Anatoliy Swishchuk - Random Motions in Markov and Semi-Markov Random Environments 2

1 Cover

2 Title Page Series Editor Nikolaos 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 Anatoliy Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino 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: 2020946634 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-706-4

4 Preface

5 Acknowledgments

6 Introduction I.1. Overview I.2. Description of the book

7 PART 1: Higher-dimensional Random Motions and Interactive Particles 1 Random Motions in Higher Dimensions 1.1. Random motion at finite speed with semi-Markov switching directions process 1.2. Random motion with uniformly distributed directions and random velocity 1.3. The distribution of random motion at non-constant velocity in semi-Markov media 1.4. Goldstein–Kac telegraph equations and random flights in higher dimensions 1.5. The jump telegraph process in Rn 2 System of Interactive Particles with Markov and Semi-Markov Switching 2.1. Description of the Markov model 2.2. Interaction of particles governed by generalized integrated telegraph processes: a semi-Markov case

8 PART 2: Financial Applications 3 Asymptotic Estimation for Application of the Telegraph Process as an Alternative to the Diffusion Process in the Black–Scholes Formula 3.1. Asymptotic expansion for the singularly perturbed random evolution in Markov media in the case of disbalance 3.2. Application: Black–Scholes formula 4 Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Markov-modulated Volatilities 4.1. Volatility derivatives 4.2. Martingale representation of a Markov process 4.3. Variance and volatility swaps for financial markets with Markov-modulated stochastic volatilities 4.4. Covariance and correlation swaps for two risky assets for financial markets with Markov-modulated stochastic volatilities 4.5. Example: variance, volatility, covariance and correlation swaps for stochastic volatility driven by two state continuous Markov chain 4.6. Numerical example 4.7. Appendix 1 5 Modeling and Pricing of Variance, Volatility, Covariance and Correlation Swaps for Financial Markets with Semi-Markov Volatilities 5.1. Introduction 5.2. Martingale representation of semi-Markov processes 5.3. Variance and volatility swaps for financial markets with semi-Markov stochastic volatilities 5.4. Covariance and correlation swaps for two risky assets in financial markets with semi-Markov stochastic volatilities 5.5. Numerical evaluation of covariance and correlation swaps with semi-Markov stochastic volatility 5.6. Appendices

9 References

10 Index

11 Summary of Volume 1

12 End User License Agreement

1 Chapter 4Table 4.1. One-step transition probability matrixTable 4.2. One-step transition probability matrix

1 Chapter 1Figure 1.1. Approximated probability density function f 3(t, x) − r(t, x) for λ =...Figure 1.2. f(t, x) for λ = 2 and v = 3, according to Garra and Orsingher (2014)...Figure 1.3. Integration of f 3(t, x) − r(t, x) and f(t, x) for λ = 2 and v = 2. T...Figure 1.4. Integration of f 3(t, x) − r(t, x) and f(t, x) for λ = 0.2 and v = 2....Figure 1.5. Integration of f 3(t, x) − r(t, x) and f(t, x) for λ = 2 and v = 0.2....Figure 1.6. Integration of f 3(t, x) and f(t, x) for λ = 0.2 and v = 0.2. The sin...

2 Chapter 3Figure 3.1. Dependence of European call option price on v (left) and λ (right)Figure 3.2. Dependence of European call option price on v and λ. For a color ver...

3 Chapter 4Figure 4.1. Variance and volatility swap prices. For a color version of this fig...Figure 4.2. Variance and volatility swap prices. For a color version of this fig...

1 Cover

2 Table of Contents

3 Title page

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 Anatoliy Pogorui, Anatoliy Swishchuk and Ramón M. Rodríguez-Dagnino 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: 2020946634 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-78630-706-4

5 Preface

6 Acknowledgments

7 Introduction

8 Begin Reading

9 References

10 Index

11 Summary of Volume 1

12 End User License Agreement

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