A computationally efficient numerical approach for multi-asset option pricing

Numerical solution of the multi-dimensional partial differential equations arising in the modelling of option pricing is a challenging problem. Mesh-free methods using global radial basis functions (RBFs) have been successfully applied to several types of such problems. However, due to the dense linear systems that need to be solved, the computational cost grows rapidly with dimension. In this paper, we propose a numerical scheme to solve the Black–Scholes equation for valuation of options prices on several underlying assets. We use the derivatives of linear combinations of multiquadric RBFs to approximate the spatial derivatives and a straightforward finite difference to approximate the time derivative. The advantages of the scheme are that it does not require solving a full matrix at each time step and the algorithm is easy to implement. The accuracy of our scheme is demonstrated on a test problem.     

A numerical method to price discrete double Barrier options under a constant elasticity of variance model with jump diffusion

We develop a numerical method to price discrete barrier options on an underlying described by the constant elasticity of variance model with jump-diffusion (CEVJD). In particular, the partial integro differential equation associated to this model is discretized in time using an operator splitting scheme whose accuracy is enhanced by repeated Richardson extrapolation. Such an approach allows us to approximate the differential terms and the jump integral by means of two different numerical techniques. Precisely, the spatial derivatives, which exist only in the weak sense, are discretized using a finite element method based on piecewise quadratic polynomials, whereas the jump integral is directly collocated at the mesh points, so that it can be easily evaluated by Simpson numerical quadrature. As shown by extensive numerical simulation, the proposed approach is very efficient from the computational standpoint, and performs significantly better than the finite difference scheme developed in Wade et al. [On smoothing of the Crank–Nicolson scheme and higher order schemes for pricing barrier options, J. Comput. Appl. Math. 204 (2007), pp. 144–158].     

Finite Difference Method for the Black–Scholes Equation Without Boundary Conditions

We present an accurate and efficient finite difference method for solving the Black–Scholes (BS) equation without boundary conditions. The BS equation is a backward parabolic partial differential equation for financial option pricing and hedging. When we solve the BS equation numerically, we typically need an artificial far-field boundary condition such as the Dirichlet, Neumann, linearity, or partial differential equation boundary condition. However, in this paper, we propose an explicit finite difference scheme which does not use a far-field boundary condition to solve the BS equation numerically. The main idea of the proposed method is that we reduce one or two computational grid points and only compute the updated numerical solution on that new grid points at each time step. By using this approach, we do not need a boundary condition. This procedure works because option pricing and computation of the Greeks use the values at a couple of grid points neighboring an interesting spot. To demonstrate the efficiency and accuracy of the new algorithm, we perform the numerical experiments such as pricing and computation of the Greeks of the vanilla call, cash-or-nothing, power, and powered options. The computational results show excellent agreement with analytical solutions.     

A meshless method for Asian style options pricing under the Merton jump-diffusion model

In this paper, we consider the partial integro-differential equation arising when a stock follows a Poisson distributed jump process, for the pricing of Asian options. We make use of the meshless radial basis functions with differential quadrature for approximating the spatial derivatives and demonstrate that the algorithm performs effectively well as compared to the commonly employed finite difference approximations. We also employ Strang splitting with the exponential time integration technique to improve temporal efficiency. Throughout the numerical experiments covered in the paper, we show how the proposed scheme can be efficiently employed for the pricing of American style Asian options under both the Black–Scholes and the Merton jump-diffusion models.     

微分方程初值问题的GDQR解法

本文用最近由Wu提出的一种数值方法-GDQR(GeneralizedDifferentialQuadra-tureRule)对工程和科学技术中常遇到的2—4阶微分方程初值问题进行了求解.部分结果与精确解或龙格-库塔方法所得结果作了对比,表明GDQR在解决常微分方程初值问题时简单方便有效.     

Normalized particle swarm optimization for complex chooser option pricing on graphics processing unit

An option is a financial instrument that derives its value from an underlying asset, for example, a stock. There are a wide range of options traded today from simple and plain (European options) to exotic (chooser options) that are very difficult to evaluate. Both buyers and sellers continue to look for efficient algorithms and faster technology to price options for better profit and to beat the competition. There are mathematical models like the Black–Scholes–Merton model used to price options approximately for simple and plain options in the form of closed form solution. However, the market is flooded with various styles of options, which are difficult to price, and hence there are many numerical techniques proposed for pricing. The computational cost for pricing complex options using these numerical techniques is exorbitant for reasonable accuracy in pricing results. Heuristic approaches such as particle swarm optimization (PSO) have been proposed for option pricing, which provide same or better results for simple options than that of numerical techniques at much less computational cost. In this study, we first map the PSO parameters to option pricing parameters. Analyzing the characteristics of PSO and option pricing, we propose a strategy to normalize some of the parameters, which helps in better understanding of the sensitivity of these and other parameters on option pricing results. We then avail of the inherent concurrency of the PSO algorithm while searching for an optimum solution, and design an algorithm for implementation on a modern state-of-the-art graphics processor unit (GPU). Our implementation makes use of the architectural features of GPU in accelerating the computing performance while maintaining accuracy on the pricing results.     

