No-Arbitrage Pricing and the Law of One Price
Summary
The document asks whether an explanation of a pricing principle in a continuous-time finance text is a proof, and how the principle relates to a later proposition. The response says the passage is explanatory: it motivates the principle and gives hints for proving the proposition, but does not itself establish a rigorous result. A proof requires precise definitions and assumptions, including a specific meaning for a price to be reasonable.
The discussion connects this idea to replication and the law of one price. If a portfolio and a derivative have identical payoffs, differing prices would allow an arbitrage, so their prices must agree under the stated no-arbitrage framing. The document does not supply the full proof or spell out its assumptions in detail; it directs readers toward a proposition whose proof is left to the reader. Its main lesson is the importance of formal definitions when turning pricing intuition into a theorem.
Key ideas
- An intuitive explanation motivates a pricing principle but does not by itself prove it.
- A rigorous proof depends on defining the principle’s terms and stating its assumptions precisely.
- A derivative and a portfolio with identical payoffs should have equal prices under the law of one price.
- Different prices for identical payoffs create an arbitrage opportunity in the no-arbitrage framework.
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Full text
# Pricing Principle 1 # Pricing Principle 1 In Tomas Björk's Arbitrage Theory in Continuous Time (or here), $\exists$ this Pricing Principle. Is the one in red supposed to be the proof of the Pricing Principle 1? Or merely an intuitive explanation? If proof, is this rigorous? Or does its proof not need to be rigorous since it is merely a Principle (In this case, I guess I am assuming Principle is synonymous with something like Rule of Thumb)? If explanation, how does one then prove Pricing Principle 1? Does it follow from Prop 2.9? If so, how does one say this exactly? The fact that other prices besides $\Pi(0;X) = V_0^h$ implies arbitrage possibility means that the fair/reasonable price of $X$ is $\Pi(0;X) = V_0^h$? It seems weird since most math textbooks usually prove statements using previous statements rather than latter ones. ## Answer by Theja Tulabandhula (score 1, accepted) https://quant.stackexchange.com/a/14068 > Is the one in red supposed to be the proof of the Pricing Principle 1? Or merely an intuitive explanation? It is not a proof. The explanation/reasoning in this paragraph lets the author state the pricing principle. It has hints on how to prove Prop 2.9 (for instance, see the line `...no difference between holding the claim and the portfolio...`). If every word in the statement of the pricing principle is precise, one could potentially prove it (starting form some set of assumptions). In particular, the word reasonable in the principle is given a specific meaning that leads to proposition 2.9, which can then be proved using the ideas from the discussion before it. This meaning is simply that the price of $X$ at $t=0$ or at $1$ being equal to the value of the corresponding replicating portfolio disallows the possibility of arbitrage and hence is reasonable. > If explanation, how does one then prove Pricing Principle 1? Does it follow from Prop 2.9? Prove it once you define every word in the priciple in a precise manner of your choosing similar to the proof of Proposition 2.9 (which is left to the reader). ## Answer by emcor (score 1) https://quant.stackexchange.com/a/14101 In general, if one can create a portfolio with the same payoff as the derivative, their prices must be equal. This is also called "Law of One Price". Here an excerpt from my script: Here EMM = Equivalent Martingale Measure (Q), NA = No-Arbitrage.
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