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Pricing American Interest Rate Derivatives by Simulation Pricing American Interest Rate Derivatives by Simulation Rami V. Tabri A Thesis in The Department of Mathematics and Statistics Presented in partial fulfillment of the requirements for the degree of Master of Science (Mathematics) at Concordia University Montreal, Quebec, Canada August 2008 ©Rami V. Tabri 2008 Library and Bibliotheque et 1*1 Archives Canada Archives Canada Published Heritage Direction du Branch Patrimoine de I'edition 395 Wellington Street 395, rue Wellington Ottawa ON K1A0N4 Ottawa ON K1A0N4 Canada Canada Your file Votre reference ISBN: 978-0-494-45530-2 Our file Notre reference ISBN: 978-0-494-45530-2 NOTICE: AVIS: The author has granted a non­ L'auteur a accorde une licence non exclusive exclusive license allowing Library permettant a la Bibliotheque et Archives and Archives Canada to reproduce, Canada de reproduire, publier, archiver, publish, archive, preserve, conserve, sauvegarder, conserver, transmettre au public communicate to the public by par telecommunication ou par Plntemet, prefer, telecommunication or on the Internet, distribuer et vendre des theses partout dans loan, distribute and sell theses le monde, a des fins commerciales ou autres, worldwide, for commercial or non­ sur support microforme, papier, electronique commercial purposes, in microform, et/ou autres formats. paper, electronic and/or any other formats. The author retains copyright L'auteur conserve la propriete du droit d'auteur ownership and moral rights in et des droits moraux qui protege cette these. this thesis. Neither the thesis Ni la these ni des extraits substantiels de nor substantial extracts from it celle-ci ne doivent etre imprimes ou autrement may be printed or otherwise reproduits sans son autorisation. reproduced without the author's permission. In compliance with the Canadian Conformement a la loi canadienne Privacy Act some supporting sur la protection de la vie privee, forms may have been removed quelques formulaires secondaires from this thesis. ont ete enleves de cette these. While these forms may be included Bien que ces formulaires in the document page count, aient inclus dans la pagination, their removal does not represent il n'y aura aucun contenu manquant. any loss of content from the thesis. Canada ABSTRACT Pricing American Interest Rate Derivatives by Simulation Rami Victor Tabri. We examine the performance of single and multifactor models of the short rate in pricing American put options on the thirty year T-bond futures contract and two, five and ten year T-note futures contracts by simulation. The models for the short rate we utilize are the one-factor, two-factor, and three-factor Vasicek and CIR models used in Babbs and Now- man [2], and Chen and Scott [9] respectively, and the three factor models of Balduzzi, Das, Ferosi, and Sundaram [3], Dai and Singleton [12], Chen [6], and the Maximal models in the Air(3) and A2r(3) subfamilies introduced in Dai and Singleton [12]. We utilize the least squares Monte Cairo algorithm developed in Longstaff and Schwartz [23] to estimate the price of the American put option on T-bond and note futures contracts, and construct a high biased estimator in order to obtain a 95% confidence interval for the true price of the American put option. Also, we approximate the optimal exercise boundaries for the American put option and early exercise premia. Since the state variables in the one-factor, two-factor, and three-factor Vasicek and CIR models have known transition distributions, our investigation focuses on the performance of these models when simulated via the Euler scheme in comparison to simulation via the transition distribution approach. For the remaining models, we compare them across con­ fidence intervals, estimated optimal exercise boundaries, and early exercise premia. With the Vasicek and CIR factor models, we found that as the number of factors increased, the early exercise premia became more non-linear across both approaches of simulation, and that the 95% for these models when simulating via the Euler scheme with our choice of high biased estimator and finite subset of basis functions, significantly increased the size of the 95% confidence intervals when compared to the confidence intervals obtained by simulation via the transition density approach. Finally the estimated early exercise bound­ aries for the CIR models when simulating via the Euler scheme were significantly different iii when simulation was conducted via the transition distribution. For the rest of the models, the Maximal model and the Dai and Singleton model in the Anr(3) subfamily had the best performance in terms of having positive early exercise premia for most of the American put options