Example of Law of One Price holds but No Arbitrage Fails

Question

I have been told that no arbitrage is a stronger assumption than than the law of one price.

In particular, Law of One Price is equivalent to the existence of a stochastic discount factor, whereas no arbitrage is equivalent to the existence of a strictly positive stochastic discount factor.

Can someone give me an example where LOOP holds, but No Arbitrage fails?

Thanks.

Answer 1

Examples where this happens are always extreme and contrived. I can think of two kinds of examples. The first is where you have an asset that for some reason has a price of zero or negative but a positive payoff in some states (perhaps no other asset pays off in that state). The second is where you have two assets with positive prices that have negative payoffs in all states (or vice versa).

You can come up with other varieties, but they're always a little weird. To show you how to do this, I'll be a little more precise. To answer this question, we need to be a little more clear about what arbitrage means. A lot of books give definitions for two kinds of arbitrage (e.g. see the asset pricing book by Darrell Duffie). Let me set up some notation first, though.

Notation

Let $p_i$ be the price of security $i$ with $p = [p1,...,p_N]'$. Let $D_{ij}$ be the payoff of security $i$ in state $j$ with $D$ being an $N \times S$ matrix, where $N$ is number of assets and $S$ is the number of states. Let $\theta_i$ be the number of securities of type $i$ that are owned, with $\theta = [\theta_1, ..., \theta_N]'$.

Some definitions

• "The law of of one price": For any portfolio $\theta$ that satisfies $D' \theta = \vec 0$, it must be that $q' \theta = 0$.

• "Arbitrage of the First Type": There exists a portfolio $\theta$ such that $p' \theta \leq 0$ and $D' \theta > \vec 0$. This means that all elements of the vector $D' \theta$ are strictly positive.

• "Arbitrage of the Second Type": There exists a portfolio $\theta$ such that $p'\theta < 0$ and $D'\theta \geq \vec 0$. This means that all elements of the vector $D' \theta$ are nonnegative.

Examples

• In this first example, consider the extreme case of the existence of only one asset. Suppose that $p = 0$ and $D=1$ (one state, but we can have more). Then the law of one price holds and arbitrage of the first type exists. This is pretty contrived, though.
• Consider the example of $p = [1 \quad 2]'$ and $$ D = \begin{bmatrix} -2 & -1 \\ -4 & - 2 \end{bmatrix}. $$ This satisfies the law of one price. However, when $\theta = [-1 \quad 0]'$ or $\theta = [0 \quad -1]'$, we have $p'\theta < 0$ and $D' \theta \geq 0$ and $\theta \neq 0$.