Master of Science Thesis A Comparison of Alternative SABR Models in a Negative Interest Rate Environment Esmée Winnubst 10338179 30th December 2016 Faculty of Economics and Business · University of Amsterdam A Comparison of Alternative SABR Models in a Negative Interest Rate Environment Master of Science Thesis For the degree of Master of Science in Financial Econometrics at the University of Amsterdam Esmée Winnubst 10338179 30th December 2016 Supervisor: Dr. N. Van Giersbergen Second Reader: Dr. S.A. Broda Faculty of Economics and Business · University of Amsterdam Statement of Originality: This document is written by Esmée Winnubst who declares to take full responsibility for the contents of this document. I declare that the text and the work presented in this document is original and that no sources other than those mentioned in the text and its references have been used in creating it. The Faculty of Economics and Business is responsible solely for supervision of completion of the work, not for the contents. Acknowledgement: I would like to thank my supervisor Noud van Giersbergen for his as- sistance during the writing of this thesis. The work in this thesis was supported by EY. I would like to thank my colleagues at EY for their support and time. Copyright c All rights reserved. Abstract The widespread used SABR model (P. S. Hagan, Kumar, Lesniewski, & Woodward, 2002) is not able to correctly price options, capture volatility curves and inter- and extrapolate market quotes in a negative interest rate environment. This thesis investigates two extensions of the SABR model that are able to work with negative rates: the normal SABR model and the free boundary SABR model. In theory both models are very promising, but both models are thus far not calibrated to actual market data. In this thesis first a closed form solution for both models is derived. The accurateness of this approximation is then tested using an arbitrage free PDE solution as benchmark. To test the two models with actual market data, first the normal ATM volatility formulas for both models are derived. Then the closed form solutions are calibrated to the swaption data for 01- 07-2005, 01-01-2008, 01-01-2016, 01-06-2016, 01-08-2016 and 01-12-2016 using the previously derived ATM formulas. The free boundary SABR model optimization ends up in a sub-optimum for the 2016 data (which contains negative rates). Therefore very inaccurate results were found around a strike price of zero. After grid searching for better start parameters without result, the calibration results of the normal SABR model were chosen as start parameters for the free boundary optimization. Now the free boundary SABR model gives better results than the normal SABR model. It can be concluded that, without using the normal SABR model for finding start parameters, the free boundary SABR model is not able to correctly model forward rates that are close to zero. Therefore this model is not a good model for financial practitioners, who need this model in all kind of situations, compared to the normal SABR model which works in every situation. Contents Abstract i 1 Introduction1 1-1 Economic Background................................ 2 1-2 Problem Statement................................. 2 1-3 Thesis Outline.................................... 2 2 Interest Rate Options5 2-1 Interest rate derivatives............................... 5 2-2 Bond options.................................... 6 2-3 Caps and Floors................................... 6 2-4 Swaptions...................................... 6 2-5 Implied Volatility.................................. 6 3 Literature Review9 3-1 The Black model.................................. 9 3-2 The SABR Model.................................. 10 3-2-1 Martingale Reresentation Theorem..................... 11 3-2-2 Description of the Model.......................... 11 3-2-3 Option pricing with the SABR model.................... 12 3-2-4 SABR approximation formula........................ 13 3-2-5 SABR in practice: the classical SABR model................ 13 3-2-6 Conclusion.................................. 15 4 Negative Interest Rate Models 17 4-1 Displaced model................................... 17 4-1-1 Drawback Displaced Model......................... 18 4-2 Stochastic Normal Model.............................. 18 4-3 The Free Boundary SABR Model.......................... 19 iv Contents 5 Partial Differential Equation Solution 23 5-1 Solving the PDE................................... 23 5-1-1 Arbitrage-free pricing within the SABR model............... 