Black-Scholes Equation Research Articles

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

High order approximation of derivatives with applications to pricing of financial derivatives

In this paper, we first compare three different methods for approximating the first and second derivatives from function values given at scattered points. Then we propose to use the most accurate derivative approximation method in a forward Euler scheme to solve the general Black–Scholes equation. We prove the scheme’s stability and error estimate. Many numerical examples applying to pricing of financial derivatives are presented to demonstrate the efficiency and accuracy of our scheme.

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.

Numerical investigation of the time-fractional Black-Scholes equation with barrier choice of regulating European option

The price variance of the associated fractal transmission mechanism was used to estimate the Black-Scholes fractional model of which a time-fractional derivative is $alpha$. In the current paper, the time-fractional Black-Scholes equation (TFBSE) that the temporal derivative is the Caputo fractional derivative is known by regulating the European option. At first, linear interpolation with a temporally $tau^{2-alpha}$ order accuracy is used for constructing the semi-discrete. Then, the spatial derivative terms are approximated with the help of the collocation approach centered on the Chebyshev polynomials of the third form (CPTF). Finally, The unconditional stability and convergence order are analyzed by applying the energy method. To show the precision of the numerical system, we solved two instances of the TFBSE. Numerical results and comparisons indicate the proposed approach is very reliable and efficient.

Computation of the unknown volatility from integral option price observations in jump–diffusion models

In this work we propose a simple and efficient algorithm to numerically approximate the time-dependent implied volatility for jump–diffusion models in option pricing that generalize the Black–Scholes equation. Here we use implicit–explicit difference schemes to compute the derivative part with fully implicit method and the integral term — in an explicit way. An average in time linearization of the diffusion term is applied, followed by a special decomposition of the unknown volatility function, which enables us to derive the implied volatility in an explicit form. Furthermore, the correctness of the algorithms is established. The presented numerical simulations demonstrate the capabilities of the current approach and confirm the robustness of the proposed methodology.

A Finite Difference-Spectral Method for Solving the European Call Option Black–Scholes Equation

In this paper, we present a novel technique based on backward-difference method and Galerkin spectral method for solving Black–Scholes equation. The main propose of this method is to reduce the solution of this problem to the solution of a system of algebraic equations. The convergence order of the proposed method is investigated. Also, we provide numerical experiment to show the validity of proposed method.

Removing non-smoothness in solving Black-Scholes equation using a perturbation method

Black-Scholes equation as one of the most celebrated mathematical models has an explicit analytical solution known as the Black-Scholes formula. Later variations of the equation, such as fractional or nonlinear Black-Scholes equations, do not have a closed form expression for the corresponding formula. In that case, one will need asymptotic expansions, such as the homotopy perturbation method, to give an approximate analytical solution. However, the solution is non-smooth at a special point. We modify the method by first performing variable transformations that push the point to infinity. As a test bed, we apply the method to the solvable Black-Scholes equation, where excellent agreement with the exact solution is obtained. We also extend our study to multi-asset basket and quanto options by reducing the cases to single-asset ones. Additionally we provide a novel analytical solution of the single-asset quanto option that is simple and different from the existing expression.

New Symmetries of Black-Scholes Equation

Lie Point symmetries and Euler’s formula for solving second order ordinary linear differential equations are used to determine symmetries for the one-dimensional Black- Scholes equation. One symmetry is utilized to determine an invariant solutions

A modification of Galerkin's method for option pricing

<p style='text-indent:20px;'>We present a novel method for solving a complicated form of a partial differential equation called the Black-Scholes equation arising from pricing European options. The novelty of this method is that we consider two terms of the equation, namely the volatility and dividend, as variables dependent on the state price. We develop a Galerkin finite element method to solve the problem. More specifically, we discretize the system along the state variable and build new basis functions which we use to approximate the solution. We establish convergence of the proposed method and numerical results are reported to show the proposed method is accurate and efficient.</p>

Numerical pricing based on fractional Black–Scholes equation with time-dependent parameters under the CEV model: Double barrier options

