论文标题
有效的倒数$ z $ - 转换和定价障碍以及带有离散监控的回溯选项
Efficient inverse $Z$-transform and pricing barrier and lookback options with discrete monitoring
论文作者
论文摘要
我们证明了简单的一般公式，可以期望随机步行的功能及其跑步极值。在其他条件下，我们使用反向$ z $转换，傅立叶/拉普拉斯反转和维也纳 - hopf分解来得出分析公式，并讨论了实现这些公式的有效数值方法。作为应用程序，计算了该过程的累积概率分布函数及其运行最大值以及以其最大值交换股票功率的期权的价格。最有效的数值方法使用反向$ z $ - 转换，SINH-加速技术和简化的梯形规则的新有效的数值实现。在具有中等特征的MAC上运行的MATLAB中的程序可以在几十毫秒中获得精度E-10，并且在ISECOND的一小部分中实现了E-10的精确度E-10。
We prove simple general formulas for expectations of functions of a random walk and its running extremum. Under additional conditions, we derive analytical formulas using the inverse $Z$-transform, the Fourier/Laplace inversion and Wiener-Hopf factorization, and discuss efficient numerical methods for realization of these formulas. As applications, the cumulative probability distribution function of the process and its running maximum and the price of the option to exchange the power of a stock for its maximum are calculated. The most efficient numerical methods use a new efficient numerical realization of the inverse $Z$-transform, the sinh-acceleration technique and simplified trapezoid rule. The program in Matlab running on a Mac with moderate characteristics achieves the precision E-10 and better in several dozen of milliseconds, and E-14 - in a fraction of a isecond.
