# The two fundamental theorems of Finance, as they relate to the martingale measure

I RECENTLY read this in an article by Battig and Jarrow, "the first fundamental theorem relates the notion of no arbitrage to the existence of an equivalent martingale measure, while the second fundamental theorem relates the notion of market completeness to the uniqueness of the equivalent martingale measure" can someone explain the difference between the two fundamental theorems? As easy and palatable as possible, preferably.

Let $\Omega$ be the outcome space at some future date and fix a specific outcome $\omega \in \Omega$. Now consider a portfolio that gives one unit of currency if $\omega$ happens and zero otherwise, i.e., with payoff $\mathbb{I}(\omega)$. Any other payoff function can be given as a linear combination of these portfolios. The price of this portfolio today is

$$q_\omega = E^\mathbb{Q}[\mathbb{I}(\omega)] = \mathbb{Q}[\omega]$$

The price is simply the probability of $\omega$ happening under an equivalent martingale measure $\mathbb{Q}[\omega]$. The no-arbitrage condition tells us that the price $q_\omega$ should be unique as long as $\mathbb{I}(\omega)$ can be replicated (FTAP1). If $\mathbb{I}(\omega)$ can be replicated for all possible $\omega \in \Omega$ (completeness), then all $\mathbb{Q}[\omega]$ should also be unique (FTAP2).

As I discussed in another answer, in the case of a stock with two possible moves, + and -, we have market completeness: there is a unique risk-neutral measure obtained from the fact that there is a unique straight line through two given points.

For a market with three possible moves (say +1, 0, -1) and just one stock, it turns out that there is more than one risk-neutral measure; i.e., the market is not complete. This corresponds to the fact that given three points in the plane $P_i(x_i,y_i)$, $1\le i\le 3$ with $x_1<x_2<x_3$, there is more than one straight line that lies between the two lines

• $\ell_{12}$ through $P_1$ and $P_2$, and
• $\ell_{13}$ through $P_1$ and $P_3$.

Namely, there are many lines that go through $P_1$ but have an intermediate slope between the slopes of $\ell_{12}$ and $\ell_{13}$.

If you add another stock to the market, it becomes complete again as far as I recall.

This all gets much more technical when you have continuous time and arbitrary real numbers as moves for the stock price. But it is still just a matter of whether the risk-neutral measure exists and is unique.