体制转换模型下金融衍生品的定价研究
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摘要
2008年全球金融危机以来,宏观经济条件与商业周期的结构变化给金融市场带来的影响引起了人们的关注。自从Hamilton(1989)将体制转换模型引入金融计量学领域以来,马尔可夫调制的体制转换模型受到了学术研究者与行业从业人员的青睐。体制转换模型通常假设调制马尔可夫链的状态代表市场经济状态,从而可以将宏观经济条件与商业周期的结构变化所带来的影响考虑进来。因此,研究体制转换模型下金融衍生品的定价是具有现实意义的。
基于马尔可夫链是连续时间、有限状态的假设,本文在前人研究成果的基础上分别考虑了体制转换扩散模型、体制转换跳扩散模型、体制转换随机利率模型、双体制转换模型与隐马尔可夫模型。由于引入马尔可夫链带来的不确定性,体制转换模型下的金融市场通常是不完备的,也就意味着该市场中存在着不唯一的等价鞅测度。如何在不完备市场中选取等价鞅测度一直以来也是备受关注的研究问题之一,本文分别采用体制转换Esscher变换与最小鞅测度两种方法选取等价鞅测度。然后,主要采用快速傅立叶变换方法推导体制转换模型下金融衍生品的解析定价公式,并通过数值分析得到相应结论。
第一章首先介绍了金融衍生品的定价模型推广。接着,介绍了一些热点问题的研究现状,主要包括不同体制转换模型下的期权定价研究,隐马尔可夫模型研究,随机利率模型研究,不完备市场下选取等价鞅测度的方法以及快速傅立叶变换的方法。然后给出了本文需要的预备知识。具体说来,本章给出了调制马尔可夫模型的数学表达式,介绍了如何通过傅立叶变换得到解析定价公式以及通过快速傅立叶变换将定价公式离散化的过程。最后,简要介绍了本文的主要研究工作。
在第二章到第四章中,本文假设体制转换模型参数是被连续时间、有限状态、可观测的马尔可夫链调制的,并考虑了不同体制转换模型下不同种类期权的定价问题。
第二章中讨论了体制转换均值回归对数正态模型下外国股权期权的定价问题。顾名思义,外国股权期权中标的资产的表示货币与结算货币是不同类型的货币。直观上理解,外国股权期权的定价问题需要研究外国股权与汇率服从的过程。第二章假设汇率服从体制转换性质的均值回归对数正态模型,探讨了外国股权期权的定价问题。按照行权价格表示货币的不同,将外国股权期权分为两大类,一类的行权价格是以外国货币表示的,记为FEOF,另外一类的行权价格是以国内货币表示的,记为FEODo为了应用快速傅立叶变换方法,本章首先推导出对数标的资产的特征函数。对于FEOF的定价,本章首先采用测度变换技巧,以外汇汇率的期望值为计价单位。FEOD的定价过程则需要将对数外国股权价格的特征函数与对数外汇汇率的特征函数进行加和。然后,通过逆傅立叶变换方法推导出FEOF期权与FEOD期权的解析定价公式。最后,本章给出了相应的数值例子和经验分析。根据敏感性分析,本章讨论了模型中参数对于期权价格的影响。经验分析中,本章采用2003年10月1日至10月17日期间日经225指数欧式看涨期权与美元／日元汇率的市场数据,估计模型参数,并比较得出体制转换均值回归对数正态模型的样本内拟合误差与样本外预测误差均小于均值回归对数正态模型相应误差的结论。
第三章考虑了体制转换跳扩散模型下幂期权的定价问题,该模型假设跳部分是由Poisson随机测度刻画的。幂期权是一种具有非线性收益的金融产品,因此,可以用来应对金融市场中存在的非线性风险,为投资者提供了一种便捷的风险管理工具。文章假定利率,标的资产的平均回报率与波动率,补偿因子均与经济状态有关。通过体制转换Esscher变换选择了一个等价鞅测度,并推导风险中性测度下对数标的资产的特征函数。然后,本章应用逆傅立叶变换推导幂期权的解析定价公式,并通过快速傅立叶变换方法将定价公式离散求解。最后,数值分析中,本章选定了两种特定的补偿因子,马尔可夫调制逆Gaussian过程和马尔可夫调制Merton跳扩散模型,探讨两种特定的补偿因子下幂期权的定价过程,并分析不同参数对幂期权价格的影响。
第四章讨论了体制转换随机利率模型下欧式期权的定价问题。本章假设短期利率服从体制转换的Hull-White模型,具体来说,利率的均值回归水平与波动率,无风险利率,标的资产的波动率以及相关系数都随着马尔可夫链状态的变化而改变。本章采用测度变换技巧将风险中性测度变换到远期测度。此时,马尔可夫链的速率矩阵也会产生相应的变化。在远期测度下,本章推导出对数标的资产的特征函数表达式,并通过快速傅立叶变换方法对期权进行定价。
第五章-第六章主要讨论体制转换模型下带有内嵌期权特征的保险产品的定价问题。
第五章研究了双体制转换模型下考虑死亡风险的权益连结年金的定价过程。双体制转换模型假设模型参数的取值随着马尔可夫链状态的变化而改变,与此同时,标的资产的价格将随着马尔可夫链状态的变化而发生一个跳。双体制转换模型的一个最主要特点是可以内生决定体制转换风险的大小,为本章提供了一种量化体制转换风险的方法。为了选取等价鞅测度,本章分别采用广义体制转换Esscher变换与最小鞅测度方法,并对这两种方法选取的等价鞅测度进行简要分析。需要注意的是,马尔可夫链的速率矩阵是随着测度变换而变化的。对于内嵌期权特征的保险产品来说,很多学者都采用期权定价技巧来研究。本章将权益连结年金产品的收益看作是一系列欧式期权收益的组合。因此,本章可以采用期权定价技巧研究相应权益连结年金产品的定价问题。通过本章的数值分析和敏感性分析,我们可以很直观地理解权益连结年金产品的定价过程及不同模型参数对产品价格的影响。
