Scholes Partial Differential Equation Research Articles

An Efficient Numerical Scheme for a Time-Fractional Black–Scholes Partial Differential Equation Derived from the Fractal Market Hypothesis

Since the early 1970s, the study of Black–Scholes (BS) partial differential equations (PDEs) under the Efficient Market Hypothesis (EMH) has been a subject of active research in financial engineering. It has now become obvious, even to casual observers, that the classical BS models derived under the EMH framework fail to account for a number of realistic price evolutions in real-time market data. An alternative approach to the EMH framework is the Fractal Market Hypothesis (FMH), which proposes better and clearer explanations of market behaviours during unfavourable market conditions. The FMH involves non-local derivatives and integral operators, as well as fractional stochastic processes, which provide better tools for explaining the dynamics of evolving market anomalies, something that classical BS models may fail to explain. In this work, using the FMH, we derive a time-fractional Black–Scholes partial differential equation (tfBS-PDE) and then transform it into a heat equation, which allows for ease of implementing a high-order numerical scheme for solving it. Furthermore, the stability and convergence properties of the numerical scheme are discussed, and overall techniques are applied to pricing European put option problems.

Convergence of the Two Point Flux Approximation method and the fitted Two Point Flux Approximation method for options pricing with local volatility function

In this paper, we deal with numerical approximations for solving the Black–Scholes Partial Differential Equation (PDE) for European and American options pricing with local volatility. This PDE is well-known to be degenerated. Local volatility model is a model where the volatility depends locally of both stock price and time. In contrast to constant volatility or time-dependent volatility models for which analytical representations of the exact solution is known for European Call options, there is no analytical solution for local volatility. The space discretization is performed using the classical finite volume method with Two-Point Flux Approximation (TPFA) and a novel scheme called Fitted Two-Point Flux Approximation (FTPFA). The Fitted Two-Point Flux Approximation (FTPFA) combines the fitted finite volume method and the standard TPFA method. More precisely the fitted finite volume method is used when the stock price approaches zero with the goal to handle the degeneracy of the PDE while the TPFA method is used on the rest of space domain. This combination yields our fitted TPFA scheme. The Euler method is used for the time discretization. We provide the rigorous convergence proofs of the two fully discretized schemes. Numerical experiments to support theoretical results are provided.

A Fast Computational Scheme for Solving the Temporal-Fractional Black–Scholes Partial Differential Equation

In this work, we propose a fast scheme based on higher order discretizations on graded meshes for resolving the temporal-fractional partial differential equation (PDE), which benefits the memory feature of fractional calculus. To avoid excessively increasing the number of discretization points, such as the standard finite difference or meshfree methods, and, at the same time, to increase the efficiency of the solver, we employ discretizations on spatially non-uniform meshes with an attention on the non-smoothness area of the underlying asset. Therefore, the PDE problem is transformed to a linear system of algebraic equations. We perform numerical simulations to observe and check the behavior of the presented scheme in contrast to the existing methods.

A Proof that Artificial Neural Networks Overcome the Curse of Dimensionality in the Numerical Approximation of Black–Scholes Partial Differential Equations

Artificial neural networks (ANNs) have very successfully been used in numerical simulations for a series of computational problems ranging from image classification/image recognition, speech recognition, time series analysis, game intelligence, and computational advertising to numerical approximations of partial differential equations (PDEs). Such numerical simulations suggest that ANNs have the capacity to very efficiently approximate high-dimensional functions and, especially, indicate that ANNs seem to admit the fundamental power to overcome the curse of dimensionality when approximating the high-dimensional functions appearing in the above named computational problems. There are a series of rigorous mathematical approximation results for ANNs in the scientific literature. Some of them prove convergence without convergence rates and some of these mathematical results even rigorously establish convergence rates but there are only a few special cases where mathematical results can rigorously explain the empirical success of ANNs when approximating high-dimensional functions. The key contribution of this article is to disclose that ANNs can efficiently approximate high-dimensional functions in the case of numerical approximations of Black-Scholes PDEs. More precisely, this work reveals that the number of required parameters of an ANN to approximate the solution of the Black-Scholes PDE grows at most polynomially in both the reciprocal of the prescribed approximation accuracy ε > 0 \varepsilon > 0 and the PDE dimension d ∈ N d \in \mathbb {N} . We thereby prove, for the first time, that ANNs do indeed overcome the curse of dimensionality in the numerical approximation of Black-Scholes PDEs.

Method of Lines for Valuation and Sensitivities of Bermudan Options

In this paper, we present a computationally efficient technique based on the Method of Lines for the approximation of the Bermudan option values via the associated partial differential equations. The method of lines converts the Black Scholes partial differential equation to a system of ordinary differential equations. The solution of the system of ordinary differential equations so obtained only requires spatial discretization and avoids discretization in time. Additionally, the exact solution of the ordinary differential equations can be obtained efficiently using the exponential matrix operation, making the method computationally attractive and straightforward to implement. An essential advantage of the proposed approach is that the associated Greeks can be computed with minimal additional computations. We illustrate, through numerical experiments, the efficacy of the proposed method in pricing and computation of the sensitivities for a European call, cash-or-nothing, powered option, and Bermudan put option.

