$ is previsible, so is its inverse. We call $B^{-1}_i$ as the discount process

11. Discounted stock process: Given the discount process $B^{-1}_i$, $Z_i = B^{-1}_i S_i$ is a process that can be observed on the same binomial tree as $S$. We call $Z_i$ the discounted stock process

12. Discounted Claim: The process $B^{-1}_T X$ is called a discounted claim

Given that the process $Y_i = E^Q[B^{-1}_T X | F_i]$ is a martingale (from the Tower property implication), and so is $Z$, the BRP implies there exist a $\phi$:

$Y_i = Y_0 + \displaystyle\sum\limits_{k = 1}^n \phi_k \Delta Z_k$

What we now need to do is to show that  the previsible process $\phi$ is something that has a physical counterpart in the the market BRP allows us price the claim $X$ – which was our objective. That we leave for the next post.