Using State Prices to Test for Arbitrage in an Incomplete Market

Summary

The document asks how to determine whether a market with a stock, a bond, and call options admits arbitrage when the option’s time-zero price is unknown. It proposes using the stock and bond alone to derive a range of state prices, then checking whether those prices can all remain positive. As an alternative, it suggests including an option and expressing state prices as a function of its unknown price, identifying the prices consistent with positive state prices.

These are posed as possible approaches rather than resolved results: the document includes no payoff matrix, numerical example, or answer confirming the proposed reasoning. Its central concept is that state prices connect security prices to state-contingent payoffs, while market incompleteness can leave multiple state-price vectors consistent with observed prices. The discussion does not establish that the proposed bounds are sufficient for the particular exercise, so applying the method requires the full payoff structure and relevant assumptions.

Key ideas

• State prices can be used to relate security prices to state-contingent payoffs.
• In an incomplete market, observed security prices may be consistent with a range of state-price vectors.
• The document proposes checking positivity of state prices as a way to reason about arbitrage.
• An unknown option price could be treated as a variable when solving for state prices.
• The proposed methods are questions, not demonstrated solutions to a specified payoff matrix.

Tags

# Determining Presence of Arbitrage


# Determining Presence of Arbitrage












I am slightly confused by part (b) of this question. My understanding is that the easiest way to determine if there is arbitrage is to compute the state prices and then look at their sign: if one or more of the state prices are nonpositive, then we have arbitrage.

We have a payoff matrix with 3 linearly independent securities, the stock, the bond, and either the first or second call option. What puzzles me is that we have prices at $t=0$ for the stock and bond but not for the call options. To my understanding, there are two ways to approach this question.

- Just use the stock and the bond (incomplete market) to compute the state prices and obtain a range for the state prices. Fix the range such that none of the state prices are 0 or negative. Then argue that if the state prices fall outside of that range there will be arbitrage.

- Use the stock, the bond, and an option and compute the state prices as a function of the unknown option price at $t=0$, $C_0$. Again, fix a range for $C_0$ such that none of the state prices are 0 or negative, and argue that whenever the option price $C_0$ lies outside of that range there will be arbitrage.

Does this sound reasonable? Does anyone have any other ideas as to how to answer this question?

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