Generalized Black-Scholes Equation Research Articles

Efficient Numerical Scheme for Generalized Black–Scholes Equations on Piecewise Uniform Shishkin-Type Mesh

Efficient Numerical Scheme for Generalized Black–Scholes Equations on Piecewise Uniform Shishkin-Type Mesh

Evolutionary Neural Networks for Option Pricing: Multi-Assets Option and Exotic Option

This paper presents a novel framework based on the evolutionary neural network to solve the generalized Black-Scholes equation arising in the financial market efficiently and accurately. We first employ evolutionary neural networks to parameterize the Partial Differential Equations (PDEs) involved in option pricing. This approach allows us to simplify the Black-Scholes PDEs and convert them into the corresponding Ordinary Differential Equations (ODEs). Thus we can use standard ODE solvers such as Euler’s method to solve the simplified ODE problem. The proposed framework is flexible and can handle various boundary conditions and terminal conditions, allowing customization based on specific market requirements and scenarios. Moreover, our method offers a reliable and deterministic solution methodology for the pricing framework, as it does not rely on stochastic training. Unlike other approaches that incorporate stochastic elements in their training process, our method eliminates the need for such training and provides consistent results. This deterministic nature enhances the reliability and stability of our approach, making it well-suited for real-world applica- tions in option pricing and financial markets. The experiments on multiple settings are carried out to illustrate the applicability and accuracy of the proposed framework.

Lattice Boltzmann Method for the Generalized Black-Scholes Equation

In this paper, an efficient lattice Boltzmann model for the generalized Black-Scholes equation governing option pricing is proposed. The Black-Scholes equation is firstly equivalently transformed into an initial value problem for a partial differential equation with a source term using the variable substitution and the derivative rules, respectively. Then, applying the multiscale Chapman-Enskog expansion, the amending function is expanded to second order to recover the convective and source terms of the macroscopic equation. The D1Q3 lattice Boltzmann model with spatial second-order accuracy is constructed, and the accuracy of the established D1Q5 model is greater than second order. The numerical simulation results demonstrate the effectiveness and numerical accuracy of the proposed models and indicate that our proposed models are suitable for solving the Black-Scholes equation. The proposed lattice Boltzmann model can also be applied to a class of partial differential equations with variable coefficients and source terms.

Option Pricing Model Driven by G-Lévy Process under the G-Expectation Framework

In this paper, we first present an option pricing model of stochastic differential equations driven by the G-Lévy process under the G-expectation framework, and prove the generalized Black-Scholes equations. Then, we present the algorithm for the time-homogeneous Poisson process versus the non-time-homogeneous Poisson process. Finally, we provide an explicit solution of generalized Black-Scholes equations and simulate it numerically with Matlab software.

Introducing and solving generalized Black–Scholes PDEs through the use of functional calculus

We introduce some families of generalized Black–Scholes equations which involve the Riemann–Liouville and Weyl space-fractional derivatives. We prove that these generalized Black–Scholes equations are well-posed in (L^1-L^infty )-interpolation spaces. More precisely, we show that the elliptic-type operators involved in these equations generate holomorphic semigroups. Then, we give explicit integral expressions for the associated solutions. In the way to obtain well-posedness, we prove a new connection between bisectorial-like operators and sectorial operators in an abstract setting. Such a connection extends the scaling property of sectorial operators to a wider family of both operators and the functions involved.

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.

Endogenous stochastic arbitrage bubbles and the Black–Scholes model

This paper develops a model that incorporates the presence of stochastic arbitrage explicitly in the Black–Scholes equation. Here, the arbitrage is generated by a stochastic bubble, which generalizes the deterministic arbitrage model obtained in the literature (Contreras et al., 2010). It is considered to be a generic stochastic dynamic for the arbitrage bubble, and a generalized Black–Scholes equation is then derived. The resulting equation is similar to that of the stochastic volatility models, but there are no undetermined parameters as the market price of risk.The proposed theory has asymptotic behaviors that are associated with the weak and strong arbitrage bubble limits. For the case where the arbitrage bubble’s volatility is zero (deterministic bubble), the weak limit corresponds to the usual Black–Scholes model. The strong limit case also give a Black–Scholes model, but the underlying asset’s mean value replaces the interest rate. When the bubble is stochastic, the theory also has weak and strong asymptotic limits that give rise to option price dynamics that are similar to the Black–Scholes model. Explicit formulas are derived for Gaussian and lognormal stochastic bubbles. Consequently, the Black–Scholes model can be considered to be a weak bubble limit of a more general stochastic model.

