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Yuichi Katsura

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Yuichi Katsura P&TcrvvCJ Efficient ValtmVmm-and Hedging of Structured Credit Products Yuichi Katsura A thesis submitted for the degree of Doctor of Philosophy of the University of London May 2005 Centre for Quantitative Finance Imperial College London nit~ýl. ABSTRACT Structured credit derivatives such as synthetic collateralized debt obligations (CDO) have been the principal growth engine in the fixed income market over the last few years. Val- uation and risk management of structured credit products by meads of Monte Carlo sim- ulations tend to suffer from numerical instability and heavy computational time. After a brief description of single name credit derivatives that underlie structured credit deriva- tives, the factor model is introduced as the portfolio credit derivatives pricing tool. This thesis then describes the rapid pricing and hedging of portfolio credit derivatives using the analytical approximation model and semi-analytical models under several specifications of factor models. When a full correlation structure is assumed or the pricing of exotic structure credit products with non-linear pay-offs is involved, the semi-analytical tractability is lost and we have to resort to the time-consuming Monte Carlo method. As a Monte Carlo variance reduction technique we extend the importance sampling method introduced to credit risk modelling by Joshi [2004] to the general n-factor problem. The large homogeneouspool model is also applied to the method of control variates as a new variance reduction tech- niques for pricing structured credit products. The combination of both methods is examined and it turns out to be the optimal choice. We also show how sensitivities of a CDO tranche can be computed efficiently by semi-analytical and Monte Carlo framework. 2 ACKNOWLEDGEMENTS First, I would like to express my thanks to the members of the Structured Credit Products Group of Mizuho International plc where I held a part-time position during most of my have time as a Ph. D. student. I am especially grateful to Kim Duncan and Neil Carthy who in given me a great opportunity to combine my academic work with more practical projects the real financial market. Without their support I would never have been able to complete this thesis. I am also grateful to my Ph. D. supervisor Professor Nicos Christofides. Last, but not least, I would like to thank my parents. I am grateful to them for their constant support, encouragement and faith in me. 3 LIST OF NOTATIONS Eý 11 A Cholesky decomposed matrix such that ajaT = a as.= fail,..., ail,..., aim} Vector of factor loadings Limit factor loadings ac,. a,,ci,..., ac,.M} composite BT Zero cupon bond with face value F that matures at time T býl EM Simple weighted factor loading for 1th industry such as 1 ajiwj ßk Sensitivity of the industry k to the global systematic factor YN S Recovery rate ET Price of equity at time T 6 Standard normally distributed idiosyncratic factor F (t) Cumulative default probability at time t FT (ti, t, ) Cumulative joint default P [TL < ti, < tn] ..., probability ..., Tn f (0, t) Instantaneous risk free forward rate r't Enlarged filtration time t . at (") Cumulative standard normal distribution function "D'1(") Inverse cumulative standard normal distribution function 4ý2, (") function p Bivariate standard normal cumulative distribution 'Dn,R ["] n- variate standard normal cumulative distribution function with correlation matrix R OX (") Characteristic function of a random variable X 9t Background filtration at time t Wt Natural filtration at time t h (") Hazard rate function 1{T>t} Survival indicator at time t KLower Attachement point of a CDO tranche KUG? Detachement point of a CDO tranche 4 cp(") Standard normal density function L (t) Cumulative portfolio loss probability at time t LMax Maximum cumulative portfolio loss tranche L (t) Percentage tranche net loss LO° (t) Large homogeneous pool model cumulative portfolio loss probability at time t Loo (t)tranche Large homogeneous pool model tranche loss probability at time t A (") Intensity function Mi ith obligor percentage net loss given default Ni (t) Default indicator 11,.;ßt} of obligor i at time t Pi (L (t) =1IV = v) Conditional standardised loss density at time t for the i=0,1, portfolio of size ..., n Pn-` (L (t) =11V = v) Recursively constructed loss distribution of the portfolio removing obligor i ptýv Probability of default by time t of obligor i conditional on the systematic factor realisation V=v RPV 01 (t, T) Risky annuity at time t with contract maturity T r (t) Risk free interest rate at time t pik Asset correlation between obligor i and j pD Default correlation between obligor i and j pintra Intra-industry asset correlation pinter Inter-industry asset correlation