非线性Black-Scholes模型下阶梯期权定价

在非线性Black-Scholes模型下,研究了阶梯期权定价问题.首先利用多尺度方法,将阶梯期权适合的偏微分方程分解成一系列常系数抛物方程;其次通过计算这些常系数抛物型方程的解,给出了修正障碍期权的近似定价公式;最后利用Feymann-Kac公式分析了近似结论的误差估计.     

非线性Black-Scholes模型下Bala期权定价

在非线性Black-Scholes模型下,研究了Bala期权定价问题.首先利用双参数摄动方法,将Bala期权适合的偏微分方程分解成一系列常系数抛物方程.其次通过计算这些常系数抛物型方程的解,给出了Bala期权的近似定价公式.最后利用Green函数分析了近似结论的误差估计.     

非线性Black-Scholes模型下几何平均亚式期权定价

在非线性Black-Scholes模型下,本文研究了几何平均亚式期权定价问题.首先利用单参数摄动方法,将亚式期权适合的偏微分方程分解成一系列常系数抛物方程.其次通过计算这些常系数抛物型方程的解,给出了几何平均亚式期权的近似定价公式.最后利用Green函数分析了近似结论的误差估计.     

A Black-Scholes option pricing model with transaction costs

We consider a boundary value problem for a nonlinear differential equation which arises in an option pricing model with transaction costs. We apply the method of upper and lower solutions in order to obtain solutions for the stationary problem. Moreover, we give conditions for the existence of solutions of the general evolution equation.     

High-order exponential spline method for pricing European options

ABSTRACT

The key purpose of the present work is to constitute an analysis of a numerical method for a degenerate partial differential equation, called the Black–Scholes equation, governing European option pricing. The method is based on exponential spline spatial discretization and an explicit finite-difference time-stepping technique. We establish the convergence and an error bound for the solutions of the fully discretized system. The numerical and graphical results elucidate that the suggested approach is very straightforward and accurate.     

A model and some evidence on pricing compound call options

The most widely accepted option pricing model, derived by Black and Scholes (B-S), studies single priced options. Nevertheless, it has important implications for the relative pricing of compound call options. Compound options are two or more option contracts on a given security with different striking prices but with each expiring on the same day.Studying the relative pricing of compound options provides insight into the efficiency of generally accepted option pricing models. Comparing prices of compound options enables us to analyze factors in option pricing that would remain hidden in studies of single options.We are not primarily concerned with efficiency of option pricing, although some of our results may bear on this issue. Our primary concerns are: (1) to determine the implications of the B-S model for compound options and (2) to explain compound option prices by a number of variables, and thus come to conclusions about option pricing generally.We found difficulty with the B-S model when attempting to explain the relative pricing of compound options. Further, from empirical tests, we found that the most important factor in explaining the relative pricing of compound options is the relative degree of leverage which is operative between the various components of a compound option set.     

A uniform asymptotic expansion for stochastic volatility model in pricing multi‐asset European options

In this paper, we consider a stochastic volatility model for pricing multi‐asset European options that are widely used in the real world, under the assumption that the volatilities are driven by different OU processes. Using the singular perturbation method for multi‐parameter and the boundary layer theory, we derive a uniform asymptotic expansion for the option prices, as well as the uniform error estimates. Copyright © 2011 John Wiley & Sons, Ltd.     

A numerical method for pricing European options with proportional transaction costs

In the paper, we propose a numerical technique based on a finite difference scheme in space and an implicit time-stepping scheme for solving the Hamilton–Jacobi–Bellman (HJB) equation arising from the penalty formulation of the valuation of European options with proportional transaction costs. We show that the approximate solution from the numerical scheme converges to the viscosity solution of the HJB equation as the mesh sizes in space and time approach zero. We also propose an iterative scheme for solving the nonlinear algebraic system arising from the discretization and establish a convergence theory for the iterative scheme. Numerical experiments are presented to demonstrate the robustness and accuracy of the method.     

A hybrid method for pricing European options based on multiple assets with transaction costs

The problem of pricing European options based on multiple assets with transaction costs is considered. These options include, for example, quality options and options on the minimum of two or more risky assets. The value of these options is the solution of a nonlinear parabolic partial differential equation subject to a final condition given by the payoff function associated with the option. A computationally efficient method to solve this final-value problem is proposed. This method is based on an asymptotic expansion of the required solution with respect to the parameters related to the transaction costs followed by the numerical solution of the linear partial differential equations obtained at each order in perturbation theory. The numerical solution of these linear problems involves an implicit finite-difference scheme for the parabolic equation and the use of the fast Fourier sine transform to solve the resulting elliptic problems. Numerical results obtained on test problems with the method proposed here are shown and discussed.     

