This introductory book emphasizes algorithms and applications, such as cryptography and error correcting codes, and is accessible to a broad audience. The presentation alternates between theory and applications in order to motivate and illustrate the mathematics. The mathematical coverage includes the basics of number theory, abstract algebra and discrete probability theory.
text is designed for an introductory probability course at the university level for sophomores, juniors, and seniors in mathematics, physical and social sciences, engineering, and computer science. It presents a thorough treatment of ideas and techniques necessary for a firm understanding of the subject. The text is also recommended for use in discrete probability courses. The material is organized so that the discrete and continuous probability discussions are presented in a separate, but parallel, manner.
In this chapter, we study the mathematical structure of a simple one-period model of a
financial market. We consider a finite number of assets. Their initial prices at time t = 0 are known, their future prices at time t = 1 are described as random variables on some
probability space. Trading takes place at time t = 0. Already in this simple model,
some basic principles of mathematical finance appear very clearly. In Section 1.2, we
single out those models which satisfy a condition of market efficiency: There are no
trading opportunities which yield a profit without any downside risk.
The binomial asset pricing model provides a powerful tool to understand arbitrage pricing theory
and probability theory. In this course, we shall use it for both these purposes.
In the binomial asset pricing model, we model stock prices in discrete time, assuming that at each
step, the stock price will change to one of two possible values. Let us begin with an initial positive
stock price S0. There are two positive numbers, d and u, with
This is the fifth book of examples from the Theory of Probability. This topic is not my favourite,
however, thanks to my former colleague, Ole Jørsboe, I somehow managed to get an idea of what it is
all about. The way I have treated the topic will often diverge from the more professional treatment.
On the other hand, it will probably also be closer to the way of thinking which is more common among
many readers, because I also had to start from scratch.
The prerequisites for the topics can e.g. be found in the Ventus: Calculus 2 series, so I shall refer the