Tight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processes

Handle URI:
http://hdl.handle.net/10754/624829
Title:
Tight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processes
Authors:
Crocce, Fabian; Häppölä, Juho; Keissling, Jonas; Tempone, Raul ( 0000-0003-1967-4446 )
Abstract:
Prices of European options whose underlying asset is driven by the L´evy process are solutions to partial integrodifferential Equations (PIDEs) that generalise the Black-Scholes equation by incorporating a non-local integral term to account for the discontinuities in the asset price. The Levy -Khintchine formula provides an explicit representation of the characteristic function of a L´evy process (cf, [6]): One can derive an exact expression for the Fourier transform of the solution of the relevant PIDE. The rapid rate of convergence of the trapezoid quadrature and the speedup provide efficient methods for evaluationg option prices, possibly for a range of parameter configurations simultaneously. A couple of works have been devoted to the error analysis and parameter selection for these transform-based methods. In [5] several payoff functions are considered for a rather general set of models, whose characteristic function is assumed to be known. [4] presents the framework and theoretical approach for the error analysis, and establishes polynomial convergence rates for approximations of the option prices. [1] presents FT-related methods with curved integration contour. The classical flat FT-methods have been, on the other hand, extended for option pricing problems beyond the European framework [3]. We present a methodology for studying and bounding the error committed when using FT methods to compute option prices. We also provide a systematic way of choosing the parameters of the numerical method, minimising the error bound and guaranteeing adherence to a pre-described error tolerance. We focus on exponential L´evy processes that may be of either diffusive or pure jump in type. Our contribution is to derive a tight error bound for a Fourier transform method when pricing options under risk-neutral Levy dynamics. We present a simplified bound that separates the contributions of the payoff and of the process in an easily processed and extensible product form that is independent of the asymptotic behaviour of the option price at extreme prices and at strike parameters. We also provide a proof for the existence of optimal parameters of the numerical computation that minimise the presented error bound.
KAUST Department:
Computer, Electrical and Mathematical Sciences & Engineering (CEMSE)
Conference/Event name:
Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)
Issue Date:
6-Jan-2016
Type:
Poster
Appears in Collections:
Posters; Conference on Advances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)

Full metadata record

DC FieldValue Language
dc.contributor.authorCrocce, Fabianen
dc.contributor.authorHäppölä, Juhoen
dc.contributor.authorKeissling, Jonasen
dc.contributor.authorTempone, Raulen
dc.date.accessioned2017-06-08T06:32:28Z-
dc.date.available2017-06-08T06:32:28Z-
dc.date.issued2016-01-06-
dc.identifier.urihttp://hdl.handle.net/10754/624829-
dc.description.abstractPrices of European options whose underlying asset is driven by the L´evy process are solutions to partial integrodifferential Equations (PIDEs) that generalise the Black-Scholes equation by incorporating a non-local integral term to account for the discontinuities in the asset price. The Levy -Khintchine formula provides an explicit representation of the characteristic function of a L´evy process (cf, [6]): One can derive an exact expression for the Fourier transform of the solution of the relevant PIDE. The rapid rate of convergence of the trapezoid quadrature and the speedup provide efficient methods for evaluationg option prices, possibly for a range of parameter configurations simultaneously. A couple of works have been devoted to the error analysis and parameter selection for these transform-based methods. In [5] several payoff functions are considered for a rather general set of models, whose characteristic function is assumed to be known. [4] presents the framework and theoretical approach for the error analysis, and establishes polynomial convergence rates for approximations of the option prices. [1] presents FT-related methods with curved integration contour. The classical flat FT-methods have been, on the other hand, extended for option pricing problems beyond the European framework [3]. We present a methodology for studying and bounding the error committed when using FT methods to compute option prices. We also provide a systematic way of choosing the parameters of the numerical method, minimising the error bound and guaranteeing adherence to a pre-described error tolerance. We focus on exponential L´evy processes that may be of either diffusive or pure jump in type. Our contribution is to derive a tight error bound for a Fourier transform method when pricing options under risk-neutral Levy dynamics. We present a simplified bound that separates the contributions of the payoff and of the process in an easily processed and extensible product form that is independent of the asymptotic behaviour of the option price at extreme prices and at strike parameters. We also provide a proof for the existence of optimal parameters of the numerical computation that minimise the presented error bound.en
dc.titleTight Error Bounds for Fourier Methods for Option Pricing for Exponential Levy Processesen
dc.typePosteren
dc.contributor.departmentComputer, Electrical and Mathematical Sciences & Engineering (CEMSE)en
dc.conference.dateJanuary 5-10, 2016en
dc.conference.nameAdvances in Uncertainty Quantification Methods, Algorithms and Applications (UQAW 2016)en
dc.conference.locationKAUSTen
kaust.authorCrocce, Fabianen
kaust.authorHäppölä, Juhoen
kaust.authorKeissling, Jonasen
kaust.authorTempone, Raulen
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