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.