Option pricing, Black-Scholes, and novel arbitrage possibilities

Analytic solutions to the Black-Scholes equation are presentedfor the case when both interest rates and the volatility ofthe underlying security are arbitrary functions of time. Severalspecial cases are examined, and novel arbitrage possibilitiesare discussed.     

A multinomial tree model for pricing credit default swap options

Among the traded credit derivatives, the market interest in credit default swap options (CDSwaptions) is enormous. We propose a multinomial tree model to price Bermudan CDSwaptions. Our basic rationale is that we distribute the occurring probability for each node in a branch proportional to the probability density function of the assumed (normal) distribution. Through this approach, without the need of solving a large number of equations simultaneously, only the first four moments are required to build an arbitrarily large N-branches tree. We also demonstrate the detailed model implementation procedure including the valuation and the estimation of critical prices through an empirical example in Tucker and Wei (J Fixed Income 15(1):88–95, 2005). Numerical results show that, in the valuation, the proposed multinomial tree model is accurate and can significantly save pricing time under the same degree of accuracy as the binomial tree model. In the estimation of critical prices, the results are less accurate than those in the valuation, but the relative errors are acceptable.     

Stochastic moment problem and hedging of generalized Black-Scholes options

In mathematical finance one is interested in the quadratic error which occurs while replacing a continuously adjusted portfolio by a discretely adjusted one. We first study higher order approximations of stochastic integrals. Then we apply the results to quantify quadratic error which occurs in estimating the discretely adjusted hedging risk in pricing European options in a generalized Black-Scholes market.     

Numerical methods for backward Markov chain driven Black-Scholes option pricing

The drift, the risk-free interest rate, and the volatility change over time horizon in realistic financial world. These frustrations break the necessary assumptions in the Black-Scholes model (BSM) in which all parameters are assumed to be constant. To better model the real markets, a modified BSM is proposed for numerically evaluating options price-changeable parameters are allowed through the backward Markov regime switching. The method of fundamental solutions (MFS) is applied to solve the modified model and price a given option. A series of numerical simulations are provided to illustrate the effect of the changing market on option pricing.     

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.     

Circulant preconditioners for pricing options

We use the normalized preconditioned conjugate gradient method with Strang’s circulant preconditioner to solve a nonsymmetric Toeplitz system A_nx=b, which arises from the discretization of a partial integro-differential equation in option pricing. By using the definition of family of generating functions introduced in [16], we prove that Strang’s circulant preconditioner leads to a superlinear convergence rate under certain conditions. Numerical results exemplify our theoretical analysis.     

European option pricing model in a stochastic and fuzzy environment

The primary goal of this paper is to price European options in the Merton＇s frame- work with underlying assets following jump-diffusion using fuzzy set theory. Owing to the vague fluctuation of the real financial market, the average jump rate and jump sizes cannot be recorded or collected accurately. So the main idea of this paper is to model the rate as a triangular fuzzy number and jump sizes as fuzzy random variables and use the property of fuzzy set to deduce two different jump-diffusion models underlying principle of rational expectations equilibrium price. Unlike many conventional models, the European option price will now turn into a fuzzy number. One of the major advantages of this model is that it allows investors to choose a reasonable European option price under an acceptable belief degree. The empirical results will serve as useful feedback information for improvements on the proposed model.     

On the pricing of American options

The problem of valuation for contingent claims that can be exercised at any time before or at maturity, such as American options, is discussed in the manner of Bensoussan [1]. We offer an approach which both simplifies and extends the results of existing theory on this topic.Research supported in part by the National Science Foundation under Grant No. NSF-DMS-84-16736 and by the Air Force Office of Scientific Research under Grant No. F49620-85-C-0144.     

Numerical solution of modified Black–Scholes equation pricing stock options with discrete dividend

This paper deals with the numerical solution of the modified Black–Scholes equation modelling the valuation of stock options with discrete dividend payments. By using a delta-defining sequence of the involved generalized Dirac delta function and applying the Mellin transform, an integral formula for the solution is obtained. Then, numerical quadrature approximations and illustrative examples are given.     

The CONV method for pricing options

In this paper, we discuss a convolution based method, the CONV method, for pricing options with early-exercise features, in which the asset prices are modeled by Lévy processes. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)