Advanced quantitative finance with c++

www.it-ebooks.info Advanced Quantitative Finance with C++ Create and implement mathematical models in C++ using Quantitative Finance Alonso Peña, Ph.D BIRMINGHAM - MUMBAI www.it-ebooks.info Advanced Quantitative Finance with C++ Copyright © 2014 Packt Publishing All rights reserved No part of this book may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, without the prior written permission of the publisher, except in the case of brief quotations embedded in critical articles or reviews Every effort has been made in the preparation of this book to ensure the accuracy of the information presented However, the information contained in this book is sold without warranty, either express or implied Neither the author, nor Packt Publishing, and its dealers and distributors will be held liable for any damages caused or alleged to be caused directly or indirectly by this book Packt Publishing has endeavored to provide trademark information about all of the companies and products mentioned in this book by the appropriate use of capitals However, Packt Publishing cannot guarantee the accuracy of this information First published: June 2014 Production reference: 1180614 Published by Packt Publishing Ltd Livery Place 35 Livery Street Birmingham B3 2PB, UK ISBN 978-1-78216-722-8 www.packtpub.com Cover image by VTR Ravi Kumar (vtrravikumar@gmail.com) [ FM-2 ] www.it-ebooks.info Credits Author Project Coordinator Alonso Peña, Ph.D Harshal Ved Reviewers Proofreader Marco Airoldi Clyde Jenkins Joseph Smidt Graphics Sheetal Aute Commissioning Editor Grant Mizen Ronak Dhruv Valentina Dsilva Acquisition Editor Disha Haria Harsha Bharwani Abhinash Sahu Content Development Editor Amit Ghodake Indexer Hemangini Bari Technical Editor Production Coordinator Humera Shaikh Kyle Albuquerque Copy Editor Cover Work Laxmi Subramanian Nilesh Bambardekar [ FM-3 ] www.it-ebooks.info About the Author Alonso Peña, Ph.D is an SDA Professor at the SDA Bocconi School of Management in Milan He has worked as a quantitative analyst in the structured products group for Thomson Reuters Risk and for Unicredit Group in London and Milan He holds a Ph.D degree from the University of Cambridge on Finite Element Analysis and the Certificate in Quantitative Finance (CQF) from 7city Learning, the U.K He has lectured and supervised graduate and post-graduate students from the universities of Oxford, Cambridge, Bocconi, Bergamo, Pavia, Castellanza, and the Politecnico di Milano His area of expertise is the pricing of financial derivatives, in particular, structured products He has publications in the fields of Quantitative Finance, applied mathematics, neuroscience, and the history of science He has been awarded the Robert J Melosh Medal—first prize for the best student paper on Finite Element Analysis, Duke University, USA; and the Rouse Ball Travelling Studentship in Mathematics, Trinity College, Cambridge He has been to the Santa Fe Institute, USA, to study complex systems in social sciences His publications include the following: • The One Factor Libor Market Model Using Monte Carlo Simulation: An Empirical Investigation • On the Role of Behavioral Finance in the Pricing of Financial Derivatives: The Case of the S&P 500 • Option Pricing with Radial Basis Functions: A Tutorial • Application of extrapolation processes to the finite element method • On the Role of Mathematical Biology in Contemporary Historiography He is currently working as a tutor for CQF (Fitch Learning) and a visiting faculty for the Indian Institute for Quantitative Finance, Mumbai He lives in Italy with his wife Marcella, his daughters Francesca and Isabel, and his son Marco [ FM-4 ] www.it-ebooks.info Acknowledgments I would like to thank many people who have made this book a reality First the magnificent support, enthusiasm, and patience of the entire team at Packt Publishing, particularly Harsha, Amit, Humera, and Harshal To Dr Pattabi Raman (Numerical Solution (U.K.) Ltd.), for his expert advice on C++ To Dr Marco Airoldi for his knowledgeable and detailed review of the book To the SDA Bocconi School of Management including my colleagues and students from the MBA, graduate, and undergraduate courses To the many persons I have been privileged to work with and to teach from the Universities of Cambridge, Oxford, Bocconi, LIUC Castellanza, Bergamo, Pavia, and Politecnico di Milano The many extraordinary quants from the Certificate in Quantitative Finance, Fitch Learning, London, as well as from Unicredit Group and Thomson Reuters Finally, to my wife, Marcella, and my children, Francesca, Isabel, and Marco—you all always remind me that "The true voyage of discovery consists not in seeking new landscapes but in having new eyes to see" (Marcel Proust) [ FM-5 ] www.it-ebooks.info About the Reviewer Marco Airoldi received his Ph.D in Theoretical Condensed Matter Physics in 1995 from the International School for Advanced Studies (SISSA) He moved definitively to finance in 1999 Marco has been chosen as the head of financial engineering in one of the top financial institutions in Italy His expertise includes the Monte Carlo simulation for option pricing and pricing system architectures [ FM-6 ] www.it-ebooks.info www.PacktPub.com Support files, eBooks, discount offers, and more You might want to visit www.PacktPub.com for support files and downloads related to your book Did you know that Packt offers eBook versions of every book published, with PDF and ePub files available? 