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In Tomas Björk's Arbitrage Theory in Continuous Time (or here), $\exists$ this Pricing Principle.

enter image description here

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.

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Please don't post links to complete e-books unless you're the copyright holder or the explicitly license allowed this. If you doubt whether this is allowed: don't. –  Bob Jansen Jul 20 at 21:29

2 Answers 2

up vote 1 down vote accepted

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).

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I already proved Prop 2.9. How does one state exactly the proof of the Pricing Principle 1? –  BCLC Jul 17 at 11:21
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How about attaching a meaning to the word reasonable? For instance, prove the following version of the pricing principle: If a claim $X$ is reachanble with a replicating portfolio $h$, then there exists an arbitrage opportunity if $\Pi(t,X) \neq V_t^h$ for some $t \in \{0,1\}$. –  Theja Jul 17 at 11:28
    
Proof attempt: Well it can't be that t=1 since X is reachable so it then follows from 2.9...QED? –  BCLC Jul 17 at 13:25

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:enter image description here

Here EMM = Equivalent Martingale Measure (Q), NA = No-Arbitrage.

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How do you find my proof attempt (see other thread) ? –  BCLC Jul 20 at 22:23
    
quant.stackexchange.com/a/14068 –  BCLC Jul 20 at 22:35
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@BCLC Principle I and Proposition 2.9 are the same, so you cant say that Principle I follows from Proposition 2.9. The text also says "more precise meaning in the proposition" so its the same. I will look up and post an image from my course script to give an intuition of the proof. –  emcor Jul 20 at 22:41
    
@theja gave an alternative version of PP1 and I proved it. Was either wrong? –  BCLC Jul 20 at 23:11
    
@BCLC Theja's idea is a correct idea for the proof, and your answer is also correct regarding $t=1$. But note $t\in(0,1)$ is in continuous time, so its all real values in the interval $(0,1)$ (not just $0$ or $1$). –  emcor Jul 20 at 23:18

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