5
$\begingroup$

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.

$\endgroup$
2
$\begingroup$

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.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.