第六章研究隐马尔可夫模型下动态保护基金的定价问题,该模型下调制马尔可夫链的状态是不可观测的。动态保护基金是规避下跌风险的一种有效工具。动态保护基金的收益可以看作是具有固定行权价格的回望看涨期权的收益与外汇汇率的乘积。类似地,可以采用期权定价技巧来对动态保护基金进行定价。本章首先通过体制转换Esscher变换选取一个等价鞅测度,然后采用偏微分方程方法对动态保护基金进行定价。由于马尔可夫链的状态是不可观测的,于是本章采用三步估计方法对隐马尔可夫模型进行参数估计。首先,基于实际市场数据,本章采用Baum-Welch算法估计马尔可夫链的参数,然后,通过Viterbi算法选择马尔可夫链的最可能路径,最后,通过最大似然估计方法对模型参数进行估计。该方法简单易行,容易理解。
Following the global financial crisis of2008, the impacts of changes in (macro)-economic conditions and business cycles have attracted increasing interests in the new millennium. Regime-switching models have been considered as a natural tool of pric-ing financial derivatives by both academic researchers and industrial practitioners since Hamilton (1989) introduced this class of models into financial econometrics. Regime-switching models typically use the states of the modulating Markov chain to represent the states of an economy, depicted by some (macro)-economic indicators. By adopting this methodology, regime-switching models can incorporate the impacts of structural changes in (macro)-economic conditions. Consequently, it is practical to consider the valuation of financial derivatives under regime-switching models.
In this thesis, the Markov chain we adopt is a continuous-time and finite-state Markov chain, either observable or unobservable, under regime-switching models. Our modelling setup includes regime-switching diffusion models, regime-switching jump-diffusion models, regime-switching stochastic interest rate models, double regime-switching models and hidden Markov models. Under regime-switching models, the additional uncertainty leads to an incomplete financial market. The selection of a pricing kernel in an incomplete market has long been discussed as more than one equivalent martingale measures may exist in an incomplete market. In this thesis, both the regime-switching Esscher transform and the minimal martingale measure approach are considered in selecting a pricing kernel under regime-switching models. To obtain analytical pricing formulae for the financial derivatives, we mainly focus on the applications of the fast Fourier transform under regime-switching models.