Polynomial differential quadrature method for numerical solution of the generalized Black-Scholes ‎equation

Polynomial differential quadrature method for numerical solution of the generalized Black-Scholes ‎equation

On Black–Scholes option pricing model with stochastic volatility: an information theoretic approach

In this article, we derive the risk-neutral measures of the stock options price and volatility by incorporating a simple constrained minimization of the Kullback measure of relative information. We obtain a second-order parabolic partial differential equation, the generalized Black–Scholes equation based on the theoretical analysis when the underlying financial asset is estimated using a stochastic volatility model. Also to investigate the analytical solution of this generalized Black–Scholes equation we have used Laplace transform homotopy perturbation method.

The chaotic Black–Scholes equation with time-dependent coefficients

In Emamirad et al. (in: Semigroups of Operators - Theory and Applications, Springer, 2014; and Proc. Amer. Math. Soc. 140:2043–2052, 2012), the Black–Scholes semigroup was studied in various Banach spaces of continuous functions regarding chaotic behavior and the null volatility limit. Here we let the volatility and the interest rates be continuous functions on the half line $$[0,\infty )$$ , and we consider the generalized Black–Scholes equation as a linear nonautonomous abstract Cauchy problem. It is shown that this problem is wellposed and an explicit formula for the corresponding strongly continuous evolution family is given. Furthermore, a hypercyclicity criterion for strongly continuous evolution families on separable Banach spaces is proved and applied to the Black–Scholes evolution family. Finally, the chaotic behavior is defined for strongly continuous evolution families and it is shown that the Black–Scholes evolution family is chaotic when the volatility and the interest rates are periodic functions.

A sixth order numerical method and its convergence for generalized Black–Scholes PDE

The main aim of this paper is to construct a new computational approach for the numerical solution of generalized Black–Scholes equation. In this approach, the temporal variable is discretized using Crank–Nicolson scheme and spatial variable is discretized using sextic B-spline collocation method. Convergence analysis of the method is discussed. The proposed method is proved to be stable and has second-order convergence with respect to time variable and sixth order convergence with respect to space variable. To illustrate the applicability and efficiency of the proposed method, we consider some benchmark problems describing European call options. Numerical results verify the orders of convergence predicted by the analysis. Numerical results reveal that the present method provides better results as compared to some existing numerical methods based on B-spline collocation approach.

An accurate solution for the generalized Black-Scholes equations governing option pricing

Today industries related to finance are essentially implementing advanced mathematical tools. In 1973, Fisher Black and Myron Scholes developed an eminent stochastic model which later coined as Black-Scholes differential equations for option pricing. This paper illustrates a convenient time integration scheme based on the generalized trapezoidal formulas (GTF $[\alpha = \frac{1}{3}]$) introduced by Chawla et al. in 1996. GTF is applied for the temporal discretization along with the classical finite difference schemes in space direction. The proposed scheme yields the (uniform) stability employing the uniform bound of the inverse operator, as well as second-order spatial accuracy and third-order temporal accuracy under reasonable conditions. Finally, the numerical illustrations and comparison with existing schemes demonstrate the stability and accuracy of the method.

Space-time kernel based numerical method for generalized Black-Scholes equation

In approximating time-dependent partial differential equations, major error always occurs in the time derivatives as compared to the spatial derivatives. In the present work the time and the spatial derivatives are both approximated using time-space radial kernels. The proposed numerical scheme avoids the time stepping procedures and produced sparse differentiation matrices. The stability and accuracy of the proposed numerical scheme is tested for the generalized Black-Scholes equation.

