问一道题：NO.PZ2016070201000041 [ FRM II ]

老师请问下 volatility smile说的是个啥啊，使用BSM mol为啥要用volatility smile with lower volatilities for out-of-the- money call anput options？ 敲出价格等于执行价格BSMmol对V的改变肯定是敏感的，后句怎么理解啊 interest rate equals the unrlying asset return? 为敲出价格远离执行价格，所以时间长也无所谓，insensitive可以理解，后句怎么理解啊interest rate is greater ththe unrlying asset return?

我对问题还是没明白，这题问的是BSM模型的局限性，而volatility smile曲线关于foreign currenoption和equity option在OTM call的曲线方向是不一样的，又怎能混为一谈表述为BSM使用的波动率过高或过低呢？

Option trars often use a volatility smile with lower volatilities for out-of-the- money call anput options when applying the BSM mol. For up-anout calls anputs, the BSM mol is insensitive to changes in implievolatility when the knock-out strike priis equto the strike prianthe interest rate equals the unrlying asset return. For wn-anout calls anputs, the BSM mol is insensitive to changes in option maturity when the knock-out strike priis greater ththe strike prianthe interest rate is greater ththe unrlying asset return. C is correct. For up-anout calls anputs anfor wn-anout calls anputs, the BSM option pricing mol is insensitive to changes in implievolatility when the knock-out strike priis equto the strike prianthe interest rate equals the unrlying asset return. The BSM mol assumes strike prices have a constant volatility, anoption trars often use a volatility smile with higher volatilities for out-of-the-money call anput options. 这道题是问用BSM来给期权定价的缺点或者局限吗？B和C想要表达的是市场上的波动率是变化的，而BSM假设波动率不变，所以有局限，对吗？只不过B说反了，C表述的情形是对的。

B不对的原因是不是因为对于期权的波动率微笑是左高右低的 所以是OTM call和ITM put的波动率小 可以使用？