Quadrature of discontinuous SDE functionals using Malliavin integration by parts

Altmayer, Martin

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URL:	https://ub-madoc.bib.uni-mannheim.de/39207
URN:	urn:nbn:de:bsz:180-madoc-392077
Document Type:	Doctoral dissertation
Year of publication:	2015
Place of publication:	Mannheim
University:	Universität Mannheim
Evaluator:	Neuenkirch, Andreas
Date of oral examination:	12 May 2015
Publication language:	English
Institution:	School of Business Informatics and Mathematics > Wirtschaftsmathematik II (Neuenkirch 2013-)
Subject:	510 Mathematics
Subject headings (SWD):	Numerische Mathematik , Malliavin-Kalkül , Stochastische Differentialgleichung
Keywords (English):	Heston model , SDE , Malliavin calculus
Abstract:	One of the major problems in mathematical finance is the pricing of options. This requires the computation of expectations of the form E(f(S_T)) with S_T being the solution to a stochastic differential equation at a specific time T and f being the payoff function of the option. A very popular choice for S is the Heston model. While in the one-dimensional case E(f(S_T)) can often be computed using methods based on PDEs or the FFT, multidimensional models typically require the use of Monte-Carlo methods. Here, the multilevel Monte-Carlo algorithm provides considerably better performance - a benefit that is however reduced if the function f is discontinuous. This thesis introduces an approach based on the integration by parts formula from Malliavin calculus to overcome this problem: The original function is replaced by a function containing its antiderivative and by a Malliavin weight term. We will prove that because the new functional is continuous, we can now apply multilevel Monte-Carlo to compute the value of the original expectation without performance reduction.