Black-Scholes Equation Research Articles

On a fast numerical solution of the nonlinear Black-Scholes equation using hyper-reduction techniques

On a fast numerical solution of the nonlinear Black-Scholes equation using hyper-reduction techniques

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.

Approximate Solution for Barrier Option Pricing Using Adaptive Differential Evolution With Learning Parameter

Black-Scholes (BS) equations, which are in the form of stochastic partial differential equations, are fundamental equations in mathematical finance, especially in option pricing. Even though there exists an analytical solution to the standard form, the equations are not straightforward to be solved numerically. The effective and efficient numerical method will be useful to solve advanced and non-standard forms of BS equations in the future. In this paper, we propose a method to solve BS equations using an approach of optimization problems, where a metaheuristic optimization algorithm is utilized to find the best-approximated solutions of the equations. Here we use the Adaptive Differential Evolution with Learning Parameter (ADELP) algorithm. The BS equations being solved are meant to find values of European option pricing that is equipped with Barrier option pricing. The result of our approximation method fits well to the analytical approximation solutions.

Hahn hybrid functions for solving distributed order fractional Black–Scholes European option pricing problem arising in financial market

The main purpose of this work is to present a new numerical method based on Hahn hybrid functions (HHFs) for solving of Black–Scholes option pricing distributed order time‐fractional partial differential equation. To this end, HHFs are introduced and their fractional integral operator with some properties of HHFs is calculated. In the next, with the help of fractional integral operator of HHFs, Gauss–Legendre quadrature formula and collocation method, distributed order time‐fractional Black–Scholes model is reduced to a system of algebraic equations. Furthermore, convergence analysis of the mentioned scheme is discussed. Finally, some test problems have been included to confirm the validity and efficiency of the mentioned numerical scheme. Moreover, Black–Scholes equations are studied through a bibliometric viewpoint.

Solving Nonlinear Partial Differential Equations via Deep Galerkin Method

This paper introduces a one-dimensional nonlinear Black-Scholes equation. Because nonlinear models are difficult to obtain analytical solutions, finite difference schemes are usually used to solve them. Numerical solutions of partial differential equations have always been a hot research topic in the frontier; the finite element method, finite body method, and other methods are commonly used in research. In general, these methods require the use of grids to approximate the definition space of PDEs. When we have a finer grid, the more accurate solution will be. However, the smaller the grid, the higher the computing cost and the larger the storage space. Therefore, the meshless method is adopted in this paper.

Solving Non-linear Kolmogorov Equations in Large Dimensions by Using Deep Learning: A Numerical Comparison of Discretization Schemes

Non-linear partial differential Kolmogorov equations are successfully used to describe a wide range of time dependent phenomena, in natural sciences, engineering or even finance. For example, in physical systems, the Allen–Cahn equation describes pattern formation associated to phase transitions. In finance, instead, the Black–Scholes equation describes the evolution of the price of derivative investment instruments. Such modern applications often require to solve these equations in high-dimensional regimes in which classical approaches are ineffective. Recently, an interesting new approach based on deep learning has been introduced byby E, Han and Jentzen [1, 2]. The main idea is to construct a deep network which is trained from the samples of discrete stochastic differential equations underlying Kolmogorov’s equation. The network is able to approximate, numerically at least, the solutions of the Kolmogorov equation with polynomial complexity in whole spatial domains. In this contribution we study variants of the deep networks by using different discretizations schemes of the stochastic differential equation. We compare the performance of the associated networks, on benchmarked examples, and show that, for some discretization schemes, improvements in the accuracy are possible without affecting the observed computational complexity.

Error analysis for physics-informed neural networks (PINNs) approximating Kolmogorov PDEs

Physics-informed neural networks approximate solutions of PDEs by minimizing pointwise residuals. We derive rigorous bounds on the error, incurred by PINNs in approximating the solutions of a large class of linear parabolic PDEs, namely Kolmogorov equations that include the heat equation and Black-Scholes equation of option pricing, as examples. We construct neural networks, whose PINN residual (generalization error) can be made as small as desired. We also prove that the total L2-error can be bounded by the generalization error, which in turn is bounded in terms of the training error, provided that a sufficient number of randomly chosen training (collocation) points is used. Moreover, we prove that the size of the PINNs and the number of training samples only grow polynomially with the underlying dimension, enabling PINNs to overcome the curse of dimensionality in this context. These results enable us to provide a comprehensive error analysis for PINNs in approximating Kolmogorov PDEs.

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.

Application of Lie symmetries to solving modified Black-Scholes equation

We perform Lie symmetry analysis on the modified Black-Scholes model described as a partial differential equation (PDE). As a result, a new complete Lie symmetry group and infinitesimal generators of the one-dimensional modified fractional Black-Scholes model are derived. Furthermore, we compute a family of exact invariant solutions that constitute the modified fractional Black-Scholes model using the associated infinitesimal generators and the corresponding similarity reduction equations. Using known solutions, more solutions are generated via group point transformations.

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.

Comment on “Removing non-smoothness in solving Black-Scholes equation using a perturbation method”

We show that the homotopy perturbation method proposed by Putri et al. (2021) [1] is simply the Taylor series of the solution of the partial differential equation about the origin of time.

An Efficient Method for Solving Fractional Black-Scholes Model with Index and Exponential Decay Kernels

The Black-Scholes equation (BSe) is fascinating in the business world for predicting the performance of financial investment valuation systems. The Caputo fractional derivative (CFD) and Caputo-Fabrizio fractional derivative operators are used in this research to analyze the BSe. The Adomian decomposition method (ADM) and the new iterative transform (NIM) approach are combined alongside the Yang transform. In addition, the convergence and uniqueness results for the aforementioned framework have been calculated. The existence and uniqueness results have been established and frequently accompanied innovative aspects of the prospective system in fixed point terminologies. To provide additional insight into such concepts, a variety of illustrations and tabulations are used. Additionally, the provided techniques regulate and modify the obtained analytical results in a really productive fashion, allowing us to modify and regulate the converging domains of the series solution in a pragmatic manner.

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.

Solution of the Black-Scholes Equation by Finite Difference Schemes

Black-Scholes (BS) equation is a popular mathematical model for determining the value of option in financial derivatives. To predict the option value during the contract of the option is a big problem. Several studies have been shown that the option price value can be determined by applying different methods. In this paper, we have discussed three finite difference methods: Explicit, Implicit and Crank-Nicolson for solving Black-Scholes equation for European call option and compared the obtained results with the exact value. It is found that the Crank-Nicolson method is more accurate and cost effective in comparison with explicit and implicit methods.

Numerical Approximations of a Nonlinear Volatility Model with European Options

Black-Scholes model plays a very significant role in the world of quantitative finance. In this paper, the focus are on both nonlinear and linear Black-Scholes (BS) equations with numerical approximations. We aim to find an effective numerical approximations for Black-Scholes model. Several models from the most relevant class of nonlinear Black-Scholes equations with European option are analyzed in this study. The problem is approached by transforming the problem into a convection-diffusion equation and later it is approximated with the help of finite difference method (Crank-Nicolson). The result of finite difference schemes (Crank-Nicolson) for several volatility models are presented, including the Risk Adjusted Pricing Methodology (RAPM), Leland’s model and the Barles’-Soner’s Model. At the same time, it is attempted to illustrate a comparison of different volatility models. In the case of linear Black-Scholes model, we approximate the model with finite difference method (FDM) and finite element method (FEM) and compare the results. All the numerical schemes are implemented in MATLAB and corresponding graphs are also presented here. GANIT J. Bangladesh Math. Soc. 42.1 (2022) 050- 068

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.