Lie-algebraic approach for pricing moving barrier options with time-dependent parameters

In this paper we apply the Lie-algebraic technique for the valuation of moving barrier options with time-dependent parameters. The value of the underlying asset is assumed to follow the constant elasticity of variance (CEV) process. By exploiting the dynamical symmetry of the pricing partial differential equations, the new approach enables us to derive the analytical kernels of the pricing formulae straightforwardly, and thus provides an efficient way for computing the prices of the moving barrier options. The method is also able to provide tight upper and lower bounds for the exact prices of CEV barrier options with fixed barriers. In view of the CEV model being empirically considered to be a better candidate in equity option pricing than the traditional Black-Scholes model, our new approach could facilitate more efficient comparative pricing and precise risk management in equity derivatives with barriers by incorporating term-structures of interest rates, volatility and dividend into the CEV option valuation model.     

连续支付红利的美式障碍期权定价的数值方法

本文利用显式差分格式为连续支付红利的向下触销型美式障碍期权定价.由于障碍的影响,定价模型的边值条件含有间断,故把结点设在障碍水平上,并在障碍附近的区域内运用局部网格加密技术,这样就可以得到较精确的期权价格.本文给出数值算例,验证算法的有效性,并分析障碍对期权价格的影响.     

Numerical valuation of discrete double barrier options

In the present paper we explore the problem for pricing discrete barrier options utilizing the Black-Scholes model for the random movement of the asset price. We postulate the problem as a path integral calculation by choosing approach that is similar to the quadrature method. Thus, the problem is reduced to the estimation of a multi-dimensional integral whose dimension corresponds to the number of the monitoring dates.We propose a fast and accurate numerical algorithm for its valuation. Our results for pricing discretely monitored one and double barrier options are in agreement with those obtained by other numerical and analytical methods in Finance and literature. A desired level of accuracy is very fast achieved for values of the underlying asset close to the strike price or the barriers.The method has a simple computer implementation and it permits observing the entire life of the option.     

Monotone iterative methods for the adaptive finite element solution of semiconductor equations

Picard, Gauss–Seidel, and Jacobi monotone iterative methods are presented and analyzed for the adaptive finite element solution of semiconductor equations in terms of the Slotboom variables. The adaptive meshes are generated by the 1-irregular mesh refinement scheme. Based on these unstructured meshes and a corresponding modification of the Scharfetter–Gummel discretization scheme, it is shown that the resulting finite element stiffness matrix is an M-matrix which together with the Shockley–Read–Hall model for the generation–recombination rate leads to an existence–uniqueness–comparison theorem with simple upper and lower solutions as initial iterates. Numerical results of simulations on a MOSFET device model are given to illustrate the accuracy and efficiency of the adaptive and monotone properties of the present methods.     

Error analysis of staggered finite difference finite volume schemes on unstructured meshes

This work combines the consistency in lower‐order differential operators with external approximations of functional spaces to obtain error estimates for finite difference finite volume schemes on unstructured nonuniform meshes. This combined approach is first applied to a one‐dimensional elliptic boundary value problem on nonuniform meshes, and a first‐order convergence rate is obtained, which agrees with the results previously reported. The approach is also applied to the staggered Marker‐and‐Cell scheme for the two‐dimensional incompressible Stokes problem on unstructured meshes. A first‐order convergence rate is obtained, which improves over a previously reported result in that it also holds on unstructured meshes. For both problems considered in this work, the convergence rate is one order higher on meshes satisfying special requirements. © 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 33: 1159–1182, 2017     

Numerical optimization methods for packing equal orthogonally oriented ellipses in a rectangular domain

Linear models are constructed for the numerical solution of the problem of packing the maximum possible number of equal ellipses of given size in a rectangular domain R. It is shown that the l _p metric can be used to determine the conditions under which ellipses with mutually orthogonal major axes (orthogonally oriented ellipses) do not intersect. In R a grid is constructed whose nodes generate a finite set T of points. It is assumed that the centers of the ellipses can be placed only at some points of T. The cases are considered when the major axes of all the ellipses are parallel to the x or y axis or the major axes of some of the ellipses are parallel to the x axis and the others, to the y axis. The problems of packing equal ellipses with centers in T are reduced to integer linear programming problems. A heuristic algorithm based on the linear models is proposed for solving the ellipse packing problems. Numerical results are presented that demonstrate the effectiveness of this approach.     

