# 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

• I am looking at (1) and (2) only. I think the correct statement of (1) is that "all components $\ge 0$ with at least one component $>0$". Check your book. In your example you have two -7 components which are negative, so in my opinion (1) is not verified. – Alex C May 26 at 2:49
• Also in (2) I thiink it should be "every component $\ge 0$". – Alex C May 26 at 2:55
• It says 'a positive amount of money in at least one of the S states' @AlexC – Landau May 26 at 8:47
• That is an incomplete statement. Check for example here people.dm.unipi.it/trevisan/didattica/2015-2016/sanminiato/… on Page 13 or in another trusted reference. One or more states have to be strictly positive and the other states have to be zero. If you can lose money in one or more states it is not a riskless arbitrage. – Alex C May 26 at 23:12
• Yes, you are right, the book is not clear in that explanation, thanks. @Alex C – Landau May 27 at 15:43