隐含波动率与微笑/偏斜

隐含波动率（IV）：把市场成交价代入定价模型（最常见 Black–Scholes）并反解得到的 $\sigma$。

• smile：同到期下 IV 随执行价呈“U 型/笑脸”。
• skew：IV 随执行价单调倾斜（股票指数常见：OTM put IV 更高）。

本质上反映真实分布的厚尾/偏度、跳跃风险、随机波动率以及供需/风险溢价等对 BS 常数波动假设的偏离。

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英文解析

Implied volatility is simply the volatility implied from the market price of an option, using some model. It is usually calculated by taking the option's price and finding the volatility in the Black- Scholes formula that returns the same price. For example, consider an at- the- money European call option with 1 year to expiry, spot value 100 and a risk free rate of $5\%$ . If the market price of this option is $6.80, then the implied volatility will be$15%$ .

Figure 8.2: Volatility smile.

If one calculates the implied volatility of options with the same expiry date but different strike prices and plots the volatilities, there is often a smile or skew shape. A volatility smile will look something like Figure 2.8.

There are many possible explanations for why some markets exhibit volatility smiles. One possible explanation is that the market is more likely to move up or down by a large amount than is assumed within the Black- Scholes model. Hence the smile reflects the market's view of the imperfections in the Black- Scholes model.

A volatility skew is similar to a smile, but it is only downward sloping, compared to the more symmetric smile. A volatility skew could look something like Figure 2.9.

Figure 8.3: Volatility skew.

For a detailed discussion of volatility smiles and skews see Chapter 18 of [6].

Here are some possible related questions:

• Which shape will the implied volatility take for equity markets, skew or smile? Explain.- When did the volatility smile first appear in the equity market and why?- If the market was pricing options incorrectly and the smile was not a persistent feature, describe an arbitrage opportunity.