Black-Scholes Equation Research Articles

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.

Optimal investment strategy to maximize the expected utility of an insurance company under Cramer-Lundberg dynamic

In this work, we examine the combined problem of optimal portfolio selection rules for an insurer in a continuous time model where the surplus of an insurance company is modelled as a compound Poisson process. The company can invest its surplus in a risk free asset and in a risky asset, governed by the Black-Scholes equation. According to utility theory, in a financial market where investors are facing uncertainty, an investor is not concerned with wealth maximization per se but with utility maximization. It is therefore possible to introduce an increasing and concave utility function $\phi(x,t)$ representing the expected utility of a risk averse investor (insurance company). Therefore, the goal of this work is not anymore to maximize the expected portfolio value or minimize the ruin probability or maximizing the expectation of the present value of all dividends paid to the shareholders up to the ruin, but to maximize the expected utility stemming from the wealth during the life contract [0,T]. In this direction, using the Dynamic Programming Principle of the problem, we obtain the Hamilton-Jacobi-Bellman equation by our optimization problem (HJB). Finally, we present numerical solutions in some cases, obtaining as optimal strategy the well known Merton's strategy.

A posteriori grid method for a time-fractional Black-Scholes equation

<abstract><p>In this paper, a posteriori grid method for solving a time-fractional Black-Scholes equation governing European options is studied. The possible singularity of the exact solution complicates the construction of the discretization scheme for the time-fractional Black-Scholes equation. The $ L1 $ method on an arbitrary grid is used to discretize the time-fractional derivative and the central difference method on a piecewise uniform grid is used to discretize the spatial derivatives. Stability properties and a posteriori error analysis for the discrete scheme are studied. Then, an adapted a posteriori grid is constructed by using a grid generation algorithm based on a posteriori error analysis. Numerical experiments show that the $ L1 $ method on an adapted a posteriori grid is more accurate than the method on the uniform grid.</p></abstract>

Nonuniform Finite Difference Scheme for the Three-Dimensional Time-Fractional Black–Scholes Equation

In this study, we present an accurate and efficient nonuniform finite difference method for the three-dimensional (3D) time-fractional Black–Scholes (BS) equation. The operator splitting scheme is used to efficiently solve the 3D time-fractional BS equation. We use a nonuniform grid for pricing 3D options. We compute the three-asset cash-or-nothing European call option and investigate the effects of the fractional-orderαin the time-fractional BS model. Numerical experiments demonstrate the efficiency and fastness of the proposed scheme.

DeLISA: Deep learning based iteration scheme approximation for solving PDEs

Solving the high dimensional partial differential equations (PDEs) with the classical numerical methods is a challenge task. As possessing the power of progressing high dimensional data, deep learning is naturally considered to solve PDEs. This paper proposes a deep learning framework based iteration scheme approximation, called DeLISA. First, we adopt the implicit multistep method and Runge-Kutta method for time iteration scheme. Then, such iteration scheme is approximated by a neural network. Due to integrating the physical information of governing equation into time iteration schemes and introducing time-dependent input, our method achieves the continuous time prediction without a mass of interior points. Here, the activation function with adaptive variable adjusts itself during the iteration process. Finally, we present numerical experiments results for some benchmark PDEs, including Burgers, Allen-Cahn, Schrödinger, carburizing and Black-Scholes equations, and verify that the proposed approach is superior to the state-of-the-art techniques on accuracy and flexibility. Moreover, the Frequency Principle is also illustrated by the changes of prediction at different iterations in this paper.

