Farkas' Lemma

\begin{equation} Ax \leq 0, c^{T}x < 0 \end{equation}

and

\begin{equation} A^{T}y + c = 0, y \geq 0 \end{equation}

are strong alteratives. This is through duality.

investment arbitrage

  1. invest \(x_{j}\) in each of \(n\) assets \(1 … n\) with prices \(p_1 … p_{n}\)
  2. suppose \(V_{ij}\) is the payoff of asset \(j\) and outcome \(i\)
  3. risk-free (cash): \(p_{1} = 1\), \(V_{i1} = 1\) for all \(i\) as our first investment

Arbitrage means there exists some \(x\) with \(p^{T}x < 0, V x \succeq 0\) (first thing is you borrow money?, and then second thing is you get more money back).

By Farkas’ Lemma, there being no arbicharnge implies there exists some \(y\) such that \(V^{T}y = p\). Recall our first column is \(1\) (from risk-free cash) making \(1^{T}y = 1\).