统计与数据科学学院知识库

回望期权是一种奇异期权，它具有强路径依赖性，其定价研究具有一定的难度和复杂性。如何对其进行合理定价一直是金融业界和学者关注的焦点，也是目前金融界理论研究的前沿问题。目前，学者们对于回望期权定价的研究大多都基于经典的Black-Scholes期权定价模型。但是，该模型中的假设条件过于严苛，譬如假定标的资产的价格变化服从几何布朗运动并且市场是无摩擦的，这并不能刻画金融资产价格的分形特征，且不能反映金融市场的实际交易情况。因此，本文采用混合次分数布朗运动驱动标的资产价格的变化，并结合Poisson过程建立混合次分数跳扩散模型，主要研究内容分为三部分：

(1) 基于混合次分数布朗运动，建立带红利的永久美式回望期权的定价模型。首先，利用Delta-对冲原理得到该期权价格所满足的偏微分方程及其边界条件。然后，运用变量代换法求解该偏微分方程，从而获得混合次分数布朗运动下，永久美式回望期权价格的闭式解与最优实施边界。最后，通过数值模拟，验证该闭式解的线性等比例放缩性质，并且讨论Hurst指数、波动率等对期权价值的影响。

(2) 引入Poisson过程，建立混合次分数跳扩散模型，得到具有交易费用的欧式回望期权定价模型。首先，利用Delta-对冲原理得到该期权价格所满足的非线性偏微分方程及其边界条件。然后，运用变量代换法对该偏微分方程进行降维，构造Crank-Nicolson格式求其数值解。最后，验证该数值方法的有效性，并讨论交易费用、波动率与无风险利率等对期权价值的影响。

(3) 选取真实的股票数据进行统计模拟分析。首先，在各大股票交易平台收集所需要的股票数据。然后，对所得数据进行简单的描述性统计分析，并对定价模型中的参数进行估计。最后，基于Monte Carlo模拟法获得不同定价模型下股票价格的模拟值，并与真实值进行对比，从而验证模型的有效性。

研究表明：与几何布朗运动、分数布朗运动、混合分数布朗运动相比，混合次分数跳扩散模型能够更好地刻画出金融资产价格的分形特征以及不连续波动，在此基础上对回望期权进行定价是合理有效的。

Lookback options are exotic options, which have strong path dependence and are difficult and complex to price. How to reasonably price the option has been the focus of the financial industry and scholars, and is currently the forefront of theoretical research in the financial sector. At present, most of the research on the pricing of lookback options is based on the classical Black-Scholes option pricing model. However, the assumptions in this model are too stringent, such as the assumption that the price change of the underlying asset follows geometric Brownian motion and the market is frictionless, which does not characterize the fractal characteristics of financial asset prices and does not reflect the actual trading situation of the financial market. Therefore, this thesis uses the mixed sub-fractional Brownian motion to drive changes in the price of the underlying asset and combines the Poisson process to build a mixed sub-fractional jump diffusion model with three main research components.

(1) The pricing model for perpetual American lookback options of stock paying continuous dividend is constructed under the mixed sub-fractional Brownian motion. Firstly, the partial differential equation satisfying the option and its boundary conditions are obtained by using the Delta hedging principle. Then, a closed form solution to the perpetual American lookback options as well as its optional exercise boundary are respectively obtained by using variable substitution method. Finally, numerical simulations are carried out to verify that the closed solution has the linear scaling property and it is discussed that the effects of Hurst index and volatility on the value of options.

(2) The pricing model of European lookback options with transaction costs is established based on the mixed sub-fractional Brownian motion and the Poisson process. Firstly, the nonlinear partial differential equation satisfying the option and its boundary conditions are obtained by using the Delta hedging principle. Then, the partial differential equation is reduced by variable substitution method, and its numerical solution is obtained by constructing a Crank-Nicolson format. Finally, the validity of the numerical method is verified, and the effects of transaction costs, volatility and risk-free interest rate on the value of the option are discussed.

(3) We select real stock data for statistical simulation analysis. Firstly, the required stock data are obtained from major stock trading platforms. Then, simple descriptive statistical analysis is performed on the data and the parameters in the pricing model are estimated. Finally, the simulated values of stock prices under different pricing models are obtained based on Monte Carlo simulations and compared with the real values to verify the validity of the mixed sub-fractional jump diffusion model.

The study result shows that the mixed sub-fractional jump diffusion model can better characterize the fractal characteristics and discontinuous fluctuations of financial asset prices than other models and that it is reasonable and effective to price lookback options on the model.