Discrete Time Models.

Summary This first chapter is dedicated to discrete time markets. We first relate the absence of arbitrage opportunities to the existence of equivalent martingale measures, i.e. of equivalent probability measures that turn discounted asset prices into martingales. These measures are the basis of the whole pricing theory. They define the price intervals for derivatives products that are acceptable for the market. When the market is complete, meaning that any source of risk can be hedged perfectly by trading liquid assets, these intervals are reduced to one single point. This unique price allows one to hedge the corresponding derivative perfectly. However, in general, these intervals are not reduced to a singleton, and only their upper-bound, the so-called super-hedging price, permits to offset all risks by using a suitable dynamic hedging strategy. We shall study in details both European and American options. At the end of the chapter, the impact of portfolio constraints will also be discussed.