分形布朗运动下有交易成本的外汇期权定价

研究了有交易成本的分形Black-Scholes外汇期权定价问题.基于汇率的分形布朗运动分布假设,运用分形布朗运动的性质和随机微积分方法,得到了欧式外汇期权价格所满足的偏微分方程.最后,建立离散时间条件下的非线性期权定价模型,并且通过解期权价格的偏微分方程给出了有交易成本的欧式外汇期权定价公式.     

次分数布朗运动下支付红利的欧式期权定价

本文主要建立了次分数布朗运动下的期权定价模型,并且考虑了支付连续红利的情形.首先利用Wick-It?积分和偏微分方法得到了期权价格所满足的偏微分方程,然后经过变量替换转化为Cauchy问题,从而得到了支付红利的次分数布朗运动环境下的欧式看涨期权定价公式,相应地根据看涨看跌定价公式,得出欧式看跌期权定价公式.最后,对定价模型中的参数进行估计,并讨论了估计量的无偏性和强收敛性.     

分数Black-Scholes模型下美式期权定价的积分方程式

为得到分数Black-Scholes模型下美式期权价格的公式,文章以看涨期权为例,应用偏微分方程法,推导期权价格的积分方程式.由于美式期权的价格可分解为欧式期权的价格和由于提前实施需要增付的期权金,而提前实施期权金与最佳实施边界的位置有关,所以为导出最佳实施边界所满足的方程,文章首先研究分数Black-Scholes方程的基本解,然后建立美式看涨期权的分解公式,推导最佳实施边界适合的非线性积分方程,从而得到美式看涨期权价格的积分方程式.美式看跌期权价格的积分方程式类似得到.     

Vasicek随机利率模型下欧式期权 定价的Mellin变换法

运用Feynman-Kac公式和偏微分方程法得到Vasicek随机利率模型下的零息债券价格公式.利用△-对冲方法建立该模型下欧式期权价值满足的偏微分方程模型,并用Mellin变换法求解该偏微分方程,最终得到欧式期权定价公式.从数值算例的结果可以看出Mellin变换法的有效性以及不同参数对期权价值的影响.     

美式看跌期权的两点Geske-Johnson近似定价法

在分数Black-Scholes模型下,应用两点Geske-Johnson定价法推导连续支付红利为常数的美式看跌期权的近似公式.首先假定期权没有提前实施,其价格为对应欧式看跌期权的价格;再将期权的实施时刻指定为两个时刻,通过中性风险定价法推导价格公式,然后利用两点Geske-Johnson定价法得到美式看跌期权价格的近似公式.最后给出一个数值算例,结果显示Hurst参数和到期日对价格的影响.     

分数维Vasicek利率模型下的欧式期权定价公式

假定股票价格和利率的运动过程服从几何分数维布朗运动,利用风险对冲技术,分数维布朗运动随机分析理论与偏微分方程方法,得到了分数维Vasicek随机利率下欧式期权所满足的定价方程,获得了波动率是对间函数的情形下欧式看涨和看跌期权的一般定价公式以及它们的平价公式.     

混合分数布朗运动环境下欧式期权定价

假设股票价格变化过程服从混合分数布朗运动,建立了混合分数布朗环境下支付连续红利的欧式股票期权的定价模型.利用混合分数布朗运动的It-公式,将支付连续红利的欧式股票期权的定价问题转化为一个偏微分方程,通过偏微分方程求解获得了混合分数布朗运动环境下支付连续红利的欧式股票看涨期权的定价公式.     

分数布朗运动驱动下带比例交易成本的期权定价

在标的资产价格服从几何分数布朗运动模型条件下,利用分数布朗运动随机分析理论和偏微分方程方法,建立了几何分数布朗运动驱动下的金融市场模型,讨论了带比例交易成本的欧式期权,并且得到了相应的期权定价公式.     

分数布朗运动下几何平均亚式权证的定价

假设股票价格变化过程服从几何分数布朗运动,建立了分数布朗运动下的亚式期权定价模型.利用分数-It-公式,推导出分数布朗运动下亚式期权的价值所满足的含有三个变量偏微分方程.然后,引进适当的组合变量,将其定解问题转化为一个与路径无关的一维微分方程问题.进一步通过随机偏微分方程方法求解出分数布朗运动下亚式期权的定价公式.最后利用权证定价原理对稀释效用做出调整后,得到分数布朗运动下亚式股本权证定价公式.<正>~~     

随机利率下服从分数跳-扩散过程的回望期权定价

文章主要研究分数CIR利率模型下,标的资产股票价格服从分数跳-扩散过程的欧式回望期权定价问题.利用无套利原理和分数It公式,建立期权定价模型,得到了期权价格所满足的偏微分方程.并利用有限差分方法,给出了微分方程隐式格式的数值解,最后通过数值实验验证了该方法的有效性,推广了已有的回望期权定价理论.     

