本研究主要討論的是在雙指數跳躍擴散過程中,雙界限選擇權在第一次跳出界限的時間的機率。其模型包含了連續面的布朗運動及離散面的雙指數分配。利用Laplace transform可以求得第一次通過的時間的分配及發生第一次跳出的時間與最終值的聯合分配。且由於雙指數跳躍過程能解決 overshoot 及 undershoot 的問題,因此能計算出第一次跳出界限的時間的機率的封閉解。 最後利用本文計算出來的 Laplace transform, 搭配 Laplace inversion 來計算出第一次跨越機率, 應用至評價雙邊觸及生效買權 (one-touch knock-in call option) , 求算選擇權的價格並與蒙地卡羅模擬法做比較。
This paper studies the first passage time over two boundaries for a double exponental jump diffusion process, which consists of a continuous part driven by a Brownian motion and a jump part with jump sizes having a double exponential distribution. Explicit solutions of the Laplace transforms, of both the distribution of the first passage times and the joint distribution of the process and its running maxima, are obtained. Because the double exponential jump diffusion process can solve the overshoot and undershoot problems, we can get closed-form solutions of the distribution of the first passage times. Finally, we use their Laplace transforms associated with a Laplace inverse algorithm to apply to pricing one-touch knock-in call option.