## BR: The Binomial Representation Theorem – III

BRP tells us that a previsible exists. Now the trick to use BRP to get a price for our claim is to do the following:

Let’s start at time . Construct a portfolio :

The value of this portfolio at time , then, becomes:

Substituting as above and rearranging gives:

Then, *here we can exploit BRP* to write , and this gives us:

Next, our strategy requires that we construct a porfolio :

That is, it *costs exactly the same* as the value of the portfolio we ended up with at time when we created at time .

This can be carried on recursively to show that our original **portfolio ** **is self-financing. **At ‘maturity’ , we end up with , i.e. our original claim. That is, the price of claim should be *exactly equal to the cost of the portfolio created originally *(no money has come in or left the system) .

The **moral of the story **is that within a binomial two-asset setting there exists a self-financing strategy that duplicates the claim that we want to price. That is, the claim can be priced no matter what happens to the path of / filtration (ironically, then, the claim in the binomial world is redundant – as it can be reconstructed from the already existing and ).

I think it’s a good way to introduce the martingale representation theorem – but I personally would have preferred that it be done using the language of delta-hedging / risk-neutralily / no-arbitrage. In a way, I think it hampers BR in formally introducing the first fundamental theorem of asset pricing (they do point towards it “…as an afterthought”). Let’s conclude this post with the formal statement then:

**First Fundamental Theorem of Asset Pricing**

No-arbitrage implies that there exists a probability measure under which the discounted price process is a martingale, and vice-versa. The measure , then, is called an equivalent martingale measure.

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