For valuing European options, a straightforward model is the well-known Black-Scholes formula. Contrary to market reality, this model assumed that interest rate and volatility are constant. To modify the Black-Scholes model, Heston and Cox-Ingersoll-Ross (CIR) offered the stochastic volatility and the stochastic interest rate models, respectively. The combination of the Heston, and the CIR models is called the Heston-Cox-Ingersoll-Ross (HCIR) model. Another essential issue that arises when purchasing or selling a good or service is the consideration of transaction costs which was ignored in the Black-Scholes technique. Leland improved the simplistic Black-Scholes strategy to take transaction costs into account. The main purpose of this paper is to apply the alternating direction implicit (ADI) method at a uniform grid for solving the HCIR model with transaction costs in the European style and comparing it with the explicit finite difference (EFD) scheme. Also, as evidence for numerical convergence, we convert the HCIR model with transaction costs to a linear PDE (HCIR) by ignoring transaction costs, then we estimate the solution of HCIR PDE using the ADI method which is a class of finite difference schemes, and compare it with analytical solution and EFD scheme. For multi-dimensional Black-Scholes equations, the ADI method, which is a category of finite difference techniques, is appropriate. When the dimensionality of the space increases, finite difference techniques frequently become more complex to perform, comprehend, and apply. Consequently, we employ the ADI approach to divide a multi-dimensional problem into several simpler, quite manageable sub-problems to overcome the dimensionality curse.

翻译：针对欧式期权定价问题，经典模型为众所周知的Black-Scholes公式。然而该模型假设利率和波动率为常数，这与市场实际情况相悖。为改进Black-Scholes模型，Heston和Cox-Ingersoll-Ross（CIR）分别提出了随机波动率模型和随机利率模型。将Heston模型与CIR模型相结合即构成Heston-Cox-Ingersoll-Ross（HCIR）模型。另一个在商品或服务交易中产生的关键问题是交易成本的考量，而Black-Scholes方法忽视了这一因素。Leland对简化的Black-Scholes策略进行了改进以纳入交易成本。本文的主要目的是在均匀网格上应用交替方向隐式（ADI）方法求解含交易成本的欧式HCIR模型，并与显式有限差分（EFD）格式进行对比。此外，为验证数值收敛性，我们通过忽略交易成本将含交易成本的HCIR模型转化为线性偏微分方程（HCIR），进而采用属于有限差分格式类别的ADI方法估计HCIR偏微分方程的解，并与解析解及EFD格式进行比较。对于多维Black-Scholes方程，作为有限差分技术分支的ADI方法具有适用性。当空间维度增加时，有限差分技术的实施、理解与应用往往愈发复杂。因此，我们采用ADI方法将多维问题分解为多个更简单且易于处理的子问题，以克服维度灾难。