Black-Scholes Equation Research Articles

Global Gauge Symmetries, Risk-Free Portfolios, and the Risk-Free Rate

We define risk-free portfolios using three gauge invariant differential operators that require such portfolios to be insensitive to price changes, to be self-financing, and to produce a zero real return so there are no risk-free profits. This definition identifies the risk-free rate as the return of an infinitely diversified portfolio rather than as an arbitrary external parameter. The risk-free rate measures the rate of global price rescaling, which is a gauge symmetry of economies. We explore the properties of risk-free rates, rederive the Black Scholes equation with a new interpretation of the risk-free rate parameter as a that background gauge field, and discuss gauge invariant discounting of cash flows.

Detection of Mispricing in the Black–Scholes PDE Using the Derivative-Free Nonlinear Kalman Filter

Financial derivatives and option pricing models are usually described with the use of stochastic differential equations and diffusion-type partial differential equations (e.g., Black---Scholes models). Considering the latter case in this paper a new filtering method for distributed parameter systems, is developed for estimating option prices variations without knowledge of initial conditions. The proposed filtering method is the so-called derivative-free nonlinear Kalman Filter and is based on a decomposition of the nonlinear partial-differential equation model into a set of ordinary differential equations with respect to time. Next, each one of the local models associated with the ordinary differential equations is transformed into a model of the linear canonical (Brunovsky) form through a change of coordinates (diffeomorphism) which is based on differential flatness theory. This transformation provides an extended description of the nonlinear dynamics of the option pricing model for which state estimation is possible by applying the standard Kalman Filter recursion. Based on the provided state estimate, validation of the Black---Scholes PDE model can be performed and the existence of inconsistent parameters in the Black---Scholes PDE model can be concluded.

Multi-asset Black–Scholes model as a variable second class constrained dynamical system

Abstract In this paper, we study the multi-asset Black–Scholes model from a structural point of view. For this, we interpret the multi-asset Black–Scholes equation as a multidimensional Schrodinger one particle equation. The analysis of the classical Hamiltonian and Lagrangian mechanics associated with this quantum model implies that, in this system, the canonical momentums cannot always be written in terms of the velocities. This feature is a typical characteristic of the constrained system that appears in the high-energy physics. To study this model in the proper form, one must apply Dirac’s method for constrained systems. The results of the Dirac’s analysis indicate that in the correlation parameters space of the multi-assets model, there exists a surface (called the Kummer surface Σ K , where the determinant of the correlation matrix is null) on which the constraint number can vary. We study in detail the cases with N = 2 and N = 3 assets. For these cases, we calculate the propagator of the multi-asset Black–Scholes equation and show that inside the Kummer Σ K surface the propagator is well defined, but outside Σ K the propagator diverges and the option price is not well defined. On Σ K the propagator is obtained as a constrained path integral and their form depends on which region of the Kummer surface the correlation parameters lie. Thus, the multi-asset Black–Scholes model is an example of a variable constrained dynamical system, and it is a new and beautiful property that had not been previously observed.

Appearances of pseudo-bosons from Black-Scholes equation

It is a well known fact that the Black-Scholes equation admits an alternative representation as a Schrödinger equation expressed in terms of a non-self-adjoint Hamiltonian. We show how pseudo-bosons, linear or not, naturally arise in this context, and how they can be used in the computation of the pricing kernel.

ON A DIVIDEND-PAYING STOCK OPTIONS PRICING MODEL (SOPM) USING CONSTANT ELASTICITY OF VARIANCE STOCHASTIC DYNAMICS

In this paper, we propose a pricing model for stock option valuation. The model is derived from the classical Black-Scholes option pricing equation via the application of the constant elasticity of variance (CEV) model with dividend yield. This modifies the Black- Scholes equation by incorporating a non-constant volatility power function of the underlying stock price, and a dividend yield parameter.

Finite difference methods for pricing American put option with rationality parameter: Numerical analysis and computing

In this paper finite difference methods for pricing American option with rationality parameter are proposed. The irrational exercise policy arising in American options is characterized in terms of a rationality parameter. The model is formulated in terms of a new nonlinear Black–Scholes equation that requires specific numerical methods. Although the solution converges to the solution of the classical American option price when the parameter tends to infinity, for finite values of the parameter the classical boundary conditions cannot apply and we propose specific ones. A logarithmic transformation is used to improve properties of the numerical solution that is constructed by explicit finite difference method. Numerical analysis provides stability conditions for the methods and its positivity. Properties of intensity function are studied from the point of view of numerical solution. Concerning the numerical methods for the original problem we propose the θ-method for time discretization, thus including explicit, fully implicit and Crank–Nicolson schemes as particular cases. The nonlinear term is treated by a Newton method. The convergence rate is illustrated by numerical examples.

