Moore-Penrose逆在期权定价中的应用研究
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摘要
自从1973年Black和Scholes[1]的开创性论文以来,未定权益的定价理论得到了迅猛发展,其中以无套利定价理论最为突出.粗略的讲,该理论表明市场无套利假设等价于存在鞅测度.如果市场是完全的,则无套利假设等价于存在唯一的鞅测度,从而,对于完全市场中的任何未定权益,都有唯一的无套利价格.然而在不完全市场中,存在许多鞅测度,对不可复制的未定权益,也就有许多无套利价格.这样,仅仅利用无套利理论无法得出未定权益唯一的价格.大量实证结果表明,实际市场是不完全的,因此研究不完全市场中未定权益的定价更加具有实际意义.
本文在利用随机过程的M-P逆求解一类线性随机方程的基础上,对不完全市场中的等价鞅测度进行了较为深入的研究,主要研究内容和结果如下.
在绪论中,主要对金融数学的发展历史,特别是对期权定价理论的研究内容、成果及目前研究热点进行了较为详尽的介绍.
第二章,首先通过把Levy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,从而得到一簇(双参数)概率测度.我们给出了这簇概率测度是等价鞅测度的充要条件.其次,在几何Levy过程模型中,利用均值修正方法构造了一个概率测度Qmn.我们证明了Qmn为等价鞅测度的充要条件是Levy过程具有Brownian运动部分,并且给出了此时Qmn关于市场概率测度的导数表达式.对于纯跳过程,我们证明了此时Qmn与市场概率测度不可能等价,但在Qmn下计算出来的欧式看涨期权价格却是无套利的.
第三章,Dzhaparidze和Spreij[2]证明了任意Rd值,可料的局部平方可积鞅的二次变差过程的M-P逆保持可料性.受此启发,我们首先证明了任意Rd×n值,可料随机过程的M-P逆仍然可料,从而推广了Dzhaparidze和Spreij[2]的结果.在此基础之上,我们进一步讨论了一类线性随机方程可料解的性质及可料解的结构.
第四章,利用第三章的结论,在扩散系数矩阵不必几乎处处满秩的条件下重新讨论了扩散模型.我们利用扩散系数矩阵的M-P逆刻画出了全体等价鞅测度,给出了扩散模型中一些等价鞅测度的具体表达式.第五章,以M-P逆的形式给出了半鞅模型中的最小鞅测度的一般表达式,并得到了扩散模型,跳扩散模型以及几何Levy过程模型的最小鞅测度.
Since the creative study of Black and Scholes[1] in 1973, theory of contingent pricing has been developed rapidly, especially in arbitrage-free pricing theory. This theory shows that the arbitrage-free market hypothesis is equivalent to the existence of martingale measures. If the market is complete, the arbitrage-free market hypothesis is equivalent to the existence of unique martingale measure. Thereby, any contingent of complete market must have a unique arbitrage-free price. However, if the market is incomplete, there may exist many equivalent martingale measures, and many different arbitrage price systems. Therefore, the valuation and hedging issues are not definite in this situation. Many empirical results show that actual market is incomplete, so studying on option pricing in incomplete is much more practical significance.
In this paper, we solve a kind of linear stochastic equations by M-P inverse of stochastic processes, and study equivalent martingale measures in incomplete market. The research contents and results as follow.
In introduction, we introduce the history of mathematical finance, especially the main research contents, results and hot topics of option pricing theory.
In Chapter 2, firstly, we extend Esscher transform to two-parameters by decom-posing Levy processes as the sum of two independent processes, and we get a cluster of (two parameters) probability measures. What's more, a necessary and sufficient con-dition for two-parameters Esscher transform measures to be equivalent martingale mea-sures is given. Secondly, a probability measure Qmo is constructed by mean correcting transform for geometric Levy process model. This paper proves Qmo is an equiva-lent martingale measure if and only if Le'vy processes has Brownian part, and gives Radon-Nikodym formula. Although Qmo can not be equivalent to physical measure for a pure jump Levy process, we show that a European call option price under Qmo is still arbitrage free.
In Chapter 3, we extend the conclusion of Dzhaparidze and Spreij[2] which says the M-P inverse of any Rd value, predictable, locally square integrable martingale is predictable, and we prove that the M-P inverse of any Rd×n value, predictable stochas-tic process is predictable. Further, the property and construction of predictable solution of linear stochastic equation are discussed.
