# Fundamental Theorem of Asset Pricing (FTAP)

In the spirit of canonical questions please state here versions of the FTAP in the following form (please only one theorem by answer) :

• Necessary definitions (or a direct link to definitions)
• Hypothesys and Context (such as existence or not of transaction costs, discrete time setting, etc...)
• Statement of the Theorem
• Reference(s) for a proof

The motivation is coming from the fact there are several versions of this theorems and having one place regrouping those differents versions would be nice.

Regards

• @TheBridge I strongly urge you to provide an example answer since you have set up a rather elaborate scheme here. – Shane Feb 9 '11 at 18:31
• @Shane : True, I could (and actually I can) do that but I feel this is weird to answer my own question. So what I do is that I'll state a FTAP after someone has done it once. Does it seems fair to you ? – TheBridge Feb 9 '11 at 20:53
• Anyway even if wrong or incomplete any version can be edited after some comments indicating where and why it should be done are properly made, don't you think ? – TheBridge Feb 9 '11 at 20:55
• @TheBridge I'd say go ahead whenever you want. As it stands, you have a small chance of someone else going first. Lead by example. – Shane Feb 9 '11 at 22:03
• @TheBridge Just to chime in on this a little further: you have no answers yet, and I think it's because the question itself isn't sufficiently clear. People don't want to work to hard to understand what's being asked. That's why I really think you will be better served by providing an initial answer, so everyone can follow the example. – Shane Feb 10 '11 at 15:48

I teach Derivative Securities in the mathematical finance program at NYU and was rather surprised to learn that there is no proof of the FTAP that is accessible to masters level students. So I wrote this. It is a simple proof for the discrete time case.

One bonus of the proof of the one period case is that it tells you how to find the arbitrage if one exists.

This question requires a comprehensive answer, perhaps beyond the confines of my input box :) Suffices here to state the following:

The First Fundamental Theorem of Asset Pricing states that in an arbitrage-free market, there exists a ("net") present value function, that is, a linear valuation rule whose value is zero when evaluated in any traded cash-flow.

This is an existence theorem, and it does not depend on the theoretical or "real" form of the market. It does not depend on discrete or continuous time modeling, as it does not depend on whether there are transaction costs, trading constraints, or missing markets. All we need to have is the assumption that we can undertake two or more trades simultaneously, that we can scale them up, and that for every given trade, we can have its "mirror" in the market - that is, that we have a linear vector space of traded cashflows.

The Second Fundamental Theorem of Asset Pricing states that when an arbitrage-free market is "complete", the linear valuation rule is unique.

It is also true that these two separate theorems with different implications, are more often than not, presented in a fused form. This can be confusing. Proofs of these facts are virtually in every graduate asset pricing book. My favourite one is Duffie's 'Dynamic Asset Pricing Theory'.