Cubic Hermite Finite Element Method for Nonlinear Black-Scholes Equation Governing European Options

2021 ◽  
Vol 2 (2) ◽  
pp. 23-38
Author(s):  
Teófilo Domingos Chihaluca
Keyword(s):  
Differential Equation ◽  
Interpolation Method ◽  
European Options ◽  
European Option ◽  
Spatial Direction ◽  
European Option Pricing ◽  
Black Scholes ◽  
Temporal Direction ◽  
Good Agreement ◽  
Hermite Finite Element

A numerical algorithm for solving a generalized Black-Scholes partial differential equation, which arises in European option pricing considering transaction costs is developed. The Crank-Nicolson method is used to discretize in the temporal direction and the Hermite cubic interpolation method to discretize in the spatial direction. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behaviour of the solutions, which is also found to be in good agreement with the exact solution.

scholarly journals Fractional Order Stochastic Differential Equation with Application in European Option Pricing

2014 ◽  
Vol 2014 ◽  
pp. 1-12 ◽  
Author(s):  
Qing Li ◽  
Yanli Zhou ◽  
Xinquan Zhao ◽  
Xiangyu Ge
Keyword(s):  
Differential Equation ◽  
Stochastic Differential Equation ◽  
Option Pricing ◽  
Memory Effect ◽  
Fractional Order ◽  
Mean Value ◽  
European Option ◽  
European Option Pricing ◽  
Proposed Model ◽  
Pricing Formula

Memory effect is an important phenomenon in financial systems, and a number of research works have been carried out to study the long memory in the financial markets. In recent years, fractional order ordinary differential equation is used as an effective instrument for describing the memory effect in complex systems. In this paper, we establish a fractional order stochastic differential equation (FSDE) model to describe the effect of trend memory in financial pricing. We, then, derive a European option pricing formula based on the FSDE model and prove the existence of the trend memory (i.e., the mean value function) in the option pricing formula when the Hurst index is between 0.5 and 1. In addition, we make a comparison analysis between our proposed model, the classic Black-Scholes model, and the stochastic model with fractional Brownian motion. Numerical results suggest that our model leads to more accurate and lower standard deviation in the empirical study.

European Option Pricing under the Double Exponential Jump Model with Stochastic Interest Rate, Stochastic Volatility and Stochastic Intensity

2014 ◽  
Vol 631-632 ◽  
pp. 1325-1328 ◽  
Author(s):  
Jin Yan Sang ◽  
Na Zhang ◽  
Ming Jian
Keyword(s):  
Stochastic Volatility ◽  
Jump Process ◽  
European Options ◽  
European Option ◽  
European Option Pricing ◽  
Stochastic Intensity ◽  
Jump Model ◽  
Double Exponential ◽  
Stochastic Interest ◽  
Form Representation

This paper explores the valuation of European options when the underlying asset follows the double exponential jump process with stochastic rate, stochastic volatility and stochastic intensity. This model better describes market characteristics, such as the volatility smile, and jump behavior. By using FFT (Fast Fourier Transform) approach, a closed form representation of the characteristic function of the process is derived for the valuation of European options. Numerical results show that the FFT method is effective and competent.

scholarly journals Studying a Tumor Growth Partial Differential Equation via the Black–Scholes Equation

Computation ◽  
2020 ◽  
Vol 8 (2) ◽  
pp. 57
Author(s):  
Winter Sinkala ◽  
Tembinkosi F. Nkalashe
Keyword(s):  
Differential Equation ◽  
Brain Tumor ◽  
Spatial Dynamics ◽  
Lie Symmetry ◽  
Financial Mathematics ◽  
European Option ◽  
Fair Price ◽  
Lie Symmetry Analysis ◽  
Partial Differential ◽  
Black Scholes

Two equations are considered in this paper—the Black–Scholes equation and an equation that models the spatial dynamics of a brain tumor under some treatment regime. We shall call the latter equation the tumor equation. The Black–Scholes and tumor equations are partial differential equations that arise in very different contexts. The tumor equation is used to model propagation of brain tumor, while the Black–Scholes equation arises in financial mathematics as a model for the fair price of a European option and other related derivatives. We use Lie symmetry analysis to establish a mapping between them and hence deduce solutions of the tumor equation from solutions of the Black–Scholes equation.

Solution of time fractional Black-Scholes European option pricing equation arising in financial market

2016 ◽  
Vol 5 (4) ◽  
Author(s):  
A.S.V. Ravi Kanth ◽  
K. Aruna
Keyword(s):  
Boundary Conditions ◽  
Option Pricing ◽  
Financial Market ◽  
Differential Transform Method ◽  
European Option ◽  
Transform Method ◽  
European Option Pricing ◽  
Black Scholes ◽  
Differential Transform ◽  
Fractional Differential

AbstractIn this paper, we present fractional differential transform method (FDTM) and modified fractional differential transform method (MFDTM) for the solution of time fractional Black-Scholes European option pricing equation. The method finds the solution without any discretization, transformation, or restrictive assumptions with the use of appropriate initial or boundary conditions. The efficiency and exactitude of the proposed methods are tested by means of three examples.

On the Black-Scholes European Option Pricing Model Robustness and Generality

2008 ◽  
Author(s):  
Hellinton Hatsuo Takada ◽  
José de Oliveira Siqueira ◽  
Marcelo de Souza Lauretto ◽  
Carlos Alberto de Bragança Pereira ◽  
Julio Michael Stern
Keyword(s):  
Option Pricing ◽  
Pricing Model ◽  
European Option ◽  
European Option Pricing ◽  
Black Scholes ◽  
Option Pricing Model ◽  
Model Robustness

scholarly journals The Approximate Analytic Solution of the Time-Fractional Black-Scholes Equation with a European Option Based on the Katugampola Fractional Derivative

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 214
Author(s):  
Sivaporn Ampun ◽  
Panumart Sawangtong
Keyword(s):  
Fractional Derivative ◽  
Perturbation Method ◽  
Homotopy Perturbation Method ◽  
Analytic Solution ◽  
Analytic Solutions ◽  
Approximate Analytic Solution ◽  
European Option ◽  
European Option Pricing ◽  
Homotopy Perturbation ◽  
Black Scholes

In the finance market, it is well known that the price change of the underlying fractal transmission system can be modeled with the Black-Scholes equation. This article deals with finding the approximate analytic solutions for the time-fractional Black-Scholes equation with the fractional integral boundary condition for a European option pricing problem in the Katugampola fractional derivative sense. It is well known that the Katugampola fractional derivative generalizes both the Riemann–Liouville fractional derivative and the Hadamard fractional derivative. The technique used to find the approximate analytic solutions of the time-fractional Black-Scholes equation is the generalized Laplace homotopy perturbation method, the combination of the generalized Laplace transform and homotopy perturbation method. The approximate analytic solution for the problem is in the form of the generalized Mittag-Leffler function. This shows that the generalized Laplace homotopy perturbation method is one of the most effective methods to construct approximate analytic solutions of the fractional differential equations. Finally, the approximate analytic solutions of the Riemann–Liouville and Hadamard fractional Black-Scholes equation with the European option are also shown.

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