Mathematical Finance Seminar
RUD 25. 1.115
Chiheb Ben Hammouda (RWTH Aachen)

Smoothing Techniques Combined with Hierarchical Approximations for Efficient Option Pricing

When approximating the expectation of a functional of a stochastic process, in particular for option pricing purposes, the performance of numerical integration methods based on deterministic quadra- ture, quasi-Monte Carlo (QMC), or multilevel Monte Carlo (MLMC) techniques may critically depend on the regularity of the integrated. To overcome this issue, we introduce in [1,2,3] different smoothing tech- niques. In the first part of the talk, we will discuss our novel numerical smoothing approach [1,2] in which we combine root-finding methods with one-dimensional integration with respect to a single well-selected variable, focusing on cases where the discretization of the asset price dynamics is necessary. We prove that, under appropriate conditions, the resulting function of the remaining variables is highly smooth, affording the improved efficiency of adaptive sparse grid quadrature (ASGQ) and QMC methods, particularly when combined with hierarchical transformations (i.e., Brownian bridge and Richardson extrapolation on the weak error). Our analysis in [1] demonstrates the advantages of combining numerical smoothing with AS- GQ and QMC methods over ASGQ and QMC methods without smoothing, and the Monte Carlo approach. Moreover, our analysis in [2] shows that our numerical smoothing improves the robustness (the kurtosis at deep levels becomes bounded) and complexity of the MLMC method. In particular, we recover the optimal MLMC complexities obtained for Lipschitz functionals.In the second part of the talk, we will discuss our efficient Fourier-based method in [3] for pricing European multi-asset options under L ́evy models. Given that the integrand in the frequency space often has higher regularity than in the physical space, we extend the one-dimensional Fourier valuation formula to the multivariate case and employ two complementary ideas. First, we smooth the Fourier integrand via an optimized choice of the damping parameters based on a proposed heuristic optimization rule. These parameters ensure integrability and control the regularity class of the integrand. Second, we use sparsification and dimension-adaptivity techniques to accelerate the convergence of the numerical quadrature in high dimensions. We demonstrate the advantages of adaptivity and our damping parameter rule on the numerical complexity of the quadrature methods. Moreover, we reveal that our approach achieves substantial computational gains compared to the Monte Carlo method for different dimensions and parameter constellations.