Why the Law of One Price Does Not Guarantee No Arbitrage
Summary
The document gives a one-period, two-asset example in which the law of one price holds while arbitrage is still possible. The assets’ initial and terminal values are arranged so that every portfolio’s terminal value is a fixed multiple of its initial value. Consequently, portfolios with the same terminal payoff also have the same initial value, satisfying the law of one price.
At the same time, the relationship between those values permits a trader to borrow to buy the risky asset and end up with a positive gain. The example illustrates why equal payoffs having equal prices is weaker than requiring the absence of arbitrage. It is a stylized mathematical illustration rather than a practical trading strategy; the closing comments note that distinctions among related no-arbitrage conditions can matter in formal theory, even when they may be treated similarly in many applications.
Key ideas
- The law of one price requires portfolios with identical future payoffs to have identical initial values.
- That condition can hold even when a portfolio offers a free gain through borrowing and asset purchases.
- The example shows that no-arbitrage implies the law of one price, but the converse need not hold.
- Related no-arbitrage and no-free-lunch conditions have distinct formal definitions in mathematical finance.
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Full text
# The relationship between no-arbitrage and the law of one price
# The relationship between no-arbitrage and the law of one price
If no-arbitrage exists, then the law of one price holds, but the existence of the law of one price does not always imply that no-arbitrage exists." To prove this, what is an example where the law of one price holds, but no-arbitrage does not? Additionally, how is this concept applied in financial engineering? And what resources would be beneficial for a mathematical proof of this concept? Thank you.
## Answer by Kevin (score 1)
https://quant.stackexchange.com/a/77530
A simple textbook example is the following. Consider a discrete market with two assets and one time step: $B_0=-0.1$, $B_1(\omega)=1 \;\forall\omega\in\Omega$ and $S_0=-0.2$ and $S_1(\omega)=2\;\forall\omega\in\Omega$. Here, $\Omega$ is the set of all outcomes. Admittedly, a simple example with $S_t=2B_t$...
Any portfolio in this market can be characterised by $\varphi=(\varphi^B,\varphi^S)$. The value process of that portfolio is \begin{align} V_0(\varphi) &= -0.1\varphi^B - 0.2\varphi^S, \\ V_1(\varphi) &= \varphi^B+2\varphi^S=-10V_0. \end{align}
This market obviously contains arbitrages. Borrow money to buy the stock, get some money now and wait one period to be even richer.
However, you'll find no violation of LOP in this market. To see this, suppose you have two self-financing trading strategies $\varphi,\psi$ with $V_1(\varphi)=V_1(\psi)$. LOP requires that $V_0(\varphi)=V_0(\psi)$, which is indeed true because $V_1(\varphi)=-10V_0(\varphi)$ and $V_1(\psi)=-10V_0(\psi)$.
While academics distinguish Law of One Price, No Arbitrage, No Free Lunch with Vanishing Risk, No Free Lunch with Bounded Risk, and No Free Lunch properties, it probably matters little in real life. Those differences are kind of technical and the concepts are almost surely identical for all practical purposes.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.