不完备市场中的几类期权定价研究
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摘要
近年来,随着金融衍生产品市场的发展,衍生产品的定价,风险管理等问题变得越来越重要.为了描述不断变化的经济环境,更多复杂的模型被提出.我们根据风险中性定价理论研究在市场不完备的情况下几类期权的定价问题.具体内容如下：
1.第一章首先介绍了期权的概念及资产定价的发展,接着,介绍了不完备市场中的期权定价,然后简要说明了信用风险模型的分类和研究状况,最后,给出了一些基本定义和定理及本论文的主要工作.
2.第二章考虑了幂函数型期权的定价问题.为了反映市场中风险资产的价格变化过程,我们首先假设风险资产的价格过程服从跳扩散模型,其次假定市场中参数随连续时间的马尔科夫链状态的转移而变化.在第一种情况下,我们假定市场利率服从Vasicek模型并且假定风险资产与市场利率是相关的,通过给定一个等价鞅测度我们得到了幂函数型期权的价格公式.在第二种情况下,我们假定市场利率,风险资产的期望收益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,给出了当标的资产价格服从马尔科夫调制的几何布朗运动时的幂函数型期权价格公式.
3.第三章讨论了约化模型下具有信用风险的几类期权的定价问题.由于不可预见的事件的发生可能会导致违约强度发生剧烈的变化,我们假设违约强度服从一个跳扩散模型.此外,我们假定期权卖方可能发生违约并且回收率是一个常数.在约化模型下,分别给出了具有信用风险的欧式期权,幂函数型期权以及交换期权的价格公式.
4.第四章研究了具有随机死亡强度的担保年金期权(Guaranteed Annuity Options)的定价问题.为了符合实际情况,我们在死亡强度中增加了“跳”的因素.我们假设死亡强度服从跳扩散模型,标的资产价格过程服从随机波动率模型,利率满足Vasicek模型,并且假设这些过程都是相关的,给出了担保年金期权的价格公式.
5.第五章在局部风险最小的标准下考虑套期保值策略以及最小鞅测度.当标的资产价格服从跳扩散模型或者马尔科夫调制的体制转换模型时,市场是不完备的,这也意味着市场中的未定权益不能通过自融资策略来套期保值.我们分别给出了跳扩散模型下内幕交易者的局部风险最小套期保值策略以及具有随机波动率的马尔科夫调制跳扩散过程下的相应策略.
综上所述,本文研究了标的资产服从不同模型时的几类期权的定价问题.获得了跳扩散模型和regime switching模型下的幂函数型期权以及随机死亡强度下的担保年金期权的价格公式.在约化模型下考虑了具有信用风险的几类期权的定价.此外,在市场不完备的情况下,考虑套期保值问题,给出了局部风险最小套期保值策略和最小鞅测度。
Recently, with the development of financial derivatives market, the pricing of deriva-tives, risk management and others have become more and more important. In order to describe the changing economic environment, more complex models are proposed. By the risk neutral pricing theorem, we investigate the pricing of several options in the incomplete market. The main contents of this thesis are listed in the following:
1. In the first chapter, the concept of options and the assets pricing are at first in-troduced. Then, we introduce the pricing of options in the incomplete market. In addition, we give a brief description of the classification and the research of the credit risk model. Finally, some basic definitions, theorems and the main results of this thesis are provided.
2. In the second chapter, we consider the pricing of the power options. In order to reflect the price process of the risky assets in the market, we first assume that the price process of the risky assets follow the jump diffusion model and then assume that the parameters in the market vary as the transition of the state of a continuous time Makov chain. In the first case, we suppose that the interest rate follow the Vasicek model, and the risky assets are correlated with the interest rate. By introducing the risk neutral measure, we obtain the pricing formula of the power options. In the second case, we assume that the interest rate, the expected return rate and the volatility are related to the state of the economy which is described by the continuous time Markov chain. Since the market under consideration is incomplete, we give an equivalent martingale measure by the regime switching Esscher transform. We get the price of the power options under the Markov-modulated geometric Brownian motion.
3. In the third chapter, we discuss the pricing of several options in a reduced form model. Since the intensity of default may fluctuate severely as unanticipated events happen, we assume that the default intensity is governed by a jump-diffusion process. In addition, we assume that the writer of the option may default and the recovery rate is a constant. In the reduced form model, we give the price of the European option, the power option and the exchange option with credit risk.
4. In the fourth chapter, the value of the guaranteed annuity options is studied with the stochastic mortality intensity. In order to conform to the actual situation, we add the "jump" to the mortality intensity. We assume that the mortality intensity follow the jump diffusion model, underlying assets follow the stochastic volatility model, interest rate is the Vasicek model which is correlated with each other. We obtain the price of the guaranteed annuity options.
5. In the fifth chapter, we investigate the hedging strategies and the minimal mar-tingale measure. When the price process of the underlying assets follow the jump diffusion model or the Markov-modulates models, the market is incomplete and the contingent claims can't be hedged by self-financing strategies. We give the lo-cally risk minimizing strategies under the jump diffusion models and the Markov modulated jump diffusion models with stochastic volatility.
In brief, this thesis discuss the pricing of several options when the underlying assets follow different models. We get the pricing of the power options under the jump diffusion process and the regime switching model, and the valuation of the guaranteed annuity options with stochastic mortality. The pricing of the options with credit risk in a reduced form model is considered. Moreover, when the market is incomplete, we consider the risk minimizing strategies and the minimal martingale measure.
