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Title: Expansion methods for high-dimensional PDEs in finance
Author: Wissmann, Rasmus
ISNI:       0000 0004 5923 3570
Awarding Body: University of Oxford
Current Institution: University of Oxford
Date of Award: 2015
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We develop expansion methods as a new computational approach towards high-dimensional partial differential equations (PDEs), particularly of such type as arising in the valuation of financial derivatives. The proposed methods are extended from [41] and use principal component analysis (PCA) of the underlying process in combination with a Taylor expansion of the value function into solutions to low-dimensional PDEs. They enable calculation of highly accurate approximate solutions with computational complexity polynomial in the number of dimensions for PDEs with a low number of dominant principal components. For the case of PDEs with constant coefficients, we show existence of expansion solutions and prove theoretical error bounds. We give a precise characterisation of when our methods can be applied and construct specific examples of a first and second order version. We provide numerical results showing that the empirically observed convergence speeds are in agreement with the theoretical predictions. For the case of PDEs with varying coefficients, we give a heuristic motivation using the Parametrix approach and empirically test the methods' accuracy for a range of variable parameter stock models. We demonstrate the applicability of our expansion methods to real-world securities pricing problems by considering path-dependent and early-exercise options in the LIBOR market model. Using the example of Bermudan swaptions and Ratchet floors, which are considered difficult benchmark problems, we give a careful analysis of the numerical accuracy and computational complexity. We are able to demonstrate that for problems with medium to high dimensionality, around 60-100, and moderate time horizons, the presented PDE methods deliver results comparable in accuracy to benchmark state-of-the-art Monte Carlo methods in similar or (significantly) faster run time.
Supervisor: Christoph, Reisinger Sponsor: Engineering and Physical Sciences Research Council ; University of Oxford ; Mathematical Institute ; St. Catherine's College ; Nomura ; German National Academic Foundation (Studienstiftung des deutschen Volkes) ; Oxford Man-Institute for Quantitative Finance ; G-Research
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: Partial differential equations ; Approximations and expansions ; Numerical analysis ; Mathematical finance ; High-dimensional PDEs ; Asymptotic expansions ; Financial derivative pricing