基于跳扩散过程的欧式交换期权定价

考虑了股票价格服从带时滞泊松跳的跳扩散模型的欧式交换期权定价问题,运用无套利理论推导出期权价值微分方程,利用变换计价单位的方法,得到交换期权的显示定价公式.     

Pricing Options under Two-Factor Markov-Modulated Stochastic Volatility Models

In this paper, we investigate the valuation of European-style call options under an extended two-factor Markov-modulated stochastic volatility model, where the first stochastic volatility component is driven by a mean-reversion square-root process and the second stochastic volatility component is modulated by a continuous-time, finite-state Markov chain. The inverse Fourier transform is adopted to obtain analytical pricing formulae. Numerical examples are given to illustrate the discretization of the pricing formulae and the implementation of our model.     

基于跳扩散过程的回望期权定价的数值算法

运用加权最小二乘蒙特卡洛模拟法(WLSM)研究标的资产服从跳扩散过程的美式回望期权定价问题,改进了Longstaff等提出的最小二乘模拟法.运用WLSM对美式回望期权进行定价,数值实验结果表明该方法具有较为显著的优势.     

Pricing Options Under a Generalized Markov-Modulated Jump-Diffusion Model

Abstract

We consider the pricing of options when the dynamics of the risky underlying asset are driven by a Markov-modulated jump-diffusion model. We suppose that the market interest rate, the drift and the volatility of the underlying risky asset switch over time according to the state of an economy, which is modelled by a continuous-time Markov chain. The measure process is defined to be a generalized mixture of Poisson random measure and encompasses a general class of processes, for example, a generalized gamma process, which includes the weighted gamma process and the inverse Gaussian process. Another interesting feature of the measure process is that jump times and jump sizes can be correlated in general. The model considered here can provide market practitioners with flexibility in modelling the dynamics of the underlying risky asset. We employ the generalized regime-switching Esscher transform to determine an equivalent martingale measure in the incomplete market setting. A system of coupled partial-differential-integral equations satisfied by the European option prices is derived. We also derive a decomposition result for an American put option into its European counterpart and early exercise premium. Simulation results of the model have been presented and discussed.     

一个条件数学期望公式在期权定价中的应用

首先证明一个条件数学期望公式,然后建立股票价格的跳过程为Poisson过程,跳跃高度为常数时股票价格过程的随机微分方程,在风险中性的假设下,找等价鞅测度.利用鞅方法和已证明的条件数学期望公式,用较简单的数学推导得到了股票价格股从跳—扩散过程的欧式期权以及复合期权的定价公式.     

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.     

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In this paper, we extend the previous Markov-modulated reflected Brownian motion model discussed in [1] to a Markov-modulated reflected jump diffusion process, where the jump component is described as a Markov-modulated compound Poisson process. We compute the joint stationary distribution of the bivariate Markov jump process. An abstract example with two states is given to illustrate how the stationary equation described as a system of ordinary integro-differential equations is solved by choosing appropriate boundary conditions. As a special case, we also give the sationary distribution for this Markov jump process but without Markovian regime-switching.     

随机利率下分数跳-扩散Ornstein-Uhlenbeck期权定价模型

假设股票价格遵循分数布朗运动和复合泊松过程驱动的随机微分方程,短期利率服从HullWhite模型,建立了随机利率情形下的分数跳-扩散Ornstein-Uhlenbeck期权定价模型,利用价格过程的实际概率测度和公平保费原理,得到了欧式看涨期权定价的解析表达式,推广了Black-Scholes模型.     

更新跳跃-扩散过程下的复合期权定价

假设关于标的股票的重大信息到达服从更新过程,并假设跳跃高度服从对数正态分布,利用期权定价的鞅方法,推导得到了股票价格服从更新跳跃-扩散过程的欧式期权以及复合期权的定价公式.     

Risk Minimizing Option Pricing for a Class of Exotic Options in a Markov-Modulated Market

We address risk minimizing option pricing in a regime switching market where the floating interest rate depends on a finite state Markov process. The growth rate and the volatility of the stock also depend on the Markov process. Using the minimal martingale measure, we show that the locally risk minimizing prices for certain exotic options satisfy a system of Black-Scholes partial differential equations with appropriate boundary conditions. We find the corresponding hedging strategies and the residual risk. We develop suitable numerical methods to compute option prices.     

