wjve 发表于 2014-1-21 13:56 
哦？为什么实际的价格分布是fat tail，就能说明atm时的volatility最小呢？
This is only an approximate result not guaranteed. e.g. can be a volatility skew. And I said ATM is the lowest just for simplicity. The lowest point should be around ATM.
The reason I have explained, if no fat tail: the volatility for say three strikes in the money at the money out of money are say 0.2,0.2,0.2, if fat tail it will be 0.3, 0.2, 0.35. ATM volatility will not necessarily stay the same. Just an example
the logic for fat tail's affection is as follows:
1. In reality, the stock distribution is fat tail (fatter than lognormal)
2. This means that the stock will go very high or very low with a higher probability compared with lognormal.
3. In this case, of course, the lognormal BS model underprice the option that is deep OTM call and put (PS: 1. since ITM, especilaly deep ITM is less sensitive to volatility we ignore the affection. 2. The affection should be analyzed case by case. It can have volatility skew. Here, I assume it is a smile)
4. But you still should use log normal and BS formula right? the only thing you can do is increase the implied volatility.
5. The implied volatility on the two ends increases and, of course, the volatility in the middle will be lower comparatively.
This is the main draw back of the BS model. It is not self-consistant. If there is a volatility smile, actually the stock price is not lognormal.
hope it's clear now.
best,
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