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Valuation of Callable Convertible Bonds Using Binomial Trees Model with Default Risk, Convertible Hedging and Arbitrage, Duration and Convexity

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Valuation of Callable Convertible Bonds Using Binomial Trees Model with Default Risk, Convertible Hedging and Arbitrage, Duration and Convexity A University of Sussex PhD thesis Available online via Sussex Research Online: http://sro.sussex.ac.uk/ This thesis is protected by copyright which belongs to the author. This thesis cannot be reproduced or quoted extensively from without first obtaining permission in writing from the Author The content must not be changed in any way or sold commercially in any format or medium without the formal permission of the Author When referring to this work, full bibliographic details including the author, title, awarding institution and date of the thesis must be given Please visit Sussex Research Online for more information and further details Valuation of Callable Convertible Bonds Using Binomial Trees Model with Default Risk, Convertible Hedging and Arbitrage, Duration and Convexity. Fahad Aldossary Submitted for the degree of Doctor of Philosophy University of Sussex January 2018 II Declaration I hereby declare that this thesis has not been and will not be, submitted in whole or in part to another University for the award of any other degree. Signature: Fahad Aldossary III UNIVERSITY OF SUSSEX FAHAD ALDOSSARY, SUBMITTED FOR DOCTOR OF PHILOSOPHY Valuation of Callable Convertible Bonds Using Binomial Trees Model with Default Risk, Convertible Hedging and Arbitrage, Duration and Convexity Abstract In this thesis, I develop a valuation model to price convertible bonds with call provision. Convertible bonds are hybrid instruments that possess both equity and debt characteristics. The purpose of this study is to build a pricing model for convertible and callable bonds and to compare the mathematical results of the model with real world market performance. I construct a two-factor valuation model, in which both the interest rate and the stock price are stochastic. I derive the partial differential equation of two stochastic variables and state the final and boundary conditions of the convertible bond using the mean reversion model on interest rate. Because it is difficult to obtain a closed solution for the American convertible bond due to its structural complexity, I use the binomial tree model to value the convertible bond by constructing the interest rate tree and stock price tree. As a convertible bond is a hybrid security of debt and equity, I combine the interest rate tree and stock price tree into one single tree. Default risk is added to the valuation tree to represent the event of a default. The model is then tested and compared with the performance of the Canadian convertible bond market. Moreover, I study the duration, convexity and Greeks of convertible bonds. These are important risk metrics in the portfolio management of the convertible bond to measure risks linked to interest rate, equity, volatility and other market factors. I investigate the partial derivative of the value of the convertible bond with respect to various parameters, such as the interest rate, stock price, volatility of the interest rate, volatility of the stock price, mean reversion of the interest rate and dividend yield of the underlying stock. A IV convertible bond arbitrage portfolio is constructed to capture the abnormal returns from the Delta hedging strategy and I describe the risks associated with these returns. The portfolio is created by matching long positions in convertible bonds, with short positions in the underlying stock to create a Delta hedged convertible bond position, which captures income and volatility. V Acknowledgements This thesis would have never been achieved without the help and blessing of God and the support of my family, friends, and supervisors. I would like to express my special thanks to my supervisor, Dr. Qi Tang, for his support, guidance, and supervision and for providing me the opportunity and encouragement to accomplish this work. Dr. Tang has provided me with great suggestions, helpful comments, and valuable academic assistance during the time of my PhD course. I would like to thank my friends and colleagues, especially Linghua Zhang for her programming support. I would like to thank my parents, brothers, and sisters for their care and prayers. Finally, I would like to thank the University of Taibah and the former Dean of the Business School, Dr. Rayan Hammad, who granted me the opportunity of this scholarship and supported me during my PhD research. VI CONTENTS Acknowledgements ........................................................................................ V 1 Introduction ............................................................................................. 1 1.1 Introduction to the convertible bond valuation model ............................ 1 1.2 Literature review ...................................................................................... 12 1.3 Data ........................................................................................................... 20 2 Interest rate model ................................................................................. 22 2.1 The Vasicek model .................................................................................... 22 2.2 Vasicek zero-coupon bond pricing ........................................................... 29 3 Stock Price Model ................................................................................... 30 3.1 CRR model ................................................................................................. 30 4 Convertible bonds pricing model involving two-stochastic factors .......... 34 4.1 Deriving the PDE for the convertible bond option .................................. 34 4.2 Conditions and solutions .......................................................................... 37 4.3 American convertible bond ...................................................................... 39 4.4 Two-stochastic-factor tree model............................................................ 40 4.4.1 Soft call and hard Call ..................................................................... 45 4.4.2 Call and convert conditions ............................................................ 46 4.5 Two-stochastic-factor tree model with default risk ............................... 47 4.5.1 Credit spread (풓풄) ........................................................................... 49 4.5.2 Default risk probability (흀) ............................................................. 50 4.6 Numerical example................................................................................... 53 4.6.1 Option-free convertible bond ......................................................... 53 4.6.2 Callable convertible bond subject to default risk .......................... 61 4.6.3 Conclusion of the numerical examples .......................................... 68 4.6.4 Further numerical examples ........................................................... 70 4.6.5 Monthly spacing numerical examples ............................................ 71 5 Duration and convexity of convertible bonds .......................................... 74 5.1 Introduction to duration and convexity .................................................. 74 5.2 Duration .................................................................................................... 75 5.3 Convexity .................................................................................................. 81 VII 5.4 Duration and convexity of the European zero-coupon convertible bond 83 5.4.1 The sensitivity of the zero-coupon bond price and duration to interest rate changes .................................................................................. 85 5.4.2 The sensitivity to the interest rate volatility 흈풓 ............................ 85 5.4.3 The impact of stock price changes on the duration and convertible bonds 87 5.4.4 The sensitivity to the stock price volatility 흈풔 ............................... 88 5.4.5 The sensitivity to the long-run rate 휽 ............................................ 88 5.4.6 The sensitivity to the mean reversion rate 풌 ................................. 89 5.4.7 Dividend yield.................................................................................. 89 5.5 Convexity .................................................................................................. 90 5.5.1 The sensitivity of convexity to stock prices. .................................. 91 5.5.2 The sensitivity of convexity to the interest rate. ........................... 92 5.6 The Greeks of the convertible bond ........................................................ 92 5.6.1 Delta of the convertible bond price ............................................... 93 5.6.2 Gamma of the convertible bond price ........................................... 95 6 Convertible Delta arbitrage .................................................................... 97 6.1 Introduction to convertible bond arbitrage ............................................ 97 6.2 Convertible bond arbitrage and portfolio construction ......................... 99 6.2.1 Delta of the binomial tree .............................................................. 99 6.2.2 Delta of the Black-Scholes model ................................................. 103 6.3 Data ......................................................................................................... 105 6.4 Results ..................................................................................................... 106 6.4.1 Results of the binomial tree method ........................................... 106 6.4.2 Results of the Black-Scholes
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