Scholes Equation Research Articles

An Analysis of the Fractional-Order Option Pricing Problem for Two Assets by the Generalized Laplace Variational Iteration Approach

An option is the right to buy or sell a good at a predetermined price in the future. For customers or financial companies, knowing an option’s pricing is crucial. It is well recognized that the Black–Scholes model is an effective tool for estimating the cost of an option. The Black–Scholes equation has an explicit analytical solution known as the Black–Scholes formula. In some cases, such as the fractional-order Black–Scholes equation, there is no closed form expression for the modified Black–Scholes equation. This article shows how to find the approximate analytic solutions for the two-dimensional fractional-order Black–Scholes equation based on the generalized Riemann–Liouville fractional derivative. The generalized Laplace variational iteration method, which incorporates the generalized Laplace transform with the variational iteration method, is the methodology used to discover the approximate analytic solutions to such an equation. The expression of the two-parameter Mittag–Leffler function represents the problem’s approximate analytical solution. Numerical investigations demonstrate that the proposed scheme is accurate and extremely effective for the two-dimensional fractional-order Black–Scholes Equation in the perspective of the generalized Riemann–Liouville fractional derivative. This guarantees that the generalized Laplace variational iteration method is one of the effective approaches for discovering approximate analytic solutions to fractional-order differential equations.

Forecasting stock options prices via the solution of an ill-posed problem for the Black–Scholes equation

In the previous paper (2016 Inverse Problems 32 015010), a new heuristic mathematical model was proposed for accurate forecasting of prices of stock options for 1–2 trading days ahead of the present one. This new technique uses the Black–Scholes equation supplied by new intervals for the underlying stock and new initial and boundary conditions for option prices. The Black–Scholes equation was solved in the positive direction of the time variable, this ill-posed initial boundary value problem was solved by the so-called quasi-reversibility method (QRM). This approach with an added trading strategy was tested on the market data for 368 stock options and good forecasting results were demonstrated. In the current paper, we use the geometric Brownian motion to provide an explanation of that effectivity using computationally simulated data for European call options. We also provide a convergence analysis for QRM. The key tool of that analysis is a Carleman estimate.

Analytical Solution of Black-Scholes Model for Pricing Barrier Option using Method of Partial Taylor Series Expansion

In this work, Black-Scholes differential equation for barrier/traditional option is solved using partial Taylor series expansion method. The developed solutions are in very good agreement with the closed-form solutions of the Black Scholes equation for the powered ML-payoff functions. Also, the analytical solutions of the new method in this present study give the same expressions as the solutions of projected differential equations and homotopy perturbation method as presented in the literature. Moreover, the reliability, speed, accuracy, and ease of application of the proposed method show its potential for wide areas of applications in science, financial mathematics, and engineering.

A spectral collocation method based on fractional Pell functions for solving time–fractional Black–Scholes option pricing model

The fractional Black–Scholes equation has been widely studied by researchers in recent years. In this article, an efficient spectral collocation method based on fractional Pell functions is proposed for solving the time–fractional Black–Scholes equation. We introduce fractional Pell functions using the transformation x→xβ(β>0) on Pell polynomials, and we look for a solution of the model as a linear combination of these functions. Using operational matrices, we approximate the fractional derivative and other terms in a convenient form of the main equation. A system of algebraic equations is obtained by collocating resultant approximate equations. Convergence analysis of the numerical method has been investigated in Sobolev space. Finally, we have demonstrated the capability of the proposed method by considering numerical experiments in the form of tables and figures.

Monotone methods in counterparty risk models with nonlinear Black–Scholes-type equations

A nonlinear Black–Scholes-type equation is studied within counterparty risk models. The classical hypothesis on the uniform Lipschitz-continuity of the nonlinear reaction function allows for an equivalent transformation of the semilinear Black–Scholes equation into a standard parabolic problem with a monotone nonlinear reaction function and an inhomogeneous linear diffusion equation. This setting allows us to construct a scheme of monotone, increasing or decreasing, iterations that converge monotonically to the true solution. As typically any numerical solution of this problem uses most computational power for computing an approximate solution to the inhomogeneous linear diffusion equation, we discuss also this question and suggest several solution methods, including those based on Monte Carlo and finite differences/elements.

Lie symmetry analysis and exact solutions of time fractional Black–Scholes equation

The Black–Scholes equation is an important analytical tool for option pricing in finance. This paper discusses the constructive methods of exact solutions to time fractional Black–Scholes equation. By constructing one-parameter Lie symmetry transformations and their corresponding group generators, time fractional Black–Scholes equation is reduced to a fractional ordinary differential equation and some group-invariant solutions are obtained. Using the invariant subspace method, the analytical representations of two forms of exact solutions of time fractional Black–Scholes equation are given constructively, and the characteristics and differences of the two exact solutions are compared in the sense of geometric figures. In this paper, the form of the equation is generalized, and more group invariant solutions and analytical solutions in the form of separated variables are obtained.

