基于几何Lévy过程的期权定价
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摘要
期权定价是金融数学理论中最基本,最重要和最具挑战性的工作之一。目前,传统的Black-Scholes期权定价模型已经相当流行,基于Black-Scholes框架的美式期权,亚式期权以及各种金融创新都得到了深入的研究和扩展。但是该模型用几何布朗运动来描述标的资产的价格运动,其不足之处是无法忽视的。为了更好地描述标的的运动状态,大量的研究工作试图将Black-Scholes模型推广到建立在Lévy过程上的模型,而这一推广带来了很多新问题亟待解决,如:市场的不完备导致鞅测度不唯一;模型的复杂程度增大导致计算相当困难等。为了解决这些问题,必须深入应用随机过程,时序,统计,数值计算和随机模拟的数学工具。本文探讨了当标的用几何Lévy过程描述时,从属市场和二维市场中期权定价的理论问题,即鞅测度的确定和选择;然后在此基础之上具体讨论了几种期权的定价。
我们首先讨论了从属市场中的期权定价。如果市场是一个公平的市场,其中不存在套利机会,那么无套利约束下一定存在这样的测度,使得标的资产价格(不存在红利时)的折现过程在该测度下是一个鞅,具有这样特征的测度称为等价鞅测度,如果市场是完备的,也就是说所有的未定权益都能被一个由标的资产和债券组成的投资组合所复制,那么等价鞅测度存在且唯一,并且未定权益的价格可以表示成为等价鞅测度下的一个数学期望的形式。Lévy系统下,市场是不完备的,在不完备的市场中,可能存在很多的等价鞅测度,由无套利原理得到的只是期权价值的一个范围,因此首先要解决的问题是如何选择和确定从属市场的鞅测度。本文选择最小熵作为对象,寻找从属市场的最小熵鞅测度(MEMM),并给出了从属市场欧式期权在该测度下的定价公式。研究表明从属市场的MEMM由一维市场的MEMM唯一确定,且Black-Scholes模型是从属市场的特例。
交换期权是一个两标的的期权,Margrabe的结论表明在Black-Scholes模型下交换期权可转换为一单资产模型,并在这一单资产模型的鞅测度下求交换期权的价值。本文探讨在Lévy系统下Margrabe的结论是否成立。我们对一特殊的二维几何Lévy过程模型进行了讨论,即假设驱动过程的连续鞅部分存在相关系数ρ,跳部分存在函数关系。首先我们给出了该二维几何Lévy过程模型的MEMM,然后对Margrabe的结论进行了验证,研究表明模型中的标的若含有跳,则Margrabe的结论不成立。
外国货币期权多是多标的期权,同交换期权的情况一样,我们假设多个标的的连续鞅部分存在相关系数ρ,跳部分存在一定的函数关系,这一假设是从模拟现实状况的观点出发以便反应汇率同各国货币价值运动的相互影响。我们对三种外国货币期权进行了研究,分别给出了期权价值满足的integro-differential方程和一种FFT近似计算方法,最后给出了一个具体实例说明标的的相关性对期权价值的影响。
可转换债券的价值等于一个普通债券加上一个隐含看涨期权,本文对这个隐含的看涨期权的价值进行了研究。分析表明这个隐含的看涨期权可看成一美式交换期权,我们可以利用测度变换将这一美式交换期权转换为一美式看跌期权,利用美式看跌期权的已有结论便可得到所求隐含看涨期权的价值。我们对双指数跳扩散模型进行了具体的分析,并运用双指数跳扩散模型下美式看跌期权的结论得到了该模型下隐含看涨期权的近似解。
以上内容的讨论都是基于一般意义下的几何Lévy过程。本文最后在一具体的几何Lévy过程模型和Black-Scholes模型下讨论基于不同平均方式的亚式期权定价。传统的亚式期权多是采用代数平均和几何平均,在本文中我们比较了基于这两种传统平均方式的亚式期权与采用调和平均的亚式期权的价值差异。利用随机模拟的方法我们发现当股票价格波动显著时采用调和平均要优于代数平均和几何平均。另外,我们还给出了调和平均亚式期权的一种近似计算方法。
In the theory of mathematical finance,option pricing is the most fundamental,important and challenging problem.The traditional Black-Scholes model has been very popular.American option,Asian option and various financial innovations in Black-Scholes framework have been studied deeply and extended.However,the asset prices are described by geometric Brownian motion and there are many shortcomings.In order to describe the underlyings' movements better,people are trying to promote the Black-Scholes model to the models based on Lévy processes.The promotion brings many new issues to be resolved,such as:there are many martingale measures since market is incomplete;the complex of model leads to the increase of calculation difficulty and so on.To solve these problems we must use the following mathematical tools deepgoingly: stochastic processes,statistics,time series,stochastic simulation and numerical calculation. In this paper,we discuss the option pricing when the underlyings are described by geometric Lévy processes.
