Picard Approximation of Stochastic Differential Equations and Application to Libor Models
AbstractThe aim of this work is to provide fast and accurate approx- imation schemes for the Monte Carlo pricing of derivatives in LIBOR market models. Standard methods can be applied to solve the stochas- tic differential equations of the successive LIBOR rates but the methods are generally slow. Our contribution is twofold. Firstly, we propose an alternative approximation scheme based on Picard iterations. This ap- proach is similar in accuracy to the Euler discretization, but with the feature that each rate is evolved independently of the other rates in the term structure. This enables simultaneous calculation of derivative prices of different maturities using parallel computing. Secondly, the product terms occurring in the drift of a LIBOR market model driven by a jump process grow exponentially as a function of the number of rates, quickly rendering the model intractable. We reduce this growth from exponen- tial to quadratic using truncated expansions of the product terms. We include numerical illustrations of the accuracy and speed of our method pricing caplets, swaptions and forward rate agreements.
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Bibliographic InfoPaper provided by School of Economics and Management, University of Aarhus in its series CREATES Research Papers with number 2010-40.
Date of creation: 16 Jul 2010
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Web page: http://www.econ.au.dk/afn/
LIBOR models; Lévy processes; Picard approximation; drift expansion; parallel computing.;
Find related papers by JEL classification:
- G12 - Financial Economics - - General Financial Markets - - - Asset Pricing; Trading Volume; Bond Interest Rates
- G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing
- C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
This paper has been announced in the following NEP Reports:
- NEP-ALL-2010-09-03 (All new papers)
- NEP-CMP-2010-09-03 (Computational Economics)
- NEP-ORE-2010-09-03 (Operations Research)
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