美式双障碍 out-in 如何估值
American-style double-barrier
题目详情
Consider an American- style double- barrier "out- in" call option. There is an out barrier above the current stock price (an "up barrier") and an in barrier below the current stock price (a "down barrier"). This option has a payoff only if all three of the following events happen: first, the stock price path includes a fall in price below the down barrier (i.e., the option is "knocked in"); second, the stock price path does not include a rise in price above the up barrier (i.e., the option is not "knocked out"); and third, the option is exercised when the stock price is above the strike (i.e., the option is in- themoney at exercise). This option is both path- dependent and American- style. Is there an easy technique for valuing the option?
解析
直接 Monte Carlo 很难处理美式提前行权；但该类产品可用障碍期权的平价分解转为可用格点法（tree/lattice）估值的 knock-out 组合。
思路：把 out-in 写成“单障碍 up-and-out 与双障碍 double-knock-out 的差”，即
其中两者都是 knock-out（路径依赖但无 in 的记忆状态），可以在树上通过扩展状态/吸收边界实现定价，并在每个节点做美式最优行权比较。
因此：