Benoîte de Saporta - Martingales and Financial Mathematics in Discrete Time

1 Cover

2 Title Page Series EditorNikolaos Limnios

3 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. 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: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Benoîte de Saporta and Mounir Zili to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021944040 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-669-2

4 Preface

5 Introduction

6 1 Elementary Probabilities and an Introduction to Stochastic Processes 1.1. Measures and σ -algebras 1.2. Probability elements 1.3. Stochastic processes 1.4. Exercises

7 2 Conditional Expectation 2.1. Conditional probability with respect to an event 2.2. Conditional expectation 2.3. Geometric interpretation 2.4. Conditional expectation and independence 2.5. Exercises

8 3 Random Walks 3.1. Trajectories of the random walk 3.2. Asymptotic behavior 3.3. The Gambler’s ruin 3.4. Exercises

9 4 Martingales 4.1. Definition 4.2. Martingale transform 4.3. The Doob decomposition 4.4. Stopping time 4.5. Stopped martingales 4.6. Exercises

10 5 Financial Markets 5.1. Financial assets 5.2. Investment strategies 5.3. Arbitrage 5.4. The Cox, Ross and Rubinstein model 5.5. Exercises 5.6. Practical work

11 6 European Options 6.1. Definition 6.2. Complete markets 6.3. Valuation and hedging 6.4. Cox, Ross and Rubinstein model 6.5. Exercises 6.6. Practical work: Simulating the value of a call option

12 7 American Options 7.1. Definition 7.2. Optimal stopping 7.3. Application to American options 7.4. The Cox, Ross and Rubinstein model 7.5. Exercises 7.6. Practical work

13 8 Solutions to Exercises and Practical Work8.1. Solutions to exercises in Chapter 1 8.2. Solutions to exercises in Chapter 2 8.3. Solutions to exercises in Chapter 3 8.4. Solutions to exercises in Chapter 4 8.5. Solutions to exercises in Chapter 5 8.6. Solutions to the practical exercises in Chapter 5 8.7. Solutions to exercises in Chapter 6 8.8. Solution to the practical exercise in Chapter 6 (section 6.6) 8.9. Solution to exercises in Chapter 7 8.10. Solution to the practical exercise in Chapter 7 (section 7.6)

14 References

15 Index

16 End User License Agreement

1 Chapter 3Figure 3.1. Graphical representation of a trajectory of a random walk between 0 ...Figure 3.2. Two paths from (1, 1) to (5, 3). For a color version of this figure,...Figure 3.3. A path from (0, 2) to (11, 1) passing through 0 (the unbroken blue l...

2 Chapter 8Figure 8.1. Possible trajectories for the random walk of four steps starting fro...Figure 8.2. Possible paths from (0, 0) to (3, 1). For a color version of this fi...Figure 8.3. Event tree for the financial market in Exercise 5.1 Figure 8.4. Event tree for the financial market in Exercise 5.3 Figure 8.5. Trajectories of the risky asset for the Cox, Ross and Rubinstein mod...Figure 8.6. Trajectories of the risky asset (blue) and the risk-free asset (gray...Figure 8.7. Trajectory of the logarithm of the wealth for the optimal strategy (...Figure 8.8. Trajectories of the expectation of the logarithm of the wealth for t...Figure 8.9. Trajectories of the wealth for the investment-withdrawal strategy in...Figure 8.10. Trajectories of the wealth for the investment-withdrawal strategy i...Figure 8.11. Trajectories of the logarithm of the cumulative withdrawal for the ...Figure 8.12. Trajectories of the expectation of the cumulative sum of the logari...Figure 8.13. Trajectories of the risky asset (blue) and the value of the Europea...Figure 8.14. Trajectories of the payoff for the American option with maturity da...Figure 8.15. Trajectories for the payoff (red) and for the value (blue) of an Am...

1 Chapter 8Table 8.1. Distribution of the random variable S Table 8.2. Distribution of the random variable X 7Table 8.3. The distribution of the random variable X 6Table 8.4. Values for the probability p of satisfying all moviegoers based on th...Table 8.5. Trajectories of the payoff (Z) and of the value (U) of the American o...Table 8.6. Trajectories of the payoff (Z) and of the value (U) of the American o...

1 Cover

2 Table of Contents

3 Title Page Series EditorNikolaos Limnios

4 Copyright First published 2021 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc. 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: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK www.iste.co.uk John Wiley & Sons, Inc. 111 River Street Hoboken, NJ 07030 USA www.wiley.com © ISTE Ltd 2021 The rights of Benoîte de Saporta and Mounir Zili to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988. Library of Congress Control Number: 2021944040 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-669-2

5 Preface

6 Introduction

7 Begin Reading

8 References

9 Index

10 End User License Agreement

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