  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.
