The student will work on designing new numerical methods based on hierarchical adaptive sparse grids quadratures combined with Fourier techniques for efficient pricing of high-dimensional (multi-assets) European Options. Specifically, the student will contemplate the possibility of finding a heuristic framework for an optimal choice of the integration contour (damping parameter) which controls the analyticity of the integrand in the Fourier space and hence accelerate the performance of the quadrature methods. He will also develop a systematic comparison between hierarchical deterministic quadrature methods, Tensor Product (TP) quadrature, Smolyak (SM) Sparse Grids quadrature, and Adaptive Sparse Grids (ASG) quadrature to numerically evaluate the option price under different pricing dynamics, Geometric Brownian Motion (GBM), Variance Gamma (VG) and Normal Inverse Gaussian (NIG) for different multi-asset payoff functions such as Basket Call/Put and Rainbow options. The student is also asked to elaborate a comparison in terms of computational complexity against the quadrature methods for different dimensions, and various combination of parameter sets within the mentioned pricing models.