Finite Difference Method for the Multi-Asset Black–Scholes Equations

Journal of Function Spaces

et al. 2021

Self Cite

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.

“…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.…”

Section: Numerical Solutionsmentioning

confidence: 99%

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

Kim

Journal of Function Spaces

et al. 2021

Self Cite

“…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:…”

Section: Numerical Solution Algorithmmentioning

confidence: 99%

An Unconditionally Stable Positivity-Preserving Scheme for the One-Dimensional Fisher–Kolmogorov–Petrovsky–Piskunov Equation

Kim

Discrete Dynamics in Nature and Society

et al. 2021

Self Cite

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.

“…Let u(x, t) be the value at time t of a contingent claim on a single interest rate with u(x, T) = Φ(x). We consider the following HW PDE with respect to x(t) in Equation ( 4) and t: 2 . Equation ( 5) can be derived by the usual no-arbitrage argument applying Ito's lemma.…”

Section: Hw Pde For a Single Interest Ratementioning

confidence: 99%

“…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]).…”

Section: Introductionmentioning

confidence: 99%

Finite Difference Method for the Hull–White Partial Differential Equations

Yang

2020

Mathematics

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.