This chapter provides the tools which are used in option pricing. The field of stochastic processes in continuous time which are defined as solutions of stochastic differential equations plays an important role. To illustrate these notions we use repeatedly approximations by stochastic processes in discrete time and refer to the results from Chapter 4.
A stochastic process in continuous time
consists of chronologically ordered random variables, but here the
variable
is continuous, i.e.
is a positive real number.
Stock prices are actually processes in discrete time. But to derive the Black-Scholes equation they are approximated by continuous time processes which are easier to handle analytically. However the simulation on a computer of such processes or the numerical computation of say American options, is carried out by means of discrete time approximations. We therefore switch the time scale between discrete and continuous depending on what is more convenient for the actual computation.