Two asset-barrier option under stochastic volatility

Financial products which depend on hitting times for two underlying assets have become very popular in the last decade. Three common examples are double-digital barrier options, two-asset barrier spread options and double lookback options. Analytical expressions for the joint distribution of the endpoints and the maximum and/or minimum values of two assets are essential in order to obtain quasi-closed form solutions for the price of these derivatives. Earlier authors derived quasi-closed form pricing expressions in the context of constant volatility and correlation. More recently solutions were provided in the presence of a common stochastic volatility factor but with restricted correlations due to the use of a method of images. In this article, we generalize this finding by allowing any value for the correlation. In this context, we derive closed-form expressions for some two-asset barrier options.     

Exponential Rosenbrock integrators for option pricing

In this paper, we are concerned with the time integration of differential equations modeling option pricing. In particular, we consider the Black-Scholes equation for American options. As an alternative to existing methods, we present exponential Rosenbrock integrators. These integrators require the evaluation of the exponential and related functions of the Jacobian matrix. The resulting methods have good stability properties. They are fully explicit and do not require the numerical solution of linear systems, in contrast to standard integrators. We have implemented some numerical experiments in Matlab showing the reliability of the new method.     

American option valuation in a stochastic volatility model with transaction costs

In the present paper we analyse the American option valuation problem in a stochastic volatility model when transaction costs are taken into account. We shall show that it can be formulated as a singular stochastic optimal control problem, proving the existence and uniqueness of the viscosity solution for the associated Hamilton–Jacobi–Bellman partial differential equation. Moreover, after performing a dimensionality reduction through a suitable choice of the utility function, we shall provide a numerical example illustrating how American options prices can be computed in the present modelling framework.     

Discrete time modeling of mean-reverting stochastic processes for real option valuation

In this paper the recombining binomial lattice approach for modeling real options and valuing managerial flexibility is generalized to address a common issue in many practical applications, underlying stochastic processes that are mean-reverting. Binomial lattices were first introduced to approximate stochastic processes for valuation of financial options, and they provide a convenient framework for numerical analysis. Unfortunately, the standard approach to constructing binomial lattices can result in invalid probabilities of up and down moves in the lattice when a mean-reverting stochastic process is to be approximated. There have been several alternative methods introduced for modeling mean-reverting processes, including simulation-based approaches and trinomial trees, however they unfortunately complicate the numerical analysis of valuation problems. The approach developed in this paper utilizes a more general binomial approximation methodology from the existing literature to model simple homoskedastic mean-reverting stochastic processes as recombining lattices. This approach is then extended to model dual correlated one-factor mean-reverting processes. These models facilitate the evaluation of options with early-exercise characteristics, as well as multiple concurrent options.     

Multifactor Heston's stochastic volatility model for European option pricing

In this study, we extend the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254] by incorporating a slow varying factor of volatility. The resulting model can be viewed as a multifactor extension of the Heston model with two additional factors driving the volatility levels. An asymptotic analysis consisting of singular and regular perturbation expansions is developed to obtain an approximation to European option prices. We also find explicit expressions for some essential functions that are available only in integral formulas in the work of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254]. This finding basically leads to considerable reduction in computational time for numerical calculation as well as calibration problems. An accuracy result of the asymptotic approximation is also provided. For numerical illustration, the multifactor Heston model is calibrated to index options on the market, and we find that the resulting implied volatility surfaces fit the market data better than those produced by the multiscale stochastic volatility model of [Fouque J‐P, Lorig MJ, SIAM J Financial Math. 2011;2(1):221‐254], particularly for long‐maturity call options.     

On the valuation of constant barrier options under spectrally one-sided exponential Lévy models and Carr''s approximation for American puts

This paper provides a general framework for pricing options with a constant barrier under spectrally one-sided exponential Lévy model, and uses it to implement of Carr's approximation for the value of the American put under this model. Simple analytic approximations for the exercise boundary and option value are obtained.     

An efficient ETD method for pricing American options under stochastic volatility with nonsmooth payoffs

Based on Cox and Matthews Exponential Time Differencing (ETD) approach, a fourth–order strongly–stable method having real distinct poles is developed and applied to solve American options under stochastic volatility with nonsmooth payoffs. A computationally efficient version of the method is constructed using partial fraction splitting technique. This approach requires to solve several backward Euler‐type linear systems at each time step. Numerical experiments are presented to demonstrate the computational efficiency, accuracy, and reliability of the method. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013     

Local discontinuous Galerkin methods with explicit-implicit-null time discretizations for solving nonlinear diffusion problems

In this paper,we discuss the local discontinuous Galerkin methods coupled with two specific explicitimplicit-null time discretizations for solving one-dimensional nonlinear diffusion problems Ut=(a(U)Ux)x.The basic idea is to add and subtract two equal terms a0 Uxx the right-hand side of the partial differential equation,then to treat the term a0 Uxx implicitly and the other terms(a(U)Ux)x-a0 Uxx explicitly.We give stability analysis for the method on a simplified model by the aid of energy analysis,which gives a guidance for the choice of a0,i.e.,a0≥max{a(u)}/2 to ensure the unconditional stability of the first order and second order schemes.The optimal error estimate is also derived for the simplified model,and numerical experiments are given to demonstrate the stability,accuracy and performance of the schemes for nonlinear diffusion equations.     

