Valuation of American Continuous-Installment Options

We present three approaches to value American continuous-installment options written on assets without dividends or with continuous dividend yield. In an American continuous-installment option, the premium is paid continuously instead of up-front. At or before maturity, the holder may terminate payments by either exercising the option or stopping the option contract. Under the usual assumptions, we are able to construct an instantaneous riskless dynamic hedging portfolio and derive an inhomogeneous Black–Scholes partial differential equation for the initial value of this option. This key result allows us to derive valuation formulas for American continuous-installment options using the integral representation method and consequently to obtain closed-form formulas by approximating the optimal stopping and exercise boundaries as multipiece exponential functions. This process is compared to the finite difference method to solve the inhomogeneous Black–Scholes PDE and a Monte Carlo approach.     

Parameter estimation approach to the free boundary for the pricing of an american call option

In this paper, we consider a free boundary problem which arises in the pricing of an American call option. The free boundary represents the optimal exercise price as a function of time before a maturity date. We are developing a parameter estimation technique to obtain both the optimal exercise curve of an American call option and its price. For the numerical solution of a forward problem, a time marching finite element method is adopted. Numerical experiment shows the convergence property of the approximation scheme.     

Augmented Lagrangian method applied to American option pricing

The American option pricing problem is originally formulated as a stochastic optimal stopping time problem. It is also equivalent to a variational inequality problem or a complementarity problem involving the Black-Scholes partial differential operator. In this paper, the corresponding variational inequality problem is discretized by using a fitted finite volume method. Based on the discretized form, an algorithm is developed by applying augmented Lagrangian method (ALM) to the valuation of the American option. Convergence properties of ALM are considered. By empirical numerical experiments, we conclude that ALM is more effective than penalty method and Lagrangian method, and comparable with the projected successive overrelaxation method (PSOR). Furthermore, numerical results show that ALM is more robust in terms of computation time under changes in market parameters: interest rate and volatility.     

A new numerical method on American option pricing

Mathematically, the Black-Scholes model of American option pricing is a free boundary problem of partial differential equation. It is well known that this model is a nonlinear problem, and it has no closed form solution. We can only obtain an approximate solution by numerical method, but the precision and stability are hard to control, because the singularity at the exercise boundary near expiration date has a great effect on precision and stability for numerical method. We propose a new numerical method, FDA method, to solve the American option pricing problem, which combines advantages the Semi-Analytical Method and the Front-Fixed Difference Method. Using the FDA method overcomes the difficulty resulting from the singularity at the terminal of optimal exercise boundary. A large amount of calculation shows that the FDA method is more accurate and stable than other numerical methods.     

Pricing American options under jump-diffusion models using local weak form meshless techniques

Recently, several numerical methods have been proposed for pricing options under jump-diffusion models but very few studies have been conducted using meshless methods [R. Chan and S. Hubbert, A numerical study of radial basis function based methods for options pricing under the one dimension jump-diffusion model, Tech. Rep., 2010; A. Saib, D. Tangman, and M. Bhuruth, A new radial basis functions method for pricing American options under Merton's jump-diffusion model, Int. J. Comput. Math. 89 (2012), pp. 1164–1185]. Indeed, only a strong form of meshless methods have been employed in these lectures. We propose the local weak form meshless methods for option pricing under Merton and Kou jump-diffusion models. Predominantly in this work we will focus on meshless local Petrov–Galerkin, local boundary integral equation methods based on moving least square approximation and local radial point interpolation based on Wendland's compactly supported radial basis functions. The key feature of this paper is applying a Richardson extrapolation technique on American option which is a free boundary problem to obtain a fixed boundary problem. Also the implicit–explicit time stepping scheme is employed for the time derivative which allows us to obtain a spars and banded linear system of equations. Numerical experiments are presented showing that the presented approaches are extremely accurate and fast.     