considered. IV Acknowledgements I would like to acknowledge the financial support I received from the Institut de Finance Mathematiques de Montreal (IFM2) as a master's research fellowship, my supervisor Dr. C. Hyndman, and the Department of Mathematics, Concordia University. IV Contents List of Figures x List of Tables xi Introduction 1 1 Mathematical Finance 1 1.1 Introduction and Motivation 1 1.2 Modeling Uncertainty 2 1.2.1 Introduction and Motivation 2 1.2.2 Mathematical Formulation of Uncertainty 5 1.3 Basic Market Model 5 1.3.1 Terminology 6 1.3.2 Results 8 1.4 Fundamental Pricing Formulae 10 1.4.1 Zero-Coupon Bond Price 10 1.4.2 Zero-Coupon Bond Futures Price 11 1.4.3 The Pricing of a European Option on a Zero-Coupon Bond 12 1.4.4 The Pricing of European Option on a Zero-Coupon Bond Futures . 14 1.4.5 The Pricing of Coupon-Bearing Bonds 16 1.4.6 The Coupon bond futures Price 16 v 1.4.7 The Pricing of European Options on Coupon-Bearing Bonds .... 17 1.4.8 The Pricing of European Options on Coupon-Bearing Bond Futures 18 1.5 Computation of Pricing Formulae 19 1.5.1 Computing the Price of a Contingent Claim 20 1.5.2 Computing Futures Prices 24 1.6 Concluding points 25 2 Affine Term Structure Models 26 2.1 Introduction 26 2.2 Concepts and Terminology 26 2.3 One-Factor Short Rate Models 29 2.3.1 The Pricing of Zero-Coupon Bonds 32 2.3.2 Zero-Coupon Bond Futures Price 35 2.3.3 Pricing European Options on Zero-Coupon Bonds 37 2.3.4 Pricing European Options on Zero-Couopon Bond Futures 43 2.4 Two-Factor Short Rate Models 46 2.4.1 Some General Remarks 46 2.4.2 The Pricing of Zero-Coupon Bonds . 49 2.4.3 Zero-Coupon Bond Futures Price 53 2.4.4 The Pricing of European Options on Zero-Coupon Bonds 55 2.4.5 European Bond Futures Price 59 2.5 Three-Factor Models 61 2.5.1 The Pricing of Zero-Coupon Bonds 70 2.5.2 The Zero-Coupon Bond Futures Price 77 2.5.3 The Pricing of a European Option on a Zero-Coupon Bond 82 2.5.4 The price of a European Option on Zero-Coupon Bond Futures . 85 vi 3 American Interest Rate Derivatives 88 3.1 Finite Expiring American Put Option 88 3.2 Practical Dynamic Programming 95 3.3 Properties of the LSM Estimator 99 3.3.1 Interleaving Property 99 3.3.2 Convergence Results 101 4 Case Studies 105 4.1 Introduction 105 4.2 The T-Bond and T-Note Futures Contracts 106 4.3 High Biased Estimator 109 4.4 The Environment 112 4.4.1 Underlying Assets 112 4.4.2 Basis Functions 112 4.4.3 Approximating the Optimal Exercise Boundary 115 4.4.4 General Remarks on Simulation 116 4.4.5 The Parameters of the Models 117 4.5 Results 118 4.5.1 The CIR Models 118 4.5.2 Models in the Alr{3) and A2r(3) Families 120 4.5.3 The Single, Two, and Three Factor Vasicek Models 123 5 Conclusions 135 5.1 Summary 135 5.2 Conclusions 137 5.3 Future Research 139 Bibliography 139 vn A Probability and Stochastic Calculus 143 A.l Concepts and Terminology 143 A.1.1 Rudimentary Probability . 143 A.l.2 Stochastic Calculus 147 B Monte Carlo Methods 150 B. 1 Introduction and Motivation 150 B.l.l Monte Carlo Simulation . 151 B.2 Variance Reduction Techniques 154 B.2.1 Antithetic Sampling 155 B.2.2 Importance Sampling 157 B.3 Simulating SDEs 158 C Limit Theorems 161 D Parameter Estimates 162 D.l Estimates for the Single, Two and Three Factor Vasicek Models 162 D.2 Estimates for the Single, Two and Three Factor CIR Models 163 D.3 Estimates for the models in the A\r (3) and A2r (3) Families 163 Vlll List of Figures 1.1 Sample path of short term interest rate 4 2.1 The functions A, Bi: B2, B3 76 2.2 The function £^(75) in the BDFS model obtained using Maple 8 77 4.1 Plots of 95% confidence intervals using the high and low biased estimators, and LSM estimators for the prices of the all the American put options, using the two factor CIR model where simulation in conducted via both the Euler scheme and transition distribution approaches 125 4.2 The estimated early exercise premia for the the American options for all strikes and bond maturities considered, when considering single two and three factor models of the short rate simulated via the Euler scheme and transition distribution approach 126 4.3 Approximate optimal exercise boundaries for the two factor CIR model using the Euler scheme 127 4.4 Approximate optimal exercise boundaries for the two factor CIR model using the transition distribution. 128 4.5 Plots of 95% confidence intervals using the high and low biased estimators, and LSM estimators for the prices of the all the American put options, using the maximal model in the A2r (3) family 129 ix 4.6 The estimated early exercise premia for the the American options for all strikes and bond maturities considered for the models in the Air(3) and A2r(3) Families 130 4.7 Approximate optimal exercise boundaries for the Air(3)Max model.
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