Statewise Payoffs Required for an Arbitrage Portfolio

Summary

The question compares two proposed arbitrage tests based on initial portfolio cost and payoffs across possible states. It notices that a portfolio costing nothing or less may pay positively in one state while losing in another, and that positive payoffs in every state can occur even when the initial cost is positive. Those examples expose a missing condition in the first stated test and a need to compare cost and payoff on consistent timing terms.

Under the standard finite-state definition, an arbitrage has nonpositive initial cost and nonnegative payoff in every state, with either a strictly negative cost or a strictly positive payoff in at least one state. Thus, a positive payoff in just one state is insufficient if another state has a loss. Likewise, uniformly positive future payoffs alone do not establish arbitrage when the portfolio has a positive initial cost; the cost must be accounted for in the payoff comparison or discounted consistently. The passage gives no specific market model or assumptions about friction, so the standard test may need adjustment for transaction costs and financing constraints.

Key ideas

• An arbitrage portfolio must avoid negative payoffs in every possible state.
• At least one state must have a strictly positive payoff, or the initial cost must be strictly negative.
• A positive payoff in one state does not qualify if another state has a loss.
• Initial costs and future payoffs must be compared on a consistent timing basis.

Tags

# Issue involving arbitrage conditions


# Issue involving arbitrage conditions












In my book it's written that if one of these two conditions is verified then you can make an arbitrage. The two conditions are: $$1) \left\{ \begin{array}{c} \mathbf{q} \bullet\mathbf{n} \le0 \\ \mathbf{Y}=D\mathbf{n} \ \ has \ at \ least \ one \ component >0 \\ \end{array} \right. $$ $$2) \left\{ \begin{array}{c} \mathbf{q} \bullet\mathbf{n} <0 \\ \mathbf{Y}=D\mathbf{n} \ \ has \ at \ least \ one \ component \ge 0 \\ \end{array} \right. $$ Where: $\mathbf{n}$ is the fraction of each title one has bought at t=0; $\mathbf{Y}$ is the portfolio payoff in each of the 'S' possible states of the world at t=1; $D$ is the matrix of the dividends of the N titles in the 'S' states; $\mathbf{q}$ is the price of each titles; Further in the book there is also written that an arbitrage occurs when: $$3) \mathbf{Y}>\mathbf{q} \bullet\mathbf{n}$$ where it stays for each component of the vector $\mathbf{Y}$. So I'm a bit confused since for example if I have: $$ \left\{ \begin{array}{c} \mathbf{q} \bullet\mathbf{n} =-5$ \\ \mathbf{Y}=[+10$ ;-7$ ;-7$] \\ \end{array} \right. $$ then $1)$ is verified but not $3)$. While if $3)$ is verified for example: $$ \left\{ \begin{array}{c} \mathbf{q} \bullet\mathbf{n} =5$ \\ \mathbf{Y}=[+10$ ;7$ ;7$] \\ \end{array} \right. $$ then $1)$ is not true in general. So I don't understand exactly when you have arbitrage. Is the book wrong? Thanks

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