There are two future states and three securities with the associated payments matrix (states by securities) The first security current price is 9, the second security current price is 5 and the third security current price is 14. Are there any arbitrage opportunities? Hint: the calculations do not require matrix inverse.

Question

BlackTom AI · Accepted Answer

First, restating the setup in my own words: there are two future states and three traded securities. The current prices are given as P1 = 9, P2 = 5, and P3 = 14. The payments matrix (states by securities) is mentioned but not displayed in the prompt, and the question asks whether arbitrage opportunities exist.

Key point to consider: arbitrage in a finite-state, finite-asset setting typically hinges on whether there exists a portfolio weights vector w such that the portfolio’s payoffs in all states are nonnegative and at least one state has a strictly positive payoff, while the initial cost is nonpositive (or negative). Equivalently, after discounting by state prices or using payoff vectors, arbitrage would show up if the current price vector lies outside the cone generated by the payoff vectors (or, in a simpler computational sense, if there is some linear combination of the payoffs that yields a nonnegative vector with nonpositive cost).

Option-by-option analysis:
- Option 1: “Yes, there are arbitrage opportunities.” Without the explicit payoff matrix, we cannot directly verify the existence of a portfolio that produces a guaranteed nonnegative payoff with a nonpositive cost across both states. Since the payoff structure of the three securities critically determines whether such a portfolio exists, this claim requires information not provided here. Therefore, asserting arbitrage outright would be premature.
- Option 2: “The market is incomplete, and therefore it is not possible to answer this question.” In a two-state economy with three traded securities, the market could be either complete or incomplete depending on whether the payoff vectors span the state space. Completeness would mean every state-contingent payoff can be replicated by a portfolio, which requires the payoffs to sufficiently span the two-dimensional state space and the absence of arbitrage. If the payoff vectors do not span the entire space or if there is redundancy that prevents a unique state-price vector, the market would be incomplete. Since the problem does not provide the actual payoff matrix, we cannot determine completeness or the absence/presence of arbitrage. Thus, this statement correctly reflects that, given the missing data, a definitive determination cannot be made.
- Option 3: “No, all securities are fairly priced and there are no arbitrage opportunities.” This conclusion would require verification that the prices lie within the no-arbitrage set determined by the payoff vectors, i.e., there exists at least one state-price vector or a risk-neutral measure that prices all securities consistently. Without the explicit payoffs, we cannot confirm this condition. Therefore, declaring no arbitrage would be unfounded with the data at hand.

In summary, because the key piece of information—the payments matrix (state-by-security payoffs)—is not provided, the ability to check for arbitrage or to assess market completeness is blocked. The price data alone (9, 5, 14) cannot by itself confirm arbitrage without knowing how each security pays in each state. As a result, any definitive determination hinges on the missing payoff structure, and the statement that the market is incomplete (and hence cannot be answered from the given data) correctly acknowledges this limitation. The other options claim specific arbitrage or fairness conclusions that cannot be substantiated without the payoff matrix.

Note: If the payoff matrix were supplied, one would typically check linear (in)dependence of payoff vectors, attempt to construct a nonpositive-cost, nonnegative-payoff portfolio, and examine whether a state-price vector exists that prices all securities consistently. The calculations could be done through linear programming or direct cone checks, which the hint suggests can be done without explicit matrix inversion.

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