# How is the fundamental theorem of asset pricing used?

I know that a multi-period market model is complete and arbitrage free if there's a unique equivalent martingale measure. The thing is, I have absolutely no clue how to apply this theorem to a simple binomial tree. I just don't get what the two things even have to do with one another.

For example, consider:

Yes, I know that $u = 1.1$. I know that $d = 0.9$. But what does this have anything to do with the complicated theorem which talks about conditional expectations and equivalent martingale measures? I guess $q = (R - d)/(u-d)$ and $1 - q$ is this equivalent martingale measure but why? And why is it unique?

well solve for the value of $q$ that makes the value of the stock divided by the bond be a martingale. You will find that only one value does so. It is the one you posted.
We know the market model is arbitrage free if and only if there exists a martingale measure $Q$, also the Binomial Model is free of arbitrage if and only if $d\le 1+R\le u\,\,$ (Arbitrage Theory in Continuous Time).It is easy to calculate the martingale probabilities.This Condition is equivalent to saying that $1 + R$ is a convex combination of $u$ and $d$ , i.e. $$1+R=u\,{{q}_{u}}+d\,{{q}_{d}}$$ On the other hand $\,{{q}_{d}}+{{q}_{u}}=1$ then \left\{ \begin{align} & u\,{{q}_{u}}+d\,{{q}_{d}}=1+R \\ & {{q}_{u}}+\,{{q}_{d}}=1 \\ \end{align} \right. \Rightarrow \left\{ \begin{align} & {{q}_{u}}=\frac{\left| \begin{matrix} 1+R & d \\ 1 & 1 \\ \end{matrix} \right|}{\left| \begin{matrix} u & d \\ 1 & 1 \\ \end{matrix} \right|}=\frac{1+R-d}{u-d} \\ & {{q}_{d}}=\frac{\left| \begin{matrix} u & 1+R \\ 1 & 1 \\ \end{matrix} \right|}{\left| \begin{matrix} u & d \\ 1 & 1 \\ \end{matrix} \right|}=\frac{u-(1+R)}{u-d} \\ \end{align} \right.