How the Fundamental Theorems Link Arbitrage, Completeness, and Pricing
Summary
The document gives an intuitive account of the two fundamental theorems of asset pricing through state-contingent payoffs. A payoff that pays in one particular future state can be used as a building block for other payoffs. Under a risk-neutral measure, its price corresponds to the probability assigned to that state. The first theorem connects the absence of arbitrage with the existence of an equivalent martingale measure, which supports consistent pricing.
The second theorem concerns completeness: if available assets can replicate payoffs in every possible state, the equivalent martingale measure is unique. A simple discrete example contrasts a market with two possible stock moves, where one stock can yield a unique measure, with three possible moves, where one stock may leave multiple measures. Adding another asset may restore completeness in that example. The explanation is deliberately simplified; the authors note that continuous-time models and unrestricted price moves require more technical treatment.
Key ideas
- The first fundamental theorem connects no arbitrage with the existence of an equivalent martingale measure.
- The second theorem connects market completeness with uniqueness of that measure.
- State-contingent payoffs provide a simple way to understand the measure's pricing role.
- A market with more possible outcomes than replicating assets may admit multiple risk-neutral measures.
- The discrete examples illustrate the ideas but do not cover the full technical detail of continuous-time markets.
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Full text
# The two fundamental theorems of Finance, as they relate to the martingale measure
# 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.
## Answer by Freelunch (score 3)
https://quant.stackexchange.com/a/37787
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
\begin{equation} q_\omega = E^\mathbb{Q}[\mathbb{I}(\omega)] = \mathbb{Q}[\omega] \end{equation}
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).
## Answer by Bjørn Kjos-Hanssen (score 1)
https://quant.stackexchange.com/a/37686
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.