Bayer, Christian, Markus Siebenmorgen, and Raul Tempone. "Smoothing the payoff for efficient computation of Basket option prices." Quantitative Finance 18, no. 3 (2018): 491-505.
Bayer, Christian, Markus Siebenmorgen, And Raul Tempone.
Computational Finance, European option pricing, Multivariate approximation and integration, Sparse grids, Stochastic Collocation methods, Monte Carlo, and Quasi-Monte Carlo methods.
We consider the problem of pricing basket options in a multivariate Black–Scholes or Variance-Gamma model. From a numerical point of view, pricing such options correspond to moderate and high-dimensional numerical integration problems with non-smooth integrands. Due to this lack of regularity, higher-order numerical integration techniques may not be directly available, requiring the use of methods like Monte Carlo specifically designed to work for non-regular problems. We propose to use the inherent smoothing property of the density of the underlying in the above models to mollify the payoff function by means of an exact conditional expectation. The resulting conditional expectation is unbiased and yields a smooth integrand, which is amenable to the efficient use of adaptive sparse-grid cubature. Numerical examples indicate that the high-order method may perform orders of magnitude faster than Monte Carlo or Quasi-Monte Carlo methods in dimensions up to 35.