An efficient numerical approach for solving three-dimensional Black-Scholes equation with stochastic volatility

This paper develops an efficient combined interpolation/finite element approach for solving a three-dimensional Black-Scholes problem with stochastic volatility. The technique consists to approximate the time derivative by interpolation whereas the space derivatives are approximated using the finite element method. Both stability and error estimates of the new algorithm are deeply analyzed in the $L^{\infty }$-norm. The proposed method is explicit, unconditionally stable, temporal second-order accurate and fourth-order convergence in space. This result suggests that the constructed scheme is faster and more efficient than a broad range of numerical methods widely studied in the literature for the Black-Scholes models. Some numerical experiments are carried out to confirm the theoretical analysis.