Firstly,we discuss the option pricing in subordinated market.If the market is a fair market and there is no arbitrage opportunity,there is a measure under the non-arbitrage constrain such that the summation of the price process and the cumulated dividends is a martingale under this measure,which is called equivalent martingale measure.If the market is a complete market,that is to say any contingent claim is replicable by the portfolio of the underlying asset and the bond,there is a unique equivalent martingale measure such that the price of the contingent claim can be expressed by the mathematical expectation under the equivalent martingale measure.However,in Lévy system,the market is incomplete.In incomplete market,there are many equivalent martingale measures. We can only get a scope of the option's value if we value option by no arbitrage approach alone.Therefore,the first thing we must solve is how to choose and set the martingale measure.We choose the minimum entropy as the target.We give the representations of the minimal entropy martingale measure(MEMM) of subordinated market and the pricing formula of European option under this measure.Research shows that the MEMM of subordinated market can be determined by the MEMM of one-dimensional market.
Exchange option has two underlyings.Margrabe derived the exact analytic price formula for the European exchange option in the Black-Scholes framework.Margrabe suggested the valuation problem can be reduced to that of a one-asset option by treating asset two as numeraire.Then price the exchange option under the martingale measure of one-asset market.In this paper,firstly,we investigate the minimal entropy martingale measure and give the density processes for 2-dimensional geometric Lévy processes when the continuous parts of driving processes have correlation coefficient and the jump parts of diving processes have function relation.Secondly we validate whether Margrabe's conclusion is proper or not in our model.Research shows that Margrabe' conclusion is not estabilished when the model has jumps.
Many foreign currency options fall into the class of multi-state models.We discuss the foreign currency option pricing under the assumption as exchange option that the continuous martingale parts of driving processes have correlation coefficientρ;the jump parts of driving processes have mapping relations.Three kinds of foreign currency options are studied.We give the integro-differential equations and FFT approximate calculate method for these option prices.Finally,a special example shows the influence of the relevance of the underlyings to option prices.
The convertible bond carries two main components:the component as potential bond and the component as call option.We find this call option can be regarded as an American exchange option and this American exchange option can be converted to American put by the method of measure transformation.Under Lévy setting no explicite analytical expression is available.But there are many studies about the approximation of American put when the underlyings are particular Lévy processes.We can make use of these results to CB call and obtain the approximation.As an example,we give the approximation representations of CB call in double exponential jump diffusion model.
Finally,We compare the Asian options with three kinds of averaging procedures by Monte Carlo simulation and numerican illustration.We demonstrate that European Asian options with harmonic averaging behave better than Asian options with arithmetic and geometric averaging procedures for the stock with remarkable fluculation. Furthermore,approximation method fbr the valuation of harmonic average options and numerical illustrations are also given.
我们首先讨论了从属市场中的期权定价。如果市场是一个公平的市场,其中不存在套利机会,那么无套利约束下一定存在这样的测度,使得标的资产价格(不存在红利时)的折现过程在该测度下是一个鞅,具有这样特征的测度称为等价鞅测度,如果市场是完备的,也就是说所有的未定权益都能被一个由标的资产和债券组成的投资组合所复制,那么等价鞅测度存在且唯一,并且未定权益的价格可以表示成为等价鞅测度下的一个数学期望的形式。Lévy系统下,市场是不完备的,在不完备的市场中,可能存在很多的等价鞅测度,由无套利原理得到的只是期权价值的一个范围,因此首先要解决的问题是如何选择和确定从属市场的鞅测度。本文选择最小熵作为对象,寻找从属市场的最小熵鞅测度(MEMM),并给出了从属市场欧式期权在该测度下的定价公式。研究表明从属市场的MEMM由一维市场的MEMM唯一确定,且Black-Scholes模型是从属市场的特例。
交换期权是一个两标的的期权,Margrabe的结论表明在Black-Scholes模型下交换期权可转换为一单资产模型,并在这一单资产模型的鞅测度下求交换期权的价值。本文探讨在Lévy系统下Margrabe的结论是否成立。我们对一特殊的二维几何Lévy过程模型进行了讨论,即假设驱动过程的连续鞅部分存在相关系数ρ,跳部分存在函数关系。首先我们给出了该二维几何Lévy过程模型的MEMM,然后对Margrabe的结论进行了验证,研究表明模型中的标的若含有跳,则Margrabe的结论不成立。
外国货币期权多是多标的期权,同交换期权的情况一样,我们假设多个标的的连续鞅部分存在相关系数ρ,跳部分存在一定的函数关系,这一假设是从模拟现实状况的观点出发以便反应汇率同各国货币价值运动的相互影响。我们对三种外国货币期权进行了研究,分别给出了期权价值满足的integro-differential方程和一种FFT近似计算方法,最后给出了一个具体实例说明标的的相关性对期权价值的影响。
可转换债券的价值等于一个普通债券加上一个隐含看涨期权,本文对这个隐含的看涨期权的价值进行了研究。分析表明这个隐含的看涨期权可看成一美式交换期权,我们可以利用测度变换将这一美式交换期权转换为一美式看跌期权,利用美式看跌期权的已有结论便可得到所求隐含看涨期权的价值。我们对双指数跳扩散模型进行了具体的分析,并运用双指数跳扩散模型下美式看跌期权的结论得到了该模型下隐含看涨期权的近似解。
以上内容的讨论都是基于一般意义下的几何Lévy过程。本文最后在一具体的几何Lévy过程模型和Black-Scholes模型下讨论基于不同平均方式的亚式期权定价。传统的亚式期权多是采用代数平均和几何平均,在本文中我们比较了基于这两种传统平均方式的亚式期权与采用调和平均的亚式期权的价值差异。利用随机模拟的方法我们发现当股票价格波动显著时采用调和平均要优于代数平均和几何平均。另外,我们还给出了调和平均亚式期权的一种近似计算方法。