Numerical Techniques for Determining Implied Volatility in Option Pricing

One of the assumptions of the Black–Scholes model is that volatility is constant in the market. However, in reality, volatility cannot be constant. In this paper we examined an inverse problem of determining the time-dependent non-constant volatility from the observed market price of a European put/call option with a strike price.These unknown non-constant volatilities are from the time the option contract was signed to the time it was exercised (strike price) where the fluctuations of option price is known but the non-constant volatilities that was generated by option price within that time period is unknown.Although many studies have used the portfolio stocks to reconstruct volatility, no study have retrieved volatility from the option market. The aim is to recover unknown non-constant volatilities in vector form from one option contract period using manufacturer observed market data, that is, data from agreed contract period using Black–Scholes Partial Differential Equation (BSPDE). This would be done by using the chain and product rule to take the derivative with respect to volatility in the theoretical(Black-Schoes PDE) model to obtain the non-constant volatilities. By keeping the fluctuations data from observed option prices constant,we differentiate the volatility in the BSPDE to obtain a gradient equation and discretize the gradient equation with the finite difference method to obtain model simulated data. It is expected that non-constant volatility that would be recovered from the market using the gradient descent method would be matched with the non-constant market volatility from the model simulated data and if they are the same or approximately same, the volatility would have deem recovered. All these would be done in the least square sense.

Inverse Multiquadric Function to Price Financial Options under the Fractional Black–Scholes Model

The inverse multiquadric radial basis function (RBF), which is one of the most important functions in the theory of RBFs, is employed on an adaptive mesh of points for pricing a fractional Black–Scholes partial differential equation (PDE) based on the modified RL derivative. To solve this problem, discretization along space is carried out on a non-uniform grid in order to focus on the hot area, at which the initial condition of the pricing model, i.e., the payoff, has discontinuity. The L1 scheme having the convergence order 2−α is used along the time fractional variable. Then, our proposed numerical method is built by matrices of differentiations to be as efficient as possible. Computational pieces of evidence are brought forward to uphold the theoretical discussions and show how the presented method is efficient in contrast to the exiting solvers.

Numerical solution of the three-asset Black–Scholes option pricing model using an efficient hybrid method

In this paper, a numerical method for solving three-asset Black–Scholes partial differential equation is presented. The model is based on the Crank–Nicolson and the radial basis function methods. Also, the convergency of the proposed method is proved. Implementation of the proposed method is specially studied on cash-or-nothing option. The numerical results show the efficiency and high accuracy of the proposed method.

Approximation of single‐barrier options partial differential equations using feed‐forward neural network

AbstractArtificial neural networks are generally employed in the numerical solution of differential equation problems. In this article, we propose an approach that deals with the combination of the feed‐forward neural network method and the optimization technique in solving the partial differential equation arising from the valuation of barrier options. The methodology entails transforming the extended Black–Scholes partial differential equations (PDE), which defines a barrier option, into a constrained optimization problem, and then proposing a trial solution that reduces the differential equation problem to an unconstrained one. This trial function consists of the adjustable and non‐adjustable neural network parameters. We design it to be differentiable, analytic, and satisfy the initial and boundary conditions of the corresponding option pricing PDE. We compare the corresponding option values to the Monte‐Carlo simulated values, Crank–Nicolson finite‐difference values and the exact Black–Scholes prices. Numerical results presented in this research show that neural networks can sufficiently solve PDE‐related problems with sufficient precision and accuracy. Furthermore, they can be applied in the fast and accurate valuation of financial derivatives without closed analytic forms.

Analytical Solutions of Black-Scholes Partial Differential Equation of Pricing for Valuations of Financial Options using Hybrid Transformation Methods

Black–Scholes partial differential equation is a generally acceptable model in financial markets for option pricing. However, without variable transformations, the provision of symbolic solutions to the variable coefficient partial differential equation is not a straight-forward task. Moreover, the coefficients of the Black–Scholes can depend on the time and the asset price which makes the analytical solution of the Black–Scholes model very difficult to develop. In this paper, analytical solutions of the model of valuations of financial options are presented using Laplace and differential transform methods. The results of the solutions of the Laplace and differential transformation methods are compared with the results of the exact analytical solutions. Moreover, numerical examples for different options pricing are presented to establish the applications, speed and accuracy of the hybrid methods.

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.

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.

Pricing American Put Option using RBF-NN: New Simulation of Black-Scholes

Abstract The present work proposes an Artificial Neural Network framework for calculating the price and delta hedging of American put option. We consider a sequence of Radial Basis function Neural Network, where each network learns the difference of the price function according to the Gaussian basis function. Based on Black Scholes partial differential equation, we improve the superiority of Artificial Neural Network by comparing the performance and the results achieved used in classic Monte Carlo Least Square simulation with those obtained by Neural networks in one dimension. Thus, numerical result shows that the Artificial Neural Network solver can reduce the computing time significantly as well as the error training.

