Arbitrage-free methods to price European options under the SABR model

110 Pages·2015·1.08 MB·English

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