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Hedging Credit Risk Using Equity Derivatives

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Hedging Credit Risk Using Equity Derivatives Hedging Credit Risk Using Equity Derivatives Paul Teixeira School of Computational and Applied Mathematics University of the Witwatersrand A dissertation submitted for the degree of Master of Science in Advanced Mathematics of Finance 2007 Declaration I declare that this is my own, unaided work. It is being submitted for the Degree of Master of Science to the University of the Witwatersrand, Johannesburg. It has not been submitted before for any degree or examination to any other University. (Signature) (Date) Abstract Equity and credit markets are often treated as independent markets. In this dissertation our objec- tive is to hedge a position in a credit default swap with either shares or share options. Structural models enable us to link credit risk to equity risk via the ¯rm's asset value. With an extended version of the seminal Merton (1974) structural model, we value credit default swaps, shares and share options using arbitrage pricing theory. Since we are interested in hedging the change in value of a credit default swap dynamically, we use a jump-di®usion model for the ¯rm's asset value in order to model the short term credit risk dynamics more accurately. Our mathematical model does not admit an explicit solutions for credit default swaps, shares and share options, thus we use a Brownian Bridge Monte Carlo procedure to value these ¯nancial products and to compute the delta hedge ratios. These delta hedge ratios measure the sensitivity of the value of a credit default swap with respect to either share or European share option prices. We apply these delta hedge ratios to simulated and market data, to test our hedging objective. The hedge performs well for the simulated data for both cases where the hedging instrument is either shares or share options. The hedging results with market data suggests that we are able to hedge the value of a credit default swap with shares, however it is more di±cult with share options. Acknowledgements I would like to express my sincere appreciation to my supervisors, David Taylor and Lawrence Madzwara, for providing me the opportunity to do this research and for their helpful insights into aspects of the dissertation. I gratefully appreciate Dr. Jordens for editing my dissertation. I would also like to thank my parents, family and fellow students for their support and encouragement. Contents 1 Introduction 1 1.1 Background to the Credit Derivatives Market . 1 1.2 Credit Risk and Equity . 2 1.3 Credit Risk Models . 3 1.4 Research Objectives . 4 1.5 Structure of the Dissertation . 5 2 Mathematical Framework 6 2.1 Introduction . 6 2.2 Arbitrage Pricing Theory . 6 2.2.1 Financial Market Model . 6 2.2.2 Equivalent Martingale Measure . 8 2.2.3 Arbitrage-Free Pricing of Contingent Claims . 11 2.3 Financial Securities Dependent on the Default Event . 12 2.3.1 General Credit Risk Model . 12 2.3.2 Arbitrage-Free Valuation Formula . 13 2.3.3 Reduced-Form Model . 13 2.3.4 Structural Model . 14 2.4 Discussion . 14 3 Credit Default Swaps 15 3.1 Introduction . 15 3.2 Credit Risk . 15 3.3 Credit Ratings . 16 3.4 Credit Derivatives . 16 3.5 Credit Default Swap . 16 3.6 Replication-Based CDS Pricing . 18 3.6.1 Replication Instruments . 18 3.6.2 Short Positions in Bonds . 19 3.6.3 CDS Replicating Strategies . 20 3.7 Bond Price-Based CDS Pricing . 22 3.7.1 Bond Price-Based Framework . 23 3.7.2 Extensions to Situation where Defaults can happen at any Time . 24 3.7.3 CDS Pricing . 25 3.8 Generic CDS Pricing Model . 26 3.9 Discussion . 27 4 Structural Models 28 4.1 Introduction . 28 4.2 The Merton Model . 28 4.3 Extensions to the Merton Model . 33 4.3.1 Capital Structure . 33 4.3.2 Default Barrier . 34 4.3.3 Stochastic Interest Rates . 36 4.3.4 Unpredictable Default Time . 37 4.3.5 Discontinuous Firm's Asset Value Process . 37 4.4 Parameter estimation . 38 4.5 Our Model . 42 ii CONTENTS iii 4.6 Discussion . 45 5 Pricing and Estimation 46 5.1 Introduction . 46 5.2 Principles of Monte Carlo Methods . 46 5.3 Brownian Bridge . 49 5.4 A Brownian Bridge Simulation Procedure for Pricing CDS Premiums . 50 5.4.1 Model Description and the CDS Pricing Formula . 51 5.4.2 Outline of the Monte Carlo Brownian Bridge Method to Price CDS premiums 52 5.4.3 Conditional First Passage Time Distribution . 