移动平均亚式期权定价研究

RESEARCH ON MOVING AVERAGE ASIAN OPTION PRICING

移动平均期权是价格依赖于标的资产移动平均价格的奇异期权,其定价依赖于每个窗口内的标的资产的价格,随着窗口不断向前滚动,就会有无穷多个移动平均过程,而无穷多个移动平均过程是无限维的非马尔科夫问题,因此该研究具有较大的挑战性,研究成果较少.移动平均亚式期权多用于场外的能源(石油,天然气,电力)衍生品合约,因此研究其定价和数值计算方法具有一定的理论意义和实践价值.本文以移动平均亚式期权为研究对象,运用截断的拉盖尔序列的方法,通过估计近似测度,以有限维的移动平均过程近似无限维的移动平均过程.随着拉盖尔序列所取阶数的增加,有限维移动平均对无限维移动平均的近似效果将越来越好.在得到近似的有限维标的资产价格的移动平均过程后,移动平均期权的定价转化为一个美式期权最优停时问题,本文用最小二乘蒙卡解决美式移动平均期权的最优行权问题,从而给移动平均亚式期权定价,数值分析表明通过对比拉盖尔方法和蒙卡模拟的方法,拉盖尔方法的稳定性较高.而对于固定执行价格和浮动执行价格移动平均亚式期权,都存在随着窗口长度增加,期权价格上涨的递增关系,这与一般的逻辑推断吻合.数值计算结果还表明当窗口长度增加到期权整个有效期时,移动平均亚式期权就退化为相应的亚式期权.进一步证明了该方法的正确性.

Moving average option is an exotic option whose price depends on the underlying asset moving average price process.With continuous rolling window, there will be infinite moving average processes, which is an infinite dimensional non-Markov problem. Therefore it is a challenging task and has few research results. Moving average option has wide applications in energy （crude oil, natural gas and electricity） derivatives contracts on OTC market, we apply truncation of Laguerre series method by estimating approximate measure, using finite dimensional moving average process to approximate infinite dimensional moving average process. With the increase of Laguerre series＇ order number, the approximation effect of finite dimensional moving average for infinite dimensional moving average becomes better and better. After having obtained the finite dimensional moving average approximation for the under- ling asset price, the pricing for the moving average option will turn to an optimal stopping time problem for pricing American option, which will be solved by the Least Squares Monte Carlo method. Comparing Laguerre and Monte Carlo method, Laguerre method is more stable. About the option price of fixed strike price moving average option and floating strike price moving average option, the option price will increase with the increases of window length, which is accordance with logical judgment. Numerical results also show that when the window length is the option maturity, the moving average asian option will degenerate to the corresponding European Asian option, which further justifies the correctness of our method.