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On the Proxy Modelling of Risk-Neutral Default Probabilities DEGREE PROJECT IN MATHEMATICS, SECOND CYCLE, 30 CREDITS STOCKHOLM, SWEDEN 2020 On the Proxy Modelling of Risk-Neutral Default Probabilities EDVIN LUNDSTRÖM KTH ROYAL INSTITUTE OF TECHNOLOGY SCHOOL OF ENGINEERING SCIENCES On the Proxy Modelling of Risk- Neutral Default Probabilities EDVIN LUNDSTRÖM Degree Projects in Financial Mathematics (30 ECTS credits) Master's Programme in Industrial Engineering and Management KTH Royal Institute of Technology year 2020 Supervisor at Svenska Handelsbanken AB: Niklas Karlsson Supervisor at KTH: Camilla Landén Examiner at KTH: Camilla Landén TRITA-SCI-GRU 2020:071 MAT-E 2020:034 Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci Abstract Since the default of Lehman Brothers in 2008, it has become increasingly important to measure, manage and price the default risk in financial derivatives. Default risk in financial derivatives is referred to as counterparty credit risk (CCR). The price of CCR is captured in Credit Valuation Adjustment (CVA). This adjustment should in principle always enter the valuation of a derivative traded over-the-counter (OTC). To calculate CVA, one needs to know the probability of default of the counterparty. Since CVA is a price, what one needs is the risk-neutral probability of default. The typical way of obtaining risk-neutral default probabilities is to build credit curves calibrated using Credit Default Swaps (CDS). However, for a majority of a bank’s counterparties there are no CDSs liquidly traded. This constitutes a major challenge. How does one model the risk-neutral default probability in the absence of observable CDS spreads? A number of methods for constructing proxy credit curves have been proposed previ- ously. A particularly popular choice is the so-called Nomura (or cross-section) model. In studying this model, we find some weaknesses, which in some instances lead to degener- ate proxy credit curves. In this thesis we propose an altered model, where the modelling quantity is changed from the CDS spread to the hazard rate. This ensures that the obtained proxy curves are valid by construction. We find that in practice, the Nomura model in many cases gives degenerate proxy credit curves. We find no such issues for the altered model. In some cases, weseethat the differences between the models are minor. The conclusion is that the altered model is a better choice since it is theoretically sound and robust. Proxymodellering av riskneutrala fallissemangssannolikheter Sammanfattning Sedan Lehman Brothers konkurs 2008 har det blivit allt viktigare att mäta, hantera och prissätta kreditrisken i finansiella derivat. Kreditrisk i finansiella derivat benämns ofta motpartsrisk (CCR). Priset på motpartsrisk fångas i kreditvärderingsjustering (CVA). Denna justering bör i princip alltid ingå i värderingen av ett derivat som handlas över disk (eng. over-the-counter, OTC). För att beräkna CVA behöver man veta sannolikheten för fallissemang (konkurs) hos motparten. Eftersom CVA är ett pris, behöver man den riskneutrala sannolikheten för fallissemang. Det typiska tillvägagångsättet för att erhålla riskneutrala sannolikheter är att bygga kreditkurvor kalibrerade med hjälp av kreditswappar (CDS:er). För en majoritet av en banks motparter finns emellertid ingen likvid handel i CDS:er. Detta utgör en stor utmaning. Hur ska man modellera riskneutrala fallissemangssannolikheter vid avsaknad av observerbara CDS-spreadar? Ett antal metoder för att konstruera proxykreditkurvor har föreslagits tidigare. Ett särskilt populärt val är den så kallade Nomura- (eller cross-section) modellen. När vi studerar denna modell hittar vi ett par svagheter, som i vissa fall leder till degenererade proxykreditkurvor. I den här uppsatsen föreslår vi en förändrad modell, där den modeller- ade kvantiteten byts från CDS-spreaden till riskfrekvensen (eng. hazard rate). Därmed säkerställs att de erhållna proxykurvorna är giltiga, per konstruktion. Vi finner att Nomura-modellen i praktiken i många fall ger degenererade proxykred- itkurvor. Vi finner inga sådana problem för den förändrade modellen. I andra fallservi att skillnaderna mellan modellerna är små. Slutsatsen är att den förändrade modellen är ett bättre val eftersom den är teoretiskt sund och robust. Acknowledgements This thesis was written during the spring of 2020 at the Royal Institute of Technology (KTH) and Svenska Handelsbanken. I would like to thank my supervisors Camilla Landén at KTH and Niklas Karlsson at Handelsbanken for their support and guidance. I would also like to thank Fredrik Bohlin at Handelsbanken for introducing the research topic. Stockholm, May 2020 Edvin Lundström Contents List of Figures IV List of Tables V 1 Introduction 1 1.1 Background . 