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Basket Option Pricing and the Mellin Transform by Derek J. Manuge A Thesis Presented to The Faculty of Graduate Studies of The University of Guelph In partial fulﬁlment of requirements for the degree of Master of Science in Applied Mathematics Guelph, Ontario, Canada ⃝c Derek J. Manuge, December, 2013

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Abstract Basket Option Pricing and the Mellin Transform Derek J. Manuge Advisor: University of Guelph, 2013 Dr. Peter T. Kim Option pricing has been an increasingly popular area of study over the past four decades. The use of the Mellin transform in such a context, however, has not. In this work we present a general multi-asset option pricing formula in the context of Mellin transforms, extending previously known results. The analytic formula derived computes European, American, and basket options with n underlying assets driven by geometric Brownian motion. Aside from the usual given parameters, the pricing formula requires three components to compute: (i) the Mellin basket payoﬀ function, (ii) the characteristic function (or exponent) of a multivariate Brownian motion with drift, and (iii) the Mellin transform of the early exercise function. A fast discretization is solved, providing option prices at incremental values of initial asset prices. As an application, European put option prices are computed for Canadian bank stocks (n = 1) and foreign exchange rates (n = 2) with USD denomination.

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iii Acknowledgments Foremost I would like to thank my advisor, Dr. Peter T. Kim, who has been a relentless source of support and inspiration for me. I am grateful to him for his life mentorship, his profound insight regarding research problems, and for providing me with the freedom to study engaging topics without rigid direction. To the members of the examining committee: Dr. Tony Desmond, Dr. Hermann Eberl, and exam chair Dr. Rajesh Pereira. Thank you for taking the time to consume and decompose my work. I recognize the rarity of your time and appreciate the signiﬁcance of your input. To the members of the department whom had a profound impact on me: Dr. Jack Weiner for turning me on to the beauty of calculus, Dr. Dan Ashlock for keeping me turned on to the wonders of abstract math while providing entertaining venues to expand my learning, Dr. Rajesh Pereira for opening my awareness to the purity of math and the opportunity to study beneath him, Dr. Marcus Garvie for instilling the importance of numerical computation, Dr. Herb Kunze for demonstrating the applicability of analysis, Dr. Hermann Eberl for catalyzing my interest in partial diﬀerential equations, Dr. Monica Cojocaru for her hospitality, and Dr. Pal Fischer for being the coolest old person I know. To my family and friends: for the companionship, the stability, and the humanism. Thank you for gracefully ignoring the 4n language that occasionally spews from my mouth. To the funding sources during my tenure: the Department of Mathematics

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iv and Statistics, the College of Physical and Engineering Sciences, the University of Guelph, Wilfred Laurier University, and Le Centre de recherches math´ematiques. For those who have been omitted here, but have knowingly impacted my life positively. Every interaction provides an opportunity to learn; thank you for being a source of encouragement, perspective, and wisdom.

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v Table of Contents List of Figures vii 1 Introduction 1 1.1 Outline of Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 A Brief History of Options Trading . . . . . . . . . . . . . . . . . . . 2 1.3 Market Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Styles of Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Option Pricing Theory . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.6 Stochastic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.7 Itoˆ’s Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.8 Black-Scholes-Merton Equation . . . . . . . . . . . . . . . . . . . . . 15 2 Analytic Option Pricing 16 2.1 European Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . 16 2.2 European Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 2.3 American Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2.4 American Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 3 The Mellin Transform 23 3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 3.2 Deﬁnition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.3 Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 3.4 Multidimensional Mellin Transform . . . . . . . . . . . . . . . . . . . 29 4 Analytic Option Pricing using the Mellin Transform 32 4.1 Previous Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 4.2 European Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . 36 4.3 European Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4 American Call Option . . . . . . . . . . . . . . . . . . . . . . . . . . 40 4.5 American Put Option . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 5 Analytic Basket Option Pricing using the Mellin Transform 44 5.1 PDE for Multi-Asset Options . . . . . . . . . . . . . . . . . . . . . . 44 5.2 Boundary Conditions for Basket Options . . . . . . . . . . . . . . . . 47 5.3 General Analytic Solution . . . . . . . . . . . . . . . . . . . . . . . . 49 5.4 Basket Payoﬀ Function . . . . . . . . . . . . . . . . . . . . . . . . . . 52

