# Need for Binomial Representation Theorem

In some texts (e.g. Baxter & Rennie, Shreve I) the binomial model is first constructed using the usual backward induction argument, and it is concluded that by no-arbitrage the time $t$ value of a claim with time $T$ payoff $X$ is $\mathbb{E}_\mathbb{Q}[\frac{B_t}{B_T} X|\mathcal{F}_t]$, where $B_t$ is the price of a cash bond at time $t$. In other words, because we determined the price of the claim at each step, the price of the portfolio that replicates that claim must be the claim's price at each step, else arbitrage. There is no mention of self-financing strategies (SFSs) or binomial representation theorem (BRT); rather, we explicitly construct a hedging strategy that replicates the claim's payoff.

Only after we have determined this price does it seems like the concept of SFSs are introduced, with the BRT invoked to prove the existence of them. Then we use a slightly different argument to arrive at the same price: the value of a SFS that replicates $X$ is $\mathbb{E}_\mathbb{Q}[\frac{B_t}{ B_T} X|\mathcal{F}_t]$ by the BRT, and because this is a SFS that replicates $X$ this must be the price of the claim, else arbitrage.

So we have two distinct approaches to arrive at the same conclusion. My question is, what purpose does the BRT serve in the binomial model? Does it just serve as an intuition builder for the martingale representation theorem (MRT) in continuous time models, where explicit construction of the hedging strategy isn't as clear?

If that's the case, it seems the BRT is specific to the binomial model, while the MRT is model-free. Is this correct?

• Please write the question title without abreviations. Furthermore. I wonder if the question is clear to someone who does not know the book. I would like to help you if you could reformulate the question a bit. But maybe someone else can anyways. – Richard May 22 '15 at 13:32
• @Richard Okay, I tried to better phrase my question. – bcf May 22 '15 at 14:41
• I understood your question (not having read the book) and I think it's a good one. – Stanley May 23 '15 at 15:49
• The martingale representation theorem has some conditions (as you implicitly note with the word continuous) but it is indeed a more general result and in mathematical finance the binomial calculations are indeed mainly motivations to intuition. – Brian B May 24 '15 at 1:29