Pricing Inflation and Interest Rates Derivatives with Macroeconomic Foundations Gabriele Luigi Sarais Imperial College London Department of Mathematics A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy in Financial Mathematics June 4, 2015 I Gabriele Luigi Sarais declare that this thesis is my own work and has not been submitted in any form for another degree or diploma at any university or other institute. Information derived from the published and unpublished work of others has been acknowledged in the text and a list of references is given in the bibliography. The copyright of this thesis rests with the author and is made available under a Creative Commons Attribution Non-Commercial No Derivatives licence. Researchers are free to copy, distribute or transmit the thesis on the condition that they attribute it, that they do not use it for commercial purposes and that they do not alter, transform or build upon it. For any reuse or redistribution, researchers must make clear to others the licence terms of this work. 2 Abstract I develop a model to price inflation and interest rates derivatives using continuous-time dynamics linked to monetary macroeconomic models: in this approach the reaction function of the central bank, the bond market liquidity, and expectations play an important role. The model explains the effects of non-standard monetary policies (like quantitative easing or its tapering) on derivatives pricing. A first adaptation of the discrete-time macroeconomic DSGE model is proposed, and some changes are made to use it for pricing: this is respectful of the original model, but it soon becomes clear that moving to continuous time brings significant benefits. The continuous-time model is built with no-arbitrage assumptions and economic hypotheses that are inspired by the DSGE model. Interestingly, in the proposed model the short rates dynamics follow a time- varying Hull-White model, which simplifies the calibration. This result is significant from a theoretical perspective as it links the new theory proposed to a well-established model. Further, I obtain closed forms for zero-coupon and year-on-year inflation payoffs. The calibration process is fully separable, which means that it is carried out in many simple steps that do not require intensive computation. The advantages of this approach become apparent when doing risk analysis on inflation derivatives: because the model explicitly takes into account economic variables, a trader can assess the impact of a change in central bank policy on a complex book of fixed income instruments, which is not straightforward when using standard models. The analytical tractability of the model makes it a candidate to tackle more complex problems, like inflation skew and counterparty/funding valuation adjustments (known by practitioners as XVA): both problems are interesting from a theoretical and an applied point of view, and, given their computational complexity, benefit from a tractable model. In both cases the results are promising. Contents Introduction 13 1 Trading and modelling inflation 14 1.1 Inflation markets . 14 1.1.1 Priceindices ....................................... 15 1.1.2 Market participants . 15 1.1.3 Payoffs . 17 1.1.4 Inflation exotics and hybrids . 20 1.2 Arbitragepricing......................................... 22 1.2.1 Main facts . 22 1.2.2 Pricing kernels in discrete time . 25 1.2.3 Discrete-time asset pricing in the lognormal case . 27 1.2.4 Model-independent inflation relationships . 28 1.3 Inflationpricingmodels..................................... 30 1.3.1 The foreign-exchange analogy . 30 1.3.2 The Jarrow-Yildirim model . 31 1.3.3 TheRBSmodel ..................................... 32 1.3.4 The HJM approach . 32 1.3.5 The BGM-I approach . 32 1.3.6 Stochastic volatility approaches . 32 1.3.7 The Hughston-Macrina (HM) model . 33 2 Monetary macroeconomic inflation models 36 2.1 IntroductiontotheDSGEmodel ................................ 37 2.1.1 Axiomatic foundations . 38 2.1.2 Model derivation . 42 4 2.1.3 Systemstability ..................................... 47 2.2 Using the DSGE model for pricing purposes . 48 2.2.1 Arbitrage-free pricing . 48 2.3 Building the continuous-time version . 58 2.3.1 Testing the dynamics against empirical evidence . 61 2.3.2 Comparing the DSGE model with the continuous-time model . 