均值回归对期权价格的影响

若价格本身均值回归（被拉回长期均值），则远离均值的大幅偏离会被“拉回”，在很多设定下会降低长期方差/尾部概率，从而使期权（尤其长端）更便宜。

但要注意：对股票更合理的均值回归对象通常是收益率/波动率/利率而不是价格水平本身；不同模型设定（特别是漂移项与风险溢价的指定）会影响“有效波动率”，从而影响期权价格方向。

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英文解析

Mean reversion is the tendency for a variable to return to some sort of long- run mean. Interest rates are generally considered to be meanreverting: they go up, they go down, but they eventually return to some sort of long- term average. In the case of a mean- reverting stock price, the stock price would tend to be pulled back to the average if the price rises or falls very far. This may reduce volatility and make the option cheaper.

A model of mean reversion makes sense for interest rates, and for stock returns, but it is by no means clear to me that it makes sense for stock prices. Bates argues that strong mean reversion in stock prices is implausible because of speculative opportunities available from buying when $S < \bar{S}$ and selling when $S > \bar{S}$ (Bates [1995, pp. 7- 8]). Lo and Wang say that autocorrelation in asset returns can increase or decrease $\sigma$ (and the option price) and that it depends upon the specification of the drift in the model (Lo and Wang [1995, p. 105]). Mean reversion in prices can easily yield negative autocorrelation in both prices and returns at some horizon. If we have autocorrelation in returns, then we do not have a GBM, the Black- Scholes model is

invalid, and the drift term in the price process may be very important (Lo and Wang [1995]). Conversely, if we have a GBM, then we have no autocorrelation in returns and the drift term is unimportant for pricing options. 50

At short horizons (e.g., daily or weekly), stock index returns used to be positively autocorrelated (Lo and MacKinlay [1988]) but this result has all but disappeared now (Crack [2014b]). At longer horizons (e.g., three or four years), Fama and French (1988) and Poterba and Summers (1988) say that stock returns are negatively autocorrelated (i.e., mean reverting). However, evidence for this is weak (Richardson [1993]). Lo and Mackinlay (1988, p. 61) say that longer- term positive autocorrelation is not inconsistent with shorter- term negative autocorrelation (i.e., mean reversion). Peterson et al. (1992) and Lo and Wang (1995) discuss option pricing when asset returns are autocorrelated. Crack and Ledoit (2010) discuss hypothesis testing when asset returns are autocorrelated.