Need for Binomial Representation Theorem

Question

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?

Answer 1

I think I've resolved this for myself, let me know your thoughts.

The backward induction argument to arrive at a price is a very explicit construction of a replicating strategy. I think of this as a "first go" at option pricing using a particular model. Each step is very explicit, and there is less room for confusion about why this must be the correct price. Again, no mention of self-financing strategies (SFSs) and hence no mention of the binomial representation theorem (BRT).

Then, only after we have determined the no-arbitrage price do we can a second pass at what we have done. It's like we're saying, "I know we already explicitly constructed this strategy, and we already have determined the no-arbitrage price, but let's define these things called martingales and we'll see that we can rephrase what we've done in a different language: you find me a binomial process that's a martingale and I'll be able to represent any other martingale in terms of it by BRT. Then after defining a SFS you'll see SFSs exist via BRT, and that the price must again be given as the discounted expectation." We can then introduce the Fundamental Theorem of Asset Pricing.