In the theory of mathematical finance,option pricing is the most fundamental,important and challenging problem.The traditional Black-Scholes model has been very popular.American option,Asian option and various financial innovations in Black-Scholes framework have been studied deeply and extended.However,the asset prices are described by geometric Brownian motion and there are many shortcomings.In order to describe the underlyings' movements better,people are trying to promote the Black-Scholes model to the models based on Lévy processes.The promotion brings many new issues to be resolved,such as:there are many martingale measures since market is incomplete;the complex of model leads to the increase of calculation difficulty and so on.To solve these problems we must use the following mathematical tools deepgoingly: stochastic processes,statistics,time series,stochastic simulation and numerical calculation. In this paper,we discuss the option pricing when the underlyings are described by geometric Lévy processes.
Firstly,we discuss the option pricing in subordinated market.If the market is a fair market and there is no arbitrage opportunity,there is a measure under the non-arbitrage constrain such that the summation of the price process and the cumulated dividends is a martingale under this measure,which is called equivalent martingale measure.If the market is a complete market,that is to say any contingent claim is replicable by the portfolio of the underlying asset and the bond,there is a unique equivalent martingale measure such that the price of the contingent claim can be expressed by the mathematical expectation under the equivalent martingale measure.However,in Lévy system,the market is incomplete.In incomplete market,there are many equivalent martingale measures. We can only get a scope of the option's value if we value option by no arbitrage approach alone.Therefore,the first thing we must solve is how to choose and set the martingale measure.We choose the minimum entropy as the target.We give the representations of the minimal entropy martingale measure(MEMM) of subordinated market and the pricing formula of European option under this measure.Research shows that the MEMM of subordinated market can be determined by the MEMM of one-dimensional market.
Exchange option has two underlyings.Margrabe derived the exact analytic price formula for the European exchange option in the Black-Scholes framework.Margrabe suggested the valuation problem can be reduced to that of a one-asset option by treating asset two as numeraire.Then price the exchange option under the martingale measure of one-asset market.In this paper,firstly,we investigate the minimal entropy martingale measure and give the density processes for 2-dimensional geometric Lévy processes when the continuous parts of driving processes have correlation coefficient and the jump parts of diving processes have function relation.Secondly we validate whether Margrabe's conclusion is proper or not in our model.Research shows that Margrabe' conclusion is not estabilished when the model has jumps.
Many foreign currency options fall into the class of multi-state models.We discuss the foreign currency option pricing under the assumption as exchange option that the continuous martingale parts of driving processes have correlation coefficientρ;the jump parts of driving processes have mapping relations.Three kinds of foreign currency options are studied.We give the integro-differential equations and FFT approximate calculate method for these option prices.Finally,a special example shows the influence of the relevance of the underlyings to option prices.
The convertible bond carries two main components:the component as potential bond and the component as call option.We find this call option can be regarded as an American exchange option and this American exchange option can be converted to American put by the method of measure transformation.Under Lévy setting no explicite analytical expression is available.But there are many studies about the approximation of American put when the underlyings are particular Lévy processes.We can make use of these results to CB call and obtain the approximation.As an example,we give the approximation representations of CB call in double exponential jump diffusion model.
Finally,We compare the Asian options with three kinds of averaging procedures by Monte Carlo simulation and numerican illustration.We demonstrate that European Asian options with harmonic averaging behave better than Asian options with arithmetic and geometric averaging procedures for the stock with remarkable fluculation. Furthermore,approximation method fbr the valuation of harmonic average options and numerical illustrations are also given.
引文
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