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Pricing and hedging of financial derivatives using a posteriori error estimates and adaptive methods for stochastic differential equations
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
2010 (English)In: Journal of Computational and Applied Mathematics, ISSN 0377-0427, E-ISSN 1879-1778, Vol. 235, 563-592 p.Article in journal (Refereed) Published
Abstract [en]

The efficient and accurate calculation of sensitivities of the price of financial derivatives with respect to perturbations of the parameters in the underlying model, the so-called `Greeks', remains a great practical challenge in the derivative industry. This is true regardless of whether methods for partial differential equations or stochastic differential equations (Monte Carlo techniques) are being used. The computation of the `Greeks' is essential to risk management and to the hedging of financial derivatives and typically requires substantially more computing time as compared to simply pricing the derivatives. Any numerical algorithm (Monte Carlo algorithm) for stochastic differential equations produces a time-discretization error and a statistical error in the process of pricing financial derivatives and calculating the associated `Greeks'. In this article we show how a posteriori error estimates and adaptive methods for stochastic differential equations can be used to control both these errors in the context of pricing and hedging of financial derivatives. In particular, we derive expansions, with leading order terms which are computable in a posteriori form, of the time-discretization errors for the price and the associated `Greeks'. These expansions allow the user to simultaneously first control the time-discretization errors in an adaptive fashion, when calculating the price, sensitivities and hedging parameters with respect to a large number of parameters, and then subsequently to ensure that the total errors are, with prescribed probability, within tolerance.

Place, publisher, year, edition, pages
Elsevier , 2010. Vol. 235, 563-592 p.
Keyword [en]
Sensitivity analysis, Parabolic partial differential equations, Stochastic differential equations, Euler scheme, A posteriori error estimate, Adaptive algorithms, hedging, Financial derivatives
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:umu:diva-37419DOI: 10.1016/j.cam.2010.06.009ISI: 000282394500004OAI: oai:DiVA.org:umu-37419DiVA: diva2:360270
Available from: 2010-11-03 Created: 2010-11-02 Last updated: 2017-12-12Bibliographically approved
In thesis
1. The Skorohod problem and weak approximation of stochastic differential equations in time-dependent domains
Open this publication in new window or tab >>The Skorohod problem and weak approximation of stochastic differential equations in time-dependent domains
2009 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

This thesis consists of a summary and four scientific articles. All four articles consider various aspects of stochastic differential equations and the purpose of the summary is to provide an introduction to this subject and to supply the notions required in order to fully understand the articles.

In the first article we conduct a thorough study of the multi-dimensional Skorohod problem in time-dependent domains. In particular we prove the existence of cádlág solutions to the Skorohod problem with oblique reflection in time-independent domains with corners. We use this existence result to construct weak solutions to stochastic differential equations with oblique reflection in time-dependent domains. In the process of obtaining these results we also establish convergence results for sequences of solutions to the Skorohod problem and a number of estimates for solutions, with bounded jumps, to the Skorohod problem.

The second article considers the problem of determining the sensitivities of a solution to a second order parabolic partial differential equation with respect to perturbations in the parameters of the equation. We derive an approximate representation of the sensitivities and an estimate of the discretization error arising in the sensitivity approximation. We apply these theoretical results to the problem of determining the sensitivities of the price of European swaptions in a LIBOR market model with respect to perturbations in the volatility structure (the so-called ‘Greeks’).

The third article treats stopped diffusions in time-dependent graph domains with low regularity. We compare, numerically, the performance of one adaptive and three non-adaptive numerical methods with respect to order of convergence, efficiency and stability. In particular we investigate if the performance of the algorithms can be improved by a transformation which increases the regularity of the domain but, at the same time, reduces the regularity of the parameters of the diffusion.

In the fourth article we use the existence results obtained in Article I to construct a projected Euler scheme for weak approximation of stochastic differential equations with oblique reflection in time-dependent domains. We prove theoretically that the order of convergence of the proposed algorithm is 1/2 and conduct numerical simulations which support this claim.

Place, publisher, year, edition, pages
Umeå: Umeå universitet, 2009. 36 p.
Series
Doctoral thesis / Umeå University, Department of Mathematics, ISSN 1102-8300 ; 42
Keyword
Skorohod problem, weak approximation, time-dependent domain, stochastic differential equations, parabolic partial differential equations, oblique reflection, stopped diffusions, Euler scheme, adaptive methods, sensitivity analysis, financial derivatives, 'Greeks'
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:umu:diva-25429 (URN)978-91-7264-823-4 (ISBN)
Distributor:
Institutionen för matematik och matematisk statistik, 90187, Umeå
Public defence
2009-09-18, MA121, MIT-huset, Umeå universitet, Umeå, 13:15 (English)
Opponent
Supervisors
Available from: 2009-08-31 Created: 2009-08-17 Last updated: 2010-11-03Bibliographically approved

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