跨过执行价就买卖：错在哪
Spot today is 90
题目详情
金融数学题：跨过执行价就买卖：错在哪。
英文原题
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”。
英文解析
This is a tricky one in that you often find that the questioner does not really understand the problem either.
The fundamental problem is that the hedging strategy is not well defined. For example, when you first touch 100 , how do you know whether the stock will go up or down? It is not clear at that time whether 100 will be crossed so you cannot know what to hold. Thus you really need to buy at and sell at . You then lose for each crossing, and the strategy is no longer riskless.
The properties of Brownian motion actually make the problem worse. If it hits a level then it will hit it again infinitely often in an arbitrarily small amount of time. So even in retrospect, it's hard to know when the crossing occurred.
Here are some possible related questions:
Do you think this would be an effective approach in practice? Does the fact that you do not know which way the stock would go when it hits 100 make sense financially?