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The Valuation and Calibration of Convertible Bonds

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The Valuation and Calibration of Convertible Bonds The valuation and calibration of convertible bonds by Sanveer Hariparsad Submitted in partial fulfillment of the requirements for the degree Magister Scientiae in the Department of Mathematics and Applied Mathematics in the Faculty of Natural and Agricultural Sciences University of Pretoria Pretoria Feb 2009 © University of Pretoria DECLARATION I, the undersigned, hereby declare that the dissertation submitted herewith for the degree Magister Scientiae to the University of Pretoria contains my own, independent work and has not been submitted for any degree at any other university. _____________________________ (Signature of candidate) Sanveer Hariparsad ____________________Day of _________________________20________________ ii ABSTRACT The valuation and calibration and of convertible bonds Sanveer Hariparsad Magister Scientae Department of Mathematics and Applied Mathematics University of Pretoria 2009 A convertible bond (CB) is a hybrid security possessing the characteristics of both debt and equity. It gives the holder the right to convert the bond into a pre-specified number of shares (usually by the same issuer of the CB) until maturity of the bond, and may also contain additional features such as callability and putability. CB’s along with all hybrid securities are difficult to value due to their uncertain income stream. In this dissertation several convertible bond valuation models are suggested, but with particular attention to the calibration of the underlying inputs into the model and also by taking default risk into account, which is extremely important given the sub- ordination of convertibles. The models range from the basic component models that decompose the CB into a straight bond and an exchange/call option; to more sophisticated ones consisting of stochastic interest rates, default risk, volatility structures, and even some exotics such as exchangeable and inflation-linked convertibles. An important aspect often missed by CB valuation models is the presence of negative convexity for extremely low share prices. As such a credit spread function dependent upon the underlying share price is introduced into the Tsiveriotis and Fernandes, and Hung and Wang models which improve upon the accuracy of the original models. Once a reliable model has been developed it becomes necessary to take advantage of convertible arbitrage trading strategies if they exist. The typical delta hedge, gamma hedge and option strategies that many convertible hedge funds employ are explained including the underlying risks with respect to the “Greeks”. iii ACKNOWLEDGEMENTS I would like to express my deep appreciation, understanding and sincere gratitude to my supervisor Professor Eben Mare for accepting me into the Financial Engineering course under trying circumstances, but mostly for his support, patience and guidance throughout my research and coursework. I was also fortunate to seek expert advice from my fellow Fixed Income team members at Old Mutual Investment Group South Africa, and gather the necessary Bloomberg data. Finally I would like to thank my parents and brother for their ever-lasting support and encouragement. iv Contents DECLARATION ........................................................................................................... ii ABSTRACT ..................................................................................................................iii LIST OF FIGURES ....................................................................................................... 2 1. Introduction .......................................................................................................... 7 1.1 Objectives ........................................................................................................................ 7 1.2 General Introduction .................................................................................................. 8 1.3 Literature Review .................................................................................................... 10 1.3.1 Credit Risk ....................................................................................................... 12 1.3.2 Valuation Models – Lattice ............................................................................. 13 1.3.3 Optimal Call and Conversion Strategies.......................................................... 14 1.3.4 Phases of Convertible Bonds ........................................................................... 15 2. Simple Piecewise Models .................................................................................... 17 2.1 Component Model ................................................................................................... 18 2.2 Margrabe Model – European Exchange .................................................................. 20 2.3 Margrabe Model – American Exchange .................................................................. 22 2.4 Parameter Sensitivities ............................................................................................ 23 2.5 Component Model ................................................................................................... 26 2.5.1 Basic Model ..................................................................................................... 27 2.5.2 Default Risk .................................................................................................... 31 2.5.3 Coupon Bearing Bonds ................................................................................... 32 2.5.4 Optimal Conversion Time T2 .......................................................................... 33 2.5.5 Loss of Value Due to Forced Early Conversion ............................................. 35 2.5.6 Numerical Example ......................................................................................... 36 3. Binomial Trees .................................................................................................... 39 3.1 Share Price ............................................................................................................... 39 3.2 Interest Rates ........................................................................................................... 41 3.3 Calibration of Interest Rate Trees ............................................................................ 42 3.3.1 Simple Term Structure Calibration .................................................................. 43 3.3.2 Arrow Debreu Calibration ............................................................................... 43 4. Credit Risk .......................................................................................................... 47 4.1 Introducing Default Probabilities ............................................................................ 47 4.2 Calibrating Risk-Free Interest Rates (Quick Review) ............................................. 48 4.3 Calibrating Risky Interest Rates (Derivation of λµt) ................................................ 49 v 5. Equity Valuation Models ................................................................................... 52 5.1 Background to Equity Valuation Models ................................................................ 52 5.2 Quadrinomial Model ............................................................................................... 53 5.2.1 Constructing the Quadrinomial Tree ............................................................... 53 5.2.2 Correlated Rollback Value .............................................................................. 55 5.2.3 Backward Valuation Process .......................................................................... 57 5.2.4 Parameter Sensitivities ................................................................................... 58 5.3 Tsiveriotis and Fernandes ........................................................................................ 64 5.3.1 The Model ....................................................................................................... 65 5.3.2 Arbitrage Free Approach ................................................................................. 67 5.3.3 Probability of Default Approach ..................................................................... 68 5.3.4 Present Value Approach .................................................................................. 70 5.3.5 Boundary Conditions ....................................................................................... 72 5.3.6 Numerical Example ............................................................................................. 73 5.4 Ayache, Forsyth and Vetzal..................................................................................... 78 5.4.1 Basic Model without Credit Risk .................................................................... 78 5.4.2 Derivation of Risky Bond ................................................................................ 80 5.4.3 Basic Model with Credit Risk ......................................................................... 82 5.4.4 Inconsistencies with TF Model ........................................................................ 85 5.5 Goldman-Sachs........................................................................................................ 89 5.5.1 The Model ....................................................................................................... 89 5.5.2 Numerical Example ........................................................................................
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