As discussed in the introduction, there are many possible definitions of mean reversion, which seems to mean different things to different people. Scholars discuss mean reversion by assuming that prices return to a fixed level or a trend path over infinite time. The most common definition is probably as follows:
Definition 1: An asset model is mean reverting if the prices tend to some constant in long-horizon time given information available now.
Many scholars used this property of mean reversion to propose more stochastic models. For example, the GARCH model proposed by Bollerslev (1986) also assumed that there is a long-term average variance rate. Using this definition, many analysts can convince themselves that volatilities obviously mean revert without violating the trivial market efficient and arbitrage free conditions. The problem with this definition is that the level of mean reversion may be time-varying. There can be found enormous number of examples for which a trend can be viewed as the target of mean reversion, for instance, the nondurable goods index discussed in Ramsay and Silverman (2002).
Under finite time horizon and discrete observations, autocorrelation of prices or returns is a well-known attribute commonly referred to as “mean reverting”. This gives the second definition of mean reversion:
Definition 2: An asset model is mean reverting if the returns or prices are negatively auto-correlated.
The general autoregressive process of order p in discrete time can be expressed as follows:
Rt =µ+φ1Rt−1+φ2Rt−2 +L+φpRt−p +εt (1) where φi’s are autocorrelation coefficients on the i-th lagged term and εt’s are standard normal innovations. However, with different values of φi’s, the process (1) does not necessarily converge. This means that a process can be negatively auto-correlated but diverges in long-horizon time, and so Definition 2 may be contrary to Definition 1. Therefore, the first-order autoregressive process AR(1) is popularly used in financial markets or academic researches.
Then there comes to the sort of process discussed in Lee (1991):
“Under this model, which has wide intuitive appeal, a below average price in one period is likely to be followed by “compensatory” above average prices in subsequent periods.”
Rt =µ+φ
(
µ−Rt−1)
+σεt (2) whereRt is the return price in period t, µ is the unconditional mean return in a single period, ε ’s are standard normal innovations,t σ is volatility, and φ is positive coefficient that also means the negative autocorrelation coefficient. The middle term on the right-hand side, the mean reversion term, measures the deviation of this process from the mean in the previous period and adds the correction with weightφ . If the process was below the mean in the previous period, the process gets a φ -kick upward; if it was above the mean, downward. That is, equation (2) captures a concept of mean reversion that explicitly models negative autocorrelations. It has frequently been said for example that the fantastic returns achieved in the 1980s were really a catching up exercise to make up for the poor returns in the 1970s. Jegadeesh (1991) also found the evidence that the monthly returns on the equally weighted index of stocks traded on the NYSE exhibits mean reversion over the period 1926-1988, according as the series of returns is found to exhibits significant negative serial correlation.In order to assess the informal evidence for this form of mean reversion in equity markets, Exley, Mehta, and Smith (2004) looked at 100 years of equity return data in 16 countries (the decennial Dimson, Marsh, and Staunton (2003) data set, split according to returns in each decade of the last century) and found no evidence that poor (good) returns in one decade are followed by good (poor) returns in the next.
However, when the authors looked at the UK annual equity return data, they found that returns in a year are negatively correlated (-0.2) in the data set with returns in the subsequent two years. Lee’s definition refers to the 1980s “catching up” with the 1970s, so the returns in these decades were negatively auto-correlated if we looked at discrete decades. But presumably if we had just looked at annual periods we would
have found that the returns in the 1980s were generally above average and all returns in the 1970s generally below average. This would seem to maybe suggest an element of positive autocorrelation if we change the time scales. So it seems that there is also some uncertainty as to whether mean reversion is a positive or negative auto- correlated phenomenon.
Equation (2) for the asset price process is in fact an example of a called stationary process. It is sometimes confusing about the difference between stationary process and mean reversion. In fact, stationary indeed provides another view to the concept of mean reversion.
Definition 3: An asset model is mean reverting if growth rates or volatilities are stationary.
A stationary process has identical distributions over time, unconditional on the immediate past. Under suitable conditions, the sample distribution of observations over a very long time period will converge to the stationary distribution. If an observation falls high up in the tail of the stationary distribution, it is likely that the following observation will be nearer to the long term average. This can give the appearance of a force driving observations over time towards a long term mean. This mean reverting force is countered by the influence of random noise which pushes the process away from its current value. Stationary series have proved fruitful for analyzing economic quantities such as interest rate, or dividend yields because at first sight it is plausible that these have a natural long term mean level. Exley, Mehta, and Smith (2004) mentioned that there exists mean reversion phenomenon in equity risk premium by using a stationary drift (or growth rate) process. They found that the evidence for mean reversion in equity market return volatility is also strong by using the data for 3-month and 5-year implied volatilities on the FTSE 100 (approximately the last 10 years) and DJ Euro-Stoxx Indices (approximately the last 5 years).
However the corresponding asset prices can already reflect either deterministic or
stochastic mean reversion in volatility so that we do not observe any associated mean reversion in prices. Asset prices may continue to describe a random walk despite the stationary of these associated processes. Another problem with this definition is that a stationary process has the same distribution at every point in time which means that the mean price is the same over all time period. But we discussed the various functions of the mean reversion at different time scales in the introduction. And it might be allowed that the mean reversion is simple dependent.
Before proposing our definition of mean reversion, we first discuss the implication of above three definitions. The Definition 1 does not imply the other two definitions because the long-term mean can not imply the negative auto-correlated and stationary processes. But the Definition 3 implies the Definition 1 because the prices are reverting to a stationary variable under the Definition 3, which means that the expectations are the same at any time and exist a long-term mean. And the Definition 2 also implies the Definition 1 because if the process value was below the mean in the previous period, the process gets an upward; if it was above the mean, downward.
This means that the process exist a long-term constant mean. However, we can not find out any implications between Definition 2 and 3. These three definitions are popularly used in finance about mean reverting, but they may be inconsistent whether the asset prices are mean reverting.
2.2 Models Regarding Mean Reversion
Interest rates and volatilities of returns of assets are the quantities that are most frequently referred as mean reverting in finance. Traditionally there are models dedicated to describe them.
The Ornstein-Uhlenbeck (O-U) process is a well-known stochastic process with an analytic solution of mean reversion. Vasicek (1977) uses it for describing the evolution of interest rates. The model specifies that the instantaneous interest rate follows the stochastic differential equation:
dXt =κ
(
µ−Xt)
dt+σdWt (3)In equation (3), µ and κ are respectively the level and speed of mean reversion. Hull and White (1990) extended (3) with both µ and κ to be time dependent.
Another extension is the CIR (Cox, Ingersoll, and Ross, 1985) model that is also commonly used in describing interest rate. In a stochastic differential equation form, the model can be expressed as
(
t)
t tt X dt X dW
dX =κ µ− +σ (4) As the model yields the non-central χ distribution as its solution, each points of the 2 process can be guaranteed to be positive almost surely. A broader class of models is the CEV (Constant Elasticity of Variance) model proposed by Cox and Ross (1976) which can be expressed as
(
t)
t tt X dt X dW
dX =κ µ− +σ β/2 (5) Similar to the CIR model, all values of the process are positive almost surely.
Besides for modeling interest rates, these models are also used for volatilities, for example Heston (1993).
It is noted that all of these models describe mean reversion at a very short time horizon with the term κ