A unified discontinuous Galerkin framework for time integration

We introduce a new discontinuous Galerkin approach for time integration. On the basis of the method of weighted residual, numerical quadratures are employed in the finite element time discretization to account for general nonlinear ordinary differential equations. Many different conditions, including explicit, implicit, and symplectic conditions, are enforced for the test functions in the variational analysis to obtain desirable features of the resulting time‐stepping scheme. The proposed discontinuous Galerkin approach provides a unified framework to derive various time‐stepping schemes, such as low‐order one‐step methods, Runge–Kutta methods, and multistep methods. On the basis of the proposed framework, several explicit Runge–Kutta methods of different orders are constructed. The derivation of symplectic Runge–Kutta methods has also been realized. The proposed framework allows the optimization of new schemes in terms of several characteristics, such as accuracy, sparseness, and stability. The accuracy optimization is performed on the basis of an analytical form of the error estimation function for a linear test initial value problem. Schemes with higher formal order of accuracy are found to provide more accurate solutions. We have also explored the optimization potential of sparseness, which is related to the general compressive sensing in signal/imaging processing. Two critical dimensions of the stability region, that is, maximal intervals along the imaginary and negative real axes, are employed as the criteria for stability optimization. This gives the largest Courant–Friedrichs–Lewy time steps in solving hyperbolic and parabolic partial differential equations, respectively. Numerical experiments are conducted to validate the optimized time‐stepping schemes. Copyright © 2013 John Wiley & Sons, Ltd.     

Advanced MCMC methods for sampling on diffusion pathspace

The need to calibrate increasingly complex statistical models requires a persistent effort for further advances on available, computationally intensive Monte-Carlo methods. We study here an advanced version of familiar Markov-chain Monte-Carlo (MCMC) algorithms that sample from target distributions defined as change of measures from Gaussian laws on general Hilbert spaces. Such a model structure arises in several contexts: we focus here at the important class of statistical models driven by diffusion paths whence the Wiener process constitutes the reference Gaussian law. Particular emphasis is given on advanced Hybrid Monte-Carlo (HMC) which makes large, derivative-driven steps in the state space (in contrast with local-move Random-walk-type algorithms) with analytical and experimental results. We illustrate its computational advantages in various diffusion processes and observation regimes; examples include stochastic volatility and latent survival models. In contrast with their standard MCMC counterparts, the advanced versions have mesh-free mixing times, as these will not deteriorate upon refinement of the approximation of the inherently infinite-dimensional diffusion paths by finite-dimensional ones used in practice when applying the algorithms on a computer.     

Convergence rates of trinomial tree methods for option pricing under regime-switching models

Recently trinomial tree methods have been developed to option pricing under regime-switching models. Although these novel trinomial tree methods are shown to be accurate via numerical examples, it needs to give a rigorous proof of the accuracy which can theoretically guarantee the reliability of the computations. The aim of this paper is to prove the convergence rates (measure of the accuracy) of the trinomial tree methods for the option pricing under regime-switching models.     

Efficient numerical methods for pricing American options under stochastic volatility

Five numerical methods for pricing American put options under Heston's stochastic volatility model are described and compared. The option prices are obtained as the solution of a two‐dimensional parabolic partial differential inequality. A finite difference discretization on nonuniform grids leading to linear complementarity problems with M‐matrices is proposed. The projected SOR, a projected multigrid method, an operator splitting method, a penalty method, and a componentwise splitting method are considered. The last one is a direct method while all other methods are iterative. The resulting systems of linear equations in the operator splitting method and in the penalty method are solved using a multigrid method. The projected multigrid method and the componentwise splitting method lead to a sequence of linear complementarity problems with one‐dimensional differential operators that are solved using the Brennan and Schwartz algorithm. The numerical experiments compare the accuracy and speed of the considered methods. The accuracies of all methods appear to be similar. Thus, the additional approximations made in the operator splitting method, in the penalty method, and in the componentwise splitting method do not increase the error essentially. The componentwise splitting method is the fastest one. All multigrid‐based methods have similar rapid grid independent convergence rates. They are about two or three times slower that the componentwise splitting method. On the coarsest grid the speed of the projected SOR is comparable with the multigrid methods while on finer grids it is several times slower. ©John Wiley & Sons, Inc. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Probability maximization models for portfolio selection under ambiguity