A second order numerical method for the time-fractional Black–Scholes European option pricing model

In this paper, we design an efficient and accurate numerical method for solving the time-fractional Black–Scholes equation governing European options. The time-fractional Black–Scholes equation is transformed into an equivalent integro-differential equation. The numerical method for the integro-differential equation is developed by using a numerical integration scheme for time discretization and central difference formulas for space discretization. The stability and convergence of the method are analyzed. The numerical method is proved to be second-order accurate in both space and time. Richardson extrapolation is also introduced to obtain a modified version of the method that exhibits faster convergence for the time-fractional Black–Scholes equation with non-smooth initial data. Finally, two numerical examples are presented to demonstrate the efficiency and accuracy of the numerical method.

Forecasting the behaviour of fractional Black-Scholes option pricing equation by laplace perturbation iteration algorithm

Financial derivatives plays a major role in all financial deals these days. Black–Scholes option pricing model gives a risk free analysis for investing in options. In the current work, a method called the Laplace Perturbation Iteration Algorithm is being applied on Fractional Black–Scholes Equation to obtain its fractional analytical solutions in series form to analyze its results for the European and American option pricing problem, quickly and accurately. Laplace Perturbation Iteration Algorithm incorporates Laplace transform and Perturbation Iteration Algorithm and forms an iterative scheme that derives the analytical solutions without any inconvenience. The Fractional Black–Scholes Equation analytical solution is obtained as a convergent power series of conveniently computed components. Four examples have been solved and their results have been discussed in this paper, in order to check the efficacy of Laplace Perturbation Iteration Algorithm.

Spectral Solutions for Fractional Black–Scholes Equations

This paper presents a numerical method to solve accurately the fractional Black–Scholes model of pricing evolution. A fully spectral collocation technique for the two independent variables is derived. The shifted fractional Jacobi–Gauss–Radau and shifted fractional Jacobi–Gauss–Lobatto collocation techniques are utilized. Firstly, the independent variables are interpolated at the shifted fractional Jacobi nodes, and the solution of the model is approximated by means of a sequence of shifted fractional Jacobi orthogonal functions. Then, the residuals at the shifted fractional Jacobi quadrature locations are estimated. As a result, an algebraic system of equations is obtained that can be solved using any appropriate approach. The accuracy of the proposed method is demonstrated using two numerical examples. It is observed that the new technique is more accurate, efficient, and feasible than other approaches reported in the literature. Indeed, the results show the exponential convergence of the method, both for smooth and nonsmooth solutions.

PRICING EUROPEAN TWO-ASSET OPTION USING THE SPECTRAL METHOD WITH SECOND-KIND CHEBYSHEV POLYNOMIALS

The path of the Lévy process can be considered for prices of options such as a Rainbow or Basket option on two assets which leads to a 2D Black–Scholes model. The generalized model of this type of equation can be referred to as a 2D spatial-fractional Black–Scholes equation. The analytical solution of this kind is very complex and difficult and can even be said to be unattainable. For this reason, a numerical method has been proposed to solve it via the collocation method based on the Chebyshev orthogonal basis. Moreover, based on the derivatives in the called model, we approximated the derivative operator by using this type of base. Then we first obtained the temporal discrete form and finally the full-discrete form and turned it into a system of linear equations with the help of Chebyshev base roots.

A Positivity-Preserving Improved Nonstandard Finite Difference Method to Solve the Black-Scholes Equation

In this paper, we evaluate and discuss different numerical methods to solve the Black–Scholes equation, including the θ-method, the mixed method, the Richardson method, the Du Fort and Frankel method, and the MADE (modified alternating directional explicit) method. These methods produce numerical drawbacks such as spurious oscillations and negative values in the solution when the volatility is much smaller than the interest rate. The MADE method sacrifices accuracy to obtain stability for the numerical solution of the Black–Scholes equation. In the present work, we improve the MADE scheme by using non-standard finite difference discretization techniques in which we use a non-local approximation for the reaction term (we call it the MMADE method). We will discuss the sufficient conditions to be positive of the new scheme. Also, we show that the proposed method is free of spurious oscillations even in the presence of discontinuous initial conditions. To demonstrate how efficient the new scheme is, some numerical experiments are performed at the end.

Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

Derivation and Application of Some Fractional Black–Scholes Equations Driven by Fractional G-Brownian Motion

On Multilevel RBF Collocation Based on Operator Newton Iteration to Solve Nonlinear Black–Scholes Equations

On Multilevel RBF Collocation Based on Operator Newton Iteration to Solve Nonlinear Black–Scholes Equations

Efficient numerical pricing of American options based on multiple shooting method: a PDE approach

In this paper, the Black–Scholes (B-S) equation to price American options is studied which is governed by a partial differential problem. A nonlinear partial differential equation (PDE) is resulted by applying a penalty approach for this problem. To numerically solve this PDE, an equipped finite difference method with variable step size (VSS) in the time direction is designed and implemented, coupled with a boundary value solver based on a multiple shooting method (MSM) in the spatial direction. It is completely proven that the proposed approximate option prices satisfy the early exercise constraint. Moreover, the convergence of the proposed method is analysed. The extracted numerical results indicate the punctuality and effectiveness of the present scheme comparing the previous works.