Markov调制L′evy模型定价的Fourier-Cos方法

对一般的Markov调制L′evy模型,利用Fourier Cosine级数展开原理得到欧式期权价格的计算方法。进一步,为了改进期权定价的Fourier Cosine级数展开方法的计算精度, Fourier Cosine级数展开的对象进行了修正,获得了欧式期权价格的修正Fourier Cosine级数展开计算方法。此外,还将获得的方法应用于Markov调制Black-Scholes模型, Markov调制Merton跳扩散模型和Markov调制CGMY L′evy模型期权定价的计算。具体的数值计算说明：修正Fourier Cosine级数展开方法应与Fourier Cosine级数展开方法相比,收敛速度要慢一些,但准确性却有很大的提高。特别是对Markov调制纯跳模型,效果更为显著。     

Unifying pricing formula for several stochastic volatility models with jumps

In this paper, we introduce a unifying approach to option pricing under continuous‐time stochastic volatility models with jumps. For European style options, a new semi‐closed pricing formula is derived using the generalized complex Fourier transform of the corresponding partial integro‐differential equation. This approach is successfully applied to models with different volatility diffusion and jump processes. We also discuss how to price options with different payoff functions in a similar way. In particular, we focus on a log‐normal and a log‐uniform jump diffusion stochastic volatility model, originally introduced by Bates and Yan and Hanson, respectively. The comparison of existing and newly proposed option pricing formulas with respect to time efficiency and precision is discussed. We also derive a representation of an option price under a new approximative fractional jump diffusion model that differs from the aforementioned models, especially for the out‐of‐the money contracts. Copyright © 2017 John Wiley & Sons, Ltd.     

A non random walk theory of exchange rate dynamics with applications to option pricing

A two dimensional stochastic process is developed to model exchange rate dynamics. We incorporate the non random walk influence of pur–chasing power parity, to synthesise the theories of international trade and foreign currency options. Our results, which include a closed form expression for the transition density function of the exchange rate and an exact formula to price currency options, offer a theoretical framework for further study of foreign exchange markets     

混合分数布朗运动下欧式期权模糊定价研究

本文采用混合分数布朗运动来刻画标的股票价格的动态变化,以此体现金融市场的长记忆性特征。在混合分数Black-Scholes模型的基础上, 基于标的股票价格、无风险利率和波动率均是模糊数的假定下,构建了欧式期权模糊定价模型。其次,分析了金融市场长记忆性的度量指标 Hurst指数H对欧式期权模糊定价模型的影响。最后,数值实验表明:考虑长记忆性特征得到的欧式期权模糊定价模型更符合实际。     

The relative entropy in CGMY processes and its applications to finance

The CGMY market model generates infinite equivalent martingale measures (EMM). In order to price options, we need an adequate method to choose one EMM. This paper presents the relative entropy for CGMY processes, and apply it to choosing an EMM called the model preserving minimal entropy martingale measure.      

Numerical valuation of options under Kou's model

Numerical methods are developed for pricing European and American options under Kou's jump-diffusion model which assumes the price of the underlying asset to behave like a geometrical Brownian motion with a drift and jumps whose size is log-double-exponentially distributed. The price of a European option is given by a partial integro-differential equation (PIDE) while American options lead to a linear complementarity problem (LCP) with the same operator. Spatial differential operators are discretized using finite differences on nonuniform grids and time stepping is performed using the implicit Rannacher scheme. For the evaluation of the integral term easy to implement recursion formulas are derived which have optimal computational cost. When pricing European options the resulting dense linear systems are solved using a stationary iteration. Also for pricing American options similar iterations can be employed. A numerical experiment demonstrates that the described method is very efficient as accurate option prices can be computed in a few milliseconds on a PC. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

随机利率下服从分数O-U过程的欧式幂期权定价

该文考虑了利率和标的资产价格的随机性和均值回复行为,把扩展的Vasick模型和分数O-U过程进行组合,在随机利率环境下,研究了标的资产价格服从分数O-U过程的两类欧式幂期权定价问题,得到相应的定价公式,并给出了欧式幂期权的看涨.看跌平价关系.     

Pricing and Hedging of Lookback Options in Hyper-exponential Jump Diffusion Models

ABSTRACT

In this article, we consider the problem of pricing lookback options in certain exponential Lévy market models. While in the classic Black-Scholes models the price of such options can be calculated in closed form, for more general asset price model, one typically has to rely on (rather time-intense) Monte-Carlo or partial (integro)-differential equation (P(I)DE) methods. However, for Lévy processes with double exponentially distributed jumps, the lookback option price can be expressed as one-dimensional Laplace transform (cf. Kou, S. G., Petrella, G., & Wang, H. (2005). Pricing path-dependent options with jump risk via Laplace transforms. The Kyoto Economic Review, 74(9), 1–23.). The key ingredient to derive this representation is the explicit availability of the first passage time distribution for this particular Lévy process, which is well-known also for the more general class of hyper-exponential jump diffusions (HEJDs). In fact, Jeannin and Pistorius (Jeannin, M., & Pistorius, M. (2010). A transform approach to calculate prices and Greeks of barrier options driven by a class of Lévy processes. Quntitative Finance, 10(6), 629–644.) were able to derive formulae for the Laplace transformed price of certain barrier options in market models described by HEJD processes. Here, we similarly derive the Laplace transforms of floating and fixed strike lookback option prices and propose a numerical inversion scheme, which allows, like Fourier inversion methods for European vanilla options, the calculation of lookback options with different strikes in one shot. Additionally, we give semi-analytical formulae for several Greeks of the option price and discuss a method of extending the proposed method to generalized hyper-exponential (as e.g. NIG or CGMY) models by fitting a suitable HEJD process. Finally, we illustrate the theoretical findings by some numerical experiments.