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Diffusion approximation of Lévy processes with a view towardsfinance
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]

Let the (log-)prices of a collection of securities be given by a d–dimensional Lévy process Xt having infinite activity and a smooth density. The value of a European contract with payoff g(x) maturing at T is determined by E[g(XT )]. Let ¯XT be a finite activity approximation to XT , where diffusion is introduced to approximate jumps smaller than a given truncation level ε > 0. The main result of this work is a derivationof an error expansion for the resulting model error, E[g(XT )−g( ¯XT )], with computable leading order term. Our estimate depends both on the choice of truncation level ε and the contract payoff g, and it is valid even when g is not continuous. Numerical experiments confirm that the error estimate is indeed a good approximation of the model error.

Using similar techniques we indicate how to construct an adaptive truncation type approximation. Numerical experiments indicate that a substantial amount of work is to be gained from such adaptive approximation. Finally, we extend the previous model error estimates to the case of Barrier options, which have a particular path dependent structure.

National Category
Computational Mathematics
URN: urn:nbn:se:kth:diva-25078OAI: diva2:355652
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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Mathematics (Div.)Numerical Analysis, NA
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