In Chapter 4, we discuss the diffusion model under the condition which dispersion matrix is not always full rank once again. All equivalent martingale measures in the diffusion model are depicted and the concrete formula of some equivalent martingale measures are given by using of the M-P inverse of dispersion matrix.
In Chapter 5, by using of M-P inversewe, we derive general formula of the mini-mal martingale measures in semimatingale model, especially in diffusion model, jump diffusion model and geometry Levy model.
本文在利用随机过程的M-P逆求解一类线性随机方程的基础上,对不完全市场中的等价鞅测度进行了较为深入的研究,主要研究内容和结果如下.
在绪论中,主要对金融数学的发展历史,特别是对期权定价理论的研究内容、成果及目前研究热点进行了较为详尽的介绍.
第二章,首先通过把Levy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,从而得到一簇(双参数)概率测度.我们给出了这簇概率测度是等价鞅测度的充要条件.其次,在几何Levy过程模型中,利用均值修正方法构造了一个概率测度Qmn.我们证明了Qmn为等价鞅测度的充要条件是Levy过程具有Brownian运动部分,并且给出了此时Qmn关于市场概率测度的导数表达式.对于纯跳过程,我们证明了此时Qmn与市场概率测度不可能等价,但在Qmn下计算出来的欧式看涨期权价格却是无套利的.
第三章,Dzhaparidze和Spreij[2]证明了任意Rd值,可料的局部平方可积鞅的二次变差过程的M-P逆保持可料性.受此启发,我们首先证明了任意Rd×n值,可料随机过程的M-P逆仍然可料,从而推广了Dzhaparidze和Spreij[2]的结果.在此基础之上,我们进一步讨论了一类线性随机方程可料解的性质及可料解的结构.
第四章,利用第三章的结论,在扩散系数矩阵不必几乎处处满秩的条件下重新讨论了扩散模型.我们利用扩散系数矩阵的M-P逆刻画出了全体等价鞅测度,给出了扩散模型中一些等价鞅测度的具体表达式.第五章,以M-P逆的形式给出了半鞅模型中的最小鞅测度的一般表达式,并得到了扩散模型,跳扩散模型以及几何Levy过程模型的最小鞅测度.
Since the creative study of Black and Scholes[1] in 1973, theory of contingent pricing has been developed rapidly, especially in arbitrage-free pricing theory. This theory shows that the arbitrage-free market hypothesis is equivalent to the existence of martingale measures. If the market is complete, the arbitrage-free market hypothesis is equivalent to the existence of unique martingale measure. Thereby, any contingent of complete market must have a unique arbitrage-free price. However, if the market is incomplete, there may exist many equivalent martingale measures, and many different arbitrage price systems. Therefore, the valuation and hedging issues are not definite in this situation. Many empirical results show that actual market is incomplete, so studying on option pricing in incomplete is much more practical significance.
In this paper, we solve a kind of linear stochastic equations by M-P inverse of stochastic processes, and study equivalent martingale measures in incomplete market. The research contents and results as follow.
In introduction, we introduce the history of mathematical finance, especially the main research contents, results and hot topics of option pricing theory.
In Chapter 2, firstly, we extend Esscher transform to two-parameters by decom-posing Levy processes as the sum of two independent processes, and we get a cluster of (two parameters) probability measures. What's more, a necessary and sufficient con-dition for two-parameters Esscher transform measures to be equivalent martingale mea-sures is given. Secondly, a probability measure Qmo is constructed by mean correcting transform for geometric Levy process model. This paper proves Qmo is an equiva-lent martingale measure if and only if Le'vy processes has Brownian part, and gives Radon-Nikodym formula. Although Qmo can not be equivalent to physical measure for a pure jump Levy process, we show that a European call option price under Qmo is still arbitrage free.
In Chapter 3, we extend the conclusion of Dzhaparidze and Spreij[2] which says the M-P inverse of any Rd value, predictable, locally square integrable martingale is predictable, and we prove that the M-P inverse of any Rd×n value, predictable stochas-tic process is predictable. Further, the property and construction of predictable solution of linear stochastic equation are discussed.
In Chapter 4, we discuss the diffusion model under the condition which dispersion matrix is not always full rank once again. All equivalent martingale measures in the diffusion model are depicted and the concrete formula of some equivalent martingale measures are given by using of the M-P inverse of dispersion matrix.
In Chapter 5, by using of M-P inversewe, we derive general formula of the mini-mal martingale measures in semimatingale model, especially in diffusion model, jump diffusion model and geometry Levy model.
引文
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