Projection of the two-dimensional Black-Scholes equation for options with underlying stock and strike prices in two different currencies

The two-variable Black-Scholes equation is used to study the option exercise price of two different currencies. Due to the complexity of dealing with several variables, reduction methods have been implemented to deal with these problems. This paper proposes an alternative reduction by using the so-called Zwanzig projection method to one-dimension, successfully developed to study the diffusion in confined systems. In this case, the option price depends on the stock price and the exchange rate between currencies. We assume that the exchange rate between currencies will depend on the stock price through some model that bounds such dependence, which somehow influences the final option price.As a result, we find a projected one-dimensional Black-Scholes equation similar to the so-called Fick-Jacobs equation for diffusion on channels. This equation is an effective Black-Scholes equation with two different interest rates, whose solution gives rise to a modified Black-Scholes formula. The properties of this solution are shown and were graphically compared with previously found solutions, showing that the corresponding difference is bounded.

Localized kernel‐based meshless method for pricing financial options underlying fractal transmission system

The variation in the option pricing of the fractal transmission system is modelled by the time fractional Black–Scholes equation (TFBSE). This paper proposes an efficient local meshless method for the numerical simulation of the TFBSE. At the first step, a difference formula of L1 type is employed to get a semi‐discrete algorithm in the temporal variable with a accuracy of order 2 − α in the case of smooth solutions, where 0 < α ≤ 1 is the fractional‐order derivative. At the second step, a localized radial basis function finite difference is adopted to derive a full‐discrete scheme. Moreover, the unconditional stability and convergence of the proposed method are analyzed based on energy norm. The exact expressions for the weights of the first and second derivatives are used by imposing a multiquadric function generated by finite difference. The proposed technique produces linear systems with tridiagonal and diagonal matrices. Numerical experiments highlight the performance of the method.

Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Numerically Pricing Nonlinear Time-Fractional Black–Scholes Equation with Time-Dependent Parameters Under Transaction Costs

Group formalism of Lie transformations, conservation laws, exact and numerical solutions of non-linear time-fractional Black–Scholes equation

The topic of the present paper concentrates on a systematic investigation of Lie group analysis of non-linear time-fractional Black–Scholes equation including numerical approximations. Lie point symmetries of the equation are found via Riemann–Liouville derivative with four different volatility models. Also the invariant solutions as a kind of exact solutions are constructed by a useful explained method. Conservation laws for the intended equation are derived by using a modified version of Noether’s theorem. A numerical simulation by Chebyshev pseudo-spectral (CPS) method is computed for the equation. Approximate solutions together with their absolute errors are found and the graphs of solutions are plotted by the action of order of derivative α.

Testing stationarity of the detrended price return in stock markets

This paper proposes a generalized porous media equation with drift as the governing equation for stock market indexes. The proposed governing equation can be expressed as a Fokker–Planck equation (FPE) with a non-constant diffusion coefficient. The governing equation accounts for non-stationary effects and describes the time evolution of the probability distribution function (PDF) of the price return. By applying Ito’s Lemma, the FPE is associated with a stochastic differential equation (SDE) that models the time evolution of the price return in a fashion different from the classical Black–Scholes equation. Both FPE and SDE equations account for a deterministic part or trend and a stochastic part or q-Gaussian noise. The q-Gaussian noise can be decomposed into a Gaussian noise affected by a standard deviation or volatility. The presented model is validated using the S&P500 index’s data from the past 25 years per minute. We show that the price return becomes Gaussian, consequently stationary by normalizing the detrended data set. The normalization of the data is calculated by subtracting the trend and then dividing by the standard deviation of the detrended price return. The stationarity test consists of representing the power spectrum in terms of the time series’s autocorrelation. Additionally, this paper presents the multifractal analysis for the detrended and normalized price return to describe the Hurst exponent dynamics over the dataset.

Spectrally accurate option pricing under the time fractional Black-Scholes model

We 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. doi:10.1017/S1446181121000286

On a Multigrid Method for Tempered Fractional Diffusion Equations

In this paper, we develop a suitable multigrid iterative solution method for the numerical solution of second- and third-order discrete schemes for the tempered fractional diffusion equation. Our discretizations will be based on tempered weighted and shifted Grünwald difference (tempered-WSGD) operators in space and the Crank–Nicolson scheme in time. We will prove, and show numerically, that a classical multigrid method, based on direct coarse grid discretization and weighted Jacobi relaxation, performs highly satisfactory for this type of equation. We also employ the multigrid method to solve the second- and third-order discrete schemes for the tempered fractional Black–Scholes equation. Some numerical experiments are carried out to confirm accuracy and effectiveness of the proposed method.