53 5.4.4 Decomposition of the CDS Pricing Formula . 54 5.4.5 Sampling a First Hitting Time in an Inter-Jump Interval . 57 5.4.6 The Simulation Algorithm to Price the Premium of a CDS . 58 5.4.7 The Brownian Bridge Simulation Algorithm to Value a CDS . 59 5.4.8 The Brownian Bridge Simulation Algorithm to Price Equity . 60 5.4.9 Numerical Results of the Brownian Bridge Simulation Algorithm . 61 5.5 Valuing Equity Options by a Monte Carlo Linear Regression Approach . 63 5.5.1 The LSM Algorithm . 64 5.5.2 Numerical Comparison of Basis Functions . 65 5.6 Estimation of parameters . 66 5.6.1 The Nelder-Mead Simplex Method . 67 5.6.2 The Nelder-Mead Algorithm . 67 5.7 Discussion . 69 6 Numerical Analysis 70 6.1 Introduction . 70 6.2 E®ects of Credit Risk on CDS, Equity and Equity Derivatives . 70 6.3 Hedging . 71 6.4 Delta Hedging . 74 6.4.1 A Measure for the E±ciency of a Delta Hedge . 76 6.4.2 Gamma Hedge . 77 6.5 Empirical Tests . 78 6.5.1 Method . 79 6.5.2 Testing the Delta Hedging Strategy with Simulated Data . 81 6.5.3 Data . 82 6.5.4 Results . 83 6.6 Conclusion . 92 6.7 Further Research Directions . 93 A Jump Processes: Miscellaneous Results 94 A.1 It^o'sFormula for Di®usions with Jumps . 94 A.2 Girsanov Theorem for Jump-Di®usion Processes . 95 A.3 The Firm's Value Process under the Risk-Neutral Measure . 96 B First Passage Time 98 B.1 The Hitting Time of a Standard Brownian Motion Process . 98 B.1.1 Distribution of the Pair (Wt;Mt)........................ 98 ¤ B.1.2 Distribution of Mt and ¿M; b ........................... 100 B.1.3 The Distribution of the pair (Wt; mt)..................... 100 ¤ B.1.4 Distribution of mt and ¿m; b ........................... 101 B.2 The First-Hitting Time for a Brownian Motion with Drift . 101 B.3 The First-Hitting Time of a Geometric Brownian Motion Process V to a Lower Barrier b .......................................... 102 C Miscellaneous 104 D Hedging Results 105 D.1 Simulated Delta Hedge Results . 105 D.2 Empirical Delta Hedge Results . 108 D.2.1 Option Descriptions Details and Hedging Time Intervals . 131 Chapter 1 Introduction 1.1 Background to the Credit Derivatives Market The rise in credit related events (i.e. bankruptcy of ¯rms, default and deterioration of the credit quality of corporate and government bonds) throughout the ¯nancial world over the past decade has been met with a commensurate increase in interest in derivatives which depend in value solely on credit risk1. Until recently there did not exist risk management products for credit risk. The market for credit derivatives2 began slowly in the early 1990's, sparked by the need to keep pace with the increasing demand for other over-the-counter derivative products. The sheer volume of business swamping the market meant that ¯nancial institutions could no longer rely on taking collateral or making loss provisions to manage their credit risk. They strived for a way to lay o® illiquid credit exposures and to protect themselves against defaulting clients. Traditionally, credit was sourced in the new issue market and placed with the end investors, where it remained irrevocably linked to the asset with which it was originally associated. With the innovation of credit derivatives it is now possible to separate credit risk from any ¯nancial obligation (i.e. loans, bonds, swaps) and manage it just like any other asset. The credit derivative tool kit has revolutionized credit risk management and fundamentally modernised the credit market. Before the advent of credit derivatives, credit risk management meant a strategy of portfolio diversi¯cation backed by credit line limits, with occasional sale of positions in the secondary market. These strategies are ine±cient, largely because they do not separate the management of risk from the asset with which the credit risk is associated. Credit derivatives are signi¯cant as they allow the risk manager to isolate credit risk and manage it discreetly without interfering with customer relationships. They also form the ¯rst mechanism via which short sales of credit instruments can be performed with reasonable liquidity. For example, it is impossible to short-sell a bank loan, however this can be achieved synthetically by buying a credit derivative o®ering credit protection. Credit derivatives allow investors to be synthetically exposed to credit assets that were unavailable before, either because.
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