基于分数Brown运动和跳-扩散过程的亚式期权定价

发布时间：2018-04-20 11:27

本文选题：亚式期权 + 自融资交易策略　；参考：《中国矿业大学》2017年硕士论文

【摘要】：亚式期权是一种强路径依赖型奇异期权,它在到期日的收益依赖于标的资产价格在整个有效期内的平均值,从而减少了价格的波动,使得亚式期权比常规期权更受欢迎。目前对亚式期权定价问题的研究大多是建立在标准布朗运动上,并且假设标的资产价格是连续不断的,同时不需要支付交易费用。但标的资产价格呈现出一种“尖峰厚尾”的分布,且存在自相似性和长期相关性;加上实际金融市场存在大量的交易费用,因此本文将在分数跳-扩散和混合分数跳-扩散两种模型下研究带比例交易费的亚式期权定价问题。主要内容如下:(1)应用分数?Ito公式推导出混合分数跳-扩散过程的?Ito公式,并采用自融资交易策略得到亚式期权的定价模型,通过求解定价模型得到亚式看涨期权以及看跌期权的价值。最后,运用Matlab软件进行数值实验,讨论定价参数赫斯特指数、跳跃强度、股票价格等对期权价值的影响。(2)利用分数跳-扩散过程下的?Ito公式和自融资交易策略建立带交易费用的亚式期权定价模型,通过定义Leland数来简化波动率修正因子,从而简化定价模型,再运用变量替换的方法对模型进行求解,得到期权价值的解析解。数值实验直观的反映了期权价值与赫斯特指数、跳跃强度以及交易费率等的关系。(3)建立了混合分数跳-扩散过程下带交易费的亚式期权定价模型,通过降维的方法将三维问题转化为二维热传导方程,并通过对经典热传导方程的求解得到亚式看涨期权的定价公式,从而推导出看跌期权的定价公式。数值实验探究了赫斯特指数、交易费率、无风险利率以及股票价格等对期权价值的影响,并得出在一定程度上混合分数跳-扩散模型更贴近实际金融市场,比分数跳-扩散模型具有更好的稳定性。
[Abstract]:Asian option is a kind of strong path dependent singular option. Its return on maturity date depends on the average value of the underlying asset price in the whole period of validity, which reduces the fluctuation of price and makes the Asian option more popular than the conventional option. At present, most of the researches on Asian option pricing are based on the standard Brownian motion, and assume that the underlying asset price is continuous and the transaction cost is not required. But the underlying asset price shows a "peak and thick tail" distribution, and there is self-similarity and long-term correlation. In addition, there are a lot of transaction costs in the actual financial markets. In this paper, we will study the pricing of Asian options with proportional transaction costs in two models: fractional hopping diffusion and mixed fractional hopping diffusion. The main contents are as follows: (1) the mixed fractional hop-diffusion process is derived by using the fractional Ito formula, and the pricing model of Asian options is obtained by using the self-financing trading strategy. The value of Asian call option and put option is obtained by solving pricing model. Finally, using Matlab software to carry on the numerical experiment, discuss the pricing parameter Hurst index, jump intensity, The influence of stock price on the value of options. (2) by using the Fractional Leap-Diffusion formula and self-financing trading strategy, the Asian option pricing model with transaction costs is established, and the volatility correction factor is simplified by defining the Leland number. Then the pricing model is simplified and the model is solved by the method of variable substitution, and the analytical solution of option value is obtained. Numerical experiments directly reflect the relationship between option value and Hurst index, jump intensity and transaction rate, etc.) A pricing model of Asian option with transaction cost under mixed fractional hopping and diffusion process is established. The three-dimensional problem is transformed into two-dimensional heat conduction equation by dimensionality reduction method. By solving the classical heat conduction equation, the pricing formula of Asian call option is obtained, and the pricing formula of put option is deduced. The effects of Hurst index, transaction rate, risk-free interest rate and stock price on the value of options are explored, and the mixed fractional jump-diffusion model is found to be closer to the real financial market to a certain extent. It has better stability than fractional hopping diffusion model.
【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;F830.91

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