跳跃——-扩散型欧式加权几何平均价格亚式期权定价

在亚式期权定价理论的基础上, 对期权的标的资产价格引入跳跃---扩散过程进行建模, 用几何Brown运动描述其常态连续变动, 用Possion过程刻画资产价格受新信息和稀有偶发事件的冲击发生跳跃的记数过程, 用对数正态随机变量描述跳跃对应的跳跃幅度, 在模型限定下运用Ito-Skorohod微分公式和等价鞅测度变换, 导出欧式加权几何平均价格亚式期权封闭形式的解析定价公式     

跳扩散模型下的欧式障碍期权的定价

本文在标的资产价格服从跳扩散模型的假设下,运用Girsanov定理获得了价格过程的等价鞅测度,用期权定价的鞅方法得出障碍期权的定价公式.     

双随机跳扩散模型下亚式期权的定价

研究了双随机跳扩散模型下的亚式期权的定价问题.首先引入一个双随机跳扩散过程.然后通过测度变换消除了亚式期权定价中的路经依赖性问题.最后利用鞅定价方法和Ito引理得到了跳扩散模型下的亚式期权价格必须满足的一个积微分方程.通过数值求解该积微分方程就可以得到了亚式期权的价格,供投资者参考.     

随机波动模型下算术亚式期权的Monte Carlo模拟定价

在随机波动模型下,研究亚式期权的定价问题.推导出了标的资产及其随机波动模型的路径,利用对偶变量法对亚式期权进行数值模拟计算,并对随机波动模型下与B-S模型下的欧式期权和亚式期权定价结果进行比较,最后给出了具有固定敲定价格和浮动敲定价格的算术亚式期权的数值计算结果.     

跳跃-扩散过程下百慕大交换期权定价

在等价鞅测度下,利用条件期望等知识导出在风险中性定价模型中,标的资产服从跳跃-扩散过程时百慕大交换期权的解析定价公式,依此结合Richardson两点外推加速法得到美式交换期权近似解.提出的数值算例阐明提前执行特征具有重要经济价值.定价结果可以评估场外交易的金融期权价格尤其是实物期权定价.     

O-U过程模型下一种改进的亚式再装期权定价

考虑到再装期权对企业经理人的激励作用,结合将有效期内标的资产的几何平均值作为期权结算价格的思想,创建了一种改进的亚式再装股票期权模型,利用等价鞅方法,推导了该期权模型在OU过程下的定价公式.     

股票价格服从指数O-U过程的再装期权定价

期权及其定价理论是目前金融管理,金融工程研究的前沿与热点问题.本文在标的资产的价格服从指数O-U过和模型假设下,运用G irsanov定理获得了该过程的唯一等价鞅测度.用期权定价的鞅方法,得出了再装期权的定价公式.     

移动平均亚式期权定价研究

移动平均期权是价格依赖于标的资产移动平均价格的奇异期权,其定价依赖于每个窗口内的标的资产的价格,随着窗口不断向前滚动,就会有无穷多个移动平均过程,而无穷多个移动平均过程是无限维的非马尔科夫问题,因此该研究具有较大的挑战性,研究成果较少.移动平均亚式期权多用于场外的能源(石油,天然气,电力)衍生品合约,因此研究其定价和数值计算方法具有一定的理论意义和实践价值.本文以移动平均亚式期权为研究对象,运用截断的拉盖尔序列的方法,通过估计近似测度,以有限维的移动平均过程近似无限维的移动平均过程.随着拉盖尔序列所取阶数的增加,有限维移动平均对无限维移动平均的近似效果将越来越好.在得到近似的有限维标的资产价格的移动平均过程后,移动平均期权的定价转化为一个美式期权最优停时问题,本文用最小二乘蒙卡解决美式移动平均期权的最优行权问题,从而给移动平均亚式期权定价,数值分析表明通过对比拉盖尔方法和蒙卡模拟的方法,拉盖尔方法的稳定性较高.而对于固定执行价格和浮动执行价格移动平均亚式期权,都存在随着窗口长度增加,期权价格上涨的递增关系,这与一般的逻辑推断吻合.数值计算结果还表明当窗口长度增加到期权整个有效期时,移动平均亚式期权就退化为相应的亚式期权.进一步证明了该方法的正确性.     

分数布朗运动驱动的幂型期权定价模型研究

股价运动分形特征的发现,说明布朗运动作为期权定价模型的初始假定存在缺陷.本文假定标的资产价格服从几何分数布朗运动,利用分数风险中性测度下的拟鞅(quasi-martingale)定价方法重新求解分数Black-Scholes模型,进而对幂型期权进行定价.结果表明,幂型期权结果包含了Black-Scholes公式和平方期权结果,且相比标准期权价格,分数期权价格要同时取决于到期日和Hurst参数H.     

随机利率下奇异期权的定价公式

在随机利率条件下,借助于测度变换获得了复合看涨期权的一般的定价公式,同时利用鞅理论和Girsanov定理,在利率服从于扩展的Vasicek利率模型时,得到了复合看涨期权精确的定价公式.用同样的方法,考虑了预设日期的重置看涨期权的定价问题,在利率服从同样的利率模型时,获得了重置看涨期权的定价公式.数值化的结果进一步说明了当利率遵循扩展的Vasicek利率模型时,B-S看涨期权的价格关于标的资产的价格是严格单调递增的,复合看涨期权的Geske公式是可以推广到随机利率的情况.     

Option Pricing Under Jump-Diffusion Processes with Regime Switching

We study an incomplete market model, based on jump-diffusion processes with parameters that are switched at random times. The set of equivalent martingale measures is determined. An analogue of the fundamental equation for the option price is derived. In the case of the two-state hidden Markov process we obtain explicit formulae for the option prices. Furthermore, we numerically compare the results corresponding to different equivalent martingale measures.     

