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High Dimensional American Options

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High Dimensional American Options High Dimensional American Options Neil Powell Firth Brasenose College University of Oxford A thesis submitted for the degree of Doctor of Philosophy Trinity Term 2005 I dedicate this thesis to my lovely wife Ruth. Acknowledgements I acknowledge the help of both my supervisors, Dr. Jeff Dewynne and Dr. William Shaw and thank them for their advice during my time in Oxford. Also, Dr. Sam Howison, Prof. Jon Chapman, Dr. Peter Carr, Dr. Ben Hambly, Dr. Peter Howell and Prof. Terry Lyons have spared time for helpful discussions. I especially thank Dr. Gregory Lieb for allowing me leave from Dresdner Kleinwort Wasserstein to complete this research. I would like to thank Dr. Bob Johnson and Prof. Peter Ross for their help, advice, and references over the years. Andy Dickinson provided useful ideas and many cups of tea. The EPSRC and Nomura International plc made this research possible by providing funding. Finally, I would like to thank my parents for their love and encouragement over the years, and my wife, Ruth, for her unfailing and invaluable love and support throughout the course of my research. Abstract Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations and numerical techniques have been developed. Pricing multi–asset (high dimensional) American options is still more difficult. We extend the method proposed theoretically by Glasserman and Yu (2004) by employing regression basis functions that are martingales under geometric Brownian motion. This results in more accurate Monte Carlo simulations, and computationally cheap lower and upper bounds to the American option price. We have implemented these models in QuantLib, the open–source derivatives pricing library. The code for many of the models discussed in this thesis can be downloaded from quantlib.org as part of a practical pricing and risk management library. We propose a new type of multi–asset option, the “Radial Barrier Option” for which we find analytic solutions. This is a barrier style option that pays out when a barrier, which is a function of the assets and their correlations, is hit. This is a useful benchmark test case for Monte Carlo simulations and may be of use in approximating multi–asset American options. We use Laplace transforms in this analysis which can be applied to give analytic results for the hitting times of Bessel processes. We investigate the asymptotic solution of the single asset Black–Scholes– Merton equation in the case of low volatility. This analysis explains the success of some American option approximations, and has the potential to be extended to basket options. Contents 1 Introduction 1 1.1 Researchobjectives............................ 3 1.2 Contributions ............................... 3 1.2.1 Monte Carlo methods for high dimensional American options. 4 1.2.2 Radialbarrieroptions . 4 1.2.3 Besselprocesshittingtimes . 5 1.2.4 An asymptotic analysis for the American call option . 5 1.3 Organisationofthisthesis ........................ 6 2 Literature Survey 9 2.1 Introduction................................ 9 2.1.1 Multi–asset framework . 12 2.2 Analyticsolutions............................. 12 2.2.1 Europeanoptions ......................... 12 2.2.2 Barrieroptions .......................... 13 2.2.3 Perpetual American options . 14 2.2.4 Geometric average options . 14 2.2.5 Multi–assetoptions. 14 2.3 Problem formulations for American options . 15 2.3.1 Inequalities ............................ 15 2.3.2 Optimal stopping formulation . 16 2.3.3 Free boundary formulation . 17 2.3.4 Linear complementarity formulation . 18 2.3.5 Primal–dual formulation . 20 2.3.6 Early exercise premium formulation . 20 2.3.7 Integral equation formulation . 21 i ii Contents 2.3.8 Viscosity solution formulation . 22 2.3.9 Otherformulations . 22 2.3.10 Alternatestochasticprocesses . 23 2.4 Analyticrelationships. 24 2.4.1 Putcallparity........................... 24 2.4.2 Multi–assetoptions. 24 2.5 Numericalmethods ............................ 25 2.5.1 Single asset American options . 25 2.5.2 Barrieroptions .......................... 33 2.5.3 High dimensional American options . 35 2.6 Softwareissues .............................. 40 2.6.1 C++ ................................ 40 2.6.2 Designpatterns .......................... 40 2.6.3 QuantLib ............................. 41 2.6.4 Parallel computational methods . 41 2.7 Conclusions ................................ 41 3 High Dimensional American Options 43 3.1 Introduction................................ 43 3.2 Problemformulation ........................... 44 3.3 Approximatedynamicprogramming. 45 3.3.1 Regressionnow .......................... 46 3.3.2 Regressionlater.......................... 