Efficiently pricing multi-asset options is a challenging problem in computational finance. Although classical Fourier methods are extremely fast in pricing single asset options, maintaining the tractability of Fourier techniques for multi-asset option pricing is still an area of active research. Fourier methods rely on explicit knowledge of the characteristic function of the suitably stochastic price process, allowing for calculation of the option price by evaluation of multidimensional integral in the Fourier domain. The high smoothness of the integrand in the Fourier space motivates the exploration of deterministic quadrature methods that are highly efficient under certain regularity assumptions, such as, adaptive sparse grids quadrature (ASGQ), and Randomized Quasi-Monte Carlo (RQMC). However, when designing a numerical quadrature method for most of the existing Fourier pricing approaches, two key factors affecting the complexity should be carefully controlled, (i) the choice of the vector of damping parameters that ensure the Fourier-integrability and control the regularity class of the integrand, (ii) the high-dimensionality of the integration problem. To address these challenges, in the first part of this thesis we propose a rule for choosing the damping parameters, resulting in smoother integrands. Moreover, we explore the effect of sparsification and dimension-adaptivity in alleviating the curse of dimensionality. Despite the efficiency of ASGQ, the error estimates are very hard to compute. In cases where error quantification is of high priority, in the second part of this thesis, we design an RQMC-based method for the (inverse) Fourier integral computation. RQMC integration is known to be highly efficient for high-dimensional integration problems of sufficiently regular integrands, and it further allows for computation of probabilistic estimates. Nonetheless, using RQMC requires an appropriate domain transformation of the unbounded integration domain to the hypercube, which may originate in a transformed integrand with singularities at the boundaries, and consequently deteriorate the rate of convergence. To preserve the nice properties of the transformed integrand,we propose a model-dependent domain transformation to avoid these corner singularities and retain the optimal efficiency of RQMC. The effectiveness of the proposed optimal damping rule, the designed domain transformation procedure, and their combination with ASGQ and RQMC are demonstrated via several numerical experiments and computational comparisons to the MC approach and the COS method.