Positive Risk-Neutral Probabilities in an Incomplete Market

Summary

The document poses a no-arbitrage question for a market with two risky assets, one risk-free asset, and four possible future states. Each risky asset's payoffs are ordered across the states, and its current price is assumed to lie between its lowest and highest state payoff. The question asks whether these bounds suffice to guarantee strictly positive risk-neutral probabilities, or whether further conditions are needed.

The excerpt states no solution or derivation, so it does not establish an additional criterion. Its useful contribution is the setup: in an incomplete market, checking each asset's price against its individual payoff range may not by itself resolve whether a single positive probability vector prices all assets consistently. A full answer would require examining the joint payoff and price constraints; the document offers no evidence or worked example beyond the stated model.

Key ideas

• The market described has two risky assets, one risk-free asset, and four future states.
• Each risky asset's payoffs are ordered from lowest to highest across the states.
• The question tests whether individual price bounds ensure a common strictly positive risk-neutral probability vector.
• The excerpt gives no solution, so it does not specify sufficient conditions for no arbitrage.

Tags

# Additional requirement for the asset price and payoff to ensure the market is arbitrage-free


# Additional requirement for the asset price and payoff to ensure the market is arbitrage-free












Suppose we have two risky assets and one risk-free asset in the market. The market is incomplete in that there are three assets and four states. The price vector at $t_0$ is: $\boldsymbol{p_0}=[p^s_{1},p^s_{2}, 1]^\intercal$ and the payoff at $t_1$ is $\boldsymbol{p_1} = \begin{bmatrix} p_1^1 & p_1^2 &p_1^3 & p_1^4 \\ p_2^1 & p_2^2 &p_2^3 & p_2^4 \\ 1 & 1 & 1 & 1 \end{bmatrix}.$ Given that $p_{n}^{4}<p_{n}^{3}<p_{n}^{2}<p_{n}^1$ for $n=1,2$. For the market to be arbitrage-free, we already have $p_{n}^{4}<p^s_{n}<p_{n}^1$ for $ n = 1,2$. To check if there is arbitrage opportunity, I am using the theorem that the no arbitrage condition is satisfied if there exists positive risk-neutral probabilities under this setup. Are there any additional conditions that is needed to make sure there exists positive risk-neutral probabilities?

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.