Real options pricing by the finite element method

Real option pricing problems in investment project evaluation are mostly solved by the simulation-based methods, the lattice methods and by the finite difference method (FDM). Only a few applications of the finite element method (FEM) to these problems have been reported in the literature; although it seems to be an alternative tool for pricing real options.Unlike the existing finite element-based papers, in this paper we use residual formulation and provide a detailed scheme for practical implementations. The FEM is introduced and developed as a numerical method for real options pricing problems. First of all, a partial differential equation (pde) model is defined, then the problem’s domain is discretized by finite elements. The weak formulation of the pde is then obtained, and finally the solution to the real option pricing problem is found by solving an algebraic system. For benchmarking purposes, the FEM is applied to known investment and abandonment option problems found in the literature and the results are compared with those of some traditional methods. These results show a good performance of the FEM and its superiority over the FDM in terms of convergence, and over the simulation-based methods in terms of the optimal exercise policy.     

Mapping of option pricing algorithms onto heterogeneous many-core architectures

The rapid development of technologies and applications in recent years poses high demands and challenges for high-performance computing. Because of their competitive performance/price ratio, heterogeneous many-core architectures are widely used in high-performance computing areas. GPU and Xeon Phi are two popular general-purpose many-core accelerators. In this paper, we demonstrate how heterogeneous many-core architectures, powered by multi-core CPUs, CUDA-enabled GPUs and Xeon Phis can be used as an efficient computational platform to accelerate popular option pricing algorithms. In order to make full use of the compute power of this architecture, we have used a hybrid computing model which consists of two types of data parallelism: worker level and device level. The worker level data parallelism uses a distributed computing infrastructure for task distribution, while the device level data parallelism uses both the multi-core CPUs and many-core accelerators for fast option pricing calculation. Experiments show that our implementations achieve good performance and scalability on this architecture and also outperform other state-of-the-art GPU-based solutions for Monte Carlo European/American option pricing and BSDE European option pricing.     

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 new radial basis functions method for pricing American options under Merton's jump-diffusion model

A new radial basis functions (RBFs) algorithm for pricing financial options under Merton's jump-diffusion model is described. The method is based on a differential quadrature approach, that allows the implementation of the boundary conditions in an efficient way. The semi-discrete equations obtained after approximation of the spatial derivatives, using RBFs based on differential quadrature are solved, using an exponential time integration scheme and we provide several numerical tests which show the superiority of this method over the popular Crank–Nicolson method. Various numerical results for the pricing of European, American and barrier options are given to illustrate the efficiency and accuracy of this new algorithm. We also show that the option Greeks such as the Delta and Gamma sensitivity measures are efficiently computed to high accuracy.     

Optimal stopping and control with two kinds of boundary conditions: application to dynamic routeing in networks

A stochastic optimal stopping and optimal control problem concerning the dynamic routeing of a randomly perturbed flow is considered. Equations for the solution of the optimal stopping and control problem are given. These equations lead to the solution of an elliptic problem with a given condition on a ‘ free boundary’ within a given domain D and a condition given on the boundary ? D of D. An example is given and a numerical study is conducted on it     

Impulsive Synchronization of Stochastic Neural Networks via Controlling Partial States

In this paper the perceptron neural networks are applied to approximate the solution of fractional optimal control problems. The necessary (and also sufficient in most cases) optimality conditions are stated in a form of fractional two-point boundary value problem. Then this problem is converted to a Volterra integral equation. By using perceptron neural network’s ability in approximating a nonlinear function, first we propose approximating functions to estimate control, state and co-state functions which they satisfy the initial or boundary conditions. The approximating functions contain neural network with unknown weights. Using an optimization approach, the weights are adjusted such that the approximating functions satisfy the optimality conditions of fractional optimal control problem. Numerical results illustrate the advantages of the method.     

A closed-form approximation for the fractional Black–Scholes model with transaction costs

In this paper, we investigate option valuation problems under the fractional Black–Scholes model. The aim is to propose a pricing formula for the European option with transaction costs, where the costs structure contains fixed costs, a cost propositional to the volume traded, and a cost proportional to the value traded. Precisely, we provide an approximate solution of the nonlinear Hoggard–Whalley–Wilmott equation. The comparison results reveal that our approximate solutions are close to the numerical computations. Moreover, the comparison results demonstrate that the price of the European option decreases as the Hurst exponent increases.     