The empirical observations show that the constant elasticity of variance (CEV) model is a practical approach to capture the implied volatility smile phenomenon. Also, due to the appearance of long-range dependence (or long memory effect) in stock returns or volatility, the fractional Black–Scholes models became particularly important in finance. We will generalize the CEV model to a fractional-CEV model with Jumarie’s fractional model (Jumarie, 2008) to capture the long memory effect in addition to show the negative relationship between stock price and its return volatility. The asset price dynamics of this model follows from a fractional stochastic differential equation, and its volatility is a function of the underlying asset price. We derive the fractional Black–Scholes equation by using the Itoˆ Lemma and fractional Taylor’s series. Previously, Jumarie’s model was used to determine the value of European and American options. In this study, given the importance of the barrier option, we use the proposed model for pricing a European double barrier option. Since most fractional PDEs do not have closed-form analytical solutions, we solve the option pricing problem numerically. Then, we investigate the stability and convergence of the proposed scheme by applying the Fourier analysis. Finally, some numerical results are given in the last section by computing the European double barrier option.

Competitively-Issued Convertible Bank Notes in a Theory of Finance: Earl Thompson Meets Fischer Black

Abstract In this paper, I show the validity of and the relationship between two previously unrelated claims in monetary theory. The first claim, made by Earl Thompson, is that privately-issued bank notes pay a positive rate of return in a competitive equilibrium. The second claim, made by Fischer Black, is that it is possible to have a gold standard in which the gold reserves of the central bank are near zero. I show that both of these claims are correct under the assumption of complete markets and perfect commitment. The link between these claims is the Black-Scholes equation applied to convertible bank notes. In commodity-based monetary systems, bank notes are perpetual American options. I extend the model to consider the implications of a lack of commitment on the part of the bank and incomplete markets. I show that both arguments break down when banks lack commitment to redemption or markets are incomplete. I conclude with implications for macroeconomic theory.

A compact difference scheme for time-fractional Black-Scholes equation with time-dependent parameters under the CEV model: American options

‎The Black-Scholes equation is one of the most important mathematical models in option pricing theory, but this model is far from market realities and cannot show memory effect in the financial market. This paper investigates an American option based on a time-fractional Black-Scholes equation under the constant elasticity of variance (CEV) model, which parameters of interest rate and dividend yield supposed as deterministic functions of time, and the price change of the underlying asset follows a fractal transmission system. This model does not have a closed-form solution; hence, we numerically price the American option by using a compact difference scheme. Also, we compare the time-fractional Black-Scholes equation under the CEV model with its generalized Black-Scholes model as α = 1 and β = 0. Moreover, we demonstrate that the introduced difference scheme is unconditionally stable and convergent using Fourier analysis. The numerical examples illustrate the efficiency and accuracy of the introduced difference scheme.

SPECTRALLY ACCURATE OPTION PRICING UNDER THE TIME-FRACTIONAL BLACK–SCHOLES MODEL

AbstractWe propose a Legendre–Laguerre spectral approximation to price the European and double barrier options in the time-fractional framework. By choosing an appropriate basis function, the spectral discretization is used for the approximation of the spatial derivatives of the time-fractional Black–Scholes equation. For the time discretization, we consider the popular$L1$finite difference approximation, which converges with order$\mathcal {O}((\Delta \tau )^{2-\alpha })$for functions which are twice continuously differentiable. However, when using the$L1$scheme for problems with nonsmooth initial data, only the first-order accuracy in time is achieved. This low-order accuracy is also observed when solving the time-fractional Black–Scholes European and barrier option pricing problems for which the payoffs are all nonsmooth. To increase the temporal convergence rate, we therefore consider a Richardson extrapolation method, which when combined with the spectral approximation in space, exhibits higher order convergence such that high accuracies over the whole discretization grid are obtained. Compared with the traditional finite difference scheme, numerical examples clearly indicate that the spectral approximation converges exponentially over a small number of grid points. Also, as demonstrated, such high accuracies can be achieved in much fewer time steps using the extrapolation approach.

A Numerical Discussion for the European Put Option Model

The Black-Scholes equations have been increasingly popular over the last three decades since they provide more practical information for optional behaviours. Therefore, effective methods have been needed to analyze these models. This study will focus mainly on investigating the behavior of the Black-Scholes equation for the European put option pricing model. To achieve this, numerical solutions of the Black-Scholes European option pricing model are produced by three combined methods. Spatial discretization of the Black-Scholes model is performed using a fourth-order finite difference (FD4) scheme that allows a highly accurate approximation of the solutions. For the time discretization, three numerical techniques are proposed: a strong-stability preserving Runge Kutta (SSPRK3), a fourth-order Runge Kutta (RK4) and a one-step method. The results produced by the combined methods have been compared with available literature and the exact solution.