Hamilton-Jacobi方程的单调有限元格式

A monotone finite element scheme is obtained by applying the finite element method to the viscosity equation of the Hamilton-Jacobi equation on unstructured meshes. Under some constraints, we show that this scheme is monotone and its numerical solution converges to the viscosity solution of the Hamilton-Jacobi equa-tion. Numerical examples test the stability and the convergence of this scheme.     

THE LOWER APPROXIMATION OF EIGENVALUE BY LUMPED MASS FINITE ELEMENT METHOD

In the present paper, we investigate properties of lumped mass finite element method (LFEM hereinafter) eigenvalues of elliptic problems. We propose an equivalent formulation of LFEM and prove that LFEM eigenvalues are smaller than the standard finite element method (SFEM hereinafter) eigenvalues. It is shown, for model eigenvalue problems with uniform meshes, that LFEM eigenvalues are not greater than exact solutions and that they are increasing functions of the number of elements of the triangulation, and numerical examples show that this result equally holds for general problems.     

Hedging with a correlated asset: Solution of a nonlinear pricing PDE

Hedging a contingent claim with an asset which is not perfectly correlated with the underlying asset results in unhedgeable residual risk. Even if the residual risk is considered diversifiable, the option writer is faced with the problem of uncertainty in the estimation of the drift rates of the underlying and the hedging instrument. If the residual risk is not considered diversifiable, then this risk can be priced using an actuarial standard deviation principle in infinitesimal time. In both cases, these models result in the same nonlinear partial differential equation (PDE). A fully implicit, monotone discretization method is developed for solution of this pricing PDE. This method is shown to converge to the viscosity solution. Certain grid conditions are required to guarantee monotonicity. An algorithm is derived which, given an initial grid, inserts a finite number of nodes in the grid to ensure that the monotonicity condition is satisfied. At each timestep, the nonlinear discretized algebraic equations are solved using an iterative algorithm, which is shown to be globally convergent. Monte Carlo hedging examples are given to illustrate the profit and loss distribution at the expiry of the option.     

Simulation of elastic wave propagation in geological media: Intercomparison of three numerical methods

For wave propagation in heterogeneous media, we compare numerical results produced by grid-characteristic methods on structured rectangular and unstructured triangular meshes and by a discontinuous Galerkin method on unstructured triangular meshes as applied to the linear system of elasticity equations in the context of direct seismic exploration with an anticlinal trap model. It is shown that the resulting synthetic seismograms are in reasonable quantitative agreement. The grid-characteristic method on structured meshes requires more nodes for approximating curved boundaries, but it has a higher computation speed, which makes it preferable for the given class of problems.     

FINITE VOLUME NUMERICAL ANALYSIS FOR PARABOLIC EQUATION WITH ROBIN BOUNDARY CONDITION

In this paper, finite volume method on unstructured meshes is studied for a parabolic convection-diffusion problem on an open bounded set of R^d (d = 2 or 3) with Robin boundary condition. Upwinding approximations are adapted to treat both the convection term and Robin boundary condition. By directly getting start from the formulation of the finite volume scheme, numerical analysis is done. By using several discrete functional analysis techniques such as summation by parts, discrete norm inequality, et al, the stability and error estimates on the approximate solution are established, existence and uniqueness of the approximate solution and the 1st order temporal norm and L^2 and H^1 spacial norm convergence properties are obtained.     

Multi‐asset barrier options and occupation time derivatives

A general framework is formulated to price various forms of European style multi‐asset barrier options and occupation time derivatives with one state variable having the barrier feature. Based on the lognormal assumption of asset price processes, the splitting direction technique is developed for deriving the joint density functions of multi‐variate terminal asset prices with provision for single or double barriers on one of the state variables. A systematic procedure is illustrated whereby multi‐asset option price formulas can be deduced in a systematic manner as extensions from those of their one‐asset counterparts. The formulation has been applied successfully to derive the analytic price formulas of multi‐asset options with external two‐sided barriers and sequential barriers, multi‐asset step options and delayed barrier options. The successful numerical implementation of these price formulas is demonstrated.     

