【作者】 杨云锋；

【导师】 刘新平；

【作者基本信息】 陕西师范大学 ， 应用数学， 2006， 硕士

【摘要】 期权定价理论一直都是金融数学研究的核心问题之一。与投资组合理论、资本资产定价理论、市场有效性理论及代理问题一起,构成现代金融学的五大理论模块。对于传统的Black-Scholes期权定价模型,国内外学者已经作了大量研究工作,获得了许多对金融实践有指导意义的结果。然而,在现实的金融市场上,当有重大信息出现时,会导致股票价格呈现一种不连续的跳跃。而且,大量的金融实践已经充分表明,Black-Scholes期权定价模型关于标的资产价格变动规律的假设与实际存在严重的偏差。因此,众多学者放宽Black-Scholes期权定价模型的某些假设条件,提出许多新的期权定价模型。跳跃扩散模型的期权定价就是其中的一种。 本文考虑的是标的资产价格服从跳跃扩散过程,由于现实中股票价格的跳跃并不一定服从Poisson跳过程,所以将跳过程一般化是符合股票价格运动实际情况的,在此基础上也考虑了其他因素的影响。 本学位论文主要致力于跳跃扩散模型的期权定价理论问题的研究,运用鞅论,随机分析等数学工具建立更一般跳跃扩散过程的期权定价数学模型,并推导出其定价公式以及平价关系。 具体来说,主要工作如下: (1) 假设标的资产价格服从跳跃扩散过程,跳过程为比Poisson跳过程更一般的计数过程,得到该模型下欧式看涨与看跌期权的定价公式以及平价关系。 (2) 在(1)的假设下,讨论了当利率为随机变量时的期权定价问题,给出了欧式买权与卖权的定价公式以及平价关系。 (3) 建立了执行价格为随机变量的跳跃扩散过程的期权定价模型,该模型实际上是一种资产交换期权,推导出了期权定价公式。推广了Margrabe的结论。 (4) 研究了股票支付红利的跳扩散过程的欧式期权定价模型。在假定支付连续的红利率和定期支付的条件下,得到了两种情况下欧式看涨期权与看跌期权的定价公式及其它们之间的平价公式。更多还原

【Abstract】 Option pricing theory is always one of the kernel problems on financial mathematics. Together with the portfolio selection theory, the capital asset pricing theory, the effectiveness theory of market and acting issue, it is regarded as one of the five theory modules in modern finance. The domestic and foreign scholars have done a great deal of researches on Black-Scholes model and obtained many results which is instructive to financial practice. However the appearance of important information will cause the stock price to a kind of not continual jumps. A mass of finance practice has indicated that there is a serious warp between the hypothesis of Black-Scholes model about the underlying asset price and the realistic markets. Therefore, many scholars put forward many new kinds of option pricing models by relaxing some assuming conditions of Black-Scholes model. Option pricing theory with jump-diffusion is one of them.This article considers that price of underlying asset price obeys jump-diffusion process, because in the reality the stock price jumps do not necessarily obey the Poisson process, jump process generalized conforms to the actual situation of stock price movement, and other influential factors has also been considered.This dissertation is intended to study option-pricing theory with jump-diffusion, so as to establish the mathematic module of option pricing with jump-diffusion process by means of mathematical tools such as martingale theory and stochastic analysis, and deduces the option pricing equation.In detail we have made main conclusions as follows:(1) Under the hypothesis of underlying asset price being driven by a jump-diffusion process and the jump process is count process that more general than Poisson process. Using martingale method, European option and put-call parity is analyzed.(2) Under the hypothesis of underlying asset price being driven by ajump-diffusion process that is a count process discussed the option pricing when interest rate is random variable, we obtain the pricing formula of European call option.(3) Establish the option-pricing model when exercise price is random variable. The option-pricing model is options to exchange one asset to another. Pricing formula of European option is also given.(4) Considering dividend, we establish the option-pricing model with jump-diffusion process. Under the hypothesis of continuous dividend, if the continuous dividend rate is p, and regular payment dividend, we get European call and put option pricing formula and their parity.更多还原