Foundations of Financial Engineering

Two portfolios with identical future cash flows in every state of the world must have the same price today; otherwise an arbitrage exists.

A self-financing portfolio with zero cost today and non-negative payoffs in every future state, strictly positive in at least one, cannot exist in an arbitrage-free market.

If a portfolio of traded assets reproduces the payoff of a claim in every state, the claim's no-arbitrage price equals the cost of the replicating portfolio.

A market is complete if every contingent claim can be replicated by traded assets; otherwise it is incomplete and some claims have a range of arbitrage-free prices.

A market is arbitrage-free if and only if there exists at least one equivalent risk-neutral (martingale) measure ℚ under which discounted asset prices are martingales.

An arbitrage-free market is complete if and only if the equivalent risk-neutral measure ℚ is unique.