Proving that Absence of Arbitrage does not imply law of one price

Question

I am trying to prove that the Absence of arbitrage statement (AOA) does not necessarily imply the law of one price (LOP). For the definitions of these concepts I am using Cochrane's book "Asset pricing".

By definition a payoff space $X$ and a pricing function $p(x)$ leave no arbitrage opportunities if for any $x\geq0$ almost surely, and $x > 0$ with nonzero probability, $p(x) > 0$.

Equivalently: if $x$ dominates $y$ – i.e., $x\geq y$ almost surely, with $x > y$ with positive probability – then $p(x) > p(y)$.

The law of one price says that we can write $$p(ax_1+bx_2)=ap(x_1)+bp(x_2)$$

Now, I don't know how to attack this problem. Should I try to prove that positivity of prices (AOA) does not necessarily imply a linear pricing function??? Can you help me to understand what would be a good attack strategy in this case?

Thank you for your help.

Answer 1

Let $X$ be endowed with the following partial order: $y \geq x $ means that $\Bbb P(y\geq x) = 1$. The AOA condition in your case states that the pricing law $p$ is strictly inctreasing with respect to $\geq$, whereas LOP says that $p$ is a linear function. Neither if the two implies another one in general. For example, if $X = \Bbb R$ then $p(x) = x^3$ is a strictly monotone increasing function which is not linear.