ABSTRACT
This chapter introduces the concepts of arbitrage, probability measure transformations, self-financing portfolios and martingales in a simple financial market. It considers simple models of security markets and describe the basic principles of valuation of contingent claims (e.g., options, futures). The key idea behind valuation in markets with uncertainty is the notion of absence of arbitrage. The chapter shows that the right price of a security, in the sense of no arbitrage opportunities, is the discounted expected value of future cash flows. However, the expectation should be computed with respect to risk-neutral probabilities, which in general differ from the objective probabilities. The pricing of derivatives in discrete and continuous-time models is built on so-called martingale measures which basically are probability distributions that are related to the historical or objective probability distribution. The chapter also presents two simple examples of martingales.
