Stochastic Models of Financial Mathematics

By Vigirdas Mackevicius

This book presents a short introduction to continuous-time financial models. An overview of the basics of stochastic analysis precedes a focus on the Black–Scholes and interest rate models. Other topics covered include self-financing strategies, option pricing, exotic options and risk-neutral probabilities. Vasicek, Cox-Ingersoll-Ross, and Heath–Jarrow–Morton interest rate models are also explored. The author presents practitioners with a basic introduction, with more rigorous information provided for mathematicians. The reader is assumed to be familiar with the basics of probability theory. Some basic knowledge of stochastic integration and differential equations theory is preferable, although all preliminary information is given in the first part of the book. Some relatively simple theoretical exercises are also provided.

• About continuous-time stochastic models of financial mathematics
• Black-Sholes model and interest rate models
• Requiring a minimum knowledge of stochastic integration and stochastic differential equations

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 8, 2016
• ISBN: 9780081020869

Vigirdas Mackevicius

Vigirdas Mackevicius is Professor of the Department of Mathematical Analysis in the Faculty of Mathematics and Informatics of Vilnius University in Lithuania. His research interests include stochastic processes, stochastic analysis, and stochastic numerics.

Book preview

Stochastic Models of Financial Mathematics

Vigirdas Mackevičius

Optimization in Insurance and Finance Set

coordinated by

Nikolaos Limnios

Yuliya Mishura

Table of Contents

Cover image

Title page

Copyright

Preface

Notations

1: Overview of the Basics of Stochastic Analysis

Abstract

1.1 Brownian motion

1.2 Stochastic integrals

1.3 Martingales, Itô processes and general Itô’s formula

1.4 Stochastic differential equations

1.5 Change of probability: the Girsanov theorem

2: The Black–Scholes Model

Abstract

2.1 Introduction: what is an option?

2.2 Self-financing strategies

2.3 Option pricing problem: the Black–Scholes model

2.4 The Black–Scholes formula

2.5 Risk-neutral probabilities: alternative derivation of the Black–Scholes formula

2.6 American options in the Black–Scholes model

2.7 Exotic options

3: Models of Interest Rates

Abstract

3.1 Modeling principles

3.2 The Vašíček model

3.3 The Cox–Ingersoll–Ross model

3.4 The Heath–Jarrow–Morton model

Bibliography

Index

Copyright

First published 2016 in Great Britain and the United States by ISTE Press Ltd and Elsevier Ltd

Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address:

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Notices

Knowledge and best practice in this field are constantly changing. As new research and experience broaden our understanding, changes in research methods, professional practices, or medical treatment may become necessary.

Practitioners and researchers must always rely on their own experience and knowledge in evaluating and using any information, methods, compounds, or experiments described herein. In using such information or methods they should be mindful of their own safety and the safety of others, including parties for whom they have a professional responsibility.

To the fullest extent of the law, neither the Publisher nor the authors, contributors, or editors, assume any liability for any injury and/or damage to persons or property as a matter of products liability, negligence or otherwise, or from any use or operation of any methods, products, instructions, or ideas contained in the material herein.

For information on all our publications visit our website at http://store.elsevier.com/

© ISTE Press Ltd 2016

The rights of Vigirdas Mackevičius to be identified as the author of this work have been asserted by him in accordance with the Copyright, Designs and Patents Act 1988.

British Library Cataloguing-in-Publication Data

A CIP record for this book is available from the British Library

Library of Congress Cataloging in Publication Data

A catalog record for this book is available from the Library of Congress

ISBN 978-1-78548-198-7

Printed and bound in the UK and US

Preface

Vigirdas Mackevičius, Vilnius

These lecture notes are based on a graduate course given for several years at Vilnius University as part of the Master’s program Financial and Actuarial Mathematics. They are intended to give a short introduction to continuous-time financial models including Black–Scholes and interest rate models. We assume the reader to be familiar with the basics of probability theory in the scope of a standard elementary course. Some basic knowledge of stochastic integration and differential equations theory is preferable, although, formally, all the preliminary information is given in part 1 of the lecture notes.

Though a large number of books and textbooks have influenced the writing of these notes, the short reference list includes only the literature directly used by the author.

The author would like to thank a large number of master’s students of the Faculty of Mathematics and Informatics of Vilnius University. Thanks to them, the book contains significantly fewer mistakes.

September 2016

Notations

 The set of positive integers {1, 2, …}

∪{+∞}

+∪{0}

 Real line (−∞, +∞)

∪ {−∞, +∞}

+ The set of non-negative real numbers [0, +∞)

x∨ y max{x, y}

x∧ y min{x, y}

1A The indicator of a set (an event) A: 1A(x) = 1 for x ∈ A; 1A(x) = 0 for x ∈ Ac

E(X), EX The expectation (or mean) of a random variable X

Var(X), Var X The variance of a random variable X

N(a,σ²) Normal distribution with expectation a and variance σ²

X ~ N(a, σ²) A random variable X with distribution N(a, σ²)

φ The standard normal density (the probability density of X ∈ N , x

, x

 Random variables X and Y are identically distributed

X Y Random variables X and Y are independent

t The history (or the past) of Brownian motion B up to moment t (Definition 1.2)

H²[0,T] The set of adapted processes X = {Xt, t ∈ [0, T (Definition 1.3)

Ĥ²[0, T] The set of adapted processes X = {Xt, t ∈ [0, T(Definition 1.6)

 The stochastic (Itô) integral of a process Y with respect to an Itô process X (Definitions 1.5 and 1.11)

X, Y  The (quadratic) covariation of Itô processes X and Y (Definition 1.12)

X  =  X, X  The quadratic variation of an Itô process X (Definition 1.12)

1

Overview of the Basics of Stochastic Analysis

Abstract

, P).

Keywords

Brownian motion; Girsanov theorem; Integrals; Ito’s formula; Martingales; Probability; Stochastic Analysis

1.1 Brownian motion

,