Black-Scholes Partial Differential Equation Research Articles

AN OPERATOR SPLITTING METHOD FOR PRICING THE ELS OPTION

This paper presents the numerical valuation of the two-asset step-down equitylinked securities (ELS) option by using the operator-splitting method (OSM). The ELS is one of the most popular financial options. The value of ELS option can be modeled by a modified Black-Scholes partial differential equation. However, regardless of whether there is a closedform solution, it is difficult and not efficient to evaluate the solution because such a solution would be represented by multiple integrations. Thus, a fast and accurate numerical algorithm is needed to value the price of the ELS option. This paper uses a finite difference method to discretize the governing equation and applies the OSM to solve the resulting discrete equations. The OSM is very robust and accurate in evaluating finite difference discretizations. We provide a detailed numerical algorithm and computational results showing the performance of the method for two underlying asset option pricing problems such as cash-or-nothing and stepdown ELS. Final option value of two-asset step-down ELS is obtained by a weighted average value using probability which is estimated by performing a MC simulation.

Efficient implicit scheme with positivity preserving and smoothing properties

Using classical finite difference schemes often generates numerical drawbacks such as spurious oscillations in the solution of the famous Black–Scholes partial differential equation. We analyze the fully implicit scheme, frequently used numerical method in Finance, that in the presence of discontinuous payoff and low volatility arises spurious oscillations. We propose a modification of this scheme so that we guarantee a smooth numerical solution, free of spurious oscillations and satisfies the positivity requirement, as is demanded for the financial solution of the Black–Scholes equation. The method is used, within the strategy suggested by Rannacher, only in few initial time steps in the presence of discontinuous initial conditions. As a consequence, although the method is low order accurate, it returns a spurious oscillations free solution. Next, starting from the smooth initial condition obtained, any other family of arbitrary higher order schemes may be used.

A new exact solution for pricing European options in a two-state regime-switching economy

In this study, we derive a new exact solution for pricing European options in a two-state regime-switching economy. Two coupled Black–Scholes partial differential equations (PDEs) under the regime switching are solved using the Fourier Transform method. A key feature of the newly-derived solution is its simplicity in the form of a single integral with a real integrand, which leads to great computational efficiency in comparison with other closed-form solutions previously presented in the literature. Numerical examples are provided to demonstrate some interesting results obtained from our pricing formula.

Study of Black-Scholes Model and its Applications

The aim of this paper is to study the Black-Scholes option pricing model. We discuss some definitions and different derivations, which are useful for further development of Black-Scholes formula and Black-Scholes partial differential equation. As an application, we obtain the solution of the Black-Scholes equation and it is represented graphically by Maple software.

Numerical Solution for 2D European Option Pricing Using Quarter-Sweep Modified Gauss-Seidel Method

Problem statement: This study presents the numerical solution of two-dimensional European option pricing problem based on Quarter-Sweep Modified Gauss-Seidel (QSMGS) iterative method. In fact, the pricing of European option with two-underlying assets can be governed by two-dimensional Black-Scholes Partial Differential Equation (PDE). Approach: The PDE needs to be discretized by using full-, half- and quarter-sweep second-order Crank-Nicolson schemes to generate a system of linear equations. Then, the Modified Gauss-Seidel, a preconditioned iterative method is applied to solve the generated linear system. Results: In order to examine the effectiveness of QSMGS method, several numerical experiments of Full-Sweep Gauss-Seidel (FSGS), Half-Sweep Gauss-Seidel (HSGS) and Quarter-Sweep Gauss-Seidel (QSGS) methods are also included for comparison purpose. Thus, the numerical experiments show that the QSMGS iterative method is the fastest in computing as well as having the least number of iterations. In the error analysis, QSMGS method shows good and consistent results. Conclusion: Finally, it can be concluded that QSMGS method is superior in increasing the convergence rate.

Numerical solution of European call option with dividends and variable volatility

In this paper, the effect of strike price, interest rate, dividends and maturities on European call option with dividends is discussed. The volatility for the data of ONGC Ltd. listed in National Stock Exchange, India, during 03-01-2000 to 30-03-2009 is forecasted by GJR-GARCH method. The option price and Greeks are determined by solving modified Black–Scholes partial differential equation by adjusting forecasted volatility at each grid point of finite difference method. It is observed that call option premium decreases as strike price and dividend increases but it increases as rate of interest and time of maturities increases. Hence call option is more profitable for a long maturity, high interest rate and low dividend.

