ADI Finite Difference Discretization of the Heston-Hull-White PDE

Abstract

In this paper we investigate the effectiveness of Alternating Direction Implicit (ADI) time discretization schemes in the numerical solution of the three-dimensional Heston-Hull-White partial differential equation, which is semidiscretized by applying finite difference schemes on nonuniform spatial grids. We consider the Heston-Hull-White model with arbitrary correlation factors, with time-dependent mean-reversion levels, with short and long maturities, for cases where the Feller condition is satisfied and for cases where it is not. In addition, both European-style call options and up-and-out call options are considered. It is shown through extensive tests that ADI schemes, with a proper choice of their parameters, perform very well in all situations -in terms of stability, accuracy and efficiency.

Efficient Estimation of Sensitivities for Counterparty Credit Risk with the Finite Difference Monte-Carlo Method

“…S left and S right are chosen to contain the region of interest, ie, the region around the expected mean of the underlying. Following Haentjens and In't Hout (2012), for options without barriers we choose…”

“…Also, for the r d -direction, we follow Haentjens and In't Hout (2012). The grid is dense around 0 and stretched symmetrically toward the boundaries R max and R max by using a sinus hyperbolic function.…”

“…The domain to be discretized is chosen as OE0; S max OE0; V max OE R max ; R max , where S max , V max and R max are chosen to be sufficiently large to minimize the effect of the imposed boundary conditions, and such that all simulated market scenarios can be interpolated on the grid. Let s 0 < s 1 < < s m 1 , v 0 < v 1 < < v m 2 and r 0 < r 1 < < r m 3 be the discretization in the s-, v-and r d -directions, respectively, similarly to in Haentjens and In't Hout (2012). In all dimensions, the grid is chosen to be nonuniform.…”

“…We do this because, for realistic test parameters, the expected mean of the variance process is close to 0. In addition, because the Heston PDE in the v-direction is convection-dominated close to 0 and the initial condition is non-smooth, numerical stability requires a high density of points in this region (Haentjens and In't Hout 2012).…”

CVA under Bates Model with Stochastic Default Intensity

“…Solving Heston PDE to price European and American option is extensively studied in [20] [21]. We follow the scheme mentioned in [21] for discretization.…”