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论文编号：	12886
作者编号：	1120150833
上传时间：	2021/12/13 15:05:31
中文题目：	基于GARCH模型与机制转换GARCH模型的期权定价与对冲
英文题目：	Option Pricing and Hedging based on GARCH Models and Regime-Switching GARCH Models
指导老师：	王永进
中文关键字：	GARCH模型；期权对冲；局部风险最小化；机制转换；内生性；跳过程；期权定价
英文关键字：	GARCH model; option hedging; local risk minimization; regime switching; endogeneity; jump process; option pricing
中文摘要：	波动率作为风险度量的指标，在风险管理、资产配置以及资产定价等各方面都有着重要应用，对波动率进行建模已经成为了金融研究中的核心问题之一。对波动率的建模主要可以分为连续时间的随机波动率模型和离散时间的GARCH模型。这两类模型都能刻画波动率聚集效应以及收益率的尖峰厚尾分布，且能解释“波动率微笑”现象，因而都受到了广泛的关注和研究。但GARCH模型比随机波动率模型更直观，更易于理解，且在应用上更为简便，因此本文主要聚焦于GARCH模型。在中国期权市场蓬勃发展的当下，以GARCH模型为基础，对期权的定价和对冲进行研究有着重要的理论和现实意义。 本文从GARCH模型出发，首先研究了GARCH模型下的期权对冲问题。具体而言，本文在非仿射高斯GARCH动态下基于网格方法实现了欧式看涨期权的局部风险最小化对冲策略（local risk minimization, LRM）。本文同时在物理测度P和风险中性测度Q下完成了局部风险最小化对冲策略，并计算了Duan所提出的delta对冲策略。本文研究了由网格方法计算的局部风险最小化策略下的期权价格和对冲比率的收敛性，结果表明该算法收敛性较好。在数值分析部分，本文考察了各对冲策略下期权价格和对冲比率对风险溢价参数和杠杆效应参数的敏感性，并计算了各策略的一步对冲误差和整体对冲误差，以比较各策略的对冲效果。研究发现物理测度下的局部风险最小化对冲策略（P-LRM）是最优的。 然后，本文在GARCH模型的基础上引入了内生性机制转换（endogenous regime switching ）模型，并研究了该框架下的期权定价问题。本文假设GARCH过程中的参数是状态依赖的，而任一时刻的状态由一个自回归过程决定，该自回归过程和资产收益过程存在相关性。本文在风险中性测度下计算了当相关系数为0和不为0时的状态转移概率，考察了各因素的变化对状态转移概率的影响，并计算了相关系数取不同值时的欧式期权、亚式期权、障碍期权和回望期权的价格。研究发现相关系数的变化会导致各类期权价格的显著差异。 最后，本文将GARCH模型和跳过程与机制转换同时结合起来，研究了带跳的机制转换GARCH（regime-switching GARCH-jump, RS-GARCH-jump）模型下的期权定价问题。本文给出了该模型下风险中性测度的定义，推导了风险中性测度下的资产动态，并提出了一种网格算法来进行期权定价。本文对S&P 500指数进行了实证研究，证明了S&P 500指数日收益率序列中存在“跳跃”现象和方差过程的结构性变化。本文研究了基于三叉树的期权定价结果收敛到真实期权价格的收敛性，结果表明该算法具有良好的收敛性。本文比较了带跳的机制转换GARCH模型与机制转换GARCH（regime-switching GARCH, RS-GARCH）模型、GARCH-jump模型、GARCH模型、Black-Scholes（BS）模型和机制转换（Regime-Switching, RS）模型在期权定价上的实证表现，实证结果证明RS-GARCH-jump模型的定价误差最小。
英文摘要：	Volatility, as an indicator of risk measurement, has important applications in risk management, asset allocation and asset pricing. Modeling volatility has become one of the core issues in financial research. The modeling of volatility can be divided into two types: continuous-time stochastic volatility model and discrete-time GARCH model. Both types of models can describe the volatility clustering effect and leptokurtosis and fat-tail of returns, and can explain the "volatility smile" phenomenon, so they have received extensive attention and has been extensively studied. But GARCH model is more intuitive and easier to understand and apply than the stochastic volatility model. Therefore, this thesis focuses on the GARCH model. At the time when the Chinese option market is booming, it is important to study option pricing and hedging based on the GARCH models both theoretically and practically. Starting from the GARCH model, this thesis first studies the option hedging problem under the GARCH model. Specifically, this thesis implements the local risk minimization hedging strategy of European call options based on the lattice method under non-affine Gaussian GARCH dynamics. We construct locally risk-minimizing (LRM) hedge ratios under both physical and risk-neutral measures, as well as standard delta strategies. We investigate the convergence of option prices and hedges resulting from the LRM strategies relative to the number of intra-daily periods used in the lattice. Several numerical experiments are conducted to assess the sensitivity of the hedge ratios to the equity risk premium and leverage effect parameters, and to compare their performance by computing the corresponding one-period and terminal hedging errors. Our results suggest that the LRM scheme under the physical measure consistently outperforms competing hedging strategies. Then, this thesis combines the endogenous regime switching model with the GARCH model, and studies the option pricing problem under this framework. We assume that the parameters in the GARCH process are state-dependent, and the state at any time is determined by an autoregressive process. This autoregressive process is correlated with the asset return process. We calculate the state transition probability when the correlation coefficient is 0 and is not 0 under the risk-neutral measure, investigate the impact that changes in various factors can have on the state transition probability, calculate the prices of European options, Asian options, barrier options and lookback options when the correlation coefficient takes different values. The results show that changes in correlation coefficient can lead to significant differences in option prices. Finally, this thesis combines the GARCH model with the jump process and regime switching model at the same time, and studies the option pricing problem under regime-switching GARCH-jump (RS-GARCH-jump) model. We define the risk-neutral measure under this model, derive the asset dynamics under the risk-neutral measure, and propose a lattice algorithm for option pricing. We conduct several empirical studies based on the daily returns of S&P 500 index, the results show that there are jumps in the S&P 500 index series and structural changes in the variance process. We investigate the convergence of the tree-based results to the true option values and show that this algorithm exhibits good convergence. By comparing the pricing performance of RS-GARCH-jump model with regime switching GARCH (RS-GARCH) model, GARCH-jump model, GARCH model, Black-Scholes (BS) model, and Regime-Switching (RS) model, we show that the RS-GARCH-jump model performs best in explaining option prices.
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