Momentum & Mean Reversion in Discrete and Continuous Time Processes
The F/X Momentum Strategy
Our approach is based upon the idea that currencies tend to be range bound, that momentum ultimately exhausts itself and that prices tend to fall faster than they rise. The strategy seeks to exploit these characteristics with short trades that may be closed within a few hours, or continue over several days when the exhaustion pattern emerges more slowly.
Momentum Strategies
A few weeks ago I wrote an extensive post on a simple momentum strategy in E-Mini Futures. The basic idea is to buy the S&P500 E-Mini futures when the contract makes a new intraday high. This is subject to the qualification that the Internal Bar Strength fall below a selected threshold level. In order words, after a period of short-term weakness – indicated by the low reading of the Internal Bar Strength – we buy when the futures recover to make a new intraday high, suggesting continued forward momentum.
IBS is quite a useful trading indicator, which you can learn more about in the blog post:
A characteristic of momentum strategies is that they can often be applied successfully across several markets, usually with simple tweaks to the strategy parameters. As a case in point, take our Tech Momentum strategy, listed on the Systematic Strategies Algotrading platform which you can find out more about here:
This swing trading strategy applies similar momentum concepts to exploits long and short momentum effects in technology sector ETFs, focusing on the PROSHARES ULTRAPRO QQQ (TQQQ) and PROSHARES ULTRAPRO SHORT QQQ (SQQQ). Does it work? The results speak for themselves:
In four years of live trading the strategy has produced a compound annual return of 48.9%, with a Sharpe Ratio of 1.78 and Sortino Ratio of 2.98. 2018 is proving to be a banner year for the strategy, which is up by more than 48% YTD.
A very attractive feature of this momentum approach is that it is almost completely uncorrelated with the market and with a beta of just over 1 is hardly more risky than the market portfolio.
You can find out more about the Tech Momentum and other momentum strategies and how to trade them live in your own account on our Strategy Leaderboard:
A Simple Momentum Strategy
Momentum trading strategies span a diverse range of trading ideas. Often they will use indicators to determine the recent underlying trend and try to gauge the strength of the trend using measures of the rate of change in the price of the asset.
One very simple momentum concept, a strategy in S&P500 E-Mini futures, is described in the following blog post:
http://www.quantifiedstrategies.com/buy-when-sp-500-makes-new-intraday-high/
The basic idea is to buy the S&P500 E-Mini futures when the contract makes a new intraday high. This is subject to the qualification that the Internal Bar Strength fall below a selected threshold level. In order words, after a period of short-term weakness – indicated by the low reading of the Internal Bar Strength – we buy when the futures recover to make a new intraday high, suggesting continued forward momentum.
IBS is quite a useful trading indicator, which you can learn more about in these posts:
http://jonathankinlay.com/2016/06/the-internal-bar-strength-indicator/
http://jonathankinlay.com/2016/06/quick-note-internal-bar-strength-stationarity/
I have developed a version of the intraday-high strategy, using parameters to generalize it and allow for strategy optimization. The Easylanguage code for my version of the strategy is as follows:
Inputs:
nContracts(1),
ndaysHigh(5),
IBSlag(1),
IBStrigger(0.15);
Vars:
IBS(0.5);
If H[IBSlag] > L[IBSlag] then
Begin
IBS=(H[IBSlag]-C[IBSlag])/(H[IBSlag]-L[IBSlag]);
end;
If (IBS <= IBStrigger) and (H[0] >= Highest(High, ndaysHigh)) then
begin
Buy nContracts contracts this bar on close;
end;
If C[0] > H[1] then
begin
Sell all contracts this bar on close;
end;
The performance results for the strategy appear quite promising, despite the downturn in strategy profitability in 2018 to date (all performance results are net of slippage and commission):
Robustness Testing with Walk Forward Optimization
We evaluate the robustness of the strategy using the Walk Forward Optimization feature in Tradestation. Walk forward analysis is the process of optimizing a trading system using a limited set of parameters, and then testing the best optimized parameter set on out-of-sample data. This process is similar to how a trader would use an automated trading system in real live trading. The in-sample time window is shifted forward by the period covered by the out-of-sample test, and the process is repeated. At the end of the test, all of the recorded results are used to assess the trading strategy.
In other words, walk forward analysis does optimization on a training set; tests on a period after the set and then rolls it all forward and repeats the process. This gives a larger out-of-sample period and allows the system developer to see how stable the system is over time.
The image below illustrates the walk forward analysis procedure. An optimization is performed over a longer period (the in-sample data), and then the optimized parameter set is tested over a subsequent shorter period (the out-of-sample data). The optimization and testing periods are shifted forward, and the process is repeated until a suitable sample size is achieved.