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vi 5.5 Early Exercise Boundary . . . . . . . . . . . . . . . . . . . . . . . . . 53 5.6 Black-Scholes-Merton Models . . . . . . . . . . . . . . . . . . . . . . 55 5.7 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 6 Numerical Option Pricing 59 6.1 Numerical Mellin Inversion . . . . . . . . . . . . . . . . . . . . . . . . 60 6.2 Fast Fourier Transform Method . . . . . . . . . . . . . . . . . . . . . 64 6.3 Mellin Transform Method . . . . . . . . . . . . . . . . . . . . . . . . 67 6.4 Application to Equity Markets (n = 1) . . . . . . . . . . . . . . . . . 72 6.5 Application to Foreign Exchange Markets (n = 2) . . . . . . . . . . . 75 7 Conclusion 78 8 Future Work 82 Bibliography 85 9 Appendix 98 9.1 Proof of lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.2 Proof of proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 99 9.3 Proof of proposition 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 101 9.4 Proof of proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 104 9.5 Generalized Black-Scholes-Merton Equation . . . . . . . . . . . . . . 106

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vii List of Figures 1.1 The payoﬀ for a European call and put option for diﬀerent values of the asset price S, given exercise price K = $100. . . . . . . . . . . . . 6 1.2 Sample paths of an asset given by geometric Brownian motion with µ = 0.5, σ = 0.5, S0 = 100, and r, q = 0. . . . . . . . . . . . . . . . . 13 2.1 The European call option price for exercise prices K ∈ [80, 120] of the asset price S = 100 with σ = 0.5, r = 0.05, q = 0.05, and T = 30/365. 18 2.2 The European put option price for exercise prices K ∈ [80, 120] of the asset price S = 100 with σ = 0.5, r = 0.05, q = 0.05, and T = 30/365. 19 5.1 The value of a basket call (left) and put (right) payoﬀ function on S1,2 ∈ [0, 3]. For the call and put payoﬀ, K = 4 and K = 2 respectively. 54 6.1 The absolute error between the approximated Mellin payoﬀ and the original payoﬀ function. On the left is the numerical procedure of (6.1) 8 10 12 14 for N = 2 (red), N = 2 (green), N = 2 (orange), and N = 2 (purple) points. On the right is the numerical procedure of (6.4) for 14 N = 2 . In both cases, K = 1.5, S ∈ [0, 3], a = 2, M = 10 and 100 points are plotted in the line graphs. . . . . . . . . . . . . . . . . . . 62 6.2 The log absolute error between the approximated Mellin payoﬀ and the original payoﬀ function. On the left is the numerical procedure 8 10 12 of (6.1) for N = 2 (red), N = 2 (green), N = 2 (orange), and 14 N = 2 (purple) points. On the right is the numerical procedure of 14 (6.4) for N = 2 . In both cases, K = 1.5, S ∈ [0, 3], a = 2, M = 10, and 100 points are plotted in the line graphs. . . . . . . . . . . . . . . 63 6.3 The European call (left) and put (right) option price using the FFT method and BSM formula for exercise prices K ∈ [80, 120] of the asset 14 price S = 100 with σ = 0.5, r = 0.05, q = 0.05, T = 30/365, N = 2 , α = 2 and η = 0.05. Points ’o’ denote the BSM prices, while points ’x’ denote the FFT prices. . . . . . . . . . . . . . . . . . . . . . . . . . . 66 6.4 The absolute and log absolute European option price error between the FFT method and BSM formula for exercise prices K ∈ [80, 120] of the asset price S = 100 with σ = 0.5, r = 0.05, q = 0, T = 30/365, 14 N = 2 , α = 2, and η = 0.05. . . . . . . . . . . . . . . . . . . . . . . 67

