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978-3-8439-2238-8, Reihe Mathematik

Martin Altmayer Quadrature of discontinuous SDE functionals using Malliavin integration by parts

106 Seiten, Dissertation Universität Mannheim (2015), Hardcover, A5

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. This theoretical result is accompanied by numerical experiments which demonstrate that using the smoothed functional improves performance by a factor between 2 and 4.

Furthermore, the same integration by parts trick that was used to smooth the functional can be applied to derive a weak rate of convergence in the Heston model without making any smoothness assumptions on the payoff f - at the price of a rather strong condition on the model parameters.