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This book is arranged as follows: ssss1 reviews the basic concepts and notations in probability and random variables, especially discrete variables, and also provides an introduction to the concept of stochastic processes. ssss1 is dedicated to the introduction and study of the concept of conditional expectation, a key concept in the definition of martingales and the computation of financial options. ssss1 aims to introduce an interesting example of stochastic process, namely a simple symmetric random walk, with minimal formalism. This chapter may be read as a standalone chapter. ssss1 defines and characterizes the concept of a martingale in discrete time and studies certain properties.

From ssss1 onward, we focus on financial mathematics, strictly speaking. We define essential financial vocabulary such as the concept of a financial asset, investment strategy and the concept of arbitrage. We also begin to establish a link with martingales. We also introduce a typical example of a discrete financial market: the Cox, Ross and Rubinstein binomial model, which acts as a guiding thread through all the following chapters. In particular, the question of Optimal Portfolio Management in this model is studied through guided work. This is a discretized version of the famous Merton problem, originally posed in the continuous Black and Scholes [MER 69] model. ssss1 introduces and studies the first large and specific category of conditional assets, i.e. European options. This is the simplest example of a financial asset which may be exercised subject to the realization of a certain condition. We examine, in detail, the question of the pricing and hedging of these options in the general case of a discrete financial market and then in the specific case of the Cox, Ross and Rubinstein model. ssss1 is dedicated to the study of a slightly more complex family of conditional assets: American options. We once again undertake a detailed study of the question of the pricing and hedging of these options and connect them to the theory of optimal stopping, in the general case of a discrete financial market and then in the specific case of the Cox, Ross and Rubinstein model.