Tradestation enables the user to run a battery of WFO tests, using different size in-sample and out-of-sample sizes and number of runs. The outcome of each test is evaluated on several specific criteria such as the net profit and drawdown and only if the system meets all of the criteria is the test designated as a “Pass”. This gives the analyst a clear sense of the robustness of his strategy across multiple periods and sample sizes.
A WFO cluster analysis summary for the momentum strategy is illustrated below. The cluster test is designated as “Failed” overall, since the strategy failed to meet the test criteria for a preponderance of the individual walk-forward tests. The optimal parameters found in each test vary considerably over the sample periods spanning 2003-2018, giving concerns about the robustness of the strategy under changing market conditions.
Improving the Strategy
We can improve both the performance and robustness of our simple momentum strategy by combining it with several other trend and momentum indicators. One such example is illustrated in the performance charts and tables below. The strategy has performed well in both bull and bear markets and in both normal and volatile market conditions:
A WFO cluster analysis indicates that the revised momentum strategy is highly robust to the choice of sample size and strategy parameters, as it passes every test in the 30-cell WFO analysis cluster table:
Conclusion
Momentum strategies are well known and easy to develop using standard methodologies, such as the simple indicators used in this example. They tend to work well in most equity index futures markets, and in some commodity markets too. One of their big drawbacks, however, is that they typically go through periods of poor performance and need to be tested thoroughly for robustness in order to ensure satisfactory results under the full range of market conditions.
Combining Momentum and Mean Reversion Strategies
The Fama-French World
For many years now the “gold standard” in factor models has been the 1996 Fama-French 3-factor model: 
Here r is the portfolio’s expected rate of return, Rf is the risk-free return rate, and Km is the return of the market portfolio. The “three factor” β is analogous to the classical β but not equal to it, since there are now two additional factors to do some of the work. SMB stands for “Small [market capitalization] Minus Big” and HML for “High [book-to-market ratio] Minus Low”; they measure the historic excess returns of small caps over big caps and of value stocks over growth stocks. These factors are calculated with combinations of portfolios composed by ranked stocks (BtM ranking, Cap ranking) and available historical market data. The Fama–French three-factor model explains over 90% of the diversified portfolios in-sample returns, compared with the average 70% given by the standard CAPM model.
The 3-factor model can also capture the reversal of long-term returns documented by DeBondt and Thaler (1985), who noted that extreme price movements over long formation periods were followed by movements in the opposite direction. (Alpha Architect has several interesting posts on the subject, including this one).
Fama and French say the 3-factor model can account for this. Long-term losers tend to have positive HML slopes and higher future average returns. Conversely, long-term winners tend to be strong stocks that have negative slopes on HML and low future returns. Fama and French argue that DeBondt and Thaler are just loading on the HML factor.
Enter Momentum
While many anomalies disappear under tests, shorter term momentum effects (formation periods ~1 year) appear robust. Carhart (1997) constructs his 4-factor model by using FF 3-factor model plus an additional momentum factor. He shows that his 4-factor model with MOM substantially improves the average pricing errors of the CAPM and the 3-factor model. After his work, the standard factors of asset pricing model are now commonly recognized as Value, Size and Momentum.
Combining Momentum and Mean Reversion
In a recent post, Alpha Architect looks as some possibilities for combining momentum and mean reversion strategies. They examine all firms above the NYSE 40th percentile for market-cap (currently around $1.8 billion) to avoid weird empirical effects associated with micro/small cap stocks. The portfolios are formed at a monthly frequency with the following 2 variables:
- Momentum = Total return over the past twelve months (ignoring the last month)
- Value = EBIT/(Total Enterprise Value)
They form the simple Value and Momentum portfolios as follows:
- EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
- MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
- Universe VW = Value-weight returns to the universe of firms.
- SP500 = S&P 500 Total return
The results show that the top decile of Value and Momentum outperformed the index over the past 50 years. The Momentum strategy has stronger returns than value, on average, but much higher volatility and drawdowns. On a risk-adjusted basis they perform similarly. 
The researchers then form the following four portfolios:
- EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
- MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
- COMBO VW = Rank firms independently on both Value and Momentum. Add the two rankings together. Select the highest decile of firms ranked on the combined rankings. Portfolio is value-weighted.
- 50% EBIT/ 50% MOM VW = Each month, invest 50% in the EBIT VW portfolio, and 50% in the MOM VW portfolio. Portfolio is value-weighted.
With the following results:
The main takeaways are:
- The combined ranked portfolio outperforms the index over the same time period.
- However