Essays on the modeling of risks in interest-rate and inflation markets Allan Sall Tang Andersen Advisor : Bjarne Astrup Jensen Ph.D. dissertation Department of Finance Copenhagen Business School May, 2011 Contents Contents iii Acknowledgements v Introduction 1 Summary 11 1 Modeling stochastic skewness in a Heath-Jarrow-Morton framework 19 1.1 Introduction........................... 20 1.2 Evidenceofstochasticskewness . 21 1.3 Modelingstochasticskewness . 27 1.4 SpecifyingtheModel . 33 1.5 Modelcalibration . ... .. .. .. .. .. ... .. .. .. 35 1.6 Results.............................. 38 1.7 Conclusion............................ 43 1.8 Appendix: Proofofproposition1 . 44 1.9 Appendix:MCMCdetails . 45 2 Inflation derivatives modeling using time changed L´evy processes 49 2.1 Introduction........................... 50 2.2 Inflationlinkedproducts . 52 2.3 TheJarrow-Yildirimmodel . 56 2.4 TimechangedL´evyprocesses . 58 2.5 An inflation HJM framework based on time changed L´evy processes............................. 61 2.6 Pricinginflationproducts . 68 2.7 Specificationofthetime-change . 72 2.8 Calibration ........................... 74 2.9 Conclusion............................ 79 iii 2.10 Appendix:Proofs . 81 2.11 Appendix: TheVarianceGammaprocess . 85 3 Inflation risk premia in the term structure of interest rates: Evidence from Euro area inflation swaps 87 3.1 Introduction........................... 88 3.2 Data: Inflation swap rates and the nominal term structure . 90 3.3 Inflationriskpremia: Whattheorypredicts . 97 3.4 A no-arbitrage model of nominal and inflation swap rates . 99 3.5 ModelEstimation. 102 3.6 Empiricalresults . 105 3.7 Conclusion............................ 120 3.8 Appendix: Derivation of nominal ZCB prices, real ZCB prices andInflationexpectations . 122 3.9 Appendix: MCMCestimation ofthemodel . 126 4 Affine Nelson-Siegel Models and Risk Management Per- formance 131 4.1 Introduction........................... 132 4.2 The Danish Government Bond Term Structure . 133 4.3 AffineTermStructureModels . 136 4.4 Multi-factor Cox-Ingersoll-Ross models . 138 4.5 AffineNelson-Siegelmodels . 140 4.6 ModelEstimation. 143 4.7 EmpiricalResults: In-Sample . 146 4.8 EmpiricalResults: Out-of-Sample . 153 4.9 Conclusion............................ 159 4.10 Appendix: Affine Nelson-Siegel models with stochastic volatil- ity ................................161 4.11 Appendix: MCMCdetails . 167 4.12 Appendix: DensityForecasts. 172 4.13 Appendix: Tables with out-of-sample forecast results . 177 Conclusion 203 Bibliography 205 iv Acknowledgements With the hand-in of this thesis, I have now finished my years as a Ph.D. student at the Department of Finance at Copenhagen Business School and Danmarks Nationalbank. The years as a Ph.D. student have at times been hard work, but it has most definitely been a good experience. First of all, I would like to thank my supervisor Bjarne Astrup Jensen for encouraging me to consider a position as a Ph.D. student, for providing me with suggestions and pin-pointing errors and unclear reasoning in my work during the process. I would like to thank Danmarks Nationalbank for giving me the opportu- nity to enter into their Ph.D. programme and the Capital Markets/Financial structure division at the European Central Bank for giving me the possibil- ity to spend 6 months in Frankfurt. The stay in Frankfurt both spurred an interest in and deepened my understanding of inflation markets. I would also like to thank professor Rama Cont for providing the opportunity to stay a semester at the Center for Financial Engineering, Columbia University. Fellow Ph.D. students at the Department of Finance and Danmarks Nation- albank also deserve thanks for creating a good environment, both socially and professionally. From Danmarks Nationalbank I would especially like to thank my office-mate Jannick Damgaard for making the Ph.D. a pleasant time, despite non-overlapping research interests. From the Department of Finance I would like to thank Mads Stenbo Nielsen and Jens Dick-Nielsen for always having their door open if I needed to discuss a topic. Outside the world of Ph.D. students, I would like to thank Jacob Ejsing and Kasper Ahrndt Lorenzen for providing valuable insights, especially regarding infla- tion markets. I also want to thank Jesper Lund and Fred Espen Benth for participating in my pre-defence and providing me with valuable comments for my thesis. On a personal level, I want to thank my parents for supporting me my entire life and providing abstractions outside the world of finance. Finally, I would like to thank my fianc´ee Anne-Sofie for constant encouragement v and patience. Your insight into the life of a Ph.D. student, have made my life as a, at times slightly confused, Ph.D. student significantly better. Copenhagen, May 2011 Allan Sall Tang Andersen vi Introduction The topic of this thesis is the modeling of risks in interest-rate and inflation markets. Interest-rate risk is an important issue to investors. For instance, according to BIS (2010) the notional value of over-the-counter interest-rate derivatives markets is 465,260 billion US-dollar. This corresponds to 77 percent of the