48 3.4 Comparison ................................ 50 3.5 Martingalebasisfunctions . 51 3.5.1 One dimensional martingale basis functions . 51 3.5.2 Two dimensional martingale basis functions . 54 3.6 Lowbiasedestimators .......................... 55 3.7 Highbiasedestimators .......................... 56 3.8 Implementation .............................. 58 3.9 Numericalresults ............................. 59 3.9.1 Regression precision . 59 3.9.2 Lowerandupperbounds. 60 3.10Evaluation................................. 64 Contents iii 3.11Conclusions ................................ 64 4 High Dimensional Radial Barrier Options 65 4.1 Introduction................................ 65 4.2 The multi–asset Black–Scholes–Merton equation . 66 4.2.1 Fixedboundary.......................... 67 4.2.2 Movingboundary......................... 69 4.3 The radial problem for u ......................... 70 4.4 Outerproblem............................... 70 4.4.1 Solution in n dimensions..................... 71 4.5 Innerproblem............................... 77 4.5.1 Solution in n dimensions..................... 78 4.5.2 Solutioninfivedimensions. 82 4.6 Convolutions ............................... 82 4.6.1 Outerproblem .......................... 83 4.6.2 Innerproblem........................... 85 4.7 Reversetransformations . 87 4.7.1 Singleassetcase.......................... 87 4.7.2 Generalradialbarrieroption. 88 4.8 Numericalresults ............................. 92 4.8.1 Implementation .......................... 92 4.8.2 Numericalresults ......................... 93 4.9 Qualitativebehaviour. 95 4.9.1 Threeassetoptions........................ 96 4.10Conclusions ................................ 96 5 Bessel Process Hitting Times 99 5.1 Introduction................................ 99 5.2 Definitionsandexistingresults. 100 5.2.1 Besselprocesshittingtimes . 101 5.2.2 Existingresults .......................... 102 5.3 InversionofLaplacetransforms . 103 5.3.1 Example.............................. 103 5.3.2 Hittingdown ........................... 104 5.3.3 Hittingup............................. 104 iv Contents 5.4 Asymptotics solutions for small t .................... 105 5.4.1 Hittingdown ........................... 105 5.4.2 Hittingup............................. 106 5.5 Numericalresults ............................. 107 5.6 Conclusions ................................ 109 6 An Asymptotic Analysis of an American Call Option with Small Volatility 111 6.1 Introduction................................ 111 6.1.1 Problem formulation . 111 6.1.2 Non-dimensionalisation . 112 6.2 Europeancalloption ........................... 112 6.2.1 Innersolution ........................... 113 6.3 Americancalloption ........................... 114 6.3.1 Case r>q ............................. 114 6.3.2 Case r<q ............................. 116 6.3.3 Perpetual American call option . 116 6.4 Conclusions ................................ 116 7 Future Work 119 7.1 High dimensional American options . 119 7.1.1 Martingale basis functions in Monte Carlo methods . 120 7.1.2 Exploitingstructure . 120 7.1.3 American options under other stochastic processes . 120 7.2 High dimensional radial barrier options . 121 7.3 Besselprocesshittingtimes . 121 7.4 An asymptotic analysis of an American call option with small volatility 121 7.5 Hybridproducts.............................. 122 7.6 Credit derivatives with American style features . 122 7.7 Conclusions ................................ 123 8 Conclusions 125 8.1 Contributions ............................... 125 8.1.1 Monte Carlo methods for high dimensional American options . 126 8.1.2 Radial barrier options . 126 Contents v 8.1.3 Besselprocesshittingtimes . 127 8.1.4 An asymptotic analysis for the American call option . 127 8.2 Limitations ................................ 127 8.3 Concludingremarks. .. .. .. .. .. ... .. .. .. .. .. ... 128 A Papers and Conferences 129 B Details of Laplace inversion 133 B.1 Outerproblem............................... 133 B.2 Innerproblem............................... 134 C One Dimensional Asymptotics of American Options 137 C.1 Europeanoption ............................. 137 C.1.1 Non-dimensionalisation . 138 C.1.2 Leadingordersolution . 139 C.1.3 Innersolution ........................... 140 C.1.4 Original variables . 144 C.1.5 Comparison with exact solution . 145 C.2 PerpetualAmericancalloption . 146 C.2.1 Perpetual American call solution when r>q .......... 146 C.2.2 Limit as ǫ2 0 with r>q .................... 148 → C.2.3 Limit as ǫ2 0 with r<q .................... 149 → C.2.4 Case r = q ............................. 150 C.2.5 Case q =0............................. 150 C.3 Americancalloption .. .. .. .. .. ... .. .. .. .. .. ... 151 C.3.1 Case r>q ............................
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