A new higher order compact finite difference method for generalised Black–Scholes partial differential equation: European call option

This paper presents a high order numerical method based on a uniform mesh to obtain a highly accurate result for generalized Black–Scholes equation arising in the financial market. We first semi-discretize the underlying problem in the temporal direction by using Crank–Nicolson scheme and then discretize the resulting set of equations by using a high order compact finite difference method (HOCFDM). The stability of the scheme is analyzed. Convergence of the suggested method is proved. It is shown that the method is of order O ( Δ t 2 + h 4 ) . Some numerical experiments are carried out in order to illustrate the applicability and accuracy of the proposed method and to validate the theoretical results as well. It is shown that HOCFDM solution matches very well with the exact solution. Further, the rate of convergence predicted theoretically is the same as that obtained numerically. The computational time for our method in Matlab programming language is provided.

Numerical solution of generalized Black–Scholes model

This paper presents a numerical scheme that approximates the option prices for different option styles, governed by the generalized Black–Scholes equation in its degenerate form. The proposed method uses the HODIE scheme in the spacial direction and the two-step backward differentiation formula in the temporal direction. It is proved that the method has second order convergence in space as well as in time. Numerical experiments validate the theoretical results.

Pricing Perpetual Put Options by the Black–Scholes Equation with a Nonlinear Volatility Function

We investigate qualitative and quantitative behavior of a solution of the mathematical model for pricing American style of perpetual put options. We assume the option price is a solution to the stationary generalized Black–Scholes equation in which the volatility function may depend on the second derivative of the option price itself. We prove existence and uniqueness of a solution to the free boundary problem. We derive a single implicit equation for the free boundary position and the closed form formula for the option price. It is a generalization of the well-known explicit closed form solution derived by Merton for the case of a constant volatility. We also present results of numerical computations of the free boundary position, option price and their dependence on model parameters.

Option pricing using a computational method based on reproducing kernel

One of the most important subject in financial mathematics is the option pricing. The most famous result in this area is Black–Scholes formula for pricing European options. This paper is concerned with a method for solving a generalized Black–Scholes equation in a reproducing kernel Hilbert space. Subsequently, the convergence of the proposed method is studied under some hypotheses which provide the theoretical basis of the proposed method. Furthermore, the error estimates for obtained approximation in reproducing kernel Hilbert space are presented. Finally, a numerical example is considered to illustrate the computation efficiency and accuracy of the proposed method.

The exact traveling wave solutions of a class of generalized Black-Scholes equation

In this paper, the traveling wave solutions of a class of generalized Black-Scholes equation are considered. By using the first integral method and the G'/G-expansion method, several exact traveling wave solutions of the equation are obtained.

A numerical study of the European option by the MLPG method with moving kriging interpolation.

In this paper, the meshless local Petrov–Galerkin (MLPG) method is applied for solving a generalized Black–Scholes equation in financial problems. This equation is a PDE governing the price evolution of a European call or a European put under the Black–Scholes model. The θ-weighted method and MLPG are used for discretizing the governing equation in time variable and option pricing, respectively. We show that the spectral radius of amplification matrix with the discrete operator is less than 1. This ensures that this numerical scheme is stable. Numerical experiments are performed with time varying volatility and the results are compared with the analytical and the numerical results of other methods.

Quintic B-spline collocation approach for solving generalized Black–Scholes equation governing option pricing

We develop a numerical algorithm for solving generalized Black–Scholes partial differential equation, which arises in European option pricing. The method comprises the horizontal method of lines for time integration and θ-method, θ∈[1/2,1] (θ=1 corresponds to the back-ward Euler method and θ=1/2 corresponds to the Crank–Nicolson method) to discretize in temporal direction and the quintic B-spline collocation method in uniform spatial direction. The convergence analysis and stability of proposed method are discussed in detail, it is justifying that the approximate solution converges to the exact solution of orders O(k+h3) for the back-ward Euler method and O(k2+h3) for the Crank–Nicolson method, where k and h are mesh sizes in the time and space directions, respectively. The proposed method is also shown to be unconditionally stable. This scheme applied on some test examples, the numerical results illustrate the efficiency of the method and confirm the theoretical behavior of the rates of convergence. Results shown by this method are found to be in good agreement with the known exact solutions. The produced results are also seen to be more accurate than some available results given in the literature.