Free surface Nvier-Stokes flows with simultaneous heat transfer and solidification/melting

A mathematical model to analyse some key aspects of the metal cast process is described involving the filling of the mould by liquid metal and simultaneously, undergoing both cooling and solidification (re-melting) phase change. A computational solution procedure based upon a finite volume discretisation approach, on both structured and unstructured meshes, is described. The overall flow solution procedure is based on the pressure correction algorithm SIMPLE suitably adapted to: (a) solve for the free surface with minimal smearing by the SEA algorithm, and (b) solve for the solidification/melting phase change using an enthalpy conservation algorithm developed by Voller, but with its root in the work of Crank many years ago.Dedicated to Professor J. Crank on the occasion of his 80th birthday     

Risk-neutral valuation of power barrier options

Barrier options are standard exotic options traded in the financial market. These instruments are different from the vanilla options as the payoff of the option depends on whether the underlying asset price reaches a predetermined barrier level, during the life of the option. In this work, we extend the vanilla call barrier options to power call barrier options where the underlying asset price is raised to a constant power, within the standard Black–Scholes framework. It is demonstrated that the pricing of the power barrier options can be obtained from standard barrier options by a transformation which involves the power contract and a adjusted barrier. Numerical results are considered.     

Automatic parallel generation of tetrahedral grids by using a domain decomposition approach

An algorithm for the automatic parallel generation of three-dimensional unstructured grids based on geometric domain decomposition is proposed. A software package based on this algorithm is described. Examples of generating meshes for some application problems on a multiprocessor computer are presented. It is shown that the parallel algorithm can significantly (by a factor of several tens) reduce the mesh generation time. Moreover, it can easily generate meshes with as many as 5 × 10⁷ elements, which can hardly be generated sequentially. Issues concerning the speedup and the improvement of the efficiency of the computations and of the quality of the resulting meshes are discussed.     

A volume‐consistent discrete formulation of aggregation population balance equations

In this paper, a new finite volume scheme for the numerical solution of the pure aggregation population balance equation, or Smoluchowski equation, on non‐uniform meshes is derived. The main feature of the new method is its simple mathematical structure and high accuracy with respect to the number density distribution as well as its moments. The new method is compared with the existing schemes given by Filbet and Laurençot (SIAM J. Sci. Comput., 25 (2004), pp. 2004–2028) and Forestier and Mancini (SIAM J. Sci. Comput., 34 (2012), pp. B840–B860) for selected benchmark problems. It is shown that the new scheme preserves all the advantages of a conventional finite volume scheme and predicts higher‐order moments as well as number density distribution with high accuracy. Copyright © 2015 John Wiley & Sons, Ltd.     

ROBUSTNESS OF AN UPWIND FINITE DIFFERENCE SCHEME FOR SEMILINEAR CONVECTION-DIFFUSION PROBLEMS WITH BOUNDARY TURNING POINTS

We consider a singularly perturbed semilinear convection-diffusion problem with a boundary layer of attractive turning-point type. It is shown that its solution can be decomposed into a regular solution component and a layer component. This decomposi-tion is used to analyse the convergence of an upwinded finite difference scheme on Shishkin meshes.     

Multigrid for American option pricing with stochastic volatility

The paper describes an implicit finite difference approach to the pricing of American options on assets with a stochastic volatility. A multigrid procedure is described for the fast iterative solution of the discrete linear complementarity problems that result. The accuracy and performance of this approach is improved considerably by a strike-price related analytic transformation of asset prices and adaptive time-stepping.     

Perturbational Results for Diffraction of Water-waves by Nearly-vertical Barriers

An exact solution (Ursell, 1947) is known for the diffractionof surface waves by a thin, flat, two-dimensional barrier, extendingdownwards from the surface to a fixed submerged point. We considerperturbations to this result caused by changing the shape ofthe barrier slightly. By using an integral equation formulation,first-order corrections to the reflection and transmission coefficientsare given in terms of two integrals of the shape. The methodcan be used both for thin, shaped barriers and for symmetric,slender shapes. A similar technique is used for barriers whichoscillate with the frequency of the incoming wave. A comparisonis made between this theory and some rival methods, and it isfound to be superior in many respects.