In stock investing, mean reversion refers to the tendency of the stock price moving towards to its mean or average over time. In particular, when the current price is less than the mean, the stock is considered under valued and expected to rise. On the other hand, when the current price is above its average, the stock price is over valued and is expected to fall. Mean reversion trading is about trading strategies to buy or sell the stock when its performance has greatly deviated from its average.

Mean reversion trading is based on more scientific methods determining stock buy and sell points than traditional charting approaches. It is mainly because precise numerical values can be derived from historical data to identify these buy/sell threshold levels. Mean reversion often arise in sideways markets, e.g., DJIA (1960-1980) as shown above. Other household name examples of mean reverting stocks include MSFT (2000-2010), WMT (2000-2010), Additional studies supporting the mean reversion stock returns can be found in Ref. [1].

Mean reversion can be generated by mathematical simulations. For example, taking *X(t)* to be a process governing by

Here the corresponding mean equals 2, rate of mean reversion 0.8, volatility 50%, and the noise *W(t)* a standard Brownian motion. A sample path is given in Figure 1 below.

Consider the case that the stock price is given by *S(t)=exp(X(t)) *and that the net position can be either flat (no stock holding) or long (with one share of stock holding) at any time. In addition, a slippage cost is added to each transactions. In this case, a set of differential equations (Hamilton-Jacobi-Bellman equations) can be solved to obtain *X1* and *X2* with *X1<X2*. Here *X1* is the **low** and *X2* the **high**. One should buy if the price is less than or equal to *exp(X1)* and sell if the price is greater than or equal to *exp(X2).*

To illustrate, let us consider a numerical example. Take the discount to be *0.5*, slippage cost 1%, then we can obtain *X1=1.331, X2=1.631.* For mathematical details, we refer to [1]. These two levels are shown in green lines in Figure 1. Clearly, two main factors affect the overall return: The probability for the price to go from *X1* to *X2* and the frequency for the price to travel from *X1* to *X2.*

In practice, mean reversion trading rules are often used in conjunction with trend following strategies. In addition to more mathematical issues such as model calibration and prediction, one has to decide *a prior* a stop loss level to implement the trading rules to prevent substantial losses.

**References:**

[1] H. Zhang and Q. Zhang, Trading a mean-reverting asset: Buy low and sell high, Automatica, Vol. 44, pp. 1511-1518, (2008).