Notes on the BENCHOP implementations for the FDNU method

This text describes the FD-NU method and its implementation for the BENCHOP-project. 1 Spatial discretizations For example, under the Black-Scholes model European option prices u satisfy the PDE ut(s, t) + 1 2 σsuss(s, t) + rsus(s, t)− ru(s, t) = 0, s > 0, t ∈ [0, T ), (1) where σ and r are the volatility and interest rate, respectively. We employ quadratically refined grids defined by si = [( i γn − 1 ) ∣∣∣∣ i γn − 1 ∣∣∣∣+ 1]K, i = 0, 1, . . . , n, where K is the strike price. The constant γ is chosen to be 4 10 except for the barrier options and under the Merton model. For the European spread options, the grids for the both spatial directions are given by the above formula with K = 100. For the Heston model, the variance grid is defined by vj = ( j nv )2 , j = 0, 1, . . . , nv. The spatial derivaties are mainly discretized using the central finite differences. Let the grid steps be denoted ∆si = si+1 − si, i = 0, 1, . . . , n− 1. Then the approroximations for the first-order and second-order spatial derivatives are us(si) ≈ 1 ∆si−1 + ∆si [ − ∆si ∆si−1 ui−1 + ( ∆si ∆si−1 − ∆si−1 ∆si ) ui + ∆si−1 ∆si ui+1 ]