随机波动率为何产生 smile

随机波动率模型中，波动率本身是随机过程，会在某些情形下变得很大，从而放大标的价格的尾部概率。

结果：$S_T$ 分布相较 BS 更“厚尾”（fat tails），远离平值的期权（尤其 OTM）比 BS 更贵；反解得到的 IV 就会随执行价上升，形成 smile/（或在带杠杆效应时形成 skew）。

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英文解析

A stochastic volatility model takes volatility to be driven by a random process. It is then possible that the volatility will become large causing large movements in the underlying asset price. These large moves give the distribution of the underlying asset fatter tails than in the Black- Scholes model. This means that options away from the money will be priced more expensively than in Black- Scholes leading to a volatility smile.

Another way to think about why the smile occurs is by looking at the second derivative of the option price with respect to the volatility. In the BlackScholes world our Vega sensitivity (how good our volatility hedge is) will be zero. However, a stochastic volatility model will not necessarily have a non- zero derivative of Vega, therefore there is Vega convexity which is related to the volatility smile.