Black-Scholes Partial Differential Equation Research Articles

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.

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.

A Physics informed neural network approach for solving time fractional Black-Scholes partial differential equations

A Physics informed neural network approach for solving time fractional Black-Scholes partial differential equations

Existence of Weak Solution for Black-Scholes Partial Differential Equation and Application of Energy Estimate Theorem in Sobolev Space

Purpose:This paper aims to solve the BS second-order parabolic equation in Sobolev spaces to obtain weak solutions for financial applications, extending previous work in this field.Methodology:This paper constructs a set of functions that transforms the Black-Scholes partial differential equation into weak formulations. This study focuses on the Mathematics of finance, particularly the evolution of option pricing. An option's underlying asset involves agreements to buy or sell at a future strike price. The Black-Scholes (BS) equation, widely used in financial applications, models this.Findings:The analytical solutions, existence, uniqueness and other estimates were also obtained in weak form using boundary conditions to establish the effects of their financial implications in Sobolev spaces.Implication:The problem's regularity conditions were considered, and the coefficients and boundary of the domain are all smooth functions. To this end, this paper illustrates the definitions and assumptions that led to valuable assertions.

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.

On a sparse and stable solver on graded meshes for solving high-dimensional parabolic pricing PDEs

The purpose of this work is to investigate the multi-dimensional Black-Scholes partial differential equation with variable coefficients numerically. The problem is of practical importance due to option pricing at the presence of multi assets. Since by increasing the dimension, the curse of dimensionality restricts the computations, the proposed solver will be constructed based on sparse arrays. Toward this goal, fourth-order finite difference approximations on graded meshes are introduced and then employed through semi-discretization. Then a sixth-order Runge-Kutta solver is employed for finding the resolution of the derived set of ordinary differential equations. The stability of the proposed scheme is furnished in detail as well. Numerical testings are given to uphold the accuracy and efficacy of the proposed procedure.

Parameter estimation for time-fractional Black-Scholes equation with S &P 500 index option

This paper aims to estimate the parameters of the time-fractional Black-Scholes (TFBS) partial differential equation with the Caputo fractional derivative by using the real option prices of the S &P 500 index options. First, the numerical solution is obtained by developing a high-order scheme with order (3-alpha ) for the time discretisation. Some theoretical analyses such as stability and convergence are presented in order to verify the efficiency and accuracy of the proposed scheme. Secondly, we employ a modified hybrid Nelder-Mead simplex search and particle swarm optimization (MH-NMSS-PSO) to identify the fractional order alpha and implied volatility sigma of the TFBS equation, and explore the financial meanings of alpha under extreme stock market conditions such as the Covid-19 and the 2008 global financial crisis. We analyse the values of alpha and compare the mean squared errors of both the TFBS model and the BS model. Our empirical results show that alpha may be regarded as a market fluctuation indicator for classifying financial environments, and the TFBS model is more capable of fitting real option data compared with the BS model, especially for put options during the economic downturn. In addition, we find and discuss an interesting relation between alpha and sigma from both the TFBS model and the BS model in three expressions, which could be an open problem for further research.

A Numerical Approximation on Black-Scholes Equation of Option Pricing

This paper considered the notion of European option which is geared towards solving analytical and numerical solutions. In particular, we examined the Black-Scholes closed form solution and modified Black-Scholes (MBS) partial differential equation using Crank-Nicolson finite difference method. These partial differential equations were approximated to obtain Call and Put option prices. The explicit price of both options is found accordingly. The numerical solutions were compared to the closed form prices of Black-Scholes formula. More so, comparisons of other parameters were discussed for the purpose of investment plans. The computational results shows: increase in stock volatility increases the value of options, when the initial stock price is equal to its strike price the values of call option is higher than the put option. This informs the investor about the behavior of stock prices for the purpose of decision making. Finally, all simulation results presented graphically using MATLAB.

An Efficient Numerical Method for Pricing Double-Barrier Options on an Underlying Stock Governed by a Fractal Stochastic Process

After the discovery of the fractal structures of financial markets, enormous effort has been dedicated to finding accurate and stable numerical schemes to solve fractional Black-Scholes partial differential equations. This work, therefore, proposes a numerical scheme for pricing double-barrier options, written on an underlying stock whose dynamics are governed by a non-standard fractal stochastic process. The resultant model is time-fractional and is herein referred to as a time-fractional Black-Scholes model. The presence of the time-fractional derivative helps to capture the time-decaying effects of the underlying stock while capturing the globalized change in underlying prices and barriers. In this paper, we present the construction of the proposed scheme, analyse it in terms of its stability and convergence, and present two numerical examples of pricing double knock-in barrier-option problems. The results suggest that the proposed scheme is unconditionally stable and convergent with order O(h2+k2).

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.

Nonlinear diffusion with the p-Laplacian in a Black-Scholes-type model

We present a new nonlinear version of the well-known Black-Scholes model for option pricing in financial mathematics. The nonlinear Black-Scholes partial differential equation is based on the quasilinear diffusion term with the p-Laplace operator \(\Delta_p\) for \(1 < p < \infty\). The existence and uniqueness of a weak solution in a weighted Sobolev space is proved, first, by methods for nonlinear parabolic problems using the Gel'fand triplet and, alternatively, by a method based on nonlinear semigroups. Finally, possible choices of other weighted Sobolev spaces are discussed to produce a function space setting more realistic in financial mathematics. See also https://ejde.math.txstate.edu/special/02/t1/abstr.html

Application Of Natural Decomposition Method For Solution Of Fractional Black-Scholes Equation

The Black-Scholes equation is a partial differential equation that can model the European call option price problem. This equation can be of the order of natural numbers or fractional. The aim of this paper is to find a solution to the fractional order Black-Scholes partial differential equation. The method used to find solutions to these equations is the Natural decomposition method. Two numerical examples are presented in this paper. The results show that the Natural decomposition method is effective and easy to use to solve the fractional Black-Scholes equation.

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.

A Solution to Linear Black-Scholes Second-order Parabolic equation in sobolev space

In this paper, a set of functions were constructed that transforms Black-Scholes partial differential equation into weak formulations. The analytical solutions: existence, uniqueness and other estimates were also obtained in weak form with the use of boundary conditions to establish the effects of its financial implications in Sobolev spaces. The regularity conditions of the problem were considered which the coefficients, the boundary of the domain are all smooth functions. To this end, the definitions, assumptions which paved way to useful assertions are illustrated in this paper.

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.