23 5-1-2 Boundary conditions............................. 25 5-2 Option pricing with positive rates.......................... 27 5-2-1 Transformation of the PDE......................... 27 5-2-2 Discretization of the PDE.......................... 29 5-2-3 Pricing options............................... 31 5-3 Option pricing with negative rates......................... 31 5-3-1 Modeling negative interest rates with the normal SABR model...... 31 5-3-2 Modeling negative interest rates with the free boundary SABR model.. 33 6 Closed Form Solution Results 35 6-1 PDF plotting..................................... 35 6-2 Performance of the Approximation Formula.................... 38 6-2-1 Test 1.................................... 38 6-2-2 Test 2.................................... 40 6-2-3 Conclusion approximation formulas..................... 42 7 Methodology 43 7-1 Dataset....................................... 43 7-2 Calibration...................................... 50 7-2-1 Hagan calibration.............................. 50 7-2-2 Calibration in this thesis........................... 50 7-3 Measurement of fit................................. 51 8 Market Data Results 53 8-1 Parameter Calibration................................ 53 8-2 Market Fit...................................... 54 8-2-1 Normal market conditions.......................... 54 8-2-2 Excited market conditions.......................... 57 8-2-3 Negative rate conditions.......................... 57 9 Conclusion 65 A Parameter Calibration Results 67 List of Figures 1-1 Central bank’s interest rates............................ 1 3-1 Implied volatility smile................................ 11 5-1 Y (z) and transformed F (z) ............................. 29 6-1 Probability density function obtained by solving the PDE for the classical SABR. 36 6-2 Probability density function obtained by solving the PDE for the normal SABR. 36 6-3 Probability density function obtained by solving the PDE for the free boundary SABR 37 6-4 Probability density function obtained by solving the PDE for the free boundary SABR with varying β ................................ 37 6-5 Probability density function obtained by solving the PDE, test 1......... 39 6-6 Volatility smile, test 1................................ 39 6-7 Absolute difference option prices PDE and approximation, test 1......... 40 6-8 Probability density function obtained by solving the PDE, test 2......... 41 6-9 Volatility smile, test 2................................ 41 6-10 Absolute difference option prices PDE and approximation, test 2......... 42 7-1 Volatility surface for a EUR swaption on 01-07-2005................ 44 7-2 Volatility surface for a EUR swaption on 01-01-2008................ 45 7-3 Volatility surface for a EUR swaption on 01-01-2016................ 46 7-4 Volatility surface for a EUR swaption on 01-06-2016................ 47 7-5 Volatility surface for a EUR swaption on 01-08-2016................ 48 7-6 Volatility surface for a EUR swaption on 01-12-2016................ 49 7-7 Strike surface for a EUR swaption on 01-08-2016................. 49 8-1 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-07-2005......... 55 vi List of Figures 8-2 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2008......... 56 8-3 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-01-2016......... 58 8-4 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-06-2016......... 59 8-5 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-08-2016......... 60 8-6 Implied normal volatility smiles generated by calibrated normal SABR and free boundary SABR models, compared to the market for 01-12-2016......... 61 List of Tables 8-1 RMSE compared for normal SABR and free boundary SABR for 01-07-2005... 54 8-2 RMSE compared for normal SABR and free boundary SABR for 01-01-2008... 57 8-3 RMSE compared for normal SABR and free boundary SABR for 01-01-2016... 62 8-4 RMSE compared for normal SABR and free boundary SABR for 01-06-2016... 62 8-5 RMSE compared for normal SABR and free boundary SABR for 01-08-2016... 63 8-6 RMSE compared for normal SABR and free boundary SABR for 01-12-2016... 63 A-1 Swaption quotes for EUR, 01-07-2005, maturity: 1Y Normal SABR....... 68 A-2 Swaption quotes for EUR, 01-07-2005, maturity: 5Y Normal SABR....... 68 A-3 Swaption quotes for EUR, 01-07-2005, maturity: 10Y Normal SABR....... 68 A-4 Swaption quotes for EUR, 01-07-2005, maturity: 1Y Free Boundary SABR.... 69 A-5 Swaption quotes for EUR, 01-07-2005, maturity: 5Y Free Boundary SABR.... 69 A-6 Swaption quotes for EUR, 01-07-2005, maturity: 10Y Free Boundary