基于马氏链近似的随机波动Lévy模型下路径依赖型期权定价研究

As an important risk hedging instrument, a derivative security is often expressed as a combination of a number of options contracts. And most of the derivative securities are path-dependent in the current financial market. Thus, path-dependent options pricing has a significant impact on financial risk management and prevention. Since a pricing model has to capture the actual trends of asset prices fluctuation, existing pricing models are very complex. There are not fast and efficient pricing methods. From the perspective of the asset prices distribution, this project will explore path-dependent options pricing under stochastic volatility Lévy model based on Markov chain approximation. First, this project will construct an approximating Markov chain to simplify the pricing model by dividing the range of the asset price and volatility rate and computing the transition probability from one state to the other. Second, this project will focus on the pricing of barrier and lookback options which are two typical path-dependent options using Markov chain approximation method, Fourier transform and fast Fourier transform algorithm. Ultimately, this project will realize the accurate and rapid pricing of the general path-dependent options under the model which is more in line with the law of asset prices fluctuation by summarizing general characters of different path-dependent option pricing methods. This research will provide theoretical basis and method for improving the ability of financial risk control and management in China.

衍生证券作为一种重要的风险对冲工具常可表示为若干期权合约的组合形式，且目前金融市场中的衍生证券大都具有路径依赖性，因而路径依赖型期权对于金融风险管理与防范具有重要影响。由于其定价模型必须反映资产价格实际变化趋势而极其复杂，目前还缺乏相应的快速有效的定价方法。本项目以随机波动Lévy模型下路径依赖型期权为研究对象，从资产价格分布角度，探索基于马氏链近似的路经依赖型期权定价方法。本项目首先通过对资产价格和波动率取值范围的合理划分以及各状态之间转移概率的确定，构建定价模型的近似马氏链使模型简化，然后利用马氏链近似方法结合傅立叶变换以及快速傅立叶变换算法，着重研究两类典型的路径依赖型期权障碍期权和回望期权的定价。最终，通过总结不同路径依赖型期权定价方法的共性，实现更加符合资产价格变化规律的模型下一般路径依赖型期权的准确、快速定价，为提升我国金融风险防控和管理能力提供理论依据和实现方法。

路径依赖期权具有独特的对冲和套期保值功能，是有效规避系统性风险的主要工具。其合理定价对于金融风险管理与防范具有重要意义。本项目以随机波动Lévy模型下路径依赖型期权为研究对象，从资产价格分布角度，探索基于马氏链近似的路经依赖型期权定价方法。项目取得的研究成果如下：1.针对随机波动Lévy模型能够较好拟合资产实际收益的特征，从实证分析角度，提出了一种模型校正算法，实现了定价模型的参数估计，结果表明估计算法具有较好的稳定性。2.构造了一类典型的随机波动Lévy模型的近似马氏链，证明了构造的近似模型和随机波动Lévy模型的收敛性，提出了基于马氏链近似方法的欧式障碍期权的定价算法。通过和蒙特卡洛模拟、有限差分方法比较，发现马氏链近似方法快速、有效、容易实施。3.从偏微分方程角度初步研究了美式路径依赖期权的退化抛物型变分不等式问题，定义了退化抛物型变分不等式的弱解，利用惩罚方法证明了弱解的存在性。