We propose a hybrid spatial finite-difference/pseudospectral discretization for European option-pricing problems under the Heston and Heston–Hull–White models. In the direction of the underlying asset, where the payoff profile is nonsmooth, we use a standard central second-order finite-difference scheme, whereas we use a Chebyshev collocation method in the other spatial dimensions. In the time domain, we employ alternating direction implicit schemes to efficiently decompose the system matrix into simpler one-dimensional problems. This approach allows us to compute numerical solutions, which are second-order accurate in time and exhibit spectral accuracy in the spatial domains except for the asset direction. The numerical experiments reveal that the proposed scheme outperforms the standard second-order finite-difference scheme in terms of accuracy versus runtime and shows an unconditionally stable behavior.

中文翻译：

---

Heston 和 Heston-Hull-White 偏微分方程的混合有限差分/伪谱方法

我们针对 Heston 和 Heston-Hull-White 模型下的欧洲期权定价问题提出了一种混合空间有限差分/伪谱离散化。在标的资产方向，收益曲线不平滑，我们使用标准的中心二阶有限差分格式，而我们在其他空间维度使用切比雪夫搭配方法。在时域中，我们采用交替方向隐式方案来有效地将系统矩阵分解为更简单的一维问题。这种方法允许我们计算数值解，这些解在时间上是二阶精度的，并且在除资产方向之外的空间域中表现出光谱精度。