1.第一章首先介绍了期权的概念及资产定价的发展,接着,介绍了不完备市场中的期权定价,然后简要说明了信用风险模型的分类和研究状况,最后,给出了一些基本定义和定理及本论文的主要工作.
2.第二章考虑了幂函数型期权的定价问题.为了反映市场中风险资产的价格变化过程,我们首先假设风险资产的价格过程服从跳扩散模型,其次假定市场中参数随连续时间的马尔科夫链状态的转移而变化.在第一种情况下,我们假定市场利率服从Vasicek模型并且假定风险资产与市场利率是相关的,通过给定一个等价鞅测度我们得到了幂函数型期权的价格公式.在第二种情况下,我们假定市场利率,风险资产的期望收益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,给出了当标的资产价格服从马尔科夫调制的几何布朗运动时的幂函数型期权价格公式.
3.第三章讨论了约化模型下具有信用风险的几类期权的定价问题.由于不可预见的事件的发生可能会导致违约强度发生剧烈的变化,我们假设违约强度服从一个跳扩散模型.此外,我们假定期权卖方可能发生违约并且回收率是一个常数.在约化模型下,分别给出了具有信用风险的欧式期权,幂函数型期权以及交换期权的价格公式.
4.第四章研究了具有随机死亡强度的担保年金期权(Guaranteed Annuity Options)的定价问题.为了符合实际情况,我们在死亡强度中增加了“跳”的因素.我们假设死亡强度服从跳扩散模型,标的资产价格过程服从随机波动率模型,利率满足Vasicek模型,并且假设这些过程都是相关的,给出了担保年金期权的价格公式.
5.第五章在局部风险最小的标准下考虑套期保值策略以及最小鞅测度.当标的资产价格服从跳扩散模型或者马尔科夫调制的体制转换模型时,市场是不完备的,这也意味着市场中的未定权益不能通过自融资策略来套期保值.我们分别给出了跳扩散模型下内幕交易者的局部风险最小套期保值策略以及具有随机波动率的马尔科夫调制跳扩散过程下的相应策略.
综上所述,本文研究了标的资产服从不同模型时的几类期权的定价问题.获得了跳扩散模型和regime switching模型下的幂函数型期权以及随机死亡强度下的担保年金期权的价格公式.在约化模型下考虑了具有信用风险的几类期权的定价.此外,在市场不完备的情况下,考虑套期保值问题,给出了局部风险最小套期保值策略和最小鞅测度。
Recently, with the development of financial derivatives market, the pricing of deriva-tives, risk management and others have become more and more important. In order to describe the changing economic environment, more complex models are proposed. By the risk neutral pricing theorem, we investigate the pricing of several options in the incomplete market. The main contents of this thesis are listed in the following:
1. In the first chapter, the concept of options and the assets pricing are at first in-troduced. Then, we introduce the pricing of options in the incomplete market. In addition, we give a brief description of the classification and the research of the credit risk model. Finally, some basic definitions, theorems and the main results of this thesis are provided.
2. In the second chapter, we consider the pricing of the power options. In order to reflect the price process of the risky assets in the market, we first assume that the price process of the risky assets follow the jump diffusion model and then assume that the parameters in the market vary as the transition of the state of a continuous time Makov chain. In the first case, we suppose that the interest rate follow the Vasicek model, and the risky assets are correlated with the interest rate. By introducing the risk neutral measure, we obtain the pricing formula of the power options. In the second case, we assume that the interest rate, the expected return rate and the volatility are related to the state of the economy which is described by the continuous time Markov chain. Since the market under consideration is incomplete, we give an equivalent martingale measure by the regime switching Esscher transform. We get the price of the power options under the Markov-modulated geometric Brownian motion.
3. In the third chapter, we discuss the pricing of several options in a reduced form model. Since the intensity of default may fluctuate severely as unanticipated events happen, we assume that the default intensity is governed by a jump-diffusion process. In addition, we assume that the writer of the option may default and the recovery rate is a constant. In the reduced form model, we give the price of the European option, the power option and the exchange option with credit risk.
4. In the fourth chapter, the value of the guaranteed annuity options is studied with the stochastic mortality intensity. In order to conform to the actual situation, we add the "jump" to the mortality intensity. We assume that the mortality intensity follow the jump diffusion model, underlying assets follow the stochastic volatility model, interest rate is the Vasicek model which is correlated with each other. We obtain the price of the guaranteed annuity options.
5. In the fifth chapter, we investigate the hedging strategies and the minimal mar-tingale measure. When the price process of the underlying assets follow the jump diffusion model or the Markov-modulates models, the market is incomplete and the contingent claims can't be hedged by self-financing strategies. We give the lo-cally risk minimizing strategies under the jump diffusion models and the Markov modulated jump diffusion models with stochastic volatility.
In brief, this thesis discuss the pricing of several options when the underlying assets follow different models. We get the pricing of the power options under the jump diffusion process and the regime switching model, and the valuation of the guaranteed annuity options with stochastic mortality. The pricing of the options with credit risk in a reduced form model is considered. Moreover, when the market is incomplete, we consider the risk minimizing strategies and the minimal martingale measure.
引文
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