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viii 6.5 The European put (left) and call (right) option price using the MT method and BSM formula for exercise prices K ∈ [80, 120] of the asset 14 price S = 100 with σ = 0.5, r = 0.05, q = 0.05, T = 30/365, N = 2 , a = 1 and η = 0.05. Points ’o’ denote the BSM prices, while points ’x’ denote the MT prices. . . . . . . . . . . . . . . . . . . . . . . . . . . 70 6.6 The absolute and log absolute European option price error between the MT method and BSM formula for exercise prices K ∈ [80, 120] of the asset price S = 100 with σ = 0.5, r = 0.05, q = 0, T = 30/365, 14 N = 2 , a = 1, and η = 0.05. . . . . . . . . . . . . . . . . . . . . . . 71 6.7 Implied option prices (green) and historical option prices (blue) are computed with r = 0.01, σ = 0.18, τ = 21/252, and K = 59 for the period starting June 20, 2013. . . . . . . . . . . . . . . . . . . . . . . 73 6.8 Implied option prices (green) and historical option prices (blue) are computed with r = 0.01, σ = 0.45, τ = 37/252, and K = 100 for the period starting December 22, 2012. . . . . . . . . . . . . . . . . . . . 73 6.9 Implied option prices (green) and historical option prices (blue) are computed with r = 0.01, σ = 0.32, τ = 39/252, and K = 100 starting January 22, 2013. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 6.10 Implied option prices (green) and historical option prices (blue) are computed with r = 0.01, σ = 0.18, τ = 45/252, and K = 60 for the period starting March 18, 2013. . . . . . . . . . . . . . . . . . . . . . 74 6.11 Implied option prices (green) and historical option prices (blue) are computed with r = 0.01, σ = 0.33, τ = 42/252, and K = 72 for the period starting June 20, 2013. . . . . . . . . . . . . . . . . . . . . . . 75 6.12 Implied basket option prices (green) are computed on CAD/USD and YEN/USD with rCAD = 0.01, rUSD = 0.0025, rY EN = 0, σY EN/USD = 0.1027, σCAD/USD = 0.0561, ρ12 = 0.81, τ = 30/365, and K = 1.5 for the period from January 1st, 2013 to October 1st, 2013. The contract th size of the YEN position is 1/100 the size of the EUR position. . . 76 6.13 Implied basket option prices (green) are computed on EUR/USD and YEN/USD with rCAD = 0.01, rUSD = 0.0025, rY EN = 0, σEUR/USD = 0.0695, σY EN/USD = 0.1027, ρ12 = 0.06, τ = 30/365, and K = 1.5 for the period from January 1st, 2013 to October 1st, 2013. The contract th size of the YEN position is 1/100 the size of the EUR position. . . . 77 6.14 Implied basket option prices (green) are computed on EUR/USD and CAD/USD with rCAD = 0.01, rUSD = 0.0025, rY EN = 0, σEUR/USD = 0.0695, σCAD/USD = 0.0561, ρ12 = 0.21, τ = 30/365, and K = 1.5 for the period from January 1st, 2013 to October 1st, 2013. . . . . . . . . 77

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1 Chapter 1 Introduction It is not by augmenting the capital of the country, but by rendering a greater part of that capital active and productive than would otherwise be so, that the most judicious operations of banking can increase the industry of the country. -Adam Smith 1.1 Outline of Work The primary objective of this thesis is to review the recent literature relating to Mellin transforms in the context of option pricing where assets are driven by geo- metric Brownian motion. By doing so, we attempt to ﬁll the gaps for the multi-asset basket option in an analytical and numerical framework. To achieve this, we present this thesis in the following manner. Chapter 1 introduces the necessary concepts of option pricing and provides ﬁnancial motivation for the pursuit of obtaining accurate pricing methods. We also review stochastic calculus and formulate the partial diﬀer- ential equation attributed to Black, Scholes, and Merton. In chapter 2, we ﬁnd the analytic solutions of European and American put/call options under a single asset driven by geometric Brownian motion. The solution method follows the conventional approach used in the literature; a change of variables followed by a reduction to a

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2 diﬀusion equation. Chapter 3 introduces the Mellin transform in the multivariate case, providing properties on positive functions, as well as relations required there- after. Chapter 4 demonstrates the applicability of the Mellin transform for solving European and American call/put options on a single asset, reviewing previous results from the literature. Chapter 5 extends these results to the multidimensional case, where we obtain the general analytic pricing formula, as well as expressions for the early exercise function, Mellin payoﬀ function, and Greeks of put options. Chapter 6 considers the numerical pricing problem, where we discretize the general formula to obtain a numerically fast and accurate procedure. Additional numerical Mellin inver- sion techniques are reviewed. Chapter 7 concludes the results of this work, Chapter 8 presents possible areas of future work, And As It Is Such, So Also As Such Is It Unto You. 1.2 A Brief History of Options Trading An option is a ﬁnancial security that presents its holder with the right, but not the obligation, to purchase a given amount of underlying asset at some future date. In practice, the underlying asset is often the price of a stock, commodity, foreign ex- change (FX) rate, index, or futures contract. With global issuance at over 4 billion a year, proper valuation is a signiﬁcant concern among institutional and personal 1 traders who use options to invest or mitigate risk by hedging. Despite their relatively recent market formalization by the Chicago Board of Options Exchange 1 Data queried from the Options Clearing Corporation (OCC) at www.theocc.com.

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