notional of the entire OTC derivatives market. Thus interest-rate deriva- tives is at the back-bone of the financial markets. According to ISDA (2009) 83 percent of Fortune 500 companies report using interest-rate derivatives in their risk management. Furthermore, many mortgage-based loans and pen- sion contracts contain either explicit or implicit interest-rate options. Thus a better understanding of the interest-rate derivative markets, and the risk associated with the traded products is of great value, both to financial and non-financial companies as well as individuals. The market for inflation linked products, such as bonds and swaps, is sig- nificantly smaller than the one for interest-rate derivatives. The market is also significantly newer than the nominal interest-rate market, with one of the prominent examples being US Treasury Inflation Protected Bonds (TIPS). TIPS were introduced in 1997 and with a notional outstanding of less than 50 billion US dollar. By 2010 the US Treasury has issued TIPS worth over 600 billion US dollar, see Christensen and Gillan (2011). The issuance of inflation linked bonds is not limited to the US; countries such as France, United Kingdom1, Germany and Japan also have a significant issuance of inflation linked bonds. With the increase in issuance and trading in inflation linked bonds deriva- tive markets have evolved. Inflation swaps, which can be compared to interest-rate swaps, have been traded at least since 1995 (see Barclays Cap- ital (2008)), and in recent years options on inflation have also been traded. With the increase in trading in inflation linked products academic interest 1United Kingdom was one of the first countries to issue inflation linked bonds. UK inflation linked bonds has been issued as early as in 1981. 1 INTRODUCTION 2 has spurred, but perhaps more interesting to individuals, central banks have started using the information content embedded in inflation linked products in relation to the economic analysis used to set policy rates. The goal of this Ph.D. thesis is to add a small piece to the puzzle of un- derstanding interest-rate and inflation markets. The thesis consists of four essays, two of which are focused on modeling interest-rate risk and two are focused on modeling inflations risks and risk premia. Each essay contributes to the literature in its field and can, of course, be read independently. In short, here follows a brief motivation for each paper. Modeling stochastic skewness in a Heath-Jarrow-Morton framework Several facts on interest rate behaviour are well known. First, interest-rate volatilities are obviously stochastic, and these volatilities tend to cluster in periods with low respectively high volatility (see for instance Ander- sen and Lund (1997)). Carr and Wu (2007) show that currency options have time-varying skewness. By using model-free estimates of the volatility and skewness priced in interest-rate options, it can be shown that interest- rate distributions also show time-varying skewness (see Trolle and Schwartz (2010)). The main purpose of the paper is to provide a consistent framework for modeling the stochastic volatility and skewness. Finally, calibrating the model to time-series of market data is interesting, as it shows the applica- bility of the model. Inflation derivatives modeling using time changed L´evy processes With the rise of inflation derivatives and more liquid markets, non-linear inflation contracts have been introduced. When considering time-series of inflation swap rates, the fact that changes in inflation swap rates show large sudden movements, i.e. jumps, is easily acknowledged. The stan- dard method for modeling inflation derivatives is the Gaussian forward-rate framework introduced in Jarrow and Yildirim (2003). In this paper we wish to model inflation derivatives in a no-arbitrage framework which could both include time varying volatility and jumps. We also want to provide evidence of the applicability of the model framework by using market data. 3 INTRODUCTION Inflation risk premia in the term structure of interest rates: Evidence from Euro area inflation swaps Market based inflation expectations such as inflation swap rates or the Break Even Inflation Rate, i.e. an inflation measure derived from nominal and inflation linked bonds, provide market participants with real-time mea- sures of inflation expectations. However, these measures include both an inflation expectation and an inflation risk premium. The purpose of the paper is to use a no-arbitrage model to disentangle the two components by using nominal swap rates, inflation swap rates, surveys on inflation expec- tations and CPI data. The model output can be used to interpret whether changes in inflation swap rates correspond to changes in inflation expecta- tions or inflation risk premia. Affine Nelson-Siegel Models and Risk Management Performance The ability to correctly assess the interest-rate risk one faces as an investor is a critical issue. This is obviously the case for a pension fund with a large bond portfolio, but it is also the case for government debt agencies who need to assess the risks of different issuance strategies.