Evaluation of Double Average Asian Options by the Legendre Spectral Method

In this paper, the evaluation of discretely sampled Asian options was considered by numerically solving the associated partial differential equations with the Legendre spectral method. Double average options were discussed as examples. The problem is a parabolic one on a finite domain whose equation degenerates into ordinary differential equations on the boundaries. A fully discrete scheme was established by using the Legendre spectral method in space and the Crank-Nicolson finite difference scheme in time. The stability and convergence of the scheme were analyzed. Numerical results show that the method can keep the spectral accuracy in space for such degenerate problems.

PRICES OF ASIAN OPTIONS UNDER STOCHASTIC INTEREST RATES

Asian options are the popular second generation derivative products and embedded in many structured notes to enhance upside performance.The embedded options,as a result,usually have a long duration.The movement of interest rates becomes more important in pricing such long-dated options.In this paper,the pricing of Asian options under stochastic interest rates is studied.Assuming Hull and White model for the interest rates,a closed-form formula for geometric-average options is derived.As a by-product,pricing formula is also given for plan-vanilla options under stochastic interest rates.

Pricing Formulae of Asian Options under the Fractional Brownian Motion

In this paper,the pricing formulae of the geometric average Asian call option with the fixed and floating strike price under the fractional Brownian motion(FBM)are given out by the method of partial differential equation(PDE).The call-put parity for the geometric average Asian options is given.The results are generalization of option pricing under standard Brownian motion.

High-Dimensional Regression on Sparse Grids Applied to Pricing Moving Window Asian Options

The pricing of moving window Asian option with an early exercise feature is considered a challenging problem in option pricing. The computational challenge lies in the unknown optimal exercise strategy and in the high dimensionality required for approximating the early exercise boundary. We use sparse grid basis functions in the Least Squares Monte Carlo approach to solve this “curse of dimensionality” problem. The resulting algorithm provides a general and convergent method for pricing moving window Asian options. The sparse grid technique presented in this paper can be generalized to pricing other high-dimensional, early-exercisable derivatives.

The Asian Option Pricing when Discrete Dividends Follow a Markov-Modulated Model

This paper is concerned with the pricing problem of the discrete arithmetic average Asian call option while the discrete dividends follow geometric Brownian motion. The volatility of the dividends model depends on the Markov-Modulated process. The binomial tree method, in which a more accurate factor has been used, is applied to solve the corresponding pricing problem. Finally, a numerical example with simulations is presented to demonstrate the effectiveness of the proposed method.

Binomial lattice for pricing Asian options on yields

An efficient binomial lattice for pricing Asian options on yields is established under the affine term structure model. In order to reconnect the path of the discrete lattice,the technique of D. Nelson and K. Ramaswamy is used to transform a stochastic interest rate process into a stochastic diffusion with unit volatility. By the binomial lattice and linear interpolation,the prices of Asian options on yields can be obtained. As the number of nodes in the tree structure grows linearly with the number of time steps, the computational speed is improved. The numerical experiments to verify the validity of the lattice are also provided.

EXPLICIT EXPRESSIONS FOR THE VALUATION AND HEDGING OF THE ARITHMETIC ASIAN OPTION

On the basis of a result in Yor (1992), this paper gives explicit expressions for the valuation and hedging of the arithmetic Asian option.

CONVERGENCE RATES OF MOVING MESH RANNACHER METHODS FOR PDES OF ASIAN OPTIONS PRICING

This paper studies the convergence rates of a moving mesh implicit finite difference method with interpolation for partial differential equations （PDEs） with moving boundary arising in Asian option pricing. The moving mesh scheme is based on Rnnacher timestepping approach whose idea is running the implicit Euler schemes in the initial few steps and continuing with Crank-Nicolson schemes. With graded meshes for time direction and moving meshes for space direction, the fully discretized scheme is constructed using quadratic interpolation between two consecutive time level for the PDEs with moving boundary. The second-order convergence rates in both time and space are proved and numerical examples are carried out to confirm the theoretical results.

Two Efficient Parameterized Boundaries for Vee's Asian Option Pricing PDE

In this paper, we derive two general parameterized boundaries of finite difference scheme for VeSe~＇s PDE which is used to price both fixed and floating strike Asian options. Using these two boundaries, we can deal with all kinds of situations, especially, some extreme cases, like overhigh volatility, very small volatility, etc, under which the Asian option is usually mispriced in many existing numerical methods. Numerical results show that our boundaries are pretty efficient.

AN ACCURATE BINOMIAL MODEL FOR PRICING AMERICAN ASIAN OPTION

This paper presents simple and fast algorithms for computing very tight upper and lower bounds on the prices of American Asian options in the binomial model.The authors choose two types sets of the actual arithmetic average prices,instead of the simulated values in other existing models,as the representative average prices at each node of the binomial tree.This approach simplifies effectively the computation and reduces the error caused by the linear interpolation.Numerical results show that the approach produces accurate upper and lower bounds compared to the other existing methods based on the binomial tree.

Valuation of Asian American Option Using a Modified Path Simulation Method

In this paper, we use a modified path simulation method for valuation of Asian American Options. This method is a modification of the path simulation model proposed by Tiley. We assume that the behavior of the log return of the underlying assets follows the Variance Gamma (VG) process, since its distribution is heavy tail and leptokurtic. We provide sensitivity analysis of this method and compare the obtained prices to Asian European option prices.

