体制转换模型下的期权定价
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摘要
期权定价是金融数学的核心问题之一.在期权定价和套期保值领域,传统的Black-Scholes期权定价公式虽然被广泛的应用,但是大量实证表明资产价格满足几何布朗运动并不符合实际情况.过去三十年,大量学者已经提出了许多不同的期权定价模型,这些模型包括跳扩散模型、Levy过程、随机波动率模型和GARCH模型等.近年来,马尔可夫调制的体制转换模型在期权定价中的应用已经引起了大量研究学者的兴趣.模型中连续时间马尔可夫链的状态被看作市场经济状态,经济状态的转移是由于经济和商业周期的结构变化而引起的.本文在前人研究成果的基础上,研究了体制转换模型下的期权定价问题,并提出了一个新的两状态体制转换模型.此外,由于马尔可夫调制的体制转换模型下的市场是不完备的,我们还给出了体制转换模型下的局部风险最小套期保值策略和最小鞅测度,具体内容如下：
1.第一章首先介绍了金融数学的起源和发展；接着,简要说明了不完备市场下的期权定价和期权定价模型的推广；另外,我们也介绍了体制转换模型及国内外研究现状；最后,给出了需要的预备知识及本论文的主要工作.
2.第二章考虑了体制转换模型下弱势欧式期权的定价问题.假定市场利率、风险资产的平均回报率和波动率都与市场经济状态有关,市场经济状态由一连续时间马尔科夫链来描述.由于市场是不完备的,利用regime switching Esscher变换得到了一个等价鞅测度,分别给出了当标的资产价格满足马尔科夫调制的几何布朗运动和跳扩散过程时弱势欧式期权的定价公式.
3.第三章提出了一个两状态体制转换模型.假定市场经济有两个状态,一个稳定状态和一个高波动状态.风险资产价格在不同状态下满足不同的随机过程.当市场状态是稳定状态时,风险资产价格满足几何布朗运动或马尔可夫调制的几何布朗运动；当市场状态是高波动状态时,风险资产价格满足跳扩散过程或马尔可夫调制的跳扩散过程.另外,在这个两状态体制转换模型下,分别讨论了几种期权的定价问题.
4.第四章研究了体制转换模型下巨灾看跌期权的定价问题.Jaimungal and Wang(2006)假定巨灾风险是非系统风险,没有考虑它的定价.在这一章,我们改进了他们的模型,假定市场利率、风险资产的平均回报率和波动率都与市场经济状态有关,并考虑了巨灾风险的定价和利用regime switching Esscher变换得到了简单回报的regime switching Esscher变换鞅测度和复合回报的regime switching Esscher变换鞅测度.另外,我们还讨论了这两个等价鞅测度的存在性并证明了简单回报的regime switching Esshcer变换鞅测度是风险资产价格满足马尔科夫调制几何Levy过程时的最小熵鞅测度.最后,给出了在这些等价鞅测度下巨灾看跌期权的定价公式.
5.第五章研究了体制转换模型下套期保值和期权定价问题.当标的资产价格满足马尔科夫调制的几何Levy过程时,市场是不完备的,这也意味着市场中的未定权益是不能通过自融资策略来套期保值的.我们给出了在体制转换模型下的局部风险最小套期保值策略和最小鞅测度,并考虑了当标的资产价格满足马尔科夫调制的几何Levy过程时欧式看涨、看跌和远期生效看涨期权的定价问题.
综上所述,本文研究了体制转换模型下的期权定价问题.获得了体制转换模型下弱势欧式期权和巨灾看跌期权的定价公式,提出了一个新的两状态体制转换模型,并考虑了在这个两状态体制转换模型下的期权定价问题.此外,由于马尔可夫调制的体制转换模型下的市场是不完备的,未定权益不能通过自融资策略来复制,本文考虑了体制转换模型下的套期保值问题,给出了局部风险最小套期保值策略和最小鞅测度.这些结果不仅在理论上有意义,而且对金融市场中的期权交易和套期保值有应用价值.
Option pricing is one of the core issues in mathematical finance. The traditional Black-Scholes option pricing formula has been widely used for pricing option and hedging in finance industry, but there are a large number of empirical results indicate that the asset price follows the geometric Brownian motion is not realistic. Over the past three decades, many different option valuation models have been proposed. Some of these include jump-diffusion models, Levy processes, stochastic volatility models, GARCH model and others. Recently, there has been considerable interest in applications of a regime switching model which is modulated by a continuous time Markov chain to option pricing problem. The states of the continuous time Markov chain can be interpreted as the states of the economy. The transitions of the states of the economy may be attributed to structural changes of the economy and business cycles. Based on the previous research result, this thesis discuss the option pricing under regime switching models and a new two-state regime switching model is provided. Moreover, because the market described by the Markov-modulated regime switching model is incomplete, a locally risk minimizing hedging strategy and the minimal martingale measure are also obtained under a regime switching model. The main contents of this thesis are listed in the following:
1. In the first chapter, the origin and development of mathematical finance are first introduced. Then we give a brief description of the option pricing in an incomplete market and the extension of option valuation models. In addition, we also introduce the regime switching model and present research at home and abroad. Finally, some preliminaries and the main results of this thesis are provided.
