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欧式期权和美式期权定价的数值方法进一步研究
【作者】 梁淑平；
【导师】 张鸿雁；
【作者基本信息】 中南大学 ， 运筹学与控制论， 2008， 硕士
【摘要】 本文以无套利定价原则和风险中性定价原理为基础,对欧式期权和美式期权问题进行进一步研究.本文的主要工作包括:(1)进一步讨论了欧式期权定价的相关理论和方法;(2)分析研究了美式期权定价的数值方法,通过参数的改进,提高了模型的计算精度;(3)基于G-融资策略通过套期保值方法进一步探讨了欧式期权公平价格和美式期权最佳交易时刻的问题.对于欧式期权,本文采用Black-Scholes模型得出期权定价公式,并且讨论对比了非完备市场衍生资产定价的几种数值方法.而对于美式期权,主要探讨分析了两种数值方法:二叉树方法和有限差分法.由于普遍使用的二叉树参数模型带有缺陷,比如,在某种情况下将产生负的概率等,于是本文借助于随机误差校正思想,重新构造了二叉树参数模型,新模型永远不会产生负的概率且具有很高的计算精度,因而可实际应用于各种美式看跌期权的定价,而且这个新模型可推广到三叉树等更高阶的树图方法中去.在有限差分法中,为了保证计算结果更加精确,本文采用了控制变量技术.最后,本文基于G-融资策略探讨了欧式期权和美式期权定价,并从理论上得出离散型美式期权的最佳套期交易时刻实际上是一个最优停时.
【Abstract】 On the base of the non-arbitrage principle and the risk neutral valuation, we discuss European option and American option pricing. The main wok includes the following contents.(1) Analyze relevant theories and methods of European option pricing;(2) Investigate numerical methods of American option pricing, and improve precise valuation via new parameters;(3) Study the fair price of European option and optimal time of American option through the hedging method based on G-hedging.First, we get European option pricing formula by Black-Scholes model, and several numerical methods of derivative valuation in incomplete markets are discussed and compared. Then, we mainly deal with the binomial tree method and the finite difference method for American option pricing. The binomial tree parameter model has the flaw, for instance, it will have the negative probability in some kind of situation. So we construct new binomial tree parameter model, which will never have the negative probability and have the very high computation precision with the aid of the thought of the random error. Thus it might be applied to kinds of American put option pricing; moreover the new model might be extended to the trinomial tree and the higher order trees. In the finite difference method, we adopt control variable technique to obtain precise valuation.At last, we discuss European option and American option pricing based on G- hedging, and prove optimal hedging-time of American option is a stopping-time.