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An Operator Splitting Method for Pricing American Options 283
where A is one of the operators A BS , A JD ,orA SV defined by (1), (2), and
(4), respectively.
The operator splitting method is derived from a formulation with a
Lagrange multiplier λ after a temporal discretization. In the continuous level,
the formulation with the Lagrange multiplier reads
{
(v t − Av) =λ, λ ≥ 0, v ≥ g,
(8)
λ(v − g) =0.
4 Discretizations
4.1 Spatial Discretizations
The LCPs are posed on infinite domain as there is no upper limit for the value
of the asset and also for variance in the case of Heston’s stochastic volatility
model. In order to use finite difference discretizations for the spatial derivatives,
the domain is truncated from sufficiently large values of x and y which
are denoted by X and Y , respectively. The choice of X for the Black–Scholes
model is considered in [KN00], for example. On the truncation boundaries a
suitable boundary condition needs to be posed. For the one-dimensional models
for put options, we use homogeneous Dirichlet boundary condition v =0
at x = X. For Heston’s model homogeneous Neumann boundary conditions
are posed. While these are fairly typical choices for boundary conditions there
are also other choices.
For the interval [0,X], we define subintervals [x i−1 ,x i ], i =1, 2,...,m,
where x i s satisfy 0 = x 0