分形市场中两类衍生证 券定价问题的研究-应用数学专业论文.docxVIP

山东科技大学硕士学位论文摘要 山东科技大学硕士学位论文 摘要 山东科技大学硕士学位论文摘要 山东科技大学硕士学位论文 摘要 1 1 2 2 摘 要 1973 年，两位伟大的金融理论家与实务家 Fisher Black 和 Myron Scholes 发表了 他们的著名论文“期权定价与公司债务”，给出了欧式期权定价的显式表达式，即著名的 Black-Scholes 公式。这是现代金融数学的一项具有里程碑意义的突破性成果。从此， 金融数学的研究得到了蓬勃的发展，取得了非常丰硕的成果，而且应用于金融市场，受 到广泛的欢迎。 本文在分数布朗运动积分理论的基础上，应用具有 Hurst 参数的分数 Black-Scholes 模型，对两类衍生证券的定价问题进行了研究。 首先介绍了金融数学发展的历史，特别对期权定价理论的主要研究内容和成果进行 了综述。 然后介绍了分数布朗运动的基础知识，包括分数布朗运动的定义，性质及拟－条件 期望和拟－鞅。 在此基础上讨论了分形市场中欧式权证的定价问题。关于无红利支付和有红利支付 的欧式权证，给出了定价公式及套期保值策略，还讨论了多噪声情形下的欧式权证和欧 式价差期权的定价问题。 最后讨论了分形市场中欧式外汇期权的定价问题。关于欧式外汇未定权益，欧式外 汇期权以及多噪声情形下欧式外汇期权给出了它们的定价公式。 关键词：分数布朗运动 ；未定权益；欧式权证；外汇期权 ABSTRACT In 1973, two great financial theorists and practices Fisher Black and Myron Scholes published their famous paper The Pricing of Options and Corporate Liability which gave the Black-scholes formula, an explicit formula of the pricing of European Option. This is a breakthrough of modern Mathematical Finance. From then on, the research of modern Mathematical Finance gained rapid development with tremendous achievements. Most remarkable of all, the Black-Scholes model has not been obtained plentiful results in theoretical research, but also been applied in financial market broadly. In 1990s the annual transaction volume in global derivative securities market achieved 50,000 billion dollars. In this dissertation, on the basis of the stochastic integral of the fractional Brownian motion, we studied the pricing of derivative securities. First we introduce the history of mathematical finance, especially its main research content and results. Then we introduce the definition, properties and quasi-conditional expectation and quasi- martingale of the fractional Brownian motion. On the basis we discuss the pricing of European warrant under fractal market. About European warrant without and with dividends payments, we give their pricing formulas. We then give the pricing formulas of European warrant and Spread Option under Multi-noise. At last, we discuss the pricing of European