Stochastic Processes and Financial Mathematics
(part one)
Chapter 0 Introduction
We live in a random world: we cannot be certain of tomorrow’s weather or what the price of petrol will be next year – but randomness is never ‘completely’ random. Often we know, or rather, believe that some events are likely and others are unlikely. We might think that two events are both possible, but are unlikely to occur together, and so on.
How should we handle this situation? Naturally, we would like to understand the world around us and, when possible, to anticipate what might happen in the future. This necessitates that we study the variety of random processes that we find around us.
We will see many and varied examples of random processes throughout this course, although we will tend to call them stochastic processes (with the same meaning). They reflect the wide variety of unpredictable ways in which reality behaves. We will also introduce a key concept used in the study of stochastic processes, known as a martingale.
It has become common, in both science and industry, to use highly complex models of the world around us. Such models cannot be magicked out of thin air. In fact, in much the same way as we might build a miniature space station out of individual pieces of Lego, what is required is a set of useful pieces that can be fitted together into realistic models. The theory of stochastic processes provides some of the most useful building blocks, and the models built from them are generally called stochastic models.
One industry that makes extensive use of stochastic modelling is finance. In this course, we will often use financial models to motivate and exemplify our discussion of stochastic processes.
The central question in a financial model is usually how much a particular object is worth. For example, we might ask how much we need to pay today, to have a barrel of oil delivered in six months time. We might ask for something more complicated: how much would it cost to have the opportunity, in six months time, to buy a barrel of oil, for a price that is agreed on today? We will study the Black-Scholes model and the concept of ‘arbitrage free pricing’, which provide somewhat surprising answers to this type of question.
0.1 Organization
Syllabus
These notes are for two courses: MAS352 and MAS61023. This is part one of the lecture notes. Part two will be made available when we reach Chapter 10.
Some sections of the course are included in MAS61023 but not in MAS352. These sections are marked with a \((\Delta )\) symbol. We will not cover these sections in lectures. Students taking MAS61023 should study these sections independently.
Some parts of the notes are marked with a \((\star )\) symbol, which means they are off-syllabus. These are often cases where detailed connections can be made to and from other parts of mathematics.
Problem sheets
The exercises are divided up according to the chapters of the course. Some exercises are marked as ‘challenge questions’ – these are intended to offer a serious, time consuming challenge to the best students.
Aside from challenge questions, it is expected that students will attempt all exercises (for the version of the course they are taking) and review their own solutions using the typed solutions provided at the end of these notes, in Appendices A and C.
At three points during each semester, an assignment of additional exercises will be set. About one week later, a mark scheme will be posted, and you should self-mark your solutions.
Examination
Both versions of the course will be examined in the summer sitting. Parts of the course marked with a \((\Delta )\) are examinable for MAS61023 but not for MAS352. Parts of the course marked with a \((\star )\) will not be examined (for everyone).
A formula sheet will be provided in the exam, see Appendices B (for semester 1) and E (for semester 2). Some detailed advice on revision can be found in Appendix D, attached to the second semester notes.
MAS61023 also contains a mid-year online test, which will take place towards the end of January and comprise 15% of the final mark. MAS352 is assessed entirely by exam.
Website
Further information, including the timetable, can be found on