混合分数布朗运动下的回望期权定价
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摘要
文章主要研究了当标的股票价格由混合分数布朗运动驱动,且支付固定交易费用时欧式回望看跌期权的定价问题.首先运用对冲原理得到该模型下欧式回望看跌期权价值所满足的非线性偏微分方程及其边界条件.然后通过变量替换将得到的偏微分方程进行降维.之后又通过对变换后的新方程构造Crank-Nicolson格式来求其数值解.最后讨论了该数值格式的收敛性、交易费比率、Hurst指数等对期权价值的影响.
This paper studied the pricing of European lookback options when the underlying asset followed a mixed fractional Brownian movement and the transaction costs were considered.Firstly,the nonlinear partial differential equation and its boundary condition were obtained using the hedging principle under the model.Secondly,the partial differential equation was reduced using variable substitution.Next,we found its numerical solution by constructing a Crank-Nicolson format.Lastly,the convergence of the numerical scheme was discussed.We also discussed the influence of the transaction fee ratio,Hurst index,and so on.
This paper studied the pricing of European lookback options when the underlying asset followed a mixed fractional Brownian movement and the transaction costs were considered.Firstly,the nonlinear partial differential equation and its boundary condition were obtained using the hedging principle under the model.Secondly,the partial differential equation was reduced using variable substitution.Next,we found its numerical solution by constructing a Crank-Nicolson format.Lastly,the convergence of the numerical scheme was discussed.We also discussed the influence of the transaction fee ratio,Hurst index,and so on.
引文
[1]GOLDMAN M B,SOSIN H B,GATTO M A.Path dependent options:Buy at the low,sell at the high[J].Journal of Finance,1979,34(5):1111-1127.
[2]CONZE A,VISWANATHAN.Path dependent options:The case of looklback options[J].Journal of Finance,1991,46(5):1893-1907.
[3]GARMAN M.Recollection in tranquility,in form Black-Scholes to Black Holes:New frontiers in options[J].Risk Magazine,1992:171-175.
[4]BROADIE M,KOU S G.Connecting discrete and continuous path-dependent options[J].Finance and Stochastics,1998,2:1-20.
[5]AITSAHLIA F,LAI T L.Random walk duality and the valuation of discrete lookback options[J].Applied Mathematical Finance,1998,5:227-240.
[6]袁国军,杜雪樵.有交易成本的回望期权定价研究[J].运筹与管理,2006,15(3):141-143.
[7]PETERS E E.Fractal structure in the capital markets[J].Financial Analyst Journal,1989,45(4):32-37.
[8]ROGERS L C G.Arbitrage with fractional Brownian motion[J].Mathematical Finance,1997,7(1):95-105.
[9]孙琳.分数布朗运动下带交易费用的期权定价[J].系统工程,2009,27(9):36-40.
[10]桑利恒,杜雪樵.分数布朗运动下的回望期权定价[J].合肥工业大学学报(自然科学版),2010,30(5):797-800.
[11]WANG X T.Scaling and long-range dependence in option pricingⅣ:Pricing European option with transaction costs under the fractional Black-Scholes model[J].Physica A,2010,389(4):789-796.
[12]WANG X T,ZHU E H,TANG M M,et al.Scaling and long-range dependence in option pricingⅡ:Pricing European option with transaction under the mixed Brownian fractional Brownian model[J].Physica A,2010,389(3):445-451.
[13]KABANOV Y M,SAFARIAN M M.On Leland's strategy of option pricing with transactions costs[J].Finance and Stochastics,1997,1(3):239-250.
[14]GRANDITS P,SCHACHINGERY W.Leland's approach to option pricing:The evolution of a discontinuity[J].Mathematical Finance,2001,11:347-355.
[15]MERTON R C.Continuous Time Finance[M].Oxford:Blackwell Publishers,1990.
[2]CONZE A,VISWANATHAN.Path dependent options:The case of looklback options[J].Journal of Finance,1991,46(5):1893-1907.
[3]GARMAN M.Recollection in tranquility,in form Black-Scholes to Black Holes:New frontiers in options[J].Risk Magazine,1992:171-175.
[4]BROADIE M,KOU S G.Connecting discrete and continuous path-dependent options[J].Finance and Stochastics,1998,2:1-20.
[5]AITSAHLIA F,LAI T L.Random walk duality and the valuation of discrete lookback options[J].Applied Mathematical Finance,1998,5:227-240.
[6]袁国军,杜雪樵.有交易成本的回望期权定价研究[J].运筹与管理,2006,15(3):141-143.
[7]PETERS E E.Fractal structure in the capital markets[J].Financial Analyst Journal,1989,45(4):32-37.
[8]ROGERS L C G.Arbitrage with fractional Brownian motion[J].Mathematical Finance,1997,7(1):95-105.
[9]孙琳.分数布朗运动下带交易费用的期权定价[J].系统工程,2009,27(9):36-40.
[10]桑利恒,杜雪樵.分数布朗运动下的回望期权定价[J].合肥工业大学学报(自然科学版),2010,30(5):797-800.
[11]WANG X T.Scaling and long-range dependence in option pricingⅣ:Pricing European option with transaction costs under the fractional Black-Scholes model[J].Physica A,2010,389(4):789-796.
[12]WANG X T,ZHU E H,TANG M M,et al.Scaling and long-range dependence in option pricingⅡ:Pricing European option with transaction under the mixed Brownian fractional Brownian model[J].Physica A,2010,389(3):445-451.
[13]KABANOV Y M,SAFARIAN M M.On Leland's strategy of option pricing with transactions costs[J].Finance and Stochastics,1997,1(3):239-250.
[14]GRANDITS P,SCHACHINGERY W.Leland's approach to option pricing:The evolution of a discontinuity[J].Mathematical Finance,2001,11:347-355.
[15]MERTON R C.Continuous Time Finance[M].Oxford:Blackwell Publishers,1990.