Pricing Multi-Asset Derivatives by Finite-Difference Method on a Quantum Computer

Following the recent great advance of quantum computing technology, there are growing interests in its applications to industries, including finance. In this article, we focus on derivative pricing based on solving the Black–Scholes partial differential equation by the finite-difference method (FDM), which is a suitable approach for some types of derivatives but suffers from the <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">curse of dimensionality</i> , that is, exponential growth of complexity in the case of multiple underlying assets. We propose a quantum algorithm for FDM-based pricing of multi-asset derivative with exponential speedup with respect to dimensionality compared with classical algorithms. The proposed algorithm utilizes the quantum algorithm for solving differential equations, which is based on quantum linear system algorithms. Addressing the specific issue in derivative pricing, that is, extracting the derivative price for the present underlying asset prices from the output state of the quantum algorithm, we present the whole of the calculation process and estimate its complexity. We believe that the proposed method opens the new possibility of accurate and high-speed derivative pricing by quantum computers.

A novel structure automatic-determined Fourier extreme learning machine for generalized Black–Scholes partial differential equation

This paper focuses on the numerical solutions of generalized Black–Scholes partial differential equations and proposes a novel structure automatic-determined Fourier extreme learning machine (SAD-FELM) to solve the partial differential equations. In the SAD-FELM, the Fourier extreme learning machine (FELM) is designed. The product of two Fourier polynomials is used to serve as basis functions in the hidden layer. The optimal sampling frequencies of two Fourier polynomials depend on the distribution of data samples. Then, the improved extreme learning machine algorithm is used to solve the linear equation systems derived from the generalized Black–Scholes partial differential equations. Furthermore, to determine the optimal sampling frequencies and the number of neurons, this paper designs a structure automatic-determined algorithm based on metaheuristic algorithms. The research conducts comparative experiments among the SAD-FELMs based on different metaheuristic algorithms. The experimental results show that the numerical solutions obtained by the SAD-FELM based on the hunger games search algorithm are the most accurate. Finally, compared with other methods, the superiority and feasibility of the SAD-FELM are proved.

Approach to the Delta Greek of nonlinear Black–Scholes equation governing European options

In this paper, we consider a class of a nonlinear option pricing models considering transaction costs. The focus is on the numerical investigation of the Delta equation, where the unknown is the first spatial derivative of the option value. A numerical algorithm for solving a generalized Black–Scholes partial differential equation, which arises in European option pricing considering transaction costs is studied. The Crank–Nicolson method used to discretize in the temporal direction and the FEM used to discretize in the spatial direction are discussed in terms of accuracy, convergence and efficiency. The efficiency and accuracy of the proposed method are tested numerically, and the results confirm the theoretical behavior of the solutions, which are also found to be in good agreement with the exact solution of the linear case.

The Black–Scholes equation in the presence of arbitrage

We apply Geometric Arbitrage Theory to obtain results in Mathematical Finance, which do not need stochastic differential geometry in their formulation. First, for generic market dynamics given by a subclass of multidimensional Itô processes we specify and prove the equivalence between No-Free-Lunch-with-Vanishing-Risk (NFLVR) and expected utility maximization. As a by-product, we provide a geometric characterization of the No-Unbounded-Profit-with-Bounded-Risk (NUPBR) condition given by the zero curvature (ZC) condition for this subclass of Itô processes. Finally, we extend the Black–Scholes partial differential equation to markets allowing arbitrage.

Novel ANN Method for Solving Ordinary and Time-Fractional Black–Scholes Equation

The main aim of this study is to introduce a 2-layered artificial neural network (ANN) for solving the Black–Scholes partial differential equation (PDE) of either fractional or ordinary orders. Firstly, a discretization method is employed to change the model into a sequence of ordinary differential equations (ODE). Subsequently, each of these ODEs is solved with the aid of an ANN. Adam optimization is employed as the learning paradigm since it can add the foreknowledge of slowing down the process of optimization when getting close to the actual optimum solution. The model also takes advantage of fine-tuning for speeding up the process and domain mapping to confront the infinite domain issue. Finally, the accuracy, speed, and convergence of the method for solving several types of the Black–Scholes model are reported.

Calibration of the temporally varying volatility and interest rate functions

In this study, we develop a calibration method of the temporally varying volatility and interest rate functions using the Black–Scholes (BS) partial differential equation and the observed market option prices with different strikes and expiries. The proposed method uses the piecewise linear interpolations between data points which are defined at the middle points of maturity dates. When we construct the volatility and interest rate, we use the exponential function so that the interpolated values are always positive. Numerical experiments with synthetic and real market data demonstrate the superior performance of the proposed method.

Qualitatively Stable Nonstandard Finite Difference Scheme for Numerical Solution of the Nonlinear Black–Scholes Equation

In this paper, we use a numerical method for solving the nonlinear Black–Scholes partial differential equation of the European option under transaction costs, which is based on the nonstandard discretization of the spatial derivatives. The proposed scheme, in addition to the unconditional positivity, is stable, consistent, and monotone. In order to illustrate the efficiency of the new method, numerical results have been performed by four models.