波动率微笑

Black–Scholes 假设常数波动率与对数正态分布，但真实市场通常存在：

• 厚尾/尖峰（kurtosis）
• 偏度（skew）与跳跃风险
• 随机波动率、杠杆效应
• 供需与风控约束对特定执行价的报价影响

这些因素使得不同执行价的期权需要不同的“等效波动率”才能匹配市场价格，于是把 IV 对 $K$ 作图就出现 smile（或 skew）。

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英文解析

If the Black- Scholes assumptions are correct, then the implied volatilities of options (those backed out of the Black- Scholes pricing formula given the other pricing parameters) should fall on a horizontal line when plotted against strike prices of the options used. However, the patterns that result include smiles and skewed lines depending upon the underlying asset and the time period (Hammer [1989]; Sullivan [1993]; Murphy [1994]; Derman and Kani [1994]). Before the Crash of 1987, you typically got smiles when you plotted the implied volatilities against strikes. Nowadays you are more likely to get skews, or smirks.

What is happening may be viewed in some different and related ways. Option prices are determined by supply and demand, not by theoretical formulae. The traders who are determining the option prices are implicitly modifying the Black- Scholes

assumptions to account for volatility that changes both with time and with stock price level. This is contrary to the Black and Scholes (1973) assumption of constant volatility irrespective of stock price or time to maturity. That is, traders assume $\sigma = \sigma (S(t),t)$ , whereas Black and Scholes assume $\sigma$ is just a constant. 6

If volatility is changing with both level of the underlying and time to maturity, then the distribution of future stock price is no longer lognormal. The distribution must be something different. Black- Scholes option pricing takes discounted expected payoffs relative to a lognormal distribution. As volatility changes through time, you are likely to get periods of little activity and periods of intense activity. These periods produce peakedness and fat tails respectively (together called "leptokurtosis"), in stock returns distributions. Fat tails are likely to lead to some sort of smile effect, because they increase the chance of payoffs away- from- the- money. 7

These irregularities have led to "stochastic volatility" models that account for volatility changing as a function of both time and stock price level (Hull and White [1987]; Scott[1987]; Wiggins [1987]; Hull [1997]). Applications to FOREX options include Chesney and Scott (1989) and Melino and Turnbull (1990). The effect of stochastic volatility on options values is similar to the effect of a jump component: both increase the probability that out- of- themoney options will finish in- the- money and increase the probability that inthe- money options will finish out- of- the- money (Wiggins [1987, pp. 360- 361]). Whether the smile is skewed left, skewed right, or symmetric in a stochastic volatility model depends upon the sign of the correlation between changes in volatility and changes in stock price (Hull [1997, Section 19.3]).