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Why No-Arbitrage Pricing Does Not Guarantee the Law of One Price

Article Quant Q&A · Author: Charlie
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Summary

The note distinguishes absence of arbitrage from the law of one price. Under the stated definition, absence of arbitrage requires a pricing rule to assign a higher price to a payoff that weakly dominates another payoff and strictly exceeds it with positive probability. The law of one price, by contrast, requires prices to respect linear combinations of payoffs.

A simple example shows why the first condition alone does not establish the second: on real-valued payoffs, a strictly increasing nonlinear pricing function such as the cube function preserves the ordering required by the no-arbitrage condition but is not linear. This illustrates the logical gap between monotonicity and linearity. The example is mathematical rather than a realistic market model, and it does not address additional market assumptions under which stronger pricing results might follow.

Key ideas

  • Absence of arbitrage is expressed as strict price monotonicity with respect to payoff dominance.
  • The law of one price requires linearity across payoff combinations.
  • A strictly increasing nonlinear function can satisfy monotonicity without satisfying linearity.
  • The example establishes non-implication under the stated general setup, not under every market model.

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


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












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 by Ulysses (score 2, accepted)

https://quant.stackexchange.com/a/15716

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.

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.