Arbitrage and No-Arbitrage in Finite-State Markets

State-by-State Gains and Arbitrage

In a finite-state market, a portfolio’s terminal gain can be listed state by state. This makes arbitrage a precise sign test: the gain must be nonnegative in every possible state and strictly positive in at least one state. The word “possible” matters. Throughout this tutorial, the state space is a nonempty finite set \(\Omega=\{\omega_1,\ldots,\omega_m\}\), and every state has positive physical probability under \(\mathbb{P}\). Thus an event that contains at least one state has positive probability.

Consider a one-period market with dates \(0\) and \(1\), a bond, and \(d\) risky assets. The bond’s value is used as numeraire: prices and gains below are discounted by the bond account. Write \(x_0\in\mathbb{R}^d\) for the vector of risky-asset prices at date \(0\), and \(x_1(\omega)\in\mathbb{R}^d\) for their discounted prices at date \(1\) in state \(\omega\). A portfolio holds \(\theta\in\mathbb{R}^d\) units of the risky assets; negative components represent short positions. Its initial risky-asset cost is \(\theta\cdot x_0\). To make the total initial cost zero, take the opposite amount in the bond. The resulting discounted gain in state \(\omega\) is

This is the one-period discounted-gain formula for a self-financing strategy. The initial bond position finances the risky holdings; it does not add another term to the discounted gain. The sign of \(G^\theta(\omega)\) is therefore the relevant quantity in each state.

The definition allows zero gains in some states. It does not require a positive gain in every state, and it does not count a portfolio with gain zero in every state as an arbitrage. The probabilities here are under the physical measure \(\mathbb{P}\). No risk-neutral measure \(\mathbb{Q}\) is being used to define arbitrage.

Worked Tests in Finite-State Markets

The One-Risky-Asset Criterion

For a market with one risky asset and a bond, write \(s\) for the current discounted asset price and \(x(\omega)\) for its discounted terminal price. Let \(x_{\min}=\min_{\omega\in\Omega}x(\omega)\) and \(x_{\max}=\max_{\omega\in\Omega}x(\omega)\). The next result gives a complete no-arbitrage test for this one-asset setting. It does not give a test for arbitrary multi-asset markets.

Proof. First suppose \(x(\omega)=s\) for every state. For every real position \(\theta\), the gain is \(G^\theta(\omega)=\theta(x(\omega)-s)=0\) in every state. No position is an arbitrage. Next suppose \(x_{\min}<s<x_{\max}\). If \(\theta>0\), choose a state attaining \(x_{\min}\); there the gain is \(\theta(x_{\min}-s)<0\). If \(\theta<0\), choose a state attaining \(x_{\max}\); there the gain is \(\theta(x_{\max}-s)<0\). If \(\theta=0\), all gains are zero. Thus no position is an arbitrage in either stated case.

For the converse, suppose neither stated case holds. If \(x_{\min}>s\), choose \(\theta=1\). Then \(x(\omega)-s\geq x_{\min}-s>0\) in every state, so this is an arbitrage. If \(x_{\max}<s\), choose \(\theta=-1\). Then \(-(x(\omega)-s)\geq s-x_{\max}>0\) in every state, again giving an arbitrage. It remains to consider \(x_{\min}\leq s\leq x_{\max}\). Since the constant-price case is excluded, not all terminal prices equal \(s\). If \(x_{\min}=s\), then \(x_{\max}>s\); choosing \(\theta=1\) gives gains \(x(\omega)-s\geq0\) in every state and a strictly positive gain in a state attaining \(x_{\max}\). If \(x_{\max}=s\), then \(x_{\min}<s\); choosing \(\theta=-1\) gives gains \(s-x(\omega)\geq0\) in every state and a strictly positive gain in a state attaining \(x_{\min}\). The remaining possibility is \(x_{\min}<s<x_{\max}\), which is one of the stated no-arbitrage cases and so cannot occur under the supposition that neither holds. These cases cover all possibilities, including the cases where \(s\) lies outside the interval of terminal prices. Therefore every market outside the two stated cases admits arbitrage. \(\square\)

Why the Boundary Cases Matter

The theorem’s strict inequalities are essential. If the current price equals the minimum terminal price but some terminal price is larger, buying the asset has zero gain in a state attaining the minimum and positive gain in a state attaining the maximum. That is arbitrage. If the current price equals the maximum and some terminal price is smaller, shorting the asset gives the corresponding arbitrage. Only when every terminal price equals the current price do all positions have zero gain.

A frequent error is to require strictly positive gain in every state. For the market with current price \(5\) and terminal prices \(5\) and \(9\), this incorrect test rejects the long position because its first-state gain is zero. The correct calculation is

Both entries are nonnegative and one is positive, so the position is an arbitrage. A different error is to inspect only the expected gain. The definition is a state-by-state condition, not a comparison of expected returns: one negative state gain disqualifies a portfolio even if its expected gain is positive. In multiple-asset markets, calculate the gain vector for the portfolio being tested and check every state; the one-asset theorem does not replace that check.