4
Bachelier and Black-Scholes
4.1 Introduction to Continuous Time Models
In this chapter we illustrate the theory developed in the previous chapters
by analyzing the most basic examples in continuous time. They still play an
important role in practice.
The binomial model (Example 3.3.1 and 3.3.5) was already analyzed in the
previous Chap. 3. It fits perfectly into the framework developed in Chap. 2,
i.e., it is based on a finite probability space Ω. Therefore we could rigorously
analyze it in Chap. 3.
If we consider the binomial model on a grid in arithmetic progression and
pass to the continuous time limit we arrive, similarly as L. Bachelier in 1900
[B 00] at Brownian motion. If we consider the binomial model on a grid in
geometric progression (the “Cox-Ross-Rubinstein” model as in example 3.3.5)
we arrive at geometric Brownian motion, similarly as P. Samuelson in 1965.
The latter model now is often called the “Black-Scholes” model.
We now pass to the continuous time setting. Strictly speaking, we jump
already one step ahead, as we have not yet developed the theory for the case
of processes in continuous time. But we believe that it is more important to
see the theory in action using these important examples in order to build up
some motivation for the formal treatment of the general theory which will be
developed later. We shall therefore deliberately use some heuristic arguments
which will be rigorously justified by the general theory developed later in this
book.
4.2 Models in Continuous Time
To do so, we suppose from now on that the reader is familiar with the notion
of Brownian motion and related concepts. We recall themartingale represen-
tation theoremfor Brownian motion, which is the continuous analogue to the