Numerical solution of time-fractional Black–Scholes equation

In this paper we present a numerical method for a time-fractional Black–Scholes equation, which is used for modeling the fractional structure of the financial market. The method is based on—first, discretization in time and then the weighted finite difference spatial approximation. Some properties of the spatial discretization are studied. The main difficulty (that originates from the non-local structure of the differential operator) of the algorithm is the impossibility to advance layer by layer in time. Numerical experiments are discussed.

A numerical study of the European option by the MLPG method with moving kriging interpolation.

In this paper, the meshless local Petrov–Galerkin (MLPG) method is applied for solving a generalized Black–Scholes equation in financial problems. This equation is a PDE governing the price evolution of a European call or a European put under the Black–Scholes model. The θ-weighted method and MLPG are used for discretizing the governing equation in time variable and option pricing, respectively. We show that the spectral radius of amplification matrix with the discrete operator is less than 1. This ensures that this numerical scheme is stable. Numerical experiments are performed with time varying volatility and the results are compared with the analytical and the numerical results of other methods.

A universal difference method for time-space fractional Black-Scholes equation

The fractional Black-Scholes (B-S) equation is an important mathematical model in finance engineering, and the study of its numerical methods has very significant practical applications. This paper constructs a new kind of universal difference method to solve the time-space fractional B-S equation. The universal difference method is analyzed to be stable, convergent, and uniquely solvable. Furthermore, it is proved that with numerical experiments the universal difference method is valid and efficient for solving the time-space fractional B-S equation. At the same time, numerical experiments indicate that the time-space fractional B-S equation is more consistent with the actual financial market.

PENENTUAN HARGA OPSI DAN NILAI HEDGE MENGGUNAKAN PERSAMAAN NON-LINEAR BLACK-SCHOLES

Option are contracts that give the right to sell and buy the asset at a price and a certain period of time. In addition investors use option as a means of hedge against asset owned. Many methods are used to determine the price of option, one of them by using the Black-Scholes equation. But its use these in the assumption that the value for the constant volatility. On market assumption are not appropriates, so many researchers proposed using a volatility calculation option that is non-constant Black-Scholes equation modelled using the volatility is not constant in the range so as to produce a non-linear equation of Black-Scholes. In addition to determine the value of hedge ratio. On completions of this study, for the numerical solution of non-linear Black-Scholes equation using method of explicit finite difference scheme. Option use in research us a stock YAHOO!inc. as the underlying asset. The result showed that the price of the option is calculated using non-linear Black-Scholes equation price close on the market. Therefore, it can produce hedge ration for a risk-free portfolio containing of the option and stock.

The numerical simulation of the tempered fractional Black–Scholes equation for European double barrier option

In recent years, the Finite Moment Log Stable(FMLS), KoBoL and CGMY models, which follow a jump process or a Lévy process, have become the most popular modeling frameworks in the financial field because they can capture some of the important characteristics in the dynamic process of stock price changes, such as large movements or jumps over small time steps. In this paper, we consider the numerical simulation of these three models. We construct a discrete implicit numerical scheme with second order accuracy, and provide a stability and convergence analysis of the numerical scheme. Furthermore, a fast bi-conjugate gradient stabilized method (FBi-CGSTAB) is used to reduce the storage space from O(M2) to O(M) and the computational cost from O(M3) to O(Mlog M) per iteration, where M is the number of space grid points. Some numerical examples are chosen in order to demonstrate the accuracy and efficiency of the proposed method and technique. Finally, as an application, we use the above numerical technique to price a European double-knock-out barrier option, and then the characteristics of the three fractional Black–Scholes (B–S) models are analyzed through comparison with the classical B–S model.

A numerical study for optimal portfolio regime-switching model I. 2D Black–Scholes equation with an exponential non-linear term

A system of weakly coupled semilinear parabolic equations of optimal portfolio in a regime-switching model is suggested by Valdez and Vargiolu (2013). To study the effects from the typical features of the system, i.e. the boundary degeneration, exponential non-linearity and mixed derivative, we consider a representative 2D Black–Scholes semilinear equation model. We construct and analyze sign preserving, flux limited finite difference schemes for the scalar problem. Numerical experiments are discussed.

Pitfalls of Exponential Fitting on the Black-Scholes PDE

This paper explores nine ways to apply exponential fitting methods to the Black-Scholes PDE. In particular, we will see that many ways lead to undesirable side-effects.

