No arbitration opportunity condition
Mathematical Finance : Asset Pricing

In this section, we explain the NA (No Arbitrage) which is usually assumed in order to characterize the price of a derivative on a financial market. We use the notations and definitions from the previous paragraphs.

Definition 14. An arbitrage opportunity is a self-financed portfolio $V=V^{\widehat{\theta }}$ such that its initial capital is V₀ = 0, such that P(V_T ≥ 0) = 1 (no risk of loss) and P(V_T > 0) > 0.

In this way, an arbitrage opportunity is not only risk-free, but also offers the chance of making a strictly positive gain from a zero initial investment. Note that it is possible to start a portfolio from V₀ = 0, for example with a position shorts on S⁰(θ⁰...