# Mean Reversion

### Mean Reversion

### Introduction to Mean Reversion

The two most popular types of trading strategies are momentum and mean reversion.

Momentum follows the “buy low, sell high” idea. It takes short-term position in stocks going up and selling them as soon as they show signs of going down.

In finance, mean reversion is the assumption that a stock’s price will tend to move to the average price over time. Generally speaking, we often have an intuitional impression of what will happen after encountering an abnormal event. For example, if the first person you see today is unusually tall, the next person you meet will probably be shorter. However, this intuition may not work in the financial markets. In other words, sometimes when a stock drops 10% on a given day it drops even further the next day.

Despite this, mean reversion is still a powerful concept that traders can use to find an edge and built trading strategy around. This article will discover how to use mean reversion to build trading systems.

### How to Implement the Mean Reversion?

Mean reversion requires price sequence satisfying stationarity, and we should be alert to the spurious relationship in the stock market. The cause of the spurious relationship is because of the cofounding factor or local stochastic trend. When two variables are influenced by the third variable simultaneously, the third one is the confounding factor. For example, the sales volume of ice cream and children drowning cases are increasing in summer, but it is abused to say that the high sales amount causes children drowning, and vice versa.

The local stochastic trend is more common in the stock markets. We use a plot to explain the local stochastic trend. Two independent Brownian motions time series are shown on the plot with red frames indicating the same trend of increase or decrease. Regarding blue line as dependent variable and red line as independent variable, we get the p-value of the coefficient 0.593. Thus, these two variables are uncorrelated.

When we analyze the non-stationary time series, it is likely to acquire a spurious relationship. In the stock markets, the return rate maintains the stationary but price not. Fortunately, although a single investment price does not fit the stationarity condition, we can combine two investments together to obtain a spread sequence which is stationary. This is also the intention of pair trading strategy.

### Ways to Find a Pair

Intuitively, we need to find two stocks whose company have similarities in their main business, company size even the risk factor which indicates two stocks can be from the same financial sector (like two mining stocks).

##### 1. Co-integration

We can use co-integration method to find such a pair. Cointegration is a statistical property of two or more time-series variables which indicates if a linear combination of the variables is stationary. If the long and short components fluctuate with common nonstationary factors, then the prices of the component portfolios would be co-integrated, and the pairs trading strategy would be expected to work. If the spread widens short the high stock and buy the low stock. As the spread narrows again to some equilibrium value, a profit results.

A common method to check the co-integration is Augmented Dickey-Fuller test (ADF test), where the model is

\(\Delta y_t = \alpha + \beta t+\gamma y_{t-1}+\delta_1\Delta y_{t-1}+\cdots+\delta_{p-1}\Delta y_{t-p-1} +\epsilon_t\)

where \(\alpha\) is a constant, \(\beta\) is the coefficient on a time trend and \(p\) is the lag order of the autoregressive process.

Now we introduce the unit root test with null hypothesis \(\gamma =0\) against the alternative hypothesis of \(\gamma < 0\). If the null hypothesis is rejected, then the sequence is stationary. We can use R to realize the ADF test.

##### 2. “Price Difference” Method

This method was structed by E. Gatev in paper *Pairs Trading: Performance of a Relative-Value Arbitrage Rule*. The implementation of pair trading has two stages. First one is called formation period, they form pairs over a 12-month period, and the second one called trading period aims to trade them in the next 6-month period. Both 12 months and 6 months are chosen arbitrarily.

Suppose the formation period length is \(T\), the price series of stock \(i\) is \(P_{i0}, P_{i1}, \cdots, P_{iT}\). With regard \(P_{i0}\) as the base we transfer the price series into logarithm price and then calculate the difference of square of two stocks \(i\) and \(j\) as \(D_{ij}\).

\(s_{ij}=lnP_{it}-lnP_{i0}\)

\(D_{ij} = \sum_{t=1}^T(s_{it}-s_{jt})^2\)

When all latent pair of stocks are calculated, the pair with the minimum \(D_{ij}\) will be used to trading under the rule of statistical arbitrage.

##### 3. Mean-reverting Gaussian Markov Chain Model

This method was structured by Robert J. Elliott in paper *Pairs Trading*. First, we build the spread model. Consider the state process \(\{x_k|k=0,1,2,\cdots\}\) where \(x_k\) denotes the value of some variable at time \(t_k=k\tau\) for \(k=0,1,2,\cdots\). We assume that \(\{x_k\}\) is mean reverting and fit the model

\(x_{k+1}-x_k=(a-bx_k)\tau+\sigma\sqrt\tau \epsilon_{k+1}\)

where \(\{\epsilon_k\}\) is iid Gamma \(N(0,1)\).

And the observation process \(\{y_k\}\) of \(\{x_k\}\) in Gaussian noise:

\(y_k=x_k+D\omega_k\)

where \(\{\omega_k\}\) are iid Gaussian \(N(0,1)\).

The paper regards \(\{y_k\}\) as a model for the observed spread of two securities at time \(t_k\), and assumes the observed spread is a noisy observation of some mean-reverting state process \(\{x_k\}\). The \(\{y_k\}\) could also model the returns of the spread portfolio as is often done in practice.

If \(y_k > \hat{x}_{k|k-1}=E[x_k|Y_{k-1}]\) the spread is regarded as too large, and so the trader could take a long position in the spread portfolio and profit when a correction occurs.

### Shortcomings of Mean Reversion

Even if we find a good pair of stocks whose spread shows a stationary property, it is hard to find such a spread during the trading period. This is because only when the spread deviates to a certain degree (such as 2 standard deviations) the strategy can be implemented. However, the frequency of such deviations is very low. As a result, such strategy usually cannot be executed for a long time.

Because it is difficult to satisfy the one-price theory between two stocks used for pairing, there is no logic to ensure that the price spread will always meet the mean reversion.

### Reference

[1] Galev, E E, William N. Goetzmann and K. Geert Rouwenhorst. “Pairs Trading: Performance of a Relative Value Arbitrage Rule.” (1999).

https://papers.ssrn.com/sol3/papers.cfm?abstract_id=141615

[2] Elliott, Robert J., John Van Der Hoek and W. P. Malcolm. “Pairs trading.” *Quantitative Finance* 5 (2005): 271 – 276.

http://stat.wharton.upenn.edu/~steele/Courses/434/434Context/PairsTrading/PairsTradingQFin05.pdf