【作者】 李军伟；

【导师】 文志雄；

【作者基本信息】 华中科技大学 ， 基础数学， 2009， 硕士

【摘要】 二十世纪八十年代初,由Mandelbrot所创立的分形几何理论在诸多领域都显示出了广泛的应用性,其中金融领域就是分形理论应用的一个典型领域.Mandelbrot曾经从统计学的观点发现了关于股票价格变动两条的规律,但这只是统计学意义上的规律,我们无法利用这些法则来预测股票的未来价格.为了减少股票价格变动的不确定性所带来的损失以提高收益的稳定性,人们创造了期权这一金融衍生品.随之而来的问题是如何为期权确定一个合理的价格,以使交易的双方都能接受.布莱克和舒尔斯两人经过不懈的努力,找出了为期权定价的公式,也就是现在所熟知的布莱克-舒尔斯公式,这一公式已成为金融学领域的一大经典.本文探讨了分形几何理论在期权定价中的应用,主要利用分数布朗运动证明了期权定价的分数次布莱克-舒尔斯公式,并把所得的结果与建立在标准布朗运动之上的经典结论进行比较,发现经典的结果只是分数次布莱克-舒尔斯的一种特殊形式.从这点可以进一步看出分形理论在金融分析中的重要作用.更多还原

【Abstract】 In 1980s, the fractal geometry theory founded by Mandelbrot had been greatly used in many fields, among which financial field is one of the typical. Mandelbrot has found two rules about the change of stock price from statistical analysis. But it’s just the statistical analysis, we can’t use the results to estimate the future price of stock.In order to increase the stability of yield, people create the option which is a derivative financial instrument. Then the coming problem is how to make a reasonable price for an option. With great efforts, Black and Scholes got a wonderful formula, famous for Black-Scholes formula. This paper is mainly about the applications of fractal geometry theory in the aspects of option pricing . I mainly use fractional Brownian motion to prove a fractional Black-Scholes formula for the price of an option and compare the results with the classical ones which based on the standard Brownian motion. I concludes that the classical results is just one special case of the fractional Black-Scholes formula. From this point we can see the wide usage of fractal geometry in financial fields.更多还原