Convex regularization of local volatility models from option prices: Convergence analysis and rates

We study a convex regularization of the local volatility surface identification problem for the Black–Scholes partial differential equation from prices of European call options. This is a highly nonlinear ill-posed problem which in practice is subject to different noise levels associated to bid–ask spreads and sampling errors. We analyze, in appropriate function spaces, different properties of the parameter-to-solution map that assigns to a given volatility surface the corresponding option prices. Using such properties, we show stability and convergence of the regularized solutions in terms of the Bregman distance with respect to a class of convex regularization functionals when the noise level goes to zero. We improve convergence rates available in the literature for the volatility identification problem. Furthermore, in the present context, we relate convex regularization with the notion of exponential families in Statistics. Finally, we connect convex regularization functionals with convex risk measures through Fenchel conjugation. We do this by showing that if the source condition for the regularization functional is satisfied, then convex risk measures can be constructed.

A cubic B-spline collocation method for a numerical solution of the generalized Black–Scholes equation

In this paper, the uniform cubic B-spline collocation method is implemented to find the numerical solution of the generalized Black–Scholes partial differential equation. We use the horizontal method of lines to discretize the temporal variable and the spatial variable by means of a θ -method, θ ∈ [ 1 / 2 , 1 ] ( θ = 1 corresponds to the back-ward Euler method and θ = 1 / 2 corresponds to the Crank–Nicolson method), and a cubic B-spline collocation method on uniform meshes, respectively. The method corresponding to θ = 1 is shown to be unconditionally stable and first order accurate with respect to the time variable and second order accurate with respect to the space variable while the method corresponding to θ = 1 / 2 is shown to be unconditionally stable and second order accurate with respect to both the variables. Finally, the numerical examples demonstrate the stability and accuracy of the method.

Valuation of European continuous-installment options

This paper is concerned with the valuation of European continuous-installment options where the aim is to determine the initial premium given a constant installment payment plan. The distinctive feature of this pricing problem is the determination, along with the initial premium, of an optimal stopping boundary since the option holder has the right to stop making installment payments at any time before maturity. Given that the initial premium function of this option is governed by an inhomogeneous Black–Scholes partial differential equation, we can obtain two alternative characterizations of the European continuous-installment option pricing problem, for which no closed-form solution is available. First, we formulate the pricing problem as a free boundary problem and using the integral representation method, we derive integral expressions for both the initial premium and the optimal stopping boundary. Next, we use the linear complementarity formulation of the pricing problem for determining the initial premium and the early stopping curve implicitly with a finite difference scheme. Finally, the pricing problem is posed as an optimal stopping problem and then implemented by a Monte Carlo approach.

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.

BARRIER OPTION PRICING UNDER THE VASICEK MODEL OF THE SHORT RATE

In this study, assume that the stock price obeys the stochastic differential equation driven by mixed fractional Brownian motion, and the short rate follows the Vasicek model. Then, the Black-Scholes partial differential equation is held by using fractional Ito formula. Finally, the pricing formulae of the barrier option are obtained by partial differential equation theory. The results of Black-Scholes model are generalized.

Nonlinear Black–Scholes Equations in Finance: Associated Control Problems and Properties of Solutions

We study properties of solutions to fully nonlinear versions of the standard Black–Scholes partial differential equation. These equations have been introduced in financial mathematics in order to deal with illiquid markets or with stochastic volatility. We show that typical nonlinear Black–Scholes equations can be viewed as dynamic programming equation of an associated control problem. We establish existence and comparison results and show that the equation induces a convex risk measure on the set of all continuous terminal value claims. Moreover, we study the asymptotic behavior of solutions as market frictions get “large.” Finally, the pricing of individual contracts relative to a book of derivatives is discussed.

Reduced basis for vanilla and basket options

In [3] it was shown that by writing the solution of the Black-Scholes partial differential equation on a small set of basis functions the computing time can be dramatically reduced. In this study we explore the generalization of the technique to basket options.