A numerical algorithm for computation modelling of 3D nonlinear wave equations based on exponential modified cubic B-spline differential quadrature method

In this paper, the authors proposed a method based on exponential modified cubic B-spline differential quadrature method (Expo-MCB-DQM) for the numerical simulation of three dimensional (3D) nonlinear wave equations subject to appropriate initial and boundary conditions. This work extends the idea of Tamsir et al. [An algorithm based on exponential modified cubic B-spline differential quadrature method for nonlinear Burgers’ equation, Appl. Math. Comput. 290 (2016), pp. 111–124] for 3D nonlinear wave type problems. Expo-MCB-DQM transforms the 3D nonlinear wave equation into a system of ordinary differential equations (ODEs). To solve the resulting system of ODEs, an optimal five stage and fourth-order strong stability preserving Runge–Kutta (SSP-RK54) scheme is used. Stability analysis of the proposed method is also discussed and found that the method is conditionally stable. Four test problems are considered in order to demonstrate the accuracy and efficiency of the algorithm.     

微分求积区域分裂法求解二维奇异摄动问题

本文应用微分求积法结合区域分裂法求解二维奇异摄动问题,数值实验表明,该方法简单易行,计算量少,精确度高.并且微分求积法结合区域分裂法把大型计算化成若干小型计算,避免了微分求积法导出的矩阵不是稀疏矩阵对大型计算不利的缺点.     

Fuzzy pricing of American options on stocks with known dividends and its algorithm

The path‐dependent property of American options leads to the complexity of its pricing. Based on the analysis of American options' characteristics and the influence of the stock dividend, the American call option fuzzy pricing method is discussed in this paper. Under the assumption that the price of stock, discount rate, the volatility, and interest rate are all fuzzy numbers, the fuzzy pricing formula of American option is proposed by using the Black–Scholes pricing model. Then the interpolation search algorithm is designed to solve the proposed pricing model. Finally, the validity and accuracy of this model and its algorithm have to be tested with some numerical examples. © 2010 Wiley Periodicals, Inc.     

A Numerical Method to Approximate Multi-Asset Option Pricing Under Exponential Lévy Model

In this paper, a modification of the original global radial basis functions-based differential quadrature (RBF-DQ) method is set forth and analyzed. The improved RBF-DQ method is applicable to the numerical approximation of solutions of a wide range of partial differential equations with mixed derivative terms. However, it appears to be considerably faster than the original method. In support of this contention, the multi-asset option pricing problems under exponential Lévy framework have been solved numerically by using the proposed method and compared with results obtained via the original RBF-DQ method. For accuracy achieved versus work expended, the improved method performs better.     

Radial basis function based level set interpolation and evolution for deformable modelling

We present a study in level set representation and evolution using radial basis functions (RBFs) for active contour and active surface models. It builds on recent works by others who introduced RBFs into level sets for structural topology optimisation. Here, we introduce the concept into deformable models and present a new level set formulation able to handle more complex topological changes, in particular perturbation away from the evolving front. In the conventional level set technique, the initial active contour/surface is implicitly represented by a signed distance function and periodically re-initialised to maintain numerical stability. We interpolate the initial distance function using RBFs on a much coarser grid, which provides great potential in modelling in high dimensional space. Its deformation is considered as an updating of the RBF interpolants, an ordinary differential equation (ODE) problem, instead of a partial differential equation (PDE) problem, and hence it becomes much easier to solve. Re-initialisation is found no longer necessary, in contrast to conventional finite difference method (FDM) based level set approaches. The proposed level set updating scheme is efficient and does not suffer from self-flattening while evolving, hence it avoids large numerical errors. Further, more complex topological changes are readily achievable and the initial contour or surface can be placed arbitrarily in the image. These properties are extensively demonstrated on both synthetic and real 2D and 3D data. We also present a novel active contour model, implemented with this level set scheme, based on multiscale learning and fusion of image primitives from vector-valued data, e.g. colour images, without channel separation or decomposition.     

Fast and accurate pricing of discretely monitored barrier options by numerical path integration

Barrier options are financial derivative contracts that are activated or deactivated according to the crossing of specified barriers by an underlying asset price. Exact models for pricing barrier options assume continuous monitoring of the underlying dynamics, usually a stock price. Barrier options in traded markets, however, nearly always assume less frequent observation, e.g. daily or weekly. These situations require approximate solutions to the pricing problem. We present a new approach to pricing such discretely monitored barrier options that may be applied in many realistic situations. In particular, we study daily monitored up-and-out call options of the European type with a single underlying stock. The approach is based on numerical approximation of the transition probability density associated with the stochastic differential equation describing the stock price dynamics, and provides accurate results in less than one second whenever a contract expires in a year or less. The flexibility of the method permits more complex underlying dynamics than the Black and Scholes paradigm, and its relative simplicity renders it quite easy to implement.     

