A practical finite difference method for the three-dimensional Black–Scholes equation

Super-Fast Computation for the Three-Asset Equity-Linked Securities Using the Finite Difference Method

“…From these three types of ELS results, we can confirm that the proposed explicit FDM is fast and accurate. Finally, we compare our method with the previous work [20], which used an operator splitting method (OSM) to verify that the proposed algorithm is fast and accurate. In the OSM, we solve the governing equations with multiple steps:…”

“…where Ω is the computational domain, r is the risk-free interest rate, σ x , σ y , and σ z are the volatilities of the underlying assets x, y, and z, respectively, and ρ xy , ρ yz , and ρ zx are the correlation values between each two underlying assets. More details about the equation can be found in [20].…”

“…Figure 3 shows the schematic diagram of the differences between u and v at τ = 2.5. More details regarding ELS structure can be found in [20].…”

“…For more details about the numerical procedure, please refer to [20]. For comparison, we take the Type 3 ELS with all the same parameter values.…”

“…An adjusted grid was used to better accommodate the boundary conditions of the problem. To save the computational time when pricing ELS, the authors in [20] obtained a non-uniform grid by trial and error. They used an implicit scheme which requires solving a tridiagonal system of discrete equations.…”