隐含波动率与微笑/偏斜

隐含波动率（IV）：把市场价格代入 Black–Scholes 公式，反解得到的 $\sigma$。

• Volatility smile：不同执行价的 IV 呈“U 形/笑脸”，常见于外汇等近似对称尾部分布的市场
• Volatility skew：IV 随执行价单调倾斜（如股票指数常见：看跌期权 IV 更高，反映负偏度/跳跌风险）

它们反映市场对真实分布“厚尾、偏度、跳跃、随机波动”等偏离 BS 常数波动假设的定价。

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英文解析

By definition, implied volatility is the unique value of the volatility parameter $\sigma$ from the lognormal model for the evolution of the price of an underlying that makes the Black- Scholes value of an option equal to the market price of the option. Implied volatility exists and is unique⁴ for any arbitrage- free market value of the option.

On the same asset, prices of options with multiple strikes and maturities are quoted, and implied volatilities can be computed for each of these options. If the price of the asset had a lognormal distribution as assumed in the Black- Scholes model, then

the resulting plots of implied volatility vs strike for the same maturity should be flat. In practice, they are not flat, and are often shaped as "smiles" or "skews".

An implied volatility smile occurs when the implied volatilities of deep in the money options and of deep out of the money options are higher than the implied volatilities of options close to at the money. Volatility smiles are typical for currency options.

An implied volatility skew occurs when the implied volatilities of options with large strikes are lower than the implied volatilities of at the money options (reverse skew), or when the implied volatilities of options with small strikes are lower than the implied volatilities of at the money options (forward skew). Reverse skews are typical for long dated equity and index options. Forward skews are typical for commodities options.