跳扩散模型下权益指数年金的定价

权益指数年金收益在最小保证基础上, 能参与特定权益的收益. 通常权益指数年金定价是在假设权益指数遵从Black-Scholes模式下进行的, 但是一些例外事件(比如, 重大的政治事件)的发生, 会导致价格的巨幅波动, 这个假设并不合理. 因此本文研究了权益指数在跳扩散模型下权益指数年金的定价问题. 运用Esscher变换方法得到了点对点指数收益方法下权益指数年金定价的显示解,并对结果作了敏感性分析.     

房价服从非时齐Poisson跳扩散的住房抵押贷款保证险的定价

利用期权定价理论和保险精算方法, 分析了住房抵押贷款保证险的定价问题, 给出了全额担保和部分担保两类住房抵押贷款保证险的定价公式, 其中未偿付额服从一般扩散过程, 房产价格服从带非时齐Poisson跳的扩散过程.     

有限状态Q过程波动率与跳组合情形的期权定价

引入了有限状态Q过程随机波动率与复合Poisson过程组合的资产价格动态模型,得到了该组合模型下欧式看涨期权定价的一般公式,推广了Hull和White的结论.最后通过数值模拟,充分体现了期权价格对初始时刻波动率大小的依赖.     

分数布朗运动和泊松过程共同驱动下的择好期权定价

本文讨论两资产择好期权的定价问题。在风险中性假设下,建立了两资产价格过程遵循分数布朗运动和带非时齐Poisson跳跃—扩散过程的择好期权定价模型,应用期权的保险精算法,给出了相应的择好期权的定价公式。     

基于带跳随机波动率模型的双币种重置期权定价研究

研究了外国标的资产价格,汇率及其波动率过程满足仿射跳扩散模型的双币种重置期权定价问题,其中波动率过程与标的资产,汇率相关,且具有共同跳跃风险成分.利用多维Feynman-Kac定理,Fourier逆变换等方法,获得了双币种重置期权价格的表达式.应用数值计算分析了波动率过程主要参数对期权价格的影响.数值结果表明,波动率因素以及跳跃风险参数对期权价格的影响是显著的.     

Vasicek随机利率和纯生跳扩散模型下的期权定价

在Vasicek随机利率模型且股票价格服从纯生跳扩散过程的情形下,利用测度变换的Girsanov定理找到定价鞅测度,推导出了有连续红利支付的且影响股票价格的标准Brown运动与影响利率的标准Brown运动相关时欧式股票期权的定价公式,最后给出此定价模型的一些特例以及算例.     

指数Levy跳扩散模型下一类新型期权的定价研究

在指数levy跳扩散模型下,通过在确定的两个时间点之间设置一个特定的常数障碍水平,构造出一类两时间点两资产最大或最小值障碍期权.这种新型期权具有两时间点彩虹期权与障碍期权的双重性质,使得该新型期权在未定权益定价方面的应用更为广泛.最后利用鞅方法,给出了该类期权的定价公式.     

跳扩散模型下一种创新的重置期权定价

首先在风险中性测度下建立股票价格的跳过程为Poisson过程,跳跃高度服从对数正态分布时股票价格的随机微分方程,利用期权定价的鞅方法推导得到了欧式重置看涨期权的价格以及一种创新的重置看涨期权的定价公式．最后给出了一个数值计算的例子,说明了创新的重置看涨期权价格要大于或等于传统的重置看涨期权和欧式看涨期权价格,并从理论上进行解释．     

股价遵循分数跳扩散的混合型双标的两值期权定价

应用风险中性定价原理,研究标的股价服从分数跳扩散过程的混合型双标的两值期权的定价问题,并得出定价公式,并与股价服从标准布朗运动的定价公式做出比较分析.     

Pricing of Spread Options on a Bivariate Jump Market and Stability to Model Risk

Abstract

We study the pricing of spread options and we obtain a Margrabe-type formula for a bivariate jump-diffusion model. Moreover, we study the robustness of the price to model risk, in the sense that we consider two types of bivariate jump-diffusion models: one allowing for infinite activity small jumps and one not. In the second model, an adequate continuous component describes the small variation of prices. We illustrate our computations by several examples.