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PacktLib is Packt's online digital book library Here, you can access, read and search across Packt's entire library of books Why subscribe? • Fully searchable across every book published by Packt • Copy and paste, print and bookmark content • On demand and accessible via web browser Free access for Packt account holders If you have an account with Packt at www.PacktPub.com, you can use this to access PacktLib today and view nine entirely free books Simply use your login credentials for immediate access [ FM-7 ] www.it-ebooks.info www.it-ebooks.info Table of Contents Preface 1 Chapter 1: What is Quantitative Finance? Discipline – finance (financial derivatives) Discipline – mathematics Discipline – informatics (C++ programming) The Bento Box template 10 Summary 12 Chapter 2: Mathematical Models 13 Chapter 3: Numerical Methods 33 Equity 13 Foreign exchange 17 Interest rates 20 Short rate models 20 Market models 22 Credit 25 Structural models 26 Intensity models 28 Summary 31 The Monte Carlo simulation method Algorithm of the MC method Example of the MC method The Binomial Trees method Algorithm of the BT method Example of the BT method www.it-ebooks.info 34 35 37 39 39 42 corresponds to           We populate the initial forward rates in the left-most column and advance column-by-column to the right until we have all the values we need, as shown on the right For more details on the calculation, see the book by (Pelsser 2000) Note that even though an IRS can be priced "statically" (that is, without simulation), we use this example to give an idea of what are the steps that the LMM method requires for calculation Credit In credit derivatives modeling, the underlying is credit risk Modern methodologies of credit risk measurement can be grouped into two alternative approaches—the structural approach pioneered by (Merton 1974) and a reduced form approach utilizing intensity based models to estimate stochastic hazard rates, pioneered by various authors, including (Jarrow and Turnbull 1995), (Jarrow, Lando, and Turnbull 1997), and (Duffie and Singleton 1999) [ 25 ] www.it-ebooks.info Mathematical Models Structural models The structural approach to credit risk assumes that a firm defaults when the market value of its assets is less than the obligations or debt it has to pay Structural models are, therefore, sometimes also referred to as asset value models These models look at a company's balance sheet and its capital structure to assess its creditworthiness However, one of the key problems with this approach is that the value of a company's assets is hard to observe directly The annual report only provides an accounting version of the company's real assets and not their market value For public companies, the equity is normally observable, as is its debt (Merton 1974) starts with the assumption of an extremely simplified capital structure of the following form: W ( W  'W Equation 10 In the preceding equation, V represents the value of the firm (the total of the assets of the firm), while E is its equity and D its debt The equity E is understood as the total value of the equity of the firm, which is equal to the market value of a share (stock) multiplied by the number of shares in the market For this simplified company, the debt is represented by a single zero coupon bond with maturity T At this point Merton asks the question "for a company with the preceding capital structure, when will it go in default?" Well, depends on our definition of default If we take as default the fact that the company cannot pay its obligations at some specific future time T, then this condition will be satisfied if the value of the company at time T, that is, V(T) is larger than the face value of debt D(T) At this moment in time, the bond holders will request payment and the company will be in position to cover it On the contrary, if at maturity, the value of the firm is less than the value of the debt it has to pay, it, therefore, will not be able to honor its obligations and will be in default These two scenarios can be defined mathematically as follows: LI 7 ! '7 WKHQ QR GHIDXOW LI 7  '7 WKHQ GHIDXOW [ 26 ] www.it-ebooks.info Chapter What about equity holders? They are in possession of the company's stock in the end Considering the preceding two scenarios, we then know that if the company goes in default, they receive nothing, while if the company continues to operate, they receive the difference between V(T) and D(T) at maturity, thus giving us the following equation: LI 7 ! ' WKHQ ( 7 7  ' ẵ ắ ( 7 LI 7  ' WKHQ ( 7 .. .Advanced Quantitative Finance with C++ Create and implement mathematical models in C++ using Quantitative Finance Alonso Peña, Ph.D BIRMINGHAM - MUMBAI www.it-ebooks.info Advanced Quantitative. .. Derivatives in C++ 51 Chapter 5: Foreign Exchange Derivatives with C++ 61 Chapter 6: Interest Rate Derivatives with C++ 75 Chapter 7: Credit Derivatives with C++ 89 Basic example – European Call Advanced. .. this book covers Chapter 1, What is Quantitative Finance? , gives a brief introduction to Quantitative Finance, delimits the subject to option pricing with C++, and describes the structure of