Accurate and Efficient Computations of the Greeks for Options Near Expiry Using the Black-Scholes Equations

We investigate the accurate computations for the Greeks using the numerical solutions of the Black-Scholes partial differential equation. In particular, we study the behaviors of the Greeks close to the maturity time and in the neighborhood around the strike price. The Black-Scholes equation is discretized using a nonuniform finite difference method. We propose a new adaptive time-stepping algorithm based on local truncation error. As a test problem for our numerical method, we consider a European cash-or-nothing call option. To show the effect of the adaptive stepping strategy, we calculate option price and its Greeks with various tolerances. Several numerical results confirm that the proposed method is fast, accurate, and practical in computing option price and the Greeks.

Stabilization of the multi-asset Black-Scholes PDE using differential flatness theory

A method for feedbsck control of the multi-asset Black-Scholes PDE is developed. By applying semi-discretization and a finite differences scheme the multi-asset Black-Scholes PDE is transformed into a state-space model consisting of ordinary nonlinear differential equations. For this set of differential equations it is shown that differential flatness properties hold. This enables to solve the associated control problem and to succeed stabilization of the options’ dynamics. For the local subsystems, into which the multi-asset Black-Scholes PDE is decomposed, it becomes possible to apply boundary-based feedback control. For each subsystem which is related to a nonlinear ODE, a virtual control input is computed, that can invert the subsystem’s dynamics and can eliminate the subsystem’s tracking error. From the last row of the state-space description, the control input (boundary condition) that is actually applied to the multi-asset Black-Scholes PDE system is found. The stability of the proposed control scheme is confirmed with the use of the Lyapunov method.

A Lie Algebraic and Numerical Investigation of the Black-Scholes Equation with Heston Volatility Model

This work deals with an extension of the Black-Scholes model for rating options with the Heston volatility model. A Lie-algebraic analysis of this equation is applied to reduce its order and compute some of its solutions. As a result of this method, a five-parameter family of solutions is obtained. Though, these solutions do not match the terminal and boundary conditions, they can be used for the validation of numerical schemes.

On the Solution of the Multi-Asset Black-Scholes Model: Correlations, Eigenvalues and Geometry

In this paper, the multi-asset Black-Scholes model is studied in terms of the importance that the correlation parameter space (equivalent to an N dimensional hypercube) has in the solution of the pricing problem. It is shown that inside of this hypercube there is a surface, called the Kummer surface ∑k, where the determinant of the correlation matrix ρ is zero, so the usual formula for the propagator of the N asset Black-Scholes equation is no longer valid. Worse than that, in some regions outside this surface, the determinant of ρ becomes negative, so the usual propagator becomes complex and divergent. Thus the option pricing model is not well defined for these regions outside ∑k. On the Kummer surface instead, the rank of the ρ matrix is a variable number. By using the Wei-Norman theorem, the propagator over the variable rank surface ∑k for the general N asset case is computed. Finally, the three assets case and its implied geometry along the Kummer surface is also studied in detail.

Calibration and Simulation of Arbitrage Effects in a Non-Equilibrium Quantum Black-Scholes Model by Using Semi-Classical Methods

An non-equilibrium Black-Scholes model, where the usual constant interest rate r is replaced by a stochastic time dependent rate r(t) of the form r(t)=r+f (t)W(t), accounting for market imperfections and prices non-alignment, is developed. The white noise amplitude f (t), called arbitrage bubble, generates a time dependent potential U(t) which changes the usual equilibrium dynamics of the traditional Black-Scholes model. The purpose of this article is to tackle the inverse problem, that is, is it possible to extract the time dependent potential U(t) and its associated bubble shape f (t), from the real empirical financial data? In order to give an answer to this question, the interacting Black-Scholes equation must be interpreted as a quantum Schrodinger equation with Hamiltonian operator H=H0+ U(t), where H0 is the equilibrium Black-Scholes Hamiltonian and U(t) is the interaction term. By using semi-classical considerations and the knowledge about the mispricing of the financial data, one can determinate an approximate functional form of the potential term U(t) and its associated bubble f (t), In all the studied cases, the non-equilibrium model performs a better estimation of the real data than the usual equilibrium model. It is expected that this new and simple methodology could help to improve option pricing estimations.

Numerical Solution Of Black-scholes Equation Using Bernstein Multi-scaling Functions

A numerical method for solving Black-Scholes equation is presented. The method is based upon Bernstein multi-scaling basis approximations. The properties of Bernstein multi-scaling functions are first presented. These properties together with the forward Euler and Ritz-Galerkin method are then utilized to reduce the Black-Scholes equation to the solution of algebraic equations. Illustrative example is included to demonstrate the validity and applicability of the new technique.

A Modified Explicit Method For The Black-scholes Equation With Positivity Preserving Property

A Modified Explicit Method For The Black-scholes Equation With Positivity Preserving Property