In Chapter1, we first briefly introduce the pricing models of financial derivatives. We further provide a literature review on topics including option valuation under different kinds of regime-switching models, hidden Markov models, stochastic interest rate models, approaches to determine an equivalent martingale measure in incomplete markets, and fast Fourier transform. Then the mathematical tools to be used in this thesis are introduced. More specifically, we give the mathematical representation of the modulating Markov chain and describe how to obtain analytical pricing formulae via Fourier transform and discretize the pricing formulas via the fast Fourier transform method. Finally, we give an overview of the papers to be included in this thesis.
In Chapters2-4, we consider the valuations of various options under different regime-switching models where model parameters are modulated by a continuous-time, finite-state and observable Markov chain. Analytical pricing formulae for these options are obtained via the inverse Fourier transform and calculated via the fast Fourier transform, providing an easier and neater way to calculate the option prices.
In Chapter2, we consider the valuation of foreign equity options, settling in one currency while the underlying assets are denominated in a different currency, under a Markovian regime-switching mean-reversion lognormal model. Intuitively, the valuation of the so-called foreign equity options has to deal with the joint dynamics of both the foreign equity and the exchange rate. Chapter2considers two kinds of foreign equity options, one with a strike price in the foreign currency (FEOF) and the other with a strike price in the domestic currency (FEOD), under the assumption that the foreign exchange rate follows a mean-reversion lognormal model with regime-switching. To utilize the fast Fourier transform method, the characteristic function of the logarithmic entity price is needed. For FEOF, a measure change technique has to be applied first to take the expectation of the foreign exchange rate as the numeraire. For FEOD, a simple summation of the characteristic functions of the logarithmic foreign equity price and the logarithmic foreign exchange rate are calculated. Then we derive analytical pricing formulae for both FEOF options and FEOD options.
The valuation of power options are discussed in Chapter3by considering a Marko-vian regime-switching jump-diffusion model, where a Poisson random measure is adopted to depict the jump component. Power options provide investors with a choice of financial products with nonlinear payoff functions. This feature is attractive to many investors, especially in a financial market that involves many types of nonlinear risks. A unique equivalent martingale measure is selected by adopting a version of regime-switching Ess-cher transform. Then, under the risk-neutral probability measure, a standard application of the inverse Fourier transform is applied to obtain analytical pricing formulae for the power option. In the numerical analysis, particular parametric forms of the compensator measure, including the Markov-modulated inverse Gaussian process and the Markov-modulated Merton jump-diffusion model, are considered.
Chapter4discusses the option valuation under a regime-switching stochastic interest rate model, which may increase the long-term effectiveness of the model. We start with a risk-neutral probability measure. To take the zero-coupon bond value as the numeraire, a measure change technique is applied to change the risk-neutral probability measure into a forward measure. By deriving the formulae for the characteristic function of the logarithmic return of the underlying asset, the fast Fourier transform can then be applied.
Chapters5-6consider the valuation of insurance products with embedded-option features under regime-switching models. In Chapter5, we investigate the valuation of equity-linked annuities with mortality risk under a double regime-switching model, where the modulating Markov chain is a continuous-time, finite-state, observable chain. In addition to the assumption that model parameters will change when the states of the Markov chain switch, we also assume that a jump of the price level of the underlying investment fund will be triggered when a state transition of the chain occurs. One of the main features of the double regime-switching models is to provide an endogenous way to determine the regime-switching risk. To specify a unique pricing kernel, we employ both the generalized version of regime-switching Esscher transform and the minimal martingale method to determine a unique equivalent martingale measure, respectively. Note that the transition matrix of the Markov chain will also change. There have been many works adopting the option valuation techniques to price insurance policies with embedded-option features. In Chapter5, we write the payoff of the equity-linked annuities as a combination of the payoffs of several European-style options. Consequently, we can utilize option valuation techniques to price the equity-linked annuities. Here, the technique of the fast Fourier transform applied to option valuation can be used to price such equity-linked annuities. Numerical analysis and sensitivity analysis provide us with intuitive understandings of the valuation of equity-linked annuities.
Chapter6discusses the valuation of dynamic fund protection under a hidden Markov model, where the states of the modulating Markov chain are unobservable. To protect investors from downside risk, a dynamic fund protection plan is an effective and convenient tool. The payoff function of a dynamic fund protection plan can be written as a product of the payoff of a fixed strike lookback call and the exchange rate. Consequently, option valuation techniques can also be utilized to price dynamic fund protection plans. In this chapter, the approach of partial differential equations is adopted to price dynamic fund protection plans by a three-stage estimation method. It consists of the Baum-Welch algorithm to estimate the model parameters of the hidden Markov chain, the Viterbi algorithm to select the most-probable path for the chain, and the maximum likelihood method to estimate the model parameters. The three-stage estimation method is intuitively appealing and easy to implement in practice.
