Pricing of lookback options under a mixed fractional Brownian movement
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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.
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关键词:
- 混合分数布朗运动 /
- 交易费 /
- 欧式回望期权 /
- Crank-Nicolson格式
Abstract: 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. -
图 1 回望看跌期权价值随Hurst指数变化图
Fig. 1 Values for lookback put options corresponding to different Hurst indexes
图 2 回望看跌期权随时间步长的收敛性
Fig. 2 Values for lookback put options corresponding to different time step numbers
图 3 回望看跌期权价值随到期时间和股票价格变化图
Fig. 3 Option value along with maturity and stock price in state one
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