pOO Large pool limit risk weights S (t) Survival probability at time t S (ti,..., tn) Cumulative joint P [7-1 ti, tn] survival probability > ..., Tn > T= (Tl, Tj) TN) Payment dates ..., ..., TN Last payment date tf wd Forwad CDS starting date T Default time 5 TXy Kendall's tau of random variables X and Y V Total value of firm value at time t V 1th systematic factor Vk kth industry systematic factor VSeller (, ) PV of the CDS/CDO contract (protection seller) VByer (") PV of the CDS/CDO contract (protection buyer) VRAs (, ) PV of the premium leg (annuity stream) VP°9 (") PV of the default leg (option payout) ý (Tj_1, Tj) Day count fraction for [Tj_l, T, ] time interval X CDS/CDO tranche premium per annum X (") Background stochastic process YN Global systematic factor affecting all industries Yk Industry specific systematic factor Wt Brownian motion at time t wi ith obligor weight 6 TABLE OF CONTENTS ABSTRACT 2 ..................................... ACKNOWLEDGEMENTS 3 ............................ LIST OF NOTATIONS 4 .............................. LIST OF TABLES 10 ................................. LIST OF FIGURES 11 ................................ 1 INTRODUCTION 12 ............................... 1.1 Credit Default Swaps 13 ............................ 1.2 Structured Portfolio Credit Products 15 ................... 1.3 Outline 19 ............................ ...... .. 2 SINGLE NAME CREDIT RISK MODELLING 21 ............ 2.1 Structural Models for Default Time Modelling 22 .............. 2.1.1 Merton [1974] Model 24 ........................ 2.1.2 Barrier Models 25 ........................... 2.2 Reduced Form Framework for Default Time Modelling 26 ......... 2.2.1 Default Probability Hazard Rate Function 27 and .......... 2.2.2 Stochastic Intensity Approach 28 ................... 2.3 Credit Default Swaps Continuous Time Pricing 34 - ............ 2.4 Credit Default Swaps Discrete Time Pricing 36 - .............. 2.5 Forward Starting Credit Default Swaps 39 .................. 2.6 Market Implied Default Probabilities 41 ................... 2.6.1 Bootstrapping: Determination of Default Probabilities at Credit Curve Points 43 ............................. 2.7 Summary 45 .................................. 3 MULTI-NAME CREDIT RISK MODELLING: FROM MERTON TO FACTOR MODELS 47 ........................... 3.1 Default Correlation the Bernoulli Mixture Model 49 and .......... 3.2 Modelling Portfolio Credit Risk by Factor Models 50 ............ 3.2.1 One Factor Gaussian Model 58 .................... 7 3.2.2 Industry Multi-Factors Gaussian Model 61 ............ .. 3.2.3 Inter-Intra Industry Model 63 ....... ....... ....... 3.3 Summary 67 .................................. 4 SEMI-ANALYTICAL IMPLEMENTATION OF FACTOR MOD- ELS 69 ........................................ 4.1 Semi-analytical Valuation of Synthetic Collateralized Debt Obligations 70 4.2 Fast Fourier Transform Implementation 74 ........... ....... 4.3 Conditional Recursive Implementation 77 .................. 4.4 Large HomogeneousPool Model 81 ...................... 4.5 Numerical Results 85 .............................. 4.6 Summary 87 .................................. 5 MONTE CARLO COPULA MODELS AND VARIANCE REDUC- TION TECHNIQUES 89 ............................. 5.1 Monte Carlo Generation Default Times Copula Models 91 of with ..... 5.2 Measures Dependence 93 of .......................... 5.3 Gaussian Copula Multivariate Merton Model 95 as ............. 5.4 Default Time Simulation Gaussian Copula 96 with ............. 5.5 Method Importance Sampling 98 of ...................... 5.6 Method Control Variates 104 of ........................ 5.7 Combining Importance Sampling Control Variate 110 and .......... 5.8 Numerical Results 112 .............................. 5.9 Summary 116 .................................. 6 FAST AND ACCURATE METHODS OF COMPUTING CDO TRANCHE GREEKS 118 . .................................... 6.1 Sensitivity Analysis CDO Tranches 118 of ................... 6.1.1 Individual Spread Sensitivity 119 ................ .... 6.1.2 Macro Spread Sensitivity 124 ................. ..... 6.1.3 Spread Convexity 126 .......................... 6.1.4 Jump-to-Default Risk Sensitivities 127 ................. 6.1.5 Correlation Sensitivities 129 ...................... 6.1.6 Time-Decay Sensitivity 131 ....................... 8 6.2 Semi-Analytical Computation Tranche Greeks 131 of ......... .... 6.2.1 Recursive Computation Tranche Greeks 134 of . ...... .... 6.2.2 FFT Computation Tranche Greeks 136 of .... .... ...... 6.3 Monte Carlo Computation Tranche Greeks 137 of .............. 6.3.1 Likelihood Ratio Method 138 ............. .... ..... 6.4 Summary 143 ........... .... ... ..... .... ..... 7 CONCLUSION AND FURTHER WORK 144 ................ APPENDIX A- THE MULTIVARIATE NORMAL DISTRIBUTION AND GAUSSIAN COPULA 147
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