A NEW CONSTRUCTION OF OS OF SUBALGEBRAS AND INVARIANT SOLUTION OF THE BLACK-SCHOLES EQUATION

In this manuscript, the Lie group technique is applied to construct a new OS and invariant solutions of a one-dimensional LA, which describes the symmetries properties of a nonlinear Black-Scholes model. The structure of LA depends on one parameter. We have shown a novel way to construct the so-called OS of subalgebras of the Black-Scholes equation by utilizing the given symmetries. We transform the symmetries of the Black-Scholes equation into a simple ordinary differential equation called the Lie equation, which provides us a way through which to construct a new optimal scheme of subalgebras of the Black-Scholes through applying the concept of LE. The OS which consists of minimal representatives is utilized to develop the invariant solution for the Black-Scholes equation. The fundamental use of the Lie group analysis to the differential equation is the categorization of group invariant solutions of differential equations via OS. Finally, we have utilized the OS to construct the invariant solution of the Black-Scholes equation.

Robust and accurate construction of the local volatility surface using the Black–Scholes equation

• We present an algorithm to reconstruct a local volatility (LV) function. • We reconstruct the local volatility surface using a nonlinear fitting function. • The results demonstrate the robustness and accuracy of the proposed algorithm. In this study, we develop a numerical method for the robust and accurate construction of a local volatility (LV) surface using the generalized Black–Scholes (BS) equation from the given option price data. The BS equation is a partial differential equation and has been used to model financial option pricing. Constant volatility was used in the classical BS model. However, it is well known that the constant volatility BS model is practically unsuitable because real financial market data demonstrate non-constant volatility behavior. The LV function is dependent on the asset prices and time. One of the difficulties in reconstructing an unknown LV surface is uniqueness. We extend a previous study of reconstructing time-dependent volatility, which is unique, to time- and space-dependent volatility surfaces. We propose an algorithm comprising four steps: the first step is estimating constant implied volatility; the second step is finding the influential region using the probability density function of a log-normal distribution; the third step is calculating the time-dependent volatility function; and the final step is reconstructing the LV surface. We use a finite difference method to numerically solve the BS model and a nonlinear fitting function to compute the LV surface. We perform computational experiments using synthetic and real market data. The numerical results demonstrate the robust and accurate construction of an unknown LV surface using the proposed method.

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.

Digital Option Pricing Approach Using A Homotopy Perturbation Method

An option is a financial contract between buyers and sellers. The Black-Scholes equation is the most popular mathematical equation used to analyze the option pricing. The exact solution of the Black-Scholes equation can be approached by several approximation methods, one of the method is a Homotopy Perturbation Method (HPM). The simplest type of option, digital options were analyzed using the HPM. The digital option pricing approach using the HPM is in a power series form, which in this paper is presented the solution in the fourth power. This solution is compared with the exact solution of the Black-Scholes equation for digital options. The results show that the approach using HPM is very accurate.

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.

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.

Solution of the Black-Scholes Equation: LDM and SDM

The main aim of this paper is to discuss a new way of a non-discretization method for the solution of the Black-Scholes equation. Black-Scholes is a mathematical model based on a partial differential equation. The solution of the model is of utmost importance in financial mathematics to estimate option pricing. Several analytical, numerical, and non-discretization methods are existing in the literature to solve the model. Two decomposition methods namely the Laplace decomposition method (LDM) and Sumudu decomposition method (SDM) are adopted for the present study. The results of the present techniques have closed an agreement with an approximate solution which has been obtained with the help of the Adomian Decomposition Method (ADM).