一类分数金融资产价格过程的近似解及其期权定价

• 1:betway官方app 数学与信息科学学院

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摘要(Abstract)：

为了拟合金融资产数据的长记忆性，很多学者利用分数布朗运动驱动的随机微分方程来刻画金融资产价格的变化规律.但是由于分数布朗运动驱动的模型在金融市场中会产生套利机会，这将给研究期权定价问题带来困难.鉴于此，首先采用一类具有半鞅性质的分数高斯过程对分数布朗运动进行近似，此近似关于L²(Ω)是一致收敛的.然后，利用分数高斯过程对金融资产价格进行统计建模，求得模型近似解的闭式表达式，并以分数阶Langevin模型作为特例，对近似解和原模型解的样本路径进行模拟，展示了二者的近似程度.最后，基于所构建的近似模型，得到了几何平均亚式看涨期权和看跌期权的定价公式.
In order to fit the long memory of financial asset data, many scholars use stochastic differential equations driven by fractional Brownian motion to describe the change law of financial asset prices. However, the model driven by fractional Brownian will generate arbitrage opportunities in the financial market, which is difficult for people to study the option pricing problem. In view of this, this paper first uses a class of fractional Gaussian processes with semimartingale properties to approximate the fractional Brownian motion, which is uniformly convergence under L2(Ω)是一致收敛的.然后，利用分数高斯过程对金融资产价格进行统计建模，求得模型近似解的闭式表达式，并以分数阶Langevin模型作为特例，对近似解和原模型解的样本路径进行模拟，展示了二者的近似程度.最后，基于所构建的近似模型，得到了几何平均亚式看涨期权和看跌期权的定价公式.
In order to fit the long memory of financial asset data, many scholars use stochastic differential equations driven by fractional Brownian motion to describe the change law of financial asset prices. However, the model driven by fractional Brownian will generate arbitrage opportunities in the financial market, which is difficult for people to study the option pricing problem. In view of this, this paper first uses a class of fractional Gaussian processes with semimartingale properties to approximate the fractional Brownian motion, which is uniformly convergence under L²(Ω). Then, the fractional Gauss process is used to estabilish the statistical model of the financial asset price, and the closed form expression of the the approximate solution of the model is obtained. Selecting the fractional Langevin model as a special case, simulating the sample paths of the approximate solution and the solution of original model, and the approximate degree of them are shown. Finally, based on the constructed approximate model, the pricing formulas of geometric average Asian call option and put option are obtained.

关键词(KeyWords)：分数布朗运动;L~2近似方法;半鞅;长记忆性;期权定价
fractional Brownian motion;L~2-approximate approach;semimartingale;long-memory;option pricing

基金项目(Foundation):国家自然科学基金(11971154);; 河南省重点研发与推广专项(软科学研究)项目(232400410034)

作者(Authors):王继霞;肖晓芳;
Wang Jixia;Xiao Xiaofang;College of Mathematics and Information Science, Henan Normal University;

DOI:10.16366/j.cnki.1000-2367.2023.04.009

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一类分数金融资产价格过程的近似解及其期权定价.pdf