A risk‐sensitive stochastic maximum principle for fully coupled forward‐backward stochastic differential equations with applications

In this paper, we are interested in the problem of optimal control where the system is given by a fully coupled forward‐backward stochastic differential equation with a risk‐sensitive performance functional. As a preliminary step, we use the risk neutral which is an extension of the initial control system where the admissible controls are convex, and an optimal solution exists.Then, we study the necessary as well as sufficient optimality conditions for risk sensitive performance. At the end of this work, we illustrate our main result by giving an example that deals with an optimal portfolio choice problem in financial market, specifically the model of control cash flow of a firm or project where, for instance, we can set the model of pricing and managing an insurance contract.     

Die Lösung des Anfangs-Randwertproblems für die Wärmeleitungsgleichung im ℝ3 mit einer Integralgleichungs-methode nach dem Rotheverfahrenmit einer Integralgleichungs-methode nach dem Rotheverfahren

We are looking for a solution of the initial boundary value problem for the threedimensional heat equation in a compact domain with a boundary of continous curvature. We use Rothe's line method, which works by discretisation of the time variable. For every time step there remains an elliptic boundary value problem, which is solved by means of an integral equation. The so obtained approximate solutions converge to the exact solution of the original problem. In case of a sphere we find a simple error estimate for the approximation. For two initial conditions the practical computations show, that the integral equations method yields useful results with relative small effort.     

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.     

Fast Laplace Transform Methods for Free-Boundary Problems of Fractional Diffusion Equations

In this paper we develop a fast Laplace transform method for solving a class of free-boundary fractional diffusion equations arising in the American option pricing. Instead of using the time-stepping methods, we develop the Laplace transform methods for solving the free-boundary fractional diffusion equations. By approximating the free boundary, the Laplace transform is taken on a fixed space region to replace discretizing the temporal variable. The hyperbola contour integral method is exploited to restore the option values. Meanwhile, the coefficient matrix has theoretically proven to be sectorial. Therefore, the highly accurate approximation by the fast Laplace transform method is guaranteed. The numerical results confirm that the proposed method outperforms the full finite difference methods in regard to the accuracy and complexity.     

Optimal stopping of Markov processes: Hilbert space theory,approximation algorithms, and an application to pricing high-dimensionalfinancial derivatives

The authors develop a theory characterizing optimal stopping times for discrete-time ergodic Markov processes with discounted rewards. The theory differs from prior work by its view of per-stage and terminal reward functions as elements of a certain Hilbert space. In addition to a streamlined analysis establishing existence and uniqueness of a solution to Bellman's equation, this approach provides an elegant framework for the study of approximate solutions. In particular, the authors propose a stochastic approximation algorithm that tunes weights of a linear combination of basis functions in order to approximate a value function. They prove that this algorithm converges (almost surely) and that the limit of convergence has some desirable properties. The utility of the approximation method is illustrated via a computational case study involving the pricing of a path dependent financial derivative security that gives rise to an optimal stopping problem with a 100-dimensional state space     

A Dimension Reduction Shannon-Wavelet Based Method for Option Pricing

We present a robust and highly efficient dimension reduction Shannon-wavelet method for computing European option prices and hedging parameters under a general jump-diffusion model with square-root stochastic variance and multi-factor Gaussian interest rates. Within a dimension reduction framework, the option price can be expressed as a two-dimensional integral that involves only (i) the value of the variance at the terminal time, and (ii) the time-integrated variance process conditional on this value. A Shannon wavelet inverse Fourier technique is developed to approximate the conditional density of the time-integrated variance process. Furthermore, thanks to the excellent approximation properties of Shannon wavelets, the overall pricing procedure is reduced to the evaluation of just a single integral that involves only the density of the terminal variance value. This single integral can be accurately evaluated, since the density of the variance at the terminal time is known in closed-form. We develop sharp approximation error bounds for the option price and hedging parameters. Numerical experiments confirm the robustness and impressive efficiency of the method.     

Moving boundary transformation for American call options with transaction cost: finite difference methods and computing

The pricing of American call option with transaction cost is a free boundary problem. Using a new transformation method the boundary is made to follow a certain known trajectory in time. The new transformed problem is solved by various finite difference methods, such as explicit and implicit schemes. Broyden's and Schubert's methods are applied as a modification to Newton's method in the case of nonlinearity in the equation. An alternating direction explicit method with second-order accuracy in time is used as an example in this paper to demonstrate the technique. Numerical results demonstrate the efficiency and the rate of convergence of the methods.