Delft University of Technology Faculty of Electrical Engineering, Mathematics and Computer Science Delft Institute of Applied Mathematics Arbitrage-free methods to price European options under the SABR model A thesis submitted to the Delft Institute of Applied Mathematics in partial fulfillment of the requirements for the degree MASTER OF SCIENCE in APPLIED MATHEMATICS by Zaza van der Have Delft, the Netherlands January 2015 Copyright (cid:13)c 2015 by Zaza van der Have. All rights reserved. MSc THESIS APPLIED MATHEMATICS “Arbitrage-free methods to price European options under the SABR model” Zaza van der Have Delft University of Technology Daily supervisors Responsible professor Dr. T.P.T. Dijkstra Prof.dr.ir. C.W. Oosterlee Prof.dr.ir. C.W. Oosterlee Other thesis committee members Dr.ir. R.J. Fokkink Ir. M.J. Ruijter January 2015 Delft, the Netherlands Abstract In this thesis we discuss several methods to price European options under the SABR model. In general, methods given in literature are not free of arbitrage and/or inaccurate for long maturities. This led to the development of a new pricing approach. We extend the BCOS method from one dimension to two dimensions. This extension is necessary for application of a simplification of the BCOS method, the DCOS method, to the SABR model. In this pricing method we use the characteristic function of the discrete forward process and the Fourier-based COS method. It is possible to price European options under the SABR model for multiple strikes in one computation with the DCOS method. Besides valuing European options, we can also price Bermudan and discretely monitored barrier options with this pricing approach. i Preface This thesis has been submitted for the degree of Master of Science in Applied Mathematics at Delft University of Technology, the Netherlands and was carried out for Rabobank in Utrecht. I would like to thank some people who have contributed in the process of writing this thesis. First of all, I would like to acknowledge prof.dr.ir. Kees Oosterlee, dr. Tim Dijkstra and ir. Marjon Ruijter for their advice and guidance during the project and for being part of the exa- mination committee. Furthermore, I would like to thank dr.ir. Robbert Fokkink from TU Delft forbeingpartoftheexaminationcommittee. IwouldalsoliketothankthewholePricingModel Validation (PMV) team of Rabobank for the pleasant working environment and for interesting discussions. In particular, Anton van der Stoep for his feedback on my draft. Last but cer- tainly not least, I would like to express my gratitude to my family and friends, and especially to Silvester Wulffers, who supported me through the duration of my study. Zaza van der Have Utrecht, January 2015 iii Contents Abstract i Preface iii 1 Introduction 1 2 SABR model and its pricing methods 3 2.1 The Hagan formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.1.1 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.1.2 Accuracy of the Hagan formula . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 The method of Antonov, Konikov and Spector . . . . . . . . . . . . . . . . . . . 6 2.2.1 Call price for the zero correlation SABR . . . . . . . . . . . . . . . . . . . 7 2.2.2 Mapping to zero correlation case . . . . . . . . . . . . . . . . . . . . . . . 8 2.2.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2.4 Brief analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3 Hagan’s arbitrage-free approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 2.3.1 The PDE for the conditional probability density function . . . . . . . . . 12 2.3.2 Finite difference scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 2.3.3 Arbitrage . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 2.3.4 Pricing formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.3.5 Accuracy of Hagan’s arbitrage-free method . . . . . . . . . . . . . . . . . 16 2.3.6 Brief analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16 3 The BCOS method 17 3.1 The discrete forward process and its characteristic function . . . . . . . . . . . . 17 3.2 COS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 3.3 BCOS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3.4 Approximation of the conditional expectations . . . . . . . . . . . . . . . . . . . 23 3.5 BCOS method summarized . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 3.6 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.6.1 Local error ∆-time-discretization . . . . . . . . . . . . . . . . . . . . . . . 26 3.6.2 Local error θ-method-discretization . . . . . . . . . . . . . . . . . . . . . . 27 3.6.3 Fourier errors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.6.4 Global error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 3.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 v vi CONTENTS 4 Extension of the BCOS method to two dimensions 32 4.1 The discrete forward process and its characteristic function . . . . . . . . . . . . 32 4.2 An adjusted-Predictor-Corrector scheme . . . . . . . . . . . . . . . . . . . . . . . 35 4.3 The characteristic function of the Heston model . . . . . . . . . . . . . . . . . . . 37 4.4 Two-dimensional COS method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 4.5 Two-dimensional BCOS method . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.6 Approximation of the conditional expectations . . . . . . . . . . . . . . . . . . . 42 4.7 Two-dimensional BCOS method summarized . . . . . . . . . . . . . . . . . . . . 45 4.8 Path-dependent options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.8.1 Bermudan option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 4.8.2 Discretely monitored barrier options . . . . . . . . . . . . . . . . . . . . . 47 4.9 Error analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 4.9.1 Local error ∆-time-discretization . . . . . . . . . . . . . . . . . . . . . . . 48 4.9.2 Local error θ-method-discretization . . . . . . . . . . . . . . . . . . . . . . 49 4.9.3 Global error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 4.10 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 5 One time step DCOS method 54 5.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 5.2 Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 5.3 Using the log transform for the forward variable. . . . . . . . . . . . . . . . . . . 58 5.4 Accuracy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 5.5 Brief analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 6 The DCOS method applied to the SABR model 63 6.1 The method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 6.2 Results. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65 6.3 Multiple strikes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Advantages and disadvantages . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68 6.5 Brief analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 7 Conclusion 71 7.1 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71 7.2 Outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72 References 73 A Derivation of the approximation for G(t,s) 75 B Hagan’s arbitrage-free method 78 C Itˆo-Taylor expansion 84 D Taylor schemes 89 E Characteristic function 95 F Adjusted-Predictor-Corrector schemes 98