# Modeling Momentum and Reversals

There is a large array of literature that documents the mean reverting behavior in stock prices, otherwise known as reversals. For example, Hameed, Huang, and Mian [HHM10] observe reversals in the best and worst subset of monthly performers within industries, whereas the industries themselves only exhibit momentum. (Page 1)

There is also a large array of literature about business cycles, including global cycles, country or currency cycles, and industry cycles. (Page 1)

Mean reversion as a general phenomenon is well known. Mean reversion is commonly posited for short rate models and mean reverting models are used in commodity option pricing. They are useful for modeling pairs trading, as well as analyzing the timing of entering and exiting positions [LL15]. (Page 2)

Here we show that mean reverting processes can be used to model reversals and momentum within and across industries. We analyze the model mathematically and through simulation. Simulation shows that in such a market, strategies which use reversals would outperform long-only strategies, often by a factor of 10 in high mean reversion environments. (Page 2)

Assuming the industry follows geometric Brownian motion implies that future perturbations are not impacted by past behavior. So, for example, if the underlying Brownian motion happened to trend upwards (downwards) for some period of time, the industry would continue to evolve from that higher (lower) (Page 2)

The average firm deviates around the average deterministic level, as do individual firms, albeit with much higher volatility. With lower mean reversion, the stock takes much longer to return to the industry level, and the industry does not follow the deterministic level as closely. (Page 4)

Since the stock price process is mean reverting to the industry level, and the latter is deterministic, the stock prices do not exhibit GBM volatility (namely var(log St) = σ 2t). Instead, the variance tends to a limit. (Page 4)

On the other hand, the industry itself, being the sum of the values of the companies in the industry, will have greatly dampened reversal behavior. (Page 5)

On larger time scales, since the process mean reverts to Ct, when the business cycle is booming, all of the industry stocks will exhibit momentum in that they are mean reverting to Ct which itself exhibits momentum. (Page 6)

More recently, there has been work on the relationship between market efficiency, martingale properties and the condition of a market being arbitrage free. This was first discovered by Samuelson [Sam65] and further elaborated on by Jensen [Jen78]. (Page 6)

Due to the mean reversion, it is likely that good performance will be followed by bad performance and visa-versa. This will occur roughly on a time frame of 1/a. (Page 6)

.As such, we expect that a strategy of buying the losers and selling the winners would outperform the industry. (Page 7)

The behavior as a function of the mean reversion is given in Figure 5. With a mean reversion of zero, the stocks are all martingales, so the market return is zero, as are the returns for all strategies, regardless of how you trade. Because the best/worst strategy holds fewer stocks, it’s return has a higher standard deviation than holding the market. (Page 7)

Motivated by the observation of reversals and momentum in the stock market, we considered a stock process model where each stock in an industry reverts to an underlying deterministic cyclic level. Mathematical considerations indicated that in such a model, one would observe short-term reversals and long term momentum, while the industry would just exhibit momentum, demonstrating that momentum and reversals can exist in efficient markets (as defined by Jarrow and Larsson [JL11]). (Page 8)