Risk-Neutral Probability Uniqueness and Market Completeness

Summary

The document asks whether the existence of two distinct risk-neutral probabilities implies that infinitely many exist. It invokes the finite-market result that, assuming no arbitrage, market completeness is equivalent to uniqueness of the risk-neutral probability. The attempted argument uses a one-period binomial market, but its probability condition is incorrect: probabilities for the two states must sum to one, and merely assuming otherwise does not establish two valid risk-neutral measures.

The general claim follows from convexity. If two distinct risk-neutral probability vectors satisfy the same linear pricing constraints, every convex combination of them also satisfies those constraints and remains a probability vector. This gives infinitely many risk-neutral probabilities between the two. The conclusion applies when both original vectors are valid solutions; the document's attempted example does not construct such a pair or verify no-arbitrage conditions.

Key ideas

• In a finite market without arbitrage, completeness is equivalent to uniqueness of the risk-neutral probability.
• Risk-neutral probabilities satisfy linear pricing constraints and the probability normalization condition.
• Convex combinations of two distinct valid risk-neutral probabilities are also valid solutions.
• The binomial argument given in the document fails to establish two valid probability measures.

Tags

# Risk-neutral probabilities


# Risk-neutral probabilities












I will use this theorem 3.2 from the book "Quantitative modeling of Derivative Securities" by Marco Avellandea:

Theorem 3.2 - Assume that there is no arbitrage, i.e. there exists a risk neutral probability $\pi$. Then, the market is complete if and only if there is a unique risk-neutral probability, i.e. the system of linear equation $$p = D\pi$$ has a unique solution where $D$ is are the different future scenarios in the market.

Show that in any model if there are two distinct risk-neutral probabilities, then there are infinite number of them.

Attempted solution: Consider a one-period binomial model:

State 1: $S_0 u$ with risk-neutral probability $\pi_u$

State 2: $S_0 l$ with risk-neutral probability $\pi_l$

Assume $\pi_u,\pi_l$ are two distinct risk-neutral probabilities, i.e., $\pi_u + \pi_l \neq 1$. Then, this violates theorem 3.2, the market is not complete since there cannot exist a unique risk-neutral probability, i.e. the system of linear equations $$p = D\pi$$ does not have a unique solution. Thus if there is not a unique solution then there are an infinite number under our example.

I am not sure if this suffices, any suggestions is greatly appreciated.

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