Is there arbitrage in this market?

Question

I have an incomplete market (rows are states and columns are securities) and I need to determine if there is arbitrage, and if so, construct an arbitrage strategy. A is the payoff matrix (payoffs at t=1) and S is the price vector at t=0.

I am a bit lost, because the state prices cannot be calculated explicitly (more unknowns than equations) and whatever I do, the only possible 'arbitrage portfolio' that I can get is the 0-vector, which is not really an arbitrage portfolio, in fact, what that means is that nothing should be sold nor bought.

I also tried using linear programming in R (lp function from lpSolve package) but I still got the 0-vector as the only possible answer.

Does anyone know if there is some other way to determine if there is, in fact, arbitrage in this market? Does the fact that the market is incomplete imply that there is arbitrage?

Any help or hints would be greatly appreciated.

Answer 1

An incomplete market can be free of arbitrage.

For an arbitrage, we need a portfolio vector $p\in\mathbb{R}^2$ such that $Ap\geq0_{\mathbb{R}^3}$, in the sense that all rows (states) have a non-negative payoff and that at least one row has a strictly positive payoff. In addition, $\langle p,S\rangle\leq0_{\mathbb{R}}$, i.e. the arbitrage should be costless (at time zero). I used some index at the zero to highlight the dimension of the inequalities.

(1) The time-zero condition means that $p_1+5p_2\leq 0$.

(2) The payoff condition means that $Ap=\begin{pmatrix} 2p_1 \\ p_1+p_2 \\ 2p_2\end{pmatrix}\geq\begin{pmatrix}0\\0\\0\end{pmatrix}$.

Condition (2) implies $p_1,p_2\geq0$. However, condition (1) requires at least one of the entries of $p$ to be negative. Thus, there can't exist an arbitrage strategy.

You thus indeed have an example of a market which is free of arbitrage but incomplete (for pricing theory, this means there exist infinitely many positive stochastic discount factors or infinitely many EMMs. Continuous models which allow for jumps or stochastic volatility are also incomplete yet free of arbitrage.).