scholarly journals A Robust Spline Collocation Method for Pricing American Put Options

2019 ◽  
Vol 2019 ◽  
pp. 1-11 ◽  
Author(s):  
Zhongdi Cen ◽  
Anbo Le ◽  
Aimin Xu
Keyword(s):  
Maximum Principle ◽  
Collocation Method ◽  
Discrete Maximum Principle ◽  
Original Form ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
Put Options ◽  
Spline Collocation Method ◽  
American Put Options ◽  
American Put

In this paper a robust numerical method is proposed for pricing American put options. The Black-Scholes differential operator in the original form is discretized by using a quadratic spline collocation method on a piecewise uniform mesh for the spatial discretization and the implicit Euler scheme for the time discretization. The position of collocation points is chosen so that the spline difference operator satisfies the discrete maximum principle, which guarantees that the scheme is maximum-norm stable. The error estimation is derived by applying the maximum principle to the discrete linear complementarity problem in two mesh sets. It is proved that the scheme is second-order convergent with respect to the spatial variable and first-order convergent with respect to the time variable. Numerical results demonstrate that the scheme is stable and accurate.

B-SPLINE COLLOCATION METHOD FOR A SINGULARLY PERTURBED REACTION–DIFFUSION PROBLEM USING ARTIFICIAL VISCOSITY

2009 ◽  
Vol 06 (01) ◽  
pp. 23-41 ◽  
Author(s):  
MOHAN K. KADALBAJOO ◽  
PUNEET ARORA
Keyword(s):  
Exact Solution ◽  
Collocation Method ◽  
Reaction Diffusion ◽  
Artificial Viscosity ◽  
Reaction Diffusion Equations ◽  
Singularly Perturbed ◽  
Uniform Mesh ◽  
Spline Collocation ◽  
B Spline ◽  
Spline Collocation Method

In this paper, we develop a B-spline collocation method using artificial viscosity for solving a class of singularly perturbed reaction–diffusion equations. We use the artificial viscosity to capture the exponential features of the exact solution on a uniform mesh, and use the B-spline collocation method, which leads to a tridiagonal linear system. The convergence analysis is given and the method is shown to have uniform convergence of second order. The design of an artificial viscosity parameter is confirmed to be a crucial ingredient for simulating the solution of the problem. Known test problems have been studied to demonstrate the accuracy of the method. Numerical results show the behavior of the method, with emphasis on treatment of boundary conditions. Results shown by the method are found to be in good agreement with the exact solution.

scholarly journals A Finite Difference Scheme for Pricing American Put Options under Kou's Jump-Diffusion Model

2013 ◽  
Vol 2013 ◽  
pp. 1-11 ◽  
Author(s):  
Jian Huang ◽  
Zhongdi Cen ◽  
Anbo Le
Keyword(s):  
Finite Difference ◽  
Difference Scheme ◽  
Diffusion Model ◽  
Finite Difference Scheme ◽  
Jump Diffusion ◽  
Uniform Mesh ◽  
Put Options ◽  
American Put Options ◽  
Jump Diffusion Model ◽  
American Put

We present a stable finite difference scheme on a piecewise uniform mesh along with a penalty method for pricing American put options under Kou's jump-diffusion model. By adding a penalty term, the partial integrodifferential complementarity problem arising from pricing American put options under Kou's jump-diffusion model is transformed into a nonlinear parabolic integro-differential equation. Then a finite difference scheme is proposed to solve the penalized integrodifferential equation, which combines a central difference scheme on a piecewise uniform mesh with respect to the spatial variable with an implicit-explicit time stepping technique. This leads to the solution of problems with a tridiagonal M-matrix. It is proved that the difference scheme satisfies the early exercise constraint. Furthermore, it is proved that the scheme is oscillation-free and is second-order convergent with respect to the spatial variable. The numerical results support the theoretical results.

scholarly journals Pricing Perpetual American Put Options with Asset-Dependent Discounting

2021 ◽  
Vol 14 (3) ◽  
pp. 130
Author(s):  
Jonas Al-Hadad ◽  
Zbigniew Palmowski
Keyword(s):  
Value Function ◽  
Asset Price ◽  
Stopping Times ◽  
Martingale Measure ◽  
Put Options ◽  
American Put Options ◽  
Exact Calculations ◽  
Negative Exponential ◽  
American Put ◽  
The Value Function

The main objective of this paper is to present an algorithm of pricing perpetual American put options with asset-dependent discounting. The value function of such an instrument can be described as VAPutω(s)=supτ∈TEs[e−∫0τω(Sw)dw(K−Sτ)+], where T is a family of stopping times, ω is a discount function and E is an expectation taken with respect to a martingale measure. Moreover, we assume that the asset price process St is a geometric Lévy process with negative exponential jumps, i.e., St=seζt+σBt−∑i=1NtYi. The asset-dependent discounting is reflected in the ω function, so this approach is a generalisation of the classic case when ω is constant. It turns out that under certain conditions on the ω function, the value function VAPutω(s) is convex and can be represented in a closed form. We provide an option pricing algorithm in this scenario and we present exact calculations for the particular choices of ω such that VAPutω(s) takes a simplified form.

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