Monte Carlo方法在Black-Scholes模型上的应用及方差缩减概述
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摘要
在欧式期权模型中Black-Scholes公式拥有性质较好的解析解,然而,在很多衍生模型(例如亚式期权)中,Black-Scholes公式的解析解难以求得。Monte Carlo方法作为一种典型的统计模拟算法,在物理学、经济学等多种领域中起到了十分重要的作用。本文对Black-Scholes模型进行修正使其更加适应Monte Carlo方法并进行计算,并对对偶抽样、控制变量方法缩小Monte Carlo方法的方差的可行性和效果进行分析,使其更少的模拟次数得到更精确的计算结果。这种计算方法避免了使用计算机多次进行解微分方程运算,从而提高了用计算机进行大规模定价运算的效率,为高频交易中的期权价格预测提供了可能。
Black-Scholes Formula has analytical solutions with good properties in some special situations,for instance,European options. However,the analytical solution is difficult to find in many derivative models like Asian options.As a sort of typical statistical simulation algorithm,Monte Carlo method plays very important roles in Physics,Economics and other significant academic fields.The main idea is that,numerical results of Black-Scholes Formula are obtained by Monte Carlo method through adjustment of model. Based on analyzing the feasibilities and the effects of the dual sampling method and control variables method,the variance of the numerical solution is reduced and controlled,which makes more accurate results with less simulation times. Avoiding to solve differential equations,this calculation method improves the operational efficiency of large-scale pricing by computers,providing a way to predict option prices in high frequency trading.
Black-Scholes Formula has analytical solutions with good properties in some special situations,for instance,European options. However,the analytical solution is difficult to find in many derivative models like Asian options.As a sort of typical statistical simulation algorithm,Monte Carlo method plays very important roles in Physics,Economics and other significant academic fields.The main idea is that,numerical results of Black-Scholes Formula are obtained by Monte Carlo method through adjustment of model. Based on analyzing the feasibilities and the effects of the dual sampling method and control variables method,the variance of the numerical solution is reduced and controlled,which makes more accurate results with less simulation times. Avoiding to solve differential equations,this calculation method improves the operational efficiency of large-scale pricing by computers,providing a way to predict option prices in high frequency trading.
引文
[1]DF Knuth.The Art of Computer Programming--Volume I:Fundamental Algorithms[M].ADDISON-WESLEY,1973.
[2]Hull J C,Basu S.Options,futures,and other derivatives[M].Pearson Education India,2016.
[3]Marsaglia G.The squeeze method for generating gamma variates[J].Computers&Mathematics with Applications,1977,3(4):321-325.
[4]Marsaglia G.Random variables and computers[R].BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA,1962.
(1)拟Monte Carlo法:不使用随机或者伪随机数列进行模拟,而是使用一种确定生成的超均匀分布列来进行数值积分和研究其它一些数值问题。
(2)在到期日确定期权收益时,不是采用标的资产当时的市场价格,而是用期权合同期内某段时间标的资产价格的平均值,这段时间被称为平均期,在对价格进行平均时,采用算术平均或几何平均。
[2]Hull J C,Basu S.Options,futures,and other derivatives[M].Pearson Education India,2016.
[3]Marsaglia G.The squeeze method for generating gamma variates[J].Computers&Mathematics with Applications,1977,3(4):321-325.
[4]Marsaglia G.Random variables and computers[R].BOEING SCIENTIFIC RESEARCH LABS SEATTLE WA,1962.
(1)拟Monte Carlo法:不使用随机或者伪随机数列进行模拟,而是使用一种确定生成的超均匀分布列来进行数值积分和研究其它一些数值问题。
(2)在到期日确定期权收益时,不是采用标的资产当时的市场价格,而是用期权合同期内某段时间标的资产价格的平均值,这段时间被称为平均期,在对价格进行平均时,采用算术平均或几何平均。