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In this paper, we briefly review the finite difference method (FDM) for the Black–Scholes (BS) equations for pricing derivative securities and provide the MATLAB codes in the Appendix for the one-, two-, and three-dimensional numerical implementation. The BS equation is discretized non-uniformly in space and implicitly in time. The two- and three-d...
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Context 1
... discretize Ω with a non-uniform space step Figure 1 is an example of 3D non-uniform grid. Let u n ijK = u x i , y j , z k , n∆τ , where i = 0, . . . ...
Context 2
... interp3() function, which returns interpolated value from a function of three variables at the query points, is used to determine the value at specific query point using the calculated u. The code for Algorithm 1 is as follows, Listing 4: i+1,j+1,k)-u(i+1,j-1,k)-u(i-1,j+1,k)+u(i-1,j-1,k))... ...
Citations
... and the other terms are similarly defined. Additional details can be found in [33,34]. We solve the discrete Equation (9) using the operator splitting method. ...
... Let δτ = ðΔτÞ α Γð2 − αÞ for simplicity of exposition; then we sequentially solve the following equations [34]: ...
... Note that if we sum up these three equations (14)-(16), then we obtain Equation (9). For the detailed numerical solution, algorithm with source program code of Equations (14)-(16) can be found in [34]. We use the linear boundary condition, specifically, for example, in the case of Equation (14) (see Figure 2): Journal of Function Spaces ...
In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-order α in the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.
... e Dirichlet boundary condition can also be used. en, we solve equation (4) using the omas algorithm [30]. Second, we solve the nonlinear equation using an interpolation method: ...
In this study, we present an unconditionally stable positivity-preserving numerical method for the Fisher–Kolmogorov–Petrovsky–Piskunov (Fisher–KPP) equation in the one-dimensional space. The Fisher–KPP equation is a reaction-diffusion system that can be used to model population growth and wave propagation. The proposed method is based on the operator splitting method and an interpolation method. We perform several characteristic numerical experiments. The computational results demonstrate the unconditional stability, boundedness, and positivity-preserving properties of the proposed scheme.
... Duffy (1976) investigated option pricing problems represented by a partial differential approach. Besides, to obtain high order accuracy in the solution of different option pricing models, some authors studied compact difference schemes (Zhao et al., 2007;Liao and Khaliq, 2009;Tangman et al., 2008;Düring et al., 2014;Jeong et al., 2018, Raol andGoura, 2020 (Heo et al., 2019, Kim et al., 2020, Yan et al., 2020. ...
The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyze these models. This study will focus mainly on investigating the behavior of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution.
... The finite difference method (FDM) has been widely used to compute the prices of financial derivatives numerically (Duffy [1]). Many studies on option pricing employ FDM, especially on models such as Black-Scholes partial differential equation (PDE) for equity or exchange rate derivatives (Kim et al. [2]). Among financial derivatives on various underlying assets, interest rate derivatives (IRDs), whose payoffs depend on interest rates or bond prices, have the largest trading volume in the global over-the-counter market (BIS [3]). ...
This paper reviews the finite difference method (FDM) for pricing interest rate derivatives (IRDs) under the Hull–White Extended Vasicek model (HW model) and provides the MATLAB codes for it. Among the financial derivatives on various underlying assets, IRDs have the largest trading volume and the HW model is widely used for pricing them. We introduce general backgrounds of the HW model, its associated partial differential equations (PDEs), and FDM formulation for one- and two-asset problems. The two-asset problem is solved by the basic operator splitting method. For numerical tests, one- and two-asset bond options are considered. The computational results show close values to analytic solutions. We conclude with a brief comment on the research topics for the PDE approach to IRD pricing.
... Step (3) Symmetrize the system given in Equation (19) as follows: ...
... Before we proceed to test the iterative Crank-Nicolson method for different payoffs, we show the symmetrization effect, Equation (20). The comparison between non-symmetrized system, given in Equation (19), and the symmetrized system is shown in Figure 2. The cash or nothing payoff is used with those parameters given in Equation (27). The slope for both symmetric and non-symmetric case is approximately constant, which suggests that the number of iteration per time steps and the number of iteration per spatical discretization does not grow as grid refinement. ...
... The second order of convergence can be proved by calculating the truncation error. The truncation error of iterative Crank-Nicolson discretization with non-uniform grid size is indeed second order in space and time (see Appendix in [19]). ...
The Crank–Nicolson method can be used to solve the Black–Scholes partial differential equation in one-dimension when both accuracy and stability is of concern. In multi-dimensions, however, discretizing the computational grid with a Crank–Nicolson scheme requires significantly large storage compared to the widely adopted Operator Splitting Method (OSM). We found that symmetrizing the system of equations resulting from the Crank–Nicolson discretization help us to use the standard pre-conditioner for the iterative matrix solver and reduces the number of iterations to get an accurate option values. In addition, the number of iterations that is required to solve the preconditioned system, resulting from the proposed iterative Crank–Nicolson scheme, does not grow with the size of the system. Thus, we can effectively reduce the order of complexity in multidimensional option pricing. The numerical results are compared to the one with implicit Operator Splitting Method (OSM) to show the effectiveness.
We provide an accurate, simple formula for pricing multidimensional European options. The formula is as simple as the Black-Scholes formula. Therefore, the (costly) computational methods are needless. Moreover, our method allows the calculation of the implied volatility of the underlying asset of a multidimensional option.
We devise a method to circumvent the complexity that arises from the option multi-dimensionality. That is, we transform the model to make it as simple as the one-dimensional case. Furthermore, the assumption of comonotonicity and other assumptions regarding the structure of the underlying asset become needless.
