基于混合分数布朗运动的回望期权定价

发布时间：2018-05-13 18:12

本文选题：混合分数布朗运动 + 交易费　；参考：《中国矿业大学》2017年硕士论文

【摘要】：回望期权是一种强路径依赖性期权,它在到期日的收益依赖于整个期权在有效期内的风险资产的最大值或最小值,这样使得回望期权在交割日的收益比较高,价格非常昂贵,更有研究的意义。当前学术界对回望期权的研究几乎都是建立在几何布朗运动的假设下,然而在实际的金融交易中,资产收益率的分布呈现“尖峰厚尾”的特点而且资产价格出现间断的不频繁的“跳跃”情况,使得在几何布朗运动下的研究与实际情况不符合。因此本文主要在混合分数布朗运动、混合跳扩散分数布朗运动以及混合分数布朗运动下带交易费三种模型下,分别对欧式回望看跌期权定价问题进行了研究。主要结果如下:(1)研究了标的股票价格服从混合分数布朗运动模型下的欧式回望期权定价问题。利用伊藤公式,风险对冲方法得到了在该模型下期权价格所满足的微分方程及其边界条件,最后由有限差分方法,得到了该微分方程的数值解,并且通过实例验证了该数值解的有效性。(2)研究了更符合实际金融市场变化的问题,在前面研究的基础上加入泊松过程,建立混合跳-扩散分数布朗运动下欧式回望看跌期权定价模型。首先利用伊藤公式,风险对冲方法得到了在该模型下期权价格所满足积分方程,通过泰勒展开,将积分方程转化为偏微分方程,然后对偏微分方程分别进行显示差分与隐式差分,得到相应的迭代方程。最后对偏微分方程进行降维变化,对变换后的数学模型构造显式离散格式并理论分析其稳定性和相容性,最后利用Matlab软件对该格式进行数值分析,讨论市场参数对期权价值的影响。(3)研究了当标的股票价格由混合分数布朗运动驱动,且支付固定交易费用时欧式回望期权的定价问题。首先运用对冲原理得到该模型下欧式回望看跌期权价值所满足的非线性偏微分方程及其边界条件。然而求解非线性偏微分方程组的解析解相当困难,又加上回望期权在到期日的执行价格的不确定性,为此,通过变量替换将得到的偏微分方程进行降维,然后通过对变换后的新方程构造Crank-Nicols格式来其数值解。最后讨论该数值格式的收敛性,交易费比率、Hurst指数等对期权价值的影响。
[Abstract]:The return option is a strong path dependent option, whose return on the maturity date depends on the maximum or minimum value of the risk asset during the expiration period of the option, which makes the return on the delivery date relatively high, and the price is very expensive. More research significance. At present, almost all the researches in academic circles are based on the hypothesis of geometric Brownian motion, however, in the actual financial transactions, The distribution of return rate of assets shows the characteristics of "peak and thick tail" and the intermittent "jump" of asset price, which makes the research under geometric Brownian motion inconsistent with the actual situation. Therefore, in this paper, the pricing problem of European lookback put options is studied under the three models of mixed fractional Brownian motion, mixed jump diffusion fractional Brownian motion and mixed fractional Brownian motion with transaction costs. The main results are as follows: 1) in this paper, we study the pricing of European lookback options under the mixed fractional Brownian motion model. By using the Ito formula and the risk hedging method, the differential equation and the boundary conditions of the option price under the model are obtained. Finally, the numerical solution of the differential equation is obtained by the finite difference method. The validity of the numerical solution is verified by an example. The problem of the change of financial market is studied. The Poisson process is added on the basis of the previous research. The pricing model of European lookback put options under mixed hopping-diffusion fractional Brownian motion is established. Firstly, by using Ito formula and risk hedging method, the integral equation of option price under this model is obtained, and the integral equation is transformed into partial differential equation by Taylor expansion. Then the partial differential equations are shown and implicit respectively, and the corresponding iterative equations are obtained. Finally, the partial differential equation is reduced in dimension, the explicit discrete scheme is constructed for the transformed mathematical model, and its stability and compatibility are analyzed theoretically. Finally, the numerical analysis of the scheme is carried out by using Matlab software. This paper discusses the influence of market parameters on the value of options. We study the pricing problem of European lookback options when the underlying stock price is driven by a mixed fractional Brownian motion and pays a fixed transaction cost. Firstly, the nonlinear partial differential equation and its boundary conditions for the value of European lookback put options under this model are obtained by using the hedging principle. However, it is very difficult to solve the analytical solution of nonlinear partial differential equations and the uncertainty of the executive price of the lookback option on the maturity date. Therefore, the dimension of the partial differential equation will be reduced by replacing the variables. Then the Crank-Nicols scheme is constructed for the transformed new equation to obtain its numerical solution. Finally, the convergence of the numerical scheme and the influence of the transaction cost ratio and Hurst exponent on the option value are discussed.
【学位授予单位】：中国矿业大学
【学位级别】：硕士
【学位授予年份】：2017
【分类号】：F224;F830.9

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