A Convenient Way to Characterize Equivalent Martingale Measures in Incomplete Markets

It is an empirical fact that the (empirically) relevant models for asset prices often describe markets that are incomplete in terms of their underlying assets, yielding many possible equivalent martingale measures under the no-arbitrage assumption. By using actual derivative prices, i.e., prices as observed in the market, additional information about the empirically relevant equivalent martingale measures might be obtained. In order to be able to process such information easily one needs a convenient way to represent all possible equivalent martingale measures in relation to derivative prices. In this paper we present such a convenient characterization. Conceptually, our characterization is not different from existing characterizations using, for example, Radon–Nikodym derivatives of martingale measures with respect to objective probabilities, but our characterization offers some advantages. The main advantage is that pricing derivatives is split up into two steps. The first step is solving a related complete markets pricing problem. This is a well-studied problem, so that it can easily be solved generally. In the second step a weighted average of the first step complete markets price must be calculated. Pricing under different equivalent martingale measures in the original market only differs with respect to the second step. The empirically relevant weighting can be determined by confronting the theoretical with the actually observed prices. As a byproduct we obtain a new and natural definition of idiosyncratic risk, which we show to be in line with the use of this term in the literature.To illustrate the ideas we discuss several examples. Among others we obtain the Hull–White formula for options on assets with stochastic volatility under close to minimal conditions that (for example) do not rely on a specification of the processes in terms of Itô diffusion.we relax the assumption of no-correlation between asset prices and volatilities in the Hull–White framework; we consider the case where the stochastic volatility does bear a risk-premium; we discuss pricing under stochastic interest rates; and we consider square-root type processes. All these pricing problems, and many more, can conveniently be handled using the approach based on our characterization of the equivalent martingale measures in continuous time markets that are incomplete in the underlying assets.     

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.      

Optimal Investment in a Levy Market

In this paper we consider the optimal investment problem in a market where the stock price process is modeled by a geometric Levy process (taking into account jumps). Except for the geometric Brownian model and the geometric Poissonian model, the resulting models are incomplete and there are many equivalent martingale measures. However, the model can be completed by the so-called power-jump assets. By doing this we allow investment in these new assets and we can try to maximize the expected utility of these portfolios. As particular cases we obtain the optimal portfolios based in stocks and bonds, showing that the new assets are superfluous for certain martingale measures that depend on the utility function we use.     

Coupling and option price comparisons in a jump-diffusion model

In this paper, we examine the dependence of option prices in a general jump-diffusion model on the choice of martingale pricing measure. Since the model is incomplete, there are many equivalent martingale measures. Each of these measures corresponds to a choice for the market price of diffusion risk and the market price of jump risk. Our main result is to show that for convex payoffs, the option price is increasing in the jump-risk parameter. We apply this result to deduce general inequalities, comparing the prices of contingent claims under various martingale measures, which have been proposed in the literature as candidate pricing measures.

Our proofs are based on couplings of stochastic processes. If there is only one possible jump size then we are able to utilize a second coupling to extend our results to include stochastic jump intensities.     

一类特殊半鞅模型中无套利测度的刻划

本文研究金融市场中一类特殊半鞅模型,其价格过程具有X=LD的形式,这里L是局部有界鞅,D是可料有限变差过程.对这类模型我们导出其等价鞅测度存在的充分必要条件.另外,我们将[2]中的条件/△M/≤C推广到M为局部有界鞅,得到相应的结果.     

Enlarged filtrations and indistinguishable processes

Abstract

We study modification properties of stochastic processes under different probability measures in an initially enlarged filtration setup. For this purpose, we consider several pure-jump Lévy processes under two equivalent probability measures and derive the associated martingale compensators with respect to different enlarged filtrations. As our main result, we prove that the obtained martingale processes under different probability measures in our enlarged filtration approach are indistinguishable. In addition, we provide a condition under which the pure-jump result can be carried over to the Brownian motion case. In this context, we show how indistinguishable Brownian motions under different probability measures can be constructed in an enlarged filtration framework. We finally apply our theoretical results to precipitation derivatives pricing under weather forecasts.     

双参数Esscher变换及其应用

通过把Lévy过程分解为两个独立过程之和,将Esscher变换由单参数推广到双参数,并给出了双参数Esscher变换测度为等价鞅测度的充要条件．     

A reduced form model for ESO valuation

In this paper we extend a reduced form model for the valuation of employee share options (ESOs) to incorporate employee departure, and company takeover. We also allow for performance linked vesting and other exotic features specific to ESOs. We clarify the assumptions underlying the reduced form model, and discuss their implications. We analyze the probabilistic structure of the model which includes an explicit characterization of the set of equivalent martingale measures, as well as the computation of the variance optimal martingale measure and the minimal martingale measure. Moreover, we deduce an additive decomposition of the relative entropy. Particular ESO specifications are studied emphasizing different aspects of the proposed framework. In this context, we also provide strict no-arbitrage bounds for ESO prices by applying optimal stopping. Furthermore, possible limitations of the proposed model are explored by examining departures from the crucial assumptions of no-arbitrage, i.e. by considering the effects of the employee having inside information.     

Optional decompositions under constraints

Summary. Motivated by a hedging problem in mathematical finance, El Karoui and Quenez [7] and Kramkov [14] have developed optional versions of the Doob-Meyer decomposition which hold simultaneously for all equivalent martingale measures. We investigate the general structure of such optional decompositions, both in additive and in multiplicative form, and under constraints corresponding to different classes of equivalent measures. As an application, we extend results of Karatzas and Cvitanić [3] on hedging problems with constrained portfolios. Received: 6 August 1996/In revised form: 5 March 1997