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Option is an asymmetric contract between two parties with future payoff derived from the price of underlying asset. Methods of pricing di erent types of options under more or less general assumptions have been extensively studied since the Nobel price winning works of Black and Scholes [1] and Merton [12] were published in 1973. A new way of pricing options with the use of Mellin transforms have been recently introduced by Panini and Srivastav [15] in 2004. This thesis offers a brief introduction to option pricing with Mellin transforms and a revision of some of the recent
research in this field.

1 Introduction 1
2 Preliminaries 3
2.1 Types of options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 Stochastic processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Assumptions of the Black-Scholes model . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 The Black-Scholes PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 Mellin transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Pricing European options with Mellin transforms 19
3.1 Pricing European put options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
3.2 Pricing European call options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
4 Theory behind the use of the Mellin transform 29
4.1 Dangers on the way . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
4.2 Exact pricing of European put options . . . . . . . . . . . . . . . . . . . . . . . . . . 32
4.3 An alternative approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
5 Options with general payo functions 39
5.1 Pricing power options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
5.2 A trick with power options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
6 Pricing American options 49
6.1 Free boundary problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.2 Black-Scholes PDE for American options . . . . . . . . . . . . . . . . . . . . . . . . . 50
6.3 Pricing American options with Mellin transforms . . . . . . . . . . . . . . . . . . . . 53
6.4 Numerical results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
7 Summary 61
Bibliography 61

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