Vasicek随机利率模型下基于条件矩匹配的算术平均亚式期权定价

在Vasicek随机利率模型下,本文引入了基于条件矩匹配的近似方法对算术平均型的亚式期权进行定价.该方法的基本原理是运用条件矩匹配找到伽马分布或者对数正态分布去近似在给定到期日标的资产价格的条件下的标的资产的积分的分布函数.为了在带有Vasicek随机利率的二维随机模型下运用分层近似方法,需要运用测度变换技巧去分离在期权价格公式中关于期权在到期日支付函数折现期望中的随机利率和标的资产函数,从而使得可将近似分布用于替换标的资产的积分的分布.基于用蒙特卡洛模拟得到的亚式期权的基准价格,我们通过几个数值例子测试本文提出的分层近似方法的有效性和稳健性.本文发现,分层近似方法与蒙特卡洛方法相比能极大地提高了亚式期权价格的计算速度,同时也保证了定价的准确性,并且用对数正态分布近似比用伽马分布的准确度更高.

Black-Scholes模型下亚式期权定价的一种快速傅里叶算法

针对几何平均亚式期权,利用快速傅里叶变换方法得到含特征函数的表达式,从而更快速求解亚式期权.对期权价格进行傅里叶变换,得到表达式可以用标的资产价格对数的特征函数来表示,该表达式可以输出期权价格的结果.对其进行离散化处理,再利用快速傅里叶变换来求解,以输出结果.对Black-Scholes模型下的几何亚式期权定价的表达式进行计算,得出其期权定价模型的特征函数,便可以带入快速求解出相应的期权价值.利用Black-Scholes模型在该方法求解的几何平均亚式期权价值与Black-Scholes模型下原始期权价值的数据进行对比,验证了该方法的有效性和高效性.

市场流动性影响下的亚式期权近似定价

将具有固定执行价格的亚式期权转换为欧式期权,从近似解的角度考虑算数平均和几何平均的亚式期权在流动性影响下的定价问题.引入贴现因子对流动性进行建模,利用傅里叶变换推导出流动性影响下的亚式期权无穷级数形式的近似定价公式,最后通过数值实验验证了近似公式解的高效性、准确性等性质.

跳跃扩散模型下不同标的资产算术平均的亚式期权定价

针对算术平均型亚式期权进行定价,分别采用标的资产为零息债券、平均资产和股票3种不同形式给出其相应的定价公式。基于每一种资产所拥有的鞅测度,分别采用伊藤公式和费曼卡茨公式,给出算术平均的亚式期权所服从的偏微分方程与终值条件。通过数值模拟验证该方法的有效性,从而有助于将该定价公式应用到更广阔的金融市场中。

次分数环境下标的股票有分红和配股的亚式期权定价

针对次分数环境下标的股票有连续分红且配、送股次数随机的几何平均亚式期权的定价问题,利用随机分析方法得到了几何平均亚式看涨、看跌期权的定价公式及其平价关系。数值模拟结果表明,几何平均亚式看涨、看跌期权的价格与配、送股比例,配股价和除权除息前股价呈不同的变化趋势,但均与Hurst指数成反比。该研究对丰富期权定价模型具有理论意义,同时为我国金融市场的期权投资者提供了参考依据。

基于混合次分数跳过程的亚式期权模糊定价

考虑到金融资产价格的长记忆性及跳跃现象,基于混合次分数布朗运动和泊松过程,建立了几何亚式期权定价模型;进一步考虑金融市场模糊性,引入模糊理论得到模糊定价模型.首先,得到混合次分数跳过程Ito∧公式及其股价所满足随机微分方程的解析解;其次,运用风险中性原理给出几何亚式期权的定价公式;然后,运用模糊理论构建了几何亚式模糊期权定价模型;最后,数值模拟分析了置信度和Hurst指数对模糊价格的影响,并将本文所建立模型与经典BS模型进行对比.结果表明,在相应的置信度下模糊定价模型能够给出较为合理的价格区间,有助于金融投资者的决策,从而验证了模型的合理性和实用性.

时间分数阶CEV模型下算术亚式期权的加权差分格式

针对时间分数阶CEV模型下算术亚式期权定价问题,提出了一个加权平均的差分方法.将偏微分方程的二维空间变量降到一维,得到了CEV模型下算术亚式期权定价的偏微分方程.将显式差分格式与隐式差分格式进行加权得到加权有限差分格式,并分析该差分格式的稳定性和收敛性.通过数值模拟说明了加权有限差分方法求解时间分数阶CEV模型是可行的.

混合次分数布朗跳-扩散模型下亚式幂型期权的定价

文章研究了混合次分数跳扩散模型下亚式幂型期权的定价问题。与传统的布朗运动驱动的期权定价模型相比,混合次分数跳扩散模型更好地刻画了金融时间序列的长记忆性、非平稳性和突发事件的影响。本文利用保险精算方法给出了具有固定执行价格的几何平均亚式幂型期权定价公式,并给出了一些特殊情形下的亚式期权定价公式。最后,给出了一个数值仿真例子说明定理的结论。