This paper considers several probability maximization models for multi-scenario portfolio selection problems in the case that future returns in possible scenarios are multi-dimensional random variables. In order to consider occurrence probabilities and decision makers’ predictions with respect to all scenarios, a portfolio selection problem setting a weight with flexibility to each scenario is proposed. Furthermore, by introducing aspiration levels to occurrence probabilities or future target profit and maximizing the minimum aspiration level, a robust portfolio selection problem is considered. Since these problems are formulated as stochastic programming problems due to the inclusion of random variables, they are transformed into deterministic equivalent problems introducing chance constraints based on the stochastic programming approach. Then, using a relation between the variance and absolute deviation of random variables, our proposed models are transformed into linear programming problems and efficient solution methods are developed to obtain the global optimal solution. Furthermore, a numerical example of a portfolio selection problem is provided to compare our proposed models with the basic model.     

A general framework for pricing Asian options under stochastic volatility on parallel architectures

In this paper, we present a transform-based algorithm for pricing discretely monitored arithmetic Asian options with remarkable accuracy in a general stochastic volatility framework, including affine models and time-changed Lévy processes. The accuracy is justified both theoretically and experimentally. In addition, to speed up the valuation process, we employ high-performance computing technologies. More specifically, we develop a parallel option pricing system that can be easily reproduced on parallel computers, also realized as a cluster of personal computers. Numerical results showing the accuracy, speed and efficiency of the procedure are reported in the paper.     

A stable local radial basis function method for option pricing problem under the Bates model

We propose a local mesh‐free method for the Bates–Scott option pricing model, a 2D partial integro‐differential equation (PIDE) arising in computational finance. A Wendland radial basis function (RBF) approach is used for the discretization of the spatial variables along with a linear interpolation technique for the integral operator. The resulting set of ordinary differential equations (ODEs) is tackled via a time integration method. A potential advantage of using RBFs is the small number of discrete equations that need to be solved. Computational experiments are presented to illustrate the performance of the contributed approach.     

Efficient L-stable method for parabolic problems with application to pricing American options under stochastic volatility

Efficient L-stable numerical method for semilinear parabolic problems with nonsmooth initial data is proposed and implemented to solve Heston’s stochastic volatility model based PDE for pricing American options under stochastic volatility. The proposed new method is also used to solve two asset American options pricing problem. Cox and Matthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, Journal of Computational Physics 176 (2002) 430-455] developed a class of exponential time differencing Runge-Kutta schemes (ETDRK) for nonlinear parabolic problems. Kassam and Trefethen [A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAM Journal on Scientific Computing 26 (4) (2005) 1214-1233] showed that while computing certain functions involved in the Cox-Matthews schemes, severe cancelation errors can occur which affect the accuracy and stability of the schemes. Kassam and Trefethen proposed complex contour integration technique to implement these schemes in a way that avoids these cancelation errors. But this approach creates new difficulties in choosing and evaluating the contour integrals for larger problems. We modify the ETDRK schemes using positivity preserving Padé approximations of the matrix exponential functions and construct computationally efficient parallel version using splitting technique. As a result of this approach it is required only to solve several backward Euler linear problems in serial or parallel.     

A computational analysis for mean exit time under non-Gaussian Lévy noises

Complex dynamical systems are often subject to non-Gaussian random fluctuations. The exit phenomenon, i.e., escaping from a bounded domain in state space, is an impact of randomness on the evolution of these dynamical systems. The existing work is about asymptotic estimate on mean exit time when the noise intensity is sufficiently small. In the present paper, however, the authors analyze mean exit time for arbitrary noise intensity, via numerical investigation. The mean exit time for a dynamical system, driven by a non-Gaussian, discontinuous (with jumps), α-stable Lévy motion, is described by a differential equation with nonlocal interactions. A numerical approach for solving this nonlocal problem is proposed. A computational analysis is conducted to investigate the relative importance of jump measure, diffusion coefficient and non-Gaussianity in affecting mean exit time.     

Discontinuous Galerkin methods for the chemotaxis and haptotaxis models

In this work, first we formulate and compare three different discontinuous Interior Penalty Galerkin methods for the 2D Keller–Segel chemotaxis model. Keller–Segel chemotaxis model is the important starting step in the modeling of the real biological system. We show in the numerical tests that two of the proposed methods fail to give accurate, oscillation-free solutions.     

An iterative method for pricing American options under jump-diffusion models

We propose an iterative method for pricing American options under jump-diffusion models. A finite difference discretization is performed on the partial integro-differential equation, and the American option pricing problem is formulated as a linear complementarity problem (LCP). Jump-diffusion models include an integral term, which causes the resulting system to be dense. We propose an iteration to solve the LCPs efficiently and prove its convergence. Numerical examples with Kou?s and Merton?s jump-diffusion models show that the resulting iteration converges rapidly.