基于马尔可夫链是连续时间、有限状态的假设,本文在前人研究成果的基础上分别考虑了体制转换扩散模型、体制转换跳扩散模型、体制转换随机利率模型、双体制转换模型与隐马尔可夫模型。由于引入马尔可夫链带来的不确定性,体制转换模型下的金融市场通常是不完备的,也就意味着该市场中存在着不唯一的等价鞅测度。如何在不完备市场中选取等价鞅测度一直以来也是备受关注的研究问题之一,本文分别采用体制转换Esscher变换与最小鞅测度两种方法选取等价鞅测度。然后,主要采用快速傅立叶变换方法推导体制转换模型下金融衍生品的解析定价公式,并通过数值分析得到相应结论。
第一章首先介绍了金融衍生品的定价模型推广。接着,介绍了一些热点问题的研究现状,主要包括不同体制转换模型下的期权定价研究,隐马尔可夫模型研究,随机利率模型研究,不完备市场下选取等价鞅测度的方法以及快速傅立叶变换的方法。然后给出了本文需要的预备知识。具体说来,本章给出了调制马尔可夫模型的数学表达式,介绍了如何通过傅立叶变换得到解析定价公式以及通过快速傅立叶变换将定价公式离散化的过程。最后,简要介绍了本文的主要研究工作。
在第二章到第四章中,本文假设体制转换模型参数是被连续时间、有限状态、可观测的马尔可夫链调制的,并考虑了不同体制转换模型下不同种类期权的定价问题。
第二章中讨论了体制转换均值回归对数正态模型下外国股权期权的定价问题。顾名思义,外国股权期权中标的资产的表示货币与结算货币是不同类型的货币。直观上理解,外国股权期权的定价问题需要研究外国股权与汇率服从的过程。第二章假设汇率服从体制转换性质的均值回归对数正态模型,探讨了外国股权期权的定价问题。按照行权价格表示货币的不同,将外国股权期权分为两大类,一类的行权价格是以外国货币表示的,记为FEOF,另外一类的行权价格是以国内货币表示的,记为FEODo为了应用快速傅立叶变换方法,本章首先推导出对数标的资产的特征函数。对于FEOF的定价,本章首先采用测度变换技巧,以外汇汇率的期望值为计价单位。FEOD的定价过程则需要将对数外国股权价格的特征函数与对数外汇汇率的特征函数进行加和。然后,通过逆傅立叶变换方法推导出FEOF期权与FEOD期权的解析定价公式。最后,本章给出了相应的数值例子和经验分析。根据敏感性分析,本章讨论了模型中参数对于期权价格的影响。经验分析中,本章采用2003年10月1日至10月17日期间日经225指数欧式看涨期权与美元／日元汇率的市场数据,估计模型参数,并比较得出体制转换均值回归对数正态模型的样本内拟合误差与样本外预测误差均小于均值回归对数正态模型相应误差的结论。
第三章考虑了体制转换跳扩散模型下幂期权的定价问题,该模型假设跳部分是由Poisson随机测度刻画的。幂期权是一种具有非线性收益的金融产品,因此,可以用来应对金融市场中存在的非线性风险,为投资者提供了一种便捷的风险管理工具。文章假定利率,标的资产的平均回报率与波动率,补偿因子均与经济状态有关。通过体制转换Esscher变换选择了一个等价鞅测度,并推导风险中性测度下对数标的资产的特征函数。然后,本章应用逆傅立叶变换推导幂期权的解析定价公式,并通过快速傅立叶变换方法将定价公式离散求解。最后,数值分析中,本章选定了两种特定的补偿因子,马尔可夫调制逆Gaussian过程和马尔可夫调制Merton跳扩散模型,探讨两种特定的补偿因子下幂期权的定价过程,并分析不同参数对幂期权价格的影响。
第四章讨论了体制转换随机利率模型下欧式期权的定价问题。本章假设短期利率服从体制转换的Hull-White模型,具体来说,利率的均值回归水平与波动率,无风险利率,标的资产的波动率以及相关系数都随着马尔可夫链状态的变化而改变。本章采用测度变换技巧将风险中性测度变换到远期测度。此时,马尔可夫链的速率矩阵也会产生相应的变化。在远期测度下,本章推导出对数标的资产的特征函数表达式,并通过快速傅立叶变换方法对期权进行定价。
第五章-第六章主要讨论体制转换模型下带有内嵌期权特征的保险产品的定价问题。
第五章研究了双体制转换模型下考虑死亡风险的权益连结年金的定价过程。双体制转换模型假设模型参数的取值随着马尔可夫链状态的变化而改变,与此同时,标的资产的价格将随着马尔可夫链状态的变化而发生一个跳。双体制转换模型的一个最主要特点是可以内生决定体制转换风险的大小,为本章提供了一种量化体制转换风险的方法。为了选取等价鞅测度,本章分别采用广义体制转换Esscher变换与最小鞅测度方法,并对这两种方法选取的等价鞅测度进行简要分析。需要注意的是,马尔可夫链的速率矩阵是随着测度变换而变化的。对于内嵌期权特征的保险产品来说,很多学者都采用期权定价技巧来研究。本章将权益连结年金产品的收益看作是一系列欧式期权收益的组合。因此,本章可以采用期权定价技巧研究相应权益连结年金产品的定价问题。通过本章的数值分析和敏感性分析,我们可以很直观地理解权益连结年金产品的定价过程及不同模型参数对产品价格的影响。
第六章研究隐马尔可夫模型下动态保护基金的定价问题,该模型下调制马尔可夫链的状态是不可观测的。动态保护基金是规避下跌风险的一种有效工具。动态保护基金的收益可以看作是具有固定行权价格的回望看涨期权的收益与外汇汇率的乘积。类似地,可以采用期权定价技巧来对动态保护基金进行定价。本章首先通过体制转换Esscher变换选取一个等价鞅测度,然后采用偏微分方程方法对动态保护基金进行定价。由于马尔可夫链的状态是不可观测的,于是本章采用三步估计方法对隐马尔可夫模型进行参数估计。首先,基于实际市场数据,本章采用Baum-Welch算法估计马尔可夫链的参数,然后,通过Viterbi算法选择马尔可夫链的最可能路径,最后,通过最大似然估计方法对模型参数进行估计。该方法简单易行,容易理解。