63 3 Inflation derivatives pricing with a central bank reaction function 69 3.1 Inflation model assumptions . 71 3.1.1 Probabilisticset-up ................................... 71 3.1.2 Financialinstruments .................................. 71 3.1.3 Financial market . 72 3.1.4 Economy dynamics and central bank role . 74 3.2 CTCBModelconstruction.................................... 77 3.3 Equivalent interest rates model . 81 3.4 Further analysis on the mean reversion property . 84 3.4.1 General case . 84 3.4.2 Constant mean reversion speed . 87 3.5 Pricing of vanilla interest rates derivatives . 89 3.6 Pricing zero-coupon inflation swaps and options . 93 3.7 Pricing year-on-year inflation swaps and options . 97 3.8 Single currency derivatives pricing simulation . 101 3.9 Extension to the open economy . 105 3.10 Uncertain-parameters extension . 106 4 Model calibration and applications 108 4.1 At-the-money calibration strategy . 108 4.1.1 Calibration steps: a first strategy . 109 4.1.2 Calibration steps: an alternative strategy . 112 4.1.3 Variance split and calibration to correlations . 113 4.1.4 The trade-off between smoothness and calibration accuracy . .116 4.2 At-the-money calibration results . 117 4.2.1 Technical assumptions . 117 4.2.2 Economic assumptions . 118 4.2.3 Market data . 119 5 4.2.4 Correlation targeting . 119 4.2.5 Results ..........................................120 4.3 Applications . 121 4.3.1 Derivatives risk as a function of the central bank reaction function . 121 4.3.2 Cross gammas as a function of the central bank reaction function . 121 4.3.3 Inflation book macro-hedging in the CTCB model . 122 4.3.4 Stress testing in the CTCB model . 123 5 Inflation skew modelling 124 5.1 Putting inflation skew in context . 125 5.2 Staticskewmodels .......................................128 5.2.1 Inflation option pricing under the t-distribution: a new formula . 128 5.2.2 Inflation option pricing under the SABR model . .131 5.2.3 A first static calibration contest . 132 5.2.4 Inflation option pricing under Gaussian mixtures . .134 5.3 The equity analogy: a new inflation skew metric . 138 5.4 Issues with skew and forward skew: pricing year-on-year trades . 145 5.5 Merton jump-diffusion model (JD) . 147 5.5.1 From diffusions to jump-diffusions . 147 5.5.2 The original model . 148 5.6 Some comments on the numerical implementation . 150 5.7 Time-varying jump-diffusion models (TV-JD) . 152 5.8 Uncertain-parameters time-varying jump-diffusion models (UP-TV-JD) . 158 5.8.1 Solving the IPDE . 163 5.9 Skew market calibration example . 167 6 Counterparty and funding risk aspects of inflation derivatives 171 6.1 Definitions, choices and fundamental results . 172 6.2 Some results with no wrong-way risk . 185 6.3 Creditmodellingoverview ....................................187 6.3.1 Structural versus intensity-based credit models . 187 6.3.2 Idiosyncratic versus systematic credit risk . 189 6.3.3 Marshall-Olkin models . 189 6.3.4 CDS-bond basis . 193 6.4 Multicurve modelling overview . 193 6 6.5 Credit modelling in the macroeconomic framework in a multicurve setting: the CR-MC- CTCBmodel ...........................................194 6.5.1 Assumptions . 195 6.5.2 Separable calibration . 198 6.6 Monte Carlo simulations . 199 6.6.1 Credit parameters choice . 200 6.6.2 Monte Carlo: some results . 201 Conclusions 206 Bibliography 207 Appendices 216 Appendix A DSGE model proofs 216 Appendix B Derivation of a European option pricing formula under a t-distribution 220 Appendix C Option pricing with mixtures 224 C.1 Introduction............................................224 C.2 Definitions and properties . 224 C.3 Moments of a Gaussian mixture . 226 7 List of Figures 2.1 TimeseriesofUSCPIpriceindex ............................... 62 2.2 TimeseriesofUSrealGDP.................................... 62 2.3 Time series of UK RPI price index. 62 2.4 TimeseriesofUKrealGDP. .................................. 62 2.5 Time series of US CPI inflation. 62 2.6 Time series of US real GDP growth rate. 62 2.7 Time series of UK RPI inflation. 62 2.8 Time series of UK real GDP growth rate. 62 2.9 Time series of UK real GDP growth expectations (survey by Bloomberg). 63 2.10 Time series of US inflation expectations (survey by University of Michigan). 63 2.11 Scatter plot of 5,000 Monte Carlo simulations of the DSGE.