1 1.2 Problem statement . 2 1.3 Purpose and research question . 3 1.4 Outline . 3 2 Counterparty credit risk 4 2.1 Financial risk . 4 2.1.1 Profit and loss . 5 2.2 Counterparty credit risk . 5 2.2.1 Exposure . 5 2.2.2 Loss given default . 7 2.2.3 Probability of default . 7 2.3 The typical situation . 7 2.4 Risk mitigating measures . 8 2.4.1 Netting . 8 2.4.2 Collateral . 9 2.5 CVA and DVA . 9 2.5.1 Regulatory CVA . 10 2.6 Default and spread risk . 11 3 Introduction to credit modelling 12 3.1 Credit models . 12 3.1.1 Firm value models . 12 3.1.2 Reduced form models . 13 3.2 P versus Q .................................... 14 3.2.1 Real-world default probabilities . 14 3.2.2 Risk-neutral default probabilities . 15 3.3 The credit risk premium . 15 3.4 Shortage of liquidity problem . 16 3.5 Proposed proxy methods . 17 3.5.1 Single-name . 18 3.5.2 Index . 18 3.5.3 Intersection . 19 3.5.4 Cross-section (Nomura) . 19 I 3.5.5 Third party . 20 3.5.6 Others . 20 4 Interest rate modelling 22 4.1 Bank account and discount factor . 22 4.2 Zero bonds and zero rates . 23 4.3 Forward rates . 24 5 Reduced form credit modelling 25 5.1 Default and survival probabilities . 25 5.2 The Poisson process . 26 5.2.1 Constant intensity . 27 5.2.2 Time-varying and deterministic intensity . 28 5.2.3 Time-varying and stochastic intensity . 29 5.2.4 Summary . 29 5.3 Building blocks . 30 5.3.1 Introducing the building blocks . 30 5.3.2 Pricing . 31 5.3.3 Pricing under simplifying assumptions . 31 5.4 Interest rate versus credit modelling . 32 6 Credit Default Swaps 34 6.1 Basic structure . 34 6.1.1 Big and small bang . 35 6.2 Valuation . 35 6.2.1 The premium leg . 36 6.2.2 The protection leg . 37 6.2.3 Full value and breakeven spread . 37 7 Model 38 7.1 Constructing the survival curve . 38 7.1.1 Desirable properties . 38 7.1.2 Interpolation . 39 7.1.3 Bootstrap . 40 7.2 Proxy models . 41 8 Data 46 8.1 Credit Default Swap quotes . 46 8.2 Interest rates . 48 9 Results and discussion 49 9.1 Initial results . 49 9.2 Additive models . 49 9.3 The Nomura model collapses . 51 9.4 Arbitrage in the curve . 52 9.5 Large differences . 53 9.6 Rating monotonicity . 53 9.7 Volatility . 55 II 10 Conclusion 57 10.1 Further research . 57 References 59 Appendix A Alternative characterisations 61 A.1 Hazard rate . 61 A.2 Implied quantities . 62 A.2.1 Relation to forward rates . 63 III List of Figures 2.1 The impact of the sign of the value at default . 6 2.2 Probability of default curve . 8 2.3 The typical bank setup . 8 3.1 The difference between real-world and risk-neutral default probabilities. 16 5.1 Risky ZCB paying one unit of currency if no default, zero otherwise . 30 5.2 Instrument paying one unit of currency if default before T , zero otherwise 31 6.1 Basic structure of a credit default swap . 34 6.2 CDS cash flows . 35 9.1 Survival curve and hazard rate for the proxy curve for BBB-rated European industrial companies . 50 9.2 Nomura method failing . 51 9.3 Nomura method resulting in negative hazard rates . 53 9.4 Large differences between the Nomura and the alternative method . 54 9.5 5Y survival probabilities for European Financial corporates . 55 9.6 15-year survival probabilities over time for the two proxy models. 56 IV List of Tables 5.1 Analogies between interest rate and credit modelling . 33 7.1 Sample dummy variable matrix . 43 8.1 Summary of CDS data . 48 9.1 Survival probabilities for European industrial companies rated BBB ac- cording to the two models. 50 9.2 Statistics for proxy credit curves built with different versions of the models 50 V Glossary CCR Counterparty Credit Risk CDS Credit Default Swap CVA Credit Valuation Adjustment DVA Debit Valuation Adjustment LGD Loss Given Default MTM Mark-To-Market OT C Over-The-Counter PD Probability of Default ZCB Zero Coupon Bond VI Notation General 1{· g Indicator function, with condition inside brackets R The set of real numbers Interest rate modelling.
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