Following the global financial crisis of2008, the impacts of changes in (macro)-economic conditions and business cycles have attracted increasing interests in the new millennium. Regime-switching models have been considered as a natural tool of pric-ing financial derivatives by both academic researchers and industrial practitioners since Hamilton (1989) introduced this class of models into financial econometrics. Regime-switching models typically use the states of the modulating Markov chain to represent the states of an economy, depicted by some (macro)-economic indicators. By adopting this methodology, regime-switching models can incorporate the impacts of structural changes in (macro)-economic conditions. Consequently, it is practical to consider the valuation of financial derivatives under regime-switching models.
In this thesis, the Markov chain we adopt is a continuous-time and finite-state Markov chain, either observable or unobservable, under regime-switching models. Our modelling setup includes regime-switching diffusion models, regime-switching jump-diffusion models, regime-switching stochastic interest rate models, double regime-switching models and hidden Markov models. Under regime-switching models, the additional uncertainty leads to an incomplete financial market. The selection of a pricing kernel in an incomplete market has long been discussed as more than one equivalent martingale measures may exist in an incomplete market. In this thesis, both the regime-switching Esscher transform and the minimal martingale measure approach are considered in selecting a pricing kernel under regime-switching models. To obtain analytical pricing formulae for the financial derivatives, we mainly focus on the applications of the fast Fourier transform under regime-switching models.
In Chapter1, we first briefly introduce the pricing models of financial derivatives. We further provide a literature review on topics including option valuation under different kinds of regime-switching models, hidden Markov models, stochastic interest rate models, approaches to determine an equivalent martingale measure in incomplete markets, and fast Fourier transform. Then the mathematical tools to be used in this thesis are introduced. More specifically, we give the mathematical representation of the modulating Markov chain and describe how to obtain analytical pricing formulae via Fourier transform and discretize the pricing formulas via the fast Fourier transform method. Finally, we give an overview of the papers to be included in this thesis.