2. In the second chapter, the pricing problem of vulnerable European options is con-sidered under a regime switching model. We suppose that the market interest rate, the appreciation rate and the volatility rate of the risky asset depend on the states of the economy which are modeled by a continuous time Markov chain. Since the market is incomplete, we adopt the regime switching Esscher transform to deter-mine an equivalent martingale measure and provide analytical pricing formulas of vulnerable European options when the dynamics of the risky asset is governed by a Markov-modulated geometric Brownian motion or a Markov-modulated jump dif-fusion process.
3. In the third chapter, a two-state regime switching model is provided. We consider a market which has two states, a stable state and a high volatility state. The dynamic of the risky asset price follows different stochastic processes in different states of the market, the risky asset price is driven by a geometric Brownian motion or a Markov-modulated geometric Brownian motion when the market is stable, but the risky asset price follows a jump diffusion process or a Markov-modulated jump diffusion process if the market state is high volatility. In addition, the pricing problem of several options is considered under this two-state regime switching model.
4. In the fourth chapter, the value of catastrophe put option is studied under a regime switching model. Jaimungal and Wang(2006) suppose that the catastrophe risk is unsystem risk which is not priced. In this chapter, we improve their model and assume that the market interest rate, the appreciation rate and the volatility rate of the risky asset all are depend on the states of the economy. In particular, the catastrophe risk is priced and the regime switching Esscher transform is adopted to obtain the simple return and the compound return regime switching Esscher transform martingale measures for the valuation problem in the incomplete market setting. In addition, we also discuss the existence of these two equivalent martin-gale measures and prove that the simple return regime switching Esscher transform martingale measure is the minimal entropy martingale measure when the risky as-set follows a Markov-modulated geometric Levy process. In the end, the explicit analytical formulas of catastrophe put option are derived under these equivalent martingale measures.
5. In the fifth chapter, option pricing and hedging is analyzed under a regime switching model. We suppose that the risky asset follows a Markov-modulated geometric Levy process. The market described by a Markov-modulated geometric Levy process is incomplete, it means that contingent claims can not be hedged perfectly by self-financing strategy. The locally risky minimal hedging strategy and the minimal martingale measure are provided under a regime switching model. Moreover, the pricing problem of European call option, European put option and forward starting call option is considered under a regime switching model.
In brief, this thesis discuss the option pricing under regime switching models. The pricing formulas of vulnerable European options and catastrophe put option are derived, a new two-state regime switching model is proposed and the valuation of some kinds of option is considered under this option valuation model. Moreover, the market described by a Markov-modulated regime switching model is incomplete, the contingent claim can not be replicated by self-financing strategy, this thesis consider the hedging of contingent claims under a regime switching model and provide the locally risk minimizing hedging strategy and the minimal martingale measure. These results are not only useful in theory but also significant in practice for pricing option and hedging in the financial markets.
1.第一章首先介绍了金融数学的起源和发展；接着,简要说明了不完备市场下的期权定价和期权定价模型的推广；另外,我们也介绍了体制转换模型及国内外研究现状；最后,给出了需要的预备知识及本论文的主要工作.
2.第二章考虑了体制转换模型下弱势欧式期权的定价问题.假定市场利率、风险资产的平均回报率和波动率都与市场经济状态有关,市场经济状态由一连续时间马尔科夫链来描述.由于市场是不完备的,利用regime switching Esscher变换得到了一个等价鞅测度,分别给出了当标的资产价格满足马尔科夫调制的几何布朗运动和跳扩散过程时弱势欧式期权的定价公式.
3.第三章提出了一个两状态体制转换模型.假定市场经济有两个状态,一个稳定状态和一个高波动状态.风险资产价格在不同状态下满足不同的随机过程.当市场状态是稳定状态时,风险资产价格满足几何布朗运动或马尔可夫调制的几何布朗运动；当市场状态是高波动状态时,风险资产价格满足跳扩散过程或马尔可夫调制的跳扩散过程.另外,在这个两状态体制转换模型下,分别讨论了几种期权的定价问题.
4.第四章研究了体制转换模型下巨灾看跌期权的定价问题.Jaimungal and Wang(2006)假定巨灾风险是非系统风险,没有考虑它的定价.在这一章,我们改进了他们的模型,假定市场利率、风险资产的平均回报率和波动率都与市场经济状态有关,并考虑了巨灾风险的定价和利用regime switching Esscher变换得到了简单回报的regime switching Esscher变换鞅测度和复合回报的regime switching Esscher变换鞅测度.另外,我们还讨论了这两个等价鞅测度的存在性并证明了简单回报的regime switching Esshcer变换鞅测度是风险资产价格满足马尔科夫调制几何Levy过程时的最小熵鞅测度.最后,给出了在这些等价鞅测度下巨灾看跌期权的定价公式.