Homotopy perturbation method to solve Black Scholes differential equation for ML-payoff function

The Homotopy Perturbation Method (HPM) is a semi analytical method for solving linear and non linear ordinary as well as partial differential equations. This paper contributes to the solution of Black Scholes equation for modified log payoff (ML-payoff) functions using HPM. It should be emphasized that these formulas are quite close to the closed form solutions of Black Scholes equation for ML-payoff functions which are very close to the celebrated option pricing formulas for plain vanilla payoff functions.

Computing Black Scholes with Uncertain Volatility—A Machine Learning Approach

In financial mathematics, it is a typical approach to approximate financial markets operating in discrete time by continuous-time models such as the Black Scholes model. Fitting this model gives rise to difficulties due to the discrete nature of market data. We thus model the pricing process of financial derivatives by the Black Scholes equation, where the volatility is a function of a finite number of random variables. This reflects an influence of uncertain factors when determining volatility. The aim is to quantify the effect of this uncertainty when computing the price of derivatives. Our underlying method is the generalized Polynomial Chaos (gPC) method in order to numerically compute the uncertainty of the solution by the stochastic Galerkin approach and a finite difference method. We present an efficient numerical variation of this method, which is based on a machine learning technique, the so-called Bi-Fidelity approach. This is illustrated with numerical examples.

A Nonstandard Finite Difference Method for a Generalized Black–Scholes Equation

An implicit finite difference scheme for the numerical solution of a generalized Black–Scholes equation is presented. The method is based on the nonstandard finite difference technique. The positivity property is discussed and it is shown that the proposed method is consistent, stable and also the order of the scheme respect to the space variable is two. As the Black–Scholes model relies on symmetry of distribution and ignores the skewness of the distribution of the asset, the proposed method will be more appropriate for solving such symmetric models. In order to illustrate the efficiency of the new method, we applied it on some test examples. The obtained results confirm the theoretical behavior regarding the order of convergence. Furthermore, the numerical results are in good agreement with the exact solution and are more accurate than other existing results in the literature.

Formulating black Scholes equation using a jump diffusion Heston’s model

In modern financial mathematics, accurate values are obtained by taking into account a considerable number of more realistic assumptions in logistic Black Scholes equation. The aspects considered here are cost of transactions in trading, perfect illiquid markets and risks that occur from non – protected portfolio or large investments that have a lot of impact on price of the assets, volatility, the percentage drift and the life of the portfolio itself. In modern world of finance, Jump diffusion process is used to assess the behavior of non – continuous asset when pricing of options. Since the introduction of Black – Scholes concept model that assumes volatility is constant; several studies have proposed models that address the shortcomings of Black – Scholes model. Heston’s models stands out amongst most volatility models because the process of volatility is greater the zero and follows mean reversion and this is what is observed in the market world. One of the shortcomings of Heston’s model is that it doesn’t incorporate the aspect jump diffusion process. The Black – Scholes partial differential equation that has been studied so far revolves around Geometric Brownian motion and its extensions. We therefore incorporate jump diffusion process on Heston’s model and use it to formulate a new Black – Scholes equation using the knowledge of partial differential equations.

A combination of dividend and Jump diffusion process on Heston model in deriving Black Scholes equation

The reality that exists in a stock market situation is that assets do pay dividend to owners of assets or derivative securities. Dupire, Derman and Kani built an option pricing process with a dividend yielding diffusion process but lacked the jump diffusion component. Jump diffusion process that mostly been captured in modern stock market do exhibit discontinuous behavior on pricing of assets. Introduction of Black Scholes concept equation that assumes volatility is constant led to several studies that have proposed models that address the shortcomings of Black – Scholes model. Heston’s models stands out amongst most volatility models because the process of volatility is greater than zero and exhibit mean reversion process and that is what is observed from the stock market. One of the shortcomings of Heston’s model is that it doesn’t incorporate the dividend yield jump diffusion process. Black Scholes equation revolves around Geometric Brownian motion and its extensions. We therefore incorporate dividend yield and jump diffusion process on Heston’s model and use it to formulate a new Black Scholes equation using the knowledge of partial differential equations.

Derivation of black Scholes equation using Heston’s model with dividend yielding asset

Black Scholes formula is crucial in modern applied finance. Since the introduction of Black – Scholes concept model that assumes volatility is constant; several studies have proposed models that address the shortcomings of Black-Scholes model. Heston’s models stands out amongst most volatility models because the process of volatility is positive and is a process that obeys mean reversion and this is what is observed in the real market world. One of the shortcomings of Heston’s model is that it doesn’t incorporate dividend yielding asset. Black Scholes partial differential equation revolves around Geometric Brownian motion and its extensions. We, therefore, incorporate dividend yielding assets on Heston’s model and use it to model a new Black Scholes equation using the knowledge of partial differential equations.