Quarter-Sweep Improving Modified Gauss-Seidel Method for Pricing European Option

The aim of this paper is to examine the application of the Quarter-Sweep Improving Modified Gauss-Seidel (QSIMGS) method in evaluating European option which governed by Black-Scholes partial differential equation (PDE). Quarter-sweep Crank-Nicolson approach is applied to approximate the PDE. Then, the generated linear system is solved by using the IMGS method. Some numerical experiments for a family of Gauss-Seidel (GS) methods such as Gauss-Seidel, Modified Gauss-Seidel (MGS) and Improving Modified Gauss-Seidel (IMGS) methods are performed with each full-, half-, and quarter-sweep iterations. Thus, from the numerical results obtained, we can show that the QSIMGS method is the most effective method. Keywords: Quarter-Sweep Improving Modified Gauss-Seidel method; Black-Scholes PDE; Crank-Nicolson scheme.

A new predictor–corrector scheme for valuing American puts

In this paper, we present a new numerical scheme, based on the finite difference method, to solve American put option pricing problems. Upon applying a Landau transform or the so-called front-fixing technique [19] to the Black–Scholes partial differential equation, a predictor–corrector finite difference scheme is proposed to numerically solve the nonlinear differential system. Through the comparison with Zhu’s analytical solution [35], we shall demonstrate that the numerical results obtained from the new scheme converge well to the exact optimal exercise boundary and option values. The results of our numerical examples suggest that this approach can be used as an accurate and efficient method even for pricing other types of financial derivative with American-style exercise.

THE WIENER-HOPF TECHNIQUE AND DISCRETELY MONITORED PATH-DEPENDENT OPTION PRICING

Fusai, Abrahams, and Sgarra (2006) employed the Wiener-Hopf technique to obtain an exact analytic expression for discretely monitored barrier option prices as the solution to the Black-Scholes partial differential equation. The present work reformulates this in the language of random walks and extends it to price a variety of other discretely monitored path-dependent options. Analytic arguments familiar in the applied mathematics literature are used to obtain fluctuation identities. This includes casting the famous identities of Baxter and Spitzer in a form convenient to price barrier, first-touch, and hindsight options. Analyzing random walks killed by two absorbing barriers with a modified Wiener-Hopf technique yields a novel formula for double-barrier option prices. Continuum limits and continuity correction approximations are considered. Numerically, efficient results are obtained by implementing Pade approximation. A Gaussian Black-Scholes framework is used as a simple model to exemplify the techniques, but the analysis applies to Levy processes generally.

Stability of central finite difference schemes on non-uniform grids for the Black–Scholes equation

This paper deals with stability in the numerical solution of the Black–Scholes partial differential equation. We investigate the semi-discretization on non-uniform grids with central, second-order finite difference schemes. Our stability analysis concerns the important question of whether, for the obtained semi-discrete matrices A, the norm ‖ e t A ‖ of the matrix exponential of tA ( t ⩾ 0 ) can be bounded suitably. Even though the considered semi-discretization of the Black–Scholes equation is widely known in the literature, a rigorous stability analysis for non-uniform grids appears to be lacking. In most cases, the matrices A are non-normal and an analysis based solely on the eigenvalues of A does not provide adequate estimates. In the present paper, we prove rigorous useful upper bounds on ‖ e t A ‖ for general non-uniform grids. Here we consider scaled spectral norms as well as the maximum norm. Our theoretical estimates are illustrated by ample numerical experiments, and practical conclusions about the stability of the schemes on non-uniform grids are derived. The results in this paper can directly be used in obtaining stability results also for time discretization schemes and are significant to a variety of applications beyond the Black–Scholes model.

A numerical method for European Option Pricing with transaction costs nonlinear equation

This paper deals with the construction of a finite difference scheme and the numerical analysis of its solution for a nonlinear Black–Scholes partial differential equation modelling stock option pricing in the realistic case when transaction costs arising in the hedging of portfolios are taken into account. The analysed model is the Barles–Soner one for which an appropriate fully nonlinear numerical method has not still applied. After construction of the numerical solution, consistency and stability are studied and some illustrative examples are included.

A long time asymptotic behavior of the free boundary for an American put

In this paper we obtain a long time asymptotic behavior of the optimal exercise boundary for an American put option. This is done by analyzing an integral equation for the rescaled exercise boundary derived from the corresponding Black-Scholes partial differential equation with a free boundary.

Risk Minimizing Option Pricing in a Regime Switching Market

We study option pricing in a regime switching market where the risk free interest rate, growth rate and the volatility of a stock depends on a finite state Markov chain. Using a minimal martingale measure we show that the risk minimizing option price satisfies a system of Black–Scholes partial differential equations with weak coupling.