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Computable error estimates of a finite difference scheme for option pricing in exponential Lévy models
KTH, School of Engineering Sciences (SCI), Mathematics (Dept.), Mathematics (Div.).
KTH, School of Computer Science and Communication (CSC), Numerical Analysis, NA.
(English)Manuscript (preprint) (Other academic)
Abstract [en]

Option prices in exponential L´evy models solve certain partial integrodifferential equations (PIDEs). This work focuses on a finite difference scheme that issuitable for solving such PIDEs. The scheme was introduced in [Cont and Voltchkova, SIAM J. Numer. Anal., 43(4):1596–1626, 2005]. The main results of this work are new estimates of the dominating error terms, namely the time and space discretization errors. In addition, the leading order terms of the error estimates are determined in computable form. The payoff is only assumed to satisfy an exponential growth condition, it is not assumed to be Lipschtitz continuous as in previous works.

If the underlying Lévy process has infinite jump activity, then the jumps smallerthan some ε> 0 are approximated by diffusion. The resulting diffusion approximationerror is also estimated, with leading order term in computable form, as well as its effecton the space and time discretization errors. Consequently, it is possible to determine how to jointly choose the space and time grid sizes and the parameter ε.

National Category
Computational Mathematics
URN: urn:nbn:se:kth:diva-25081OAI: diva2:355665
QC 20101007Available from: 2010-10-07 Created: 2010-10-07 Last updated: 2010-10-08Bibliographically approved
In thesis
1. Approximation and Calibration of Stochastic Processes in Finance
Open this publication in new window or tab >>Approximation and Calibration of Stochastic Processes in Finance
2010 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis is a study of approximation and calibration of stochastic processes with applications in finance. It consists of an introduction and four research papers.

The introduction is as an overview of the role of mathematics incertain areas of finance. It contains a brief introduction to the mathematicaltheory of option pricing, as well as a description of a mathematicalmodel of a financial exchange. The introduction also includessummaries of the four research papers.

In Paper I, Markov decision theory is applied to design algorithmic trading strategies in an order driven market. A high dimensional Markov chain is used to model the state and evolution of the limit order book. Trading strategies are formulated as optimal decision problems. Conditions that guarantee existence of optimal strategies are provided, as well as a value-iterative algorithm that enables numerical construction of optimal strategies. The results are illustrated with numerical experiments on high frequency data from a foreign exchange market.

Paper II focuses on asset pricing with Lévy processes. The expected value E[g(XT )] is estimated using a Monte Carlo method, when Xt is a d-dimensional Lévy process having infinite jump activity and a smooth density. Approximating jumps smaller then a parameter ε > 0 by diffusion results in a weak approximation, Xt, of Xt. The main result of the paper is an estimate of the resulting model error E[g(XT )] − E[g(T )], with a computable leading order term.

Option prices in exponential Lévy models solve certain partia lintegro-differential equations (PIDEs). A finite difference scheme suitable for solving such PIDEs is studied in Paper III. The main resultsare estimates of the time and space discretization errors, with leading order terms in computable form. If the underlying Lévy process has infinite jump activity, the jumps smaller than some ε > 0 are replacedby diffusion. The size of this diffusion approximation is estimated, as well as its effect on the space and time discretization errors. Combined, the results of the paper are enough to determine how to jointly choose the grid size and the parameter ε.

In Paper IV it is demonstrated how optimal control can be used to calibrate a jump-diffusion process to quoted option prices. The calibration problem is formulated as an optimal control problem with the model parameter as a control variable. The corresponding regularized Hamiltonian system is solved with a symplectic Euler method.

Place, publisher, year, edition, pages
Stockholm: KTH, 2010. viii, 45 p.
Trita-MAT. MA, ISSN 1401-2278 ; 10:05
National Category
urn:nbn:se:kth:diva-25090 (URN)978-91-7415-741-3 (ISBN)
Public defence
2010-10-18, Sal F3, Lindstedtsvägen 26, KTH, Stockholm, 13:00 (English)
QC 20101008Available from: 2010-10-08 Created: 2010-10-08 Last updated: 2010-10-08Bibliographically approved

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Kiessling, JonasTempone, Raúl
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