A second-order weak approximation of Heston model by discrete random variables

In our previous paper [A. Lenkšas and V. Mackevičius, Weak approximation of Heston model by discrete random variables, Math. Comput. Simul., 113:1–15, 2015], we constructed a first-order weak approximation for the solution of the Heston model that uses, at each step, generation of two discrete two-valued random variables. An extension of that result to a second-order approximation has met some serious challenges, which we have finally overcome in this paper. We construct a second-order weak approximation for the solution of the Heston model that uses, at each step, generation of two simple discrete random variables. The log-Heston equation system is split into the deterministic part, solvable explicitly, and the stochastic part that is approximated by discrete random variables. The approximation is illustrated by several option pricing simulation examples, including the comparison of the constructed approximation with well-known approximations by Andersen and Alfonsi.