The Law of One Price and Zero-Payoff Trading Strategies
Summary
The document relates the law of one price to the absence of a self-financing strategy that has a zero terminal payoff but a negative initial value. If two strategies replicate the same claim at different prices, subtracting one from the other produces a zero-payoff strategy; choosing the order of subtraction can make its initial value negative. Conversely, a negative-cost zero-payoff strategy would replicate the same terminal claim as doing nothing, contradicting the law of one price.
The argument is framed for self-financing strategies whose values and payoffs can be compared and subtracted. Its key step in the converse direction is that the difference strategy must remain admissible and that its negative can also be considered; without this, ruling out negative initial value alone would not establish that the initial value is exactly zero. The answer gives an abstract equivalence, not a model-specific market example or a treatment of frictions, constraints, or transaction costs.
Key ideas
- Two strategies with identical terminal payoffs should have the same initial value under the law of one price.
- Subtracting strategies with equal terminal payoffs creates a self-financing strategy with zero terminal value, assuming subtraction preserves admissibility.
- A negative-cost zero-payoff strategy conflicts with the law of one price because doing nothing has the same payoff at zero cost.
- To infer equal initial prices from the absence of negative-cost strategies, the framework must also allow reversing the difference strategy.
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Full text
# Law of One price and the Inconcistent pricing strategy # Law of One price and the Inconcistent pricing strategy Background Information: A market satisfies the Law of One Price if every two self-financing strategies that replicate the same claim have the same initial value. An inconsistent pricing strategy is a self-financing strategy $\phi$ with $V_T(\phi)\equiv 0$ and $V_0(\phi) < 0$. Question: > Prove the Law of One Price holds if and only if there does not exist an inconsistent pricing strategy. Attempted proof - Suppose we have two self financing strategies $\phi$ and $\psi$ that replicates some claim $X$ such that $V_0(\phi) = V_0(\psi)$. Hence we cannot satisfy the condition of $V_0(\phi) < 0$ nor $V_0(\psi) < 0$ so there is no inconsistent pricing strategy. I am not sure how to show the converse and whether this is rigorous enough. Any suggestions are greatly appreciated. ## Answer by Gordon (score 2, accepted) https://quant.stackexchange.com/a/30856 Assume the law of one price. We show that there does not exist an inconsistent pricing strategy. Suppose that $\phi$ is an inconsistent self-financing trading strategy, that is, $V_T(\phi)\equiv 0$ and $V_0(\phi) < 0$. Consider another self-financing trading strategy $\psi$ that does not nothing, that is, without holding any of the underlying assets. Then $V_T(\psi)\equiv 0$ and $V_0(\psi) = 0$. This contradicts the law of one price, since both $\phi$ and $\psi$ replicate the same claim, but the initial prices are different. On the other hand, assuming that there does not exist an inconsistent pricing strategy, we show that the law of one price holds. Consider any two self-financing trading strategies $\phi_1$ and $\phi_2$ such that $V_T(\phi_1) = V_T(\phi_2)$. Note that $\phi=\phi_1-\phi_2$ is also a self-financing trading strategy, and $V_T(\phi) = V_T(\phi_1) - V_T(\phi_2)\equiv 0$. Since there does not exist an inconsistent pricing strategy, $V_0(\phi) \equiv 0$. That is, $V_0(\phi_1) =V_0(\phi_2)$. Therefore, the law of one price holds.
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