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FFT - Introduction to Fourier Transform Methods for Option...

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Introduction to Fourier Transform Methods for Option Pricing Research Seminar in Financial Mathematics | Fall 2010 Lecture notes available online at: www.math.fsu.edu/~pgarreau Pierre Garreau Florida State University 1017 Academic Way, 208 Love Building Tallahassee, FL 32306 www.math.fsu.edu/~pgarreau Department of Mathematics | Florida State University Abstract This paper introduces Fourier Transform methods for Option Pricing. I present the pricing problem and its dual in the context of a modelisation of asset returns thanks to random walks and their limits: infinitely divisible distributions. After a review of the Black-Scholes (BS) and Merton’s Jump-diffusion (MJD) models, I will show how L´ evy processes arise naturally to model the dynamics of stocks and will focus on their characteristic function in connection with Fourier Transforms. Fast Fourier Transform methods are then discussed to price options as presented by P. Carr (1999). I then address convergence issues and will present the results obtained for BS, MJD and Variance Gamma. Key words: L´ evy processes, Fourier Transform, Option pricing, Black-Scholes, Jump-Diffusion.
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CONTENTS Contents 1 Motivations 3 2 Pricing Framework 4 2.1 The Black-Scholes Model for European Options . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Merton’s Jump-Diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Towards a general framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.1 Elements of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 2.3.2 Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 3 Fourier Transform and Option Pricing 11 3.1 Elements of theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 3.2 Numerical Fourier Transform: FFT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 4 Results 15 4.1 Black-Scholes and Merton’s jump-diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . 15 4.2 Variance Gamma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 2
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1 Motivations In the following we are interested in pricing Vanilla Options. Fourier Transform methods allow to generalize the Black-Scholes model to cases where the underlying does not follow a diffusion-type dy- namic. The following overview serves as a reminder on pricing methods. Such methods serve the main purpose of calibration, the problem of greek valuation is therefore voluntarily left aside. The market is defined on the historical probability space (Ω , F , P ) and equiped with a filtration ( F t ) t 0 representing the amount of available information up to time t . If the market is driven by a single Brownian motion, then we will write F t = σ ( W s , 0 s t ) (completed by the null sets). In the case of Merton’s jump diffusion, F t represents the knowledge of the jump sizes, the jump times and the Brownian motion. In further dimension, the filtration would be generated by all independent processes. Vanilla options give the right, but not the obligation, to buy or sell an asset S - the underlying - at a previously agreed maturity date T and price K - the strike price. Arbitrage arguments specify the price of the option at time t as u ( t,x ) = E Q bracketleftBig e R T t r ( s ) ds φ ( S t,x T ) bracketrightBig , (1.1) i.e., the expected value of the discounted payoff in a risk neutral world. This equality may be illustrated in the following example. Consider you are selling the option and forming a portfolio V t to hedge it.
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