5.第五章研究了体制转换模型下套期保值和期权定价问题.当标的资产价格满足马尔科夫调制的几何Levy过程时,市场是不完备的,这也意味着市场中的未定权益是不能通过自融资策略来套期保值的.我们给出了在体制转换模型下的局部风险最小套期保值策略和最小鞅测度,并考虑了当标的资产价格满足马尔科夫调制的几何Levy过程时欧式看涨、看跌和远期生效看涨期权的定价问题.
综上所述,本文研究了体制转换模型下的期权定价问题.获得了体制转换模型下弱势欧式期权和巨灾看跌期权的定价公式,提出了一个新的两状态体制转换模型,并考虑了在这个两状态体制转换模型下的期权定价问题.此外,由于马尔可夫调制的体制转换模型下的市场是不完备的,未定权益不能通过自融资策略来复制,本文考虑了体制转换模型下的套期保值问题,给出了局部风险最小套期保值策略和最小鞅测度.这些结果不仅在理论上有意义,而且对金融市场中的期权交易和套期保值有应用价值.
Option pricing is one of the core issues in mathematical finance. The traditional Black-Scholes option pricing formula has been widely used for pricing option and hedging in finance industry, but there are a large number of empirical results indicate that the asset price follows the geometric Brownian motion is not realistic. Over the past three decades, many different option valuation models have been proposed. Some of these include jump-diffusion models, Levy processes, stochastic volatility models, GARCH model and others. Recently, there has been considerable interest in applications of a regime switching model which is modulated by a continuous time Markov chain to option pricing problem. The states of the continuous time Markov chain can be interpreted as the states of the economy. The transitions of the states of the economy may be attributed to structural changes of the economy and business cycles. Based on the previous research result, this thesis discuss the option pricing under regime switching models and a new two-state regime switching model is provided. Moreover, because the market described by the Markov-modulated regime switching model is incomplete, a locally risk minimizing hedging strategy and the minimal martingale measure are also obtained under a regime switching model. The main contents of this thesis are listed in the following:
1. In the first chapter, the origin and development of mathematical finance are first introduced. Then we give a brief description of the option pricing in an incomplete market and the extension of option valuation models. In addition, we also introduce the regime switching model and present research at home and abroad. Finally, some preliminaries and the main results of this thesis are provided.
2. In the second chapter, the pricing problem of vulnerable European options is con-sidered under a regime switching model. We suppose that the market interest rate, the appreciation rate and the volatility rate of the risky asset depend on the states of the economy which are modeled by a continuous time Markov chain. Since the market is incomplete, we adopt the regime switching Esscher transform to deter-mine an equivalent martingale measure and provide analytical pricing formulas of vulnerable European options when the dynamics of the risky asset is governed by a Markov-modulated geometric Brownian motion or a Markov-modulated jump dif-fusion process.
3. In the third chapter, a two-state regime switching model is provided. We consider a market which has two states, a stable state and a high volatility state. The dynamic of the risky asset price follows different stochastic processes in different states of the market, the risky asset price is driven by a geometric Brownian motion or a Markov-modulated geometric Brownian motion when the market is stable, but the risky asset price follows a jump diffusion process or a Markov-modulated jump diffusion process if the market state is high volatility. In addition, the pricing problem of several options is considered under this two-state regime switching model.
4. In the fourth chapter, the value of catastrophe put option is studied under a regime switching model. Jaimungal and Wang(2006) suppose that the catastrophe risk is unsystem risk which is not priced. In this chapter, we improve their model and assume that the market interest rate, the appreciation rate and the volatility rate of the risky asset all are depend on the states of the economy. In particular, the catastrophe risk is priced and the regime switching Esscher transform is adopted to obtain the simple return and the compound return regime switching Esscher transform martingale measures for the valuation problem in the incomplete market setting. In addition, we also discuss the existence of these two equivalent martin-gale measures and prove that the simple return regime switching Esscher transform martingale measure is the minimal entropy martingale measure when the risky as-set follows a Markov-modulated geometric Levy process. In the end, the explicit analytical formulas of catastrophe put option are derived under these equivalent martingale measures.
5. In the fifth chapter, option pricing and hedging is analyzed under a regime switching model. We suppose that the risky asset follows a Markov-modulated geometric Levy process. The market described by a Markov-modulated geometric Levy process is incomplete, it means that contingent claims can not be hedged perfectly by self-financing strategy. The locally risky minimal hedging strategy and the minimal martingale measure are provided under a regime switching model. Moreover, the pricing problem of European call option, European put option and forward starting call option is considered under a regime switching model.
In brief, this thesis discuss the option pricing under regime switching models. The pricing formulas of vulnerable European options and catastrophe put option are derived, a new two-state regime switching model is proposed and the valuation of some kinds of option is considered under this option valuation model. Moreover, the market described by a Markov-modulated regime switching model is incomplete, the contingent claim can not be replicated by self-financing strategy, this thesis consider the hedging of contingent claims under a regime switching model and provide the locally risk minimizing hedging strategy and the minimal martingale measure. These results are not only useful in theory but also significant in practice for pricing option and hedging in the financial markets.
引文
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