In Chapters2-4, we consider the valuations of various options under different regime-switching models where model parameters are modulated by a continuous-time, finite-state and observable Markov chain. Analytical pricing formulae for these options are obtained via the inverse Fourier transform and calculated via the fast Fourier transform, providing an easier and neater way to calculate the option prices.
In Chapter2, we consider the valuation of foreign equity options, settling in one currency while the underlying assets are denominated in a different currency, under a Markovian regime-switching mean-reversion lognormal model. Intuitively, the valuation of the so-called foreign equity options has to deal with the joint dynamics of both the foreign equity and the exchange rate. Chapter2considers two kinds of foreign equity options, one with a strike price in the foreign currency (FEOF) and the other with a strike price in the domestic currency (FEOD), under the assumption that the foreign exchange rate follows a mean-reversion lognormal model with regime-switching. To utilize the fast Fourier transform method, the characteristic function of the logarithmic entity price is needed. For FEOF, a measure change technique has to be applied first to take the expectation of the foreign exchange rate as the numeraire. For FEOD, a simple summation of the characteristic functions of the logarithmic foreign equity price and the logarithmic foreign exchange rate are calculated. Then we derive analytical pricing formulae for both FEOF options and FEOD options.
The valuation of power options are discussed in Chapter3by considering a Marko-vian regime-switching jump-diffusion model, where a Poisson random measure is adopted to depict the jump component. Power options provide investors with a choice of financial products with nonlinear payoff functions. This feature is attractive to many investors, especially in a financial market that involves many types of nonlinear risks. A unique equivalent martingale measure is selected by adopting a version of regime-switching Ess-cher transform. Then, under the risk-neutral probability measure, a standard application of the inverse Fourier transform is applied to obtain analytical pricing formulae for the power option. In the numerical analysis, particular parametric forms of the compensator measure, including the Markov-modulated inverse Gaussian process and the Markov-modulated Merton jump-diffusion model, are considered.
Chapter4discusses the option valuation under a regime-switching stochastic interest rate model, which may increase the long-term effectiveness of the model. We start with a risk-neutral probability measure. To take the zero-coupon bond value as the numeraire, a measure change technique is applied to change the risk-neutral probability measure into a forward measure. By deriving the formulae for the characteristic function of the logarithmic return of the underlying asset, the fast Fourier transform can then be applied.
Chapters5-6consider the valuation of insurance products with embedded-option features under regime-switching models. In Chapter5, we investigate the valuation of equity-linked annuities with mortality risk under a double regime-switching model, where the modulating Markov chain is a continuous-time, finite-state, observable chain. In addition to the assumption that model parameters will change when the states of the Markov chain switch, we also assume that a jump of the price level of the underlying investment fund will be triggered when a state transition of the chain occurs. One of the main features of the double regime-switching models is to provide an endogenous way to determine the regime-switching risk. To specify a unique pricing kernel, we employ both the generalized version of regime-switching Esscher transform and the minimal martingale method to determine a unique equivalent martingale measure, respectively. Note that the transition matrix of the Markov chain will also change. There have been many works adopting the option valuation techniques to price insurance policies with embedded-option features. In Chapter5, we write the payoff of the equity-linked annuities as a combination of the payoffs of several European-style options. Consequently, we can utilize option valuation techniques to price the equity-linked annuities. Here, the technique of the fast Fourier transform applied to option valuation can be used to price such equity-linked annuities. Numerical analysis and sensitivity analysis provide us with intuitive understandings of the valuation of equity-linked annuities.
Chapter6discusses the valuation of dynamic fund protection under a hidden Markov model, where the states of the modulating Markov chain are unobservable. To protect investors from downside risk, a dynamic fund protection plan is an effective and convenient tool. The payoff function of a dynamic fund protection plan can be written as a product of the payoff of a fixed strike lookback call and the exchange rate. Consequently, option valuation techniques can also be utilized to price dynamic fund protection plans. In this chapter, the approach of partial differential equations is adopted to price dynamic fund protection plans by a three-stage estimation method. It consists of the Baum-Welch algorithm to estimate the model parameters of the hidden Markov chain, the Viterbi algorithm to select the most-probable path for the chain, and the maximum likelihood method to estimate the model parameters. The three-stage estimation method is intuitively appealing and easy to implement in practice.
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