跨过执行价就买卖：错在哪
Spot today is 90
题目详情
Suppose spot today is 90 . A call option is struck at 100 and expires in one year. There are no interest rates. Spot moves log- normally in a perfect Black- Scholes world. I claim that I can hedge the option for free. Whenever spot crosses 100 in an upwards direction I borrow 100 and buy the stock. Whenever spot crosses 100 in a downwards direction I sell the stock and repay my loan. At expiry either the option is out- of- themoney in which case I have no position or it is in- the- money and I use the 100 dollar strike to payoff my loan. Thus the option has been hedged for free. Where is the error in this argument?
解析
该“免费对冲”策略的问题在于：在连续时间、布朗路径下，“穿越 100”这一概念使得策略并不良定义，且即便用 -带宽修正，也会产生无限次交易与成本。
- 当价格第一次触到 100 时，你无法在触点当下知道接下来是向上穿越还是向下穿越，因此无法在 100 处瞬间决定持仓。
- 真实可执行只能是：上穿时在 买入、下穿时在 卖出，于是每次往返至少损失 。
- 布朗运动具有“打到某一水平后会在任意短时间内无穷次回访”的性质，导致穿越次数趋于无穷，累计损失发散。
因此该策略并非无风险复制，也不可能“免费对冲”。