ABSTRACT
This chapter discusses to applications of stochastic processes in mathematical finance, and more particularly to option pricing problems. It discusses several models such as the binomial, the Cox-Ross-Rubinstein and the celebrated Black-Scholes-Merton model. The up and down evolution of the price of a risky asset in a neutral financial market is defined in terms of a martingale with respect to some filtration such that for two random up and down variables (Un, Dn) s.t. The filtration represents the information available on the financial market. This discrete time model represents the evolution of the price between time units such as weeks, days, hours, or milliseconds. The strategy of the option writer is to find some initial portfolio value such that there exists a self-financing strategy b for which the terminal value of the portfolio is precisely the payoff function, that is,. This minimal initial value is called the price of the option.