A compact finite difference scheme for fractional Black-Scholes option pricing model

In this paper, we present a numerical technique for solving the time-fractional Black-Scholes (TFBS) equation describing European options. The time-fractional derivative is described by means of Caputo and a compact finite difference method is employed for discretization of space derivative. Stability and convergence of the fully discrete scheme are studied. Two test problems with the known analytical solutions are considered to demonstrate the efficiency and accuracy of the method and validate the theoretical result. Further, in order to demonstrate the practicability of the method, the method is applied to three European options pricing problems governed by the TFBS equations, where analytical solutions to these problems are not known. We study the effect of fractional order derivative on the solution profile corresponding to option price.

Generalized heat diffusion equations with variable coefficients and their fractalization from the Black-Scholes equation

In this study, we prove that modified diffusion equations, including the generalized Burgers’ equation with variable coefficients, can be derived from the Black-Scholes equation with a time-dependent parameter based on the propagator method known in quantum and statistical physics. The extension for the case of a local fractal derivative is also addressed and analyzed.

PDTM Approach to Solve Black Scholes Equation for Powered ML-Payoff Function

In this paper, the Projected Differential Transform Method (PDTM) has been used to solve the Black Scholes differential equation for powered Modified Log Payoff (ML-Payoff) functions, $max {S^klnbig(frac{S}{K}big),0}$ and $max{S^klnbig(frac{K}{S}big),0}, (kin mathbb{R^{+}}cup {0})$. It is the generalization of Black Scholes model for ML-Payoff functions. It can be seen that values from PDTM is quite accurate to the closed form solutions.

On high-order schemes for tempered fractional partial differential equations

In this paper, we propose third-order semi-discretized schemes in space based on the tempered weighted and shifted Grunwald difference (tempered-WSGD) operators for the tempered fractional diffusion equation. We also show stability and convergence analysis for the fully discrete scheme based a Crank–Nicolson scheme in time. A third-order scheme for the tempered Black–Scholes equation is also proposed and tested numerically. Some numerical experiments are carried out to confirm accuracy and effectiveness of these proposed methods.

Application of the Generalized Laplace Homotopy Perturbation Method to the Time-Fractional Black–Scholes Equations Based on the Katugampola Fractional Derivative in Caputo Type

In the finance market, the Black–Scholes equation is used to model the price change of the underlying fractal transmission system. Moreover, the fractional differential equations recently are accepted by researchers that fractional differential equations are a powerful tool in studying fractal geometry and fractal dynamics. Fractional differential equations are used in modeling the various important situations or phenomena in the real world such as fluid flow, acoustics, electromagnetic, electrochemistry and material science. There is an important question in finance: “Can the fractional differential equation be applied in the financial market?”. The answer is “Yes”. Due to the self-similar property of the fractional derivative, it can reply to the long-range dependence better than the integer-order derivative. Thus, these advantages are beneficial to manage the fractal structure in the financial market. In this article, the classical Black–Scholes equation with two assets for the European call option is modified by replacing the order of ordinary derivative with the fractional derivative order in the Caputo type Katugampola fractional derivative sense. The analytic solution of time-fractional Black–Scholes European call option pricing equation with two assets is derived by using the generalized Laplace homotopy perturbation method. The used method is the combination of the homotopy perturbation method and generalized Laplace transform. The analytic solution of the time-fractional Black–Scholes equation is carried out in the form of a Mittag–Leffler function. Finally, the effects of the fractional-order in the Caputo type Katugampola fractional derivative to change of a European call option price are shown.

Efficient operator splitting and spectral methods for the time-space fractional Black–Scholes equation

In this paper, we aim at developing improved L1 operator splitting method and spectral method for Black–Scholes differential systems with fractional derivatives in both time and space. We address the challenge of slow convergence in time and provide efficient approaches to handle complementarity conditions in the differential system. We design an operator splitting method to decouple the constraints together with an improved L1 approximation to accommodate nonsmooth initial data. In the space dimension, a spectral collocation method based on Legendre–Gauss–Lobatto points was employed. To show the efficiency and accuracy, we benchmark our improved L1 method with the original L1 method within the operator splitting framework. For European options, both the L1 method and improved L1 method provide the convergence order of 2−γ, where γ is fractional order in time. For American options, it is necessary to apply the improved L1 method to achieve the order of 2−γ.