No-Arbitrage and Completeness in a Binomial Model
Summary
The document discusses a textbook claim that an arbitrage-free general binomial model is complete, meaning contingent claims can be replicated by portfolios. The question focuses on the proof’s linear equations and on why the up and down factors must differ. The response argues that a genuine binomial model has two distinct stock outcomes, so the up factor exceeds the down factor; this makes the equations for the replicating portfolio uniquely solvable.
The exchange also raises a broader conceptual issue: how uniqueness of a risk-neutral measure relates to completeness. The included answer addresses the distinct-outcomes condition but does not develop that connection or give a general proof covering multi-period models. Its reasoning is therefore a narrow clarification of the one-step binomial setup, and relies on the model definition excluding identical outcomes.
Key ideas
- Completeness in the stated setting means contingent claims can be replicated by portfolios.
- The answer relies on the binomial model having distinct up and down stock outcomes.
- Distinct outcomes make the relevant linear system uniquely solvable for a replicating portfolio.
- The document raises, but does not fully explain, the relationship between unique risk-neutral measures and completeness.
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Full text
# Why does arbitrage free imply complete market? # Why does arbitrage free imply complete market? Proposition 2.10 of Tomas Bjork's "Arbitrage Theory in Continuous Time" states that if the general binomial model is free of arbitrage then it is also complete i.e. every contingent claim has a replicating portfolio. Here is the proof in question: Note that this exact question has already been asked and answered here, but I don't think that the answers are satisfactory because in the chosen answer he says that the system of linear equations only has a proof if $d < u$, but this is not true because it has a solution in the case that $d > u$ by symmetry and in the case of $d = u$ because then the two linear equations are the same. Also, the answer claims that by Proposition 2.3 $d < 1 + R < u$, but in the actual proposition these inequalities aren't strict! I'm not sure I really understand the top voted answer, but I get the feeling it is also not relevant because earlier in the chapter it says that we assume there is no bid-ask spread. As for the last answer I feel like it might be an actual answer to the question, but I don't see why a unique risk neutral measure implies completeness. The book's definition for completeness is that you can price a claim, and I don't see how this definition relates to the definition about the uniqueness of risk neutral measures? ## Answer by dm63 (score 2) https://quant.stackexchange.com/a/35573 You agree that the proposition is proven if the equations have a unique solution. You agree that there is a unique solution if u>d. Then we just have to show that u>d. But the definition of u and d is that we have a binomial model where there are two possible outcomes for the stock, a higher outcome su and a lower outcome sd. Hence u>=d by assumption , and in fact u>d because if u=d we do not have a binomial model.
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