【考MFE的进】为什么payoff不连续delta hedge就不effective?

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MFE第六版ASM P293 14.2的上面： delta hedge is not so effective for all-or-nothing options because the payoff is discontinuous.P298 14.2.3 开头：delta hedging is not so effective for gap options. The problem is that the payoff on a gap option is discontinuous at the trigger.


不明白delta hedge怎样才有效，求解释！！！

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关键词：Effective payoff Effect hedge Delta MFE

delta hedge 是针对 price small changes，因为delta为价格变化的一阶导数。如果价格发生 jump，或者 有gap，那么delta hedge 可能效果就不好了

说说我的看法：如果要做到delta hedging，就是总的portfolio对价钱的变动不sensitive，也就是总portfolio的delta为0，如果有gap或者jump的话函数不连续可导，delta在某些点不存在。所以不好做delta hedging

非线性，一阶导是变动的，不连续更不行

delta hedging, basically, means, borrow money and buy stock at time of issue (in case of call option).  and as time and stock price changes, the level of money borrowed and stock held change accorrdingly.  this leads to two results:

1.  S > K: recieve K and buy rest of stock => pay stock and repay loan
2.  S < K: sell stock and repay loan.

the level of money borrowed and stock held at expriy are different in the two cases, and the difference are usually quite big.

if S is close to K, at the time of expiry, we are not sure if case 1 and 2 will happen and therefore not possible to hedge.

hope it helps