Numerical Solution of HCIR Equation with Transaction Costs using Alternating Direction Implicit Method
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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.
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