本文研究了无风险利率改进的Black-Scholes 期权定价模型问题。利用指数函数和Ito公式的方法，获得了一种改进的Black-Scholes 期权定价模型，推广了现有Black-Scholes 期权定价模型的结果。

This paper studies the pricing model of Black-Scholes option under the changed risk-free rate, and achieves an improved Black-Scholes option pricing model by the method of the index and Ito formula. It promotes the existing Black-Scholes option pricing model.

自相似性和长期相关性等分形特性已被认为是现代金融市场最具代表的特征,这使得分数布朗运动成为数理金融研究中更为合适的工具。文章通过假定股票价格服从几何分数布朗运动,构建了It^o型分数Black-Scholes市场;接着在分数风险中性测度下,利用随机微分方程和拟鞅(quasi-martingale)定价方法给出了分数Black-Scholes期权定价模型,使得原始的Black-Scholes公式仅成为其特例;最后借助推广的分数Clark-Clone公式,给出了分数欧式期权的套期保值策略。

T he self-similarity and long-range dependence properties ,w hich are regarded as the most representa-tive properties of modern financial market ,make the fractional Brownian motion(FBM) a suitable tool in dif-ferent applications like mathematical finance .Under the hypotheses that the price follows geometric FBM ,the Ito^fractional Black-Scholes market is constructed .Using the stochastic differential equation and the quasi-martingale pricing method based on the fractional risk neutral measure ,the fractional Black-Scholes model is solved ,which makes the original Black-Scholes a specific example . Finally , by the generalized fractional Clark-Clone equation ,the hedging strategy of fractional European option is presented .

文章首先给出了Black-Scholes-Merton公式的推倒过程,并且在B-S-M公式基础上结合实际,得到支付连续红利的股票期权定价模型。在文章最后以中石油股票期权定价为例,对得到的支付连续红利的股票期权定价模型进行验证。

This paper first gives the tear down process of Black-Scholes-Merton formula, and combined with the actual on the basis of B-S-M formula. it gets the stock option pricing model which pay the continuous dividend .It gets continuous dividend stock option pricing model validated in order to petrochina stock option pricing as an example at the end of the article.

将量子概率引入到期权定价是最近几年一个新的研究趋势,也称为量子金融.为了期权定价更方便,文章建立了量子三叉树模型,同时利用量子概率建立了连续量子Black-Scholes(B-S)模型。实例应用和Matlab期权敏感性分析都验证了量子B-S优于经典B-S,从而为连续期权定价提供量子决策的途径。

Application of quantum probability in option pricing is a new trend in resent years,which also called"quantum finance". In order to make option pricing more convenient, quantum trigeminal tree and continuous quantum Black-Scholes (B-S) were established. Application example and analysis of option sensitivity show that quantum B-S is powerful than the classical, which could provide more beautiful results on option pricing.

考虑Black-Scholes模型下美式回望看跌期权的定价问题.先采用有限差分法对BlackScholes方程离散,求解期权价格,再通过Newton法求解最佳实施边界.用两种方法交替求解,得到了期权价格和最佳实施边界的数值逼近结果.数值实验验证了算法的有效性.

The authors mainly studied the numerical method for valuing American lookback put options under the Black-Scholes model.Applying the finite difference method,we obtained the discretization form of the Black-Scholes equation,which was used to solve the option value,and we got the optimal exercise boundary by Newton’s method.Solving this problem by the two method in turn,we can get the option price and the optimal exercise boundary simultaneously.Numerical experiments verify the efficiency of the method.