基于高斯过程与批量汤普森抽样的动态定价策略

考虑短期内需求不确定情况下同类型产品的定价策略研究,引入高斯过程进行需求函数的学习,利用批量汤普森算法建立基于探索-利用的两阶段学习和决策过程的定价模型。在利用提出的GP-PTS(Gaussian processparallel Thompson sampling)算法完成数值实验和某平台出行的真实数据应用后得出的结果表明：算法的精准度取决于特征是否完备,若给定一个先验且产品特征完备时,基于GP-PTS算法模拟出来的价格会取得比目前平台价格策略更好的收益,为企业在短期内进行定价决策提供良好借鉴。     

The parameterized level set method for structural topology optimization with shape sensitivity constraint factor

In recent years, the parameterized level set method (PLSM) has attracted widespread attention for its good stability, high efficiency and the smooth result of topology optimization compared with the conventional level set method. In the PLSM, the radial basis functions (RBFs) are often used to perform interpolation fitting for the conventional level set equation, thereby transforming the iteratively updating partial differential equation (PDE) into ordinary differential equations (ODEs). Hence, the RBFs play a key role in improving efficiency, accuracy and stability of the numerical computation in the PLSM for structural topology optimization, which can describe the structural topology and its change in the optimization process. In particular, the compactly supported radial basis function (CS-RBF) has been widely used in the PLSM for structural topology optimization because it enjoys considerable advantages. In this work, based on the CS-RBF, we propose a PLSM for structural topology optimization by adding the shape sensitivity constraint factor to control the step length in the iterations while updating the design variables with the method of moving asymptote (MMA). With the shape sensitivity constraint factor, the updating step length is changeable and controllable in the iterative process of MMA algorithm so as to increase the optimization speed. Therefore, the efficiency and stability of structural topology optimization can be improved by this method. The feasibility and effectiveness of this method are demonstrated by several typical numerical examples involving topology optimization of single-material and multi-material structures.

     

A Robust Numerical Scheme For Pricing American Options Under Regime Switching Based On Penalty Method

This paper is devoted to develop a robust numerical method to solve a system of complementarity problems arising from pricing American options under regime switching. Based on a penalty method, the system of complementarity problems are approximated by a set of coupled nonlinear partial differential equations (PDEs). We then introduce a fitted finite volume method for the spatial discretization along with a fully implicit time stepping scheme for the PDEs, which results in a system of nonlinear algebraic equations. We show that this scheme is consistent, stable and monotone, hence convergent. To solve the system of nonlinear equations effectively, an iterative solution method is established. The convergence of the solution method is shown. Numerical tests are performed to examine the convergence rate and verify the effectiveness and robustness of the new numerical scheme.     

Equity warrants pricing model under Fractional Brownian motion and an empirical study

In this paper, we construct equity warrants pricing model under Fractional Brownian motion, deduce the European options pricing formula with a simple method, then propose the warrants pricing formula, and extend it to cover equity warrants on a stock providing dividends. Finally, taking Changdian warrant in Chinese stock market as an example, we illustrate that the results based on the new warrants pricing formula is more accuracy than the classical results based on traditional pricing model.     

Adaptive residual subsampling methods for radial basis function interpolation and collocation problems

We construct a new adaptive algorithm for radial basis functions (RBFs) method applied to interpolation, boundary-value, and initial-boundary-value problems with localized features. Nodes can be added and removed based on residuals evaluated at a finer point set. We also adapt the shape parameters of RBFs based on the node spacings to prevent the growth of the conditioning of the interpolation matrix. The performance of the method is shown in numerical examples in one and two space dimensions with nontrivial domains.     

Numerical simulation of two-dimensional sine-Gordon solitons by differential quadrature method

During the past few decades, the idea of using differential quadrature methods for numerical solutions of partial differential equations (PDEs) has received much attention throughout the scientific community. In this article, we proposed a numerical technique based on polynomial differential quadrature method (PDQM) to find the numerical solutions of two-dimensional sine-Gordon equation with Neumann boundary conditions. The PDQM reduced the problem into a system of second-order linear differential equations. Then, the obtained system is changed into a system of ordinary differential equations and lastly, RK4 method is used to solve the obtained system. Numerical results are obtained for various cases involving line and ring solitons. The numerical results are found to be in good agreement with the exact solutions and the numerical solutions that exist in literature. It is shown that the technique is easy to apply for multidimensional problems.     

Adaptive finite differences and IMEX time-stepping to price options under Bates model

In this paper, we consider numerical pricing of European and American options under the Bates model, a model which gives rise to a partial-integro differential equation. This equation is discretized in space using adaptive finite differences while an IMEX scheme is employed in time. The sparse linear systems of equations in each time-step are solved using an LU-decomposition and an operator splitting technique is employed for the linear complementarity problems arising for American options. The integral part of the equation is treated explicitly in time which means that we have to perform matrix-vector multiplications each time-step with a matrix with dense blocks. These multiplications are accomplished through fast Fourier transforms. The great performance of the method is demonstrated through numerical experiments.     

A comparative study of numerical integration based on Haar wavelets and hybrid functions

A quadrature rule based on uniform Haar wavelets and hybrid functions is proposed to find approximate values of definite integrals. The wavelet-based algorithm can be easily extended to find numerical approximations for double, triple and improper integrals. The main advantage of this method is its efficiency and simple applicability. Error estimates of the proposed method alongside numerical examples are given to test the convergence and accuracy of the method.