American option pricing under stochastic volatility: an efficient numerical approach

This paper develops a new numerical technique to price an American option written upon an underlying asset that follows a bivariate diffusion process. The technique presented here exploits the supermartingale representation of an American option price together with a coarse approximation of its early exercise surface that is based on an efficient implementation of the least-squares Monte–Carlo algorithm (LSM) of Longstaff and Schwartz (Rev Financ Stud 14:113–147, 2001). Our approach also has the advantage of avoiding two main issues associated with LSM, namely its inherent bias and the basis functions selection problem. Extensive numerical results show that our approach yields very accurate prices in a computationally efficient manner. Finally, the flexibility of our method allows for its extension to a much larger class of optimal stopping problems than addressed in this paper.     

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.     

投影三角分解法定价带随机波动率的美式期权

考虑数值求解Heston随机波动率美式期权定价问题,通过在空间方向采用中心差分格式离散二维偏微分算子,在时间方向利用隐式交替方向格式,将美式期权定价问题转化成求解每个时间层上的若干个线性互补问题.针对一般美式期权定价模型离散得到的线性互补问题,构造出投影三角分解法进行求解,并在理论上给出算法的收敛条件.数值实验表明,所构造的数值方法对于求解美式期权定价问题是有效的,并且优于经典的投影超松弛迭代法和算子分裂方法.     

Homotopy analysis method for option pricing under stochastic volatility

In this paper, the homotopy analysis method, whose original concept comes from algebraic topology, is applied to connect the Black–Scholes option price (the good initial guess) to the option price under general stochastic volatility environment in a recursive manner. We obtain the homotopy solutions for the European vanilla and barrier options as well as the relevant convergence conditions.     

Operator splitting methods for pricing American options under stochastic volatility

We consider the numerical pricing of American options under Heston’s stochastic volatility model. The price is given by a linear complementarity problem with a two-dimensional parabolic partial differential operator. We propose operator splitting methods for performing time stepping after a finite difference space discretization. The idea is to decouple the treatment of the early exercise constraint and the solution of the system of linear equations into separate fractional time steps. With this approach an efficient numerical method can be chosen for solving the system of linear equations in the first fractional step before making a simple update to satisfy the early exercise constraint. Our analysis suggests that the Crank–Nicolson method and the operator splitting method based on it have the same asymptotic order of accuracy. The numerical experiments show that the operator splitting methods have comparable discretization errors. They also demonstrate the efficiency of the operator splitting methods when a multigrid method is used for solving the systems of linear equations.     

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     

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.     

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.     

FFT based option pricing under a mean reverting process with stochastic volatility and jumps

Numerous studies present strong empirical evidence that certain financial assets may exhibit mean reversion, stochastic volatility or jumps. This paper explores the valuation of European options when the underlying asset follows a mean reverting log-normal process with stochastic volatility and jumps. A closed form representation of the characteristic function of the process is derived for the computation of European option prices via the fast Fourier transform.     

Calibrated American option pricing by stochastic linear programming

Abstract

We propose an approach for computing the arbitrage-free interval for the price of an American option in discrete incomplete market models via linear programming. The main idea is built replicating strategies that use both the basic asset and some European derivatives available on the market for trading. This method goes under the name of calibrated option pricing and it has given significant results for European options. Here, we extend the analysis to American options showing that the arbitrage-free interval can be characterized in terms of martingale measures and that it gets significantly reduced with respect to the non-calibrated case.     

High-order compact finite difference scheme for option pricing in stochastic volatility models

We derive a new high-order compact finite difference scheme for option pricing in stochastic volatility models. The scheme is fourth order accurate in space and second order accurate in time. Under some restrictions, theoretical results like unconditional stability in the sense of von Neumann are presented. Where the analysis becomes too involved we validate our findings by a numerical study. Numerical experiments for the European option pricing problem are presented. We observe fourth order convergence for non-smooth payoff.     

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     

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.     

A spectral-collocation method for pricing perpetual American puts with stochastic volatility

Based on the Legendre pseudospectral method, we propose a numerical treatment for pricing perpetual American put option with stochastic volatility. In this simple approach, a nonlinear algebraic equation system is first derived, and then solved by the Gauss-Newton algorithm. The convergence of the current scheme is ensured by constructing a test example similar to the original problem, and comparing the numerical option prices with those produced by the classical Projected SOR (PSOR) method. The results of our numerical experiments suggest that the proposed scheme is both accurate and efficient, since the spectral accuracy can be easily achieved within a small number of iterations. Moreover, based on the numerical results, we also discuss the impact of stochastic volatility term on the prices of perpetual American puts.     

Wavelet-based option pricing: An empirical study

In this paper, we scrutinize the empirical performance of a wavelet-based option pricing model which leverages the powerful computational capability of wavelets in approximating risk-neutral moment-generating functions. We focus on the forecasting and hedging performance of the model in comparison with that of popular alternative models, including the stochastic volatility model with jumps, the practitioner Black–Scholes model and the neural network based model. Using daily index options written on the German DAX 30 index from January 2009 to December 2012, our results suggest that the wavelet-based model compares favorably with all other models except the neural network based one, especially for long-term options. Hence our novel wavelet-based option pricing model provides an excellent nonparametric alternative for valuing option prices.     

A partial differential equation connected to option pricing with stochastic volatility: Regularity results and discretization

This paper completes a previous work on a Black and Scholes equation with stochastic volatility. This is a degenerate parabolic equation, which gives the price of a European option as a function of the time, of the price of the underlying asset, and of the volatility, when the volatility is a function of a mean reverting Orstein-Uhlenbeck process, possibly correlated with the underlying asset. The analysis involves weighted Sobolev spaces. We give a characterization of the domain of the operator, which permits us to use results from the theory of semigroups. We then study a related model elliptic problem and propose a finite element method with a regular mesh with respect to the intrinsic metric associated with the degenerate operator. For the error estimate, we need to prove an approximation result.

     

An analytic pricing formula for lookback options under stochastic volatility

In this work, an analytic pricing formula for floating strike lookback options under Heston’s stochastic volatility model is derived by means of the homotopy analysis method. The fixed strike lookback options can then be priced on the basis of the results of floating strike and the put–call parity relation for lookback options.     

A fast numerical approach to option pricing with stochastic interest rate,stochastic volatility and double jumps

This study proposes a pricing model through allowing for stochastic interest rate and stochastic volatility in the double exponential jump-diffusion setting. The characteristic function of the proposed model is then derived. Fast numerical solutions for European call and put options pricing based on characteristic function and fast Fourier transform (FFT) technique are developed. Simulations show that our numerical technique is accurate, fast and easy to implement, the proposed model is suitable for modeling long-time real-market changes. The model and the proposed option pricing method are useful for empirical analysis of asset returns and risk management in firms.     

Risk premium and fair option prices under stochastic volatility: the HARA solution

We have solved the problem of finding (HARA) fair option price under a general stochastic volatility model. For a given HARA utility, the ‘risk premium’, i.e., the ‘market price of volatility risk’ is determined via a solution of a certain nonlinear PDE. Equivalently, the fair option price is determined as a solution of an uncoupled system of a non-linear PDE and a Black–Scholes type PDE. To cite this article: S.D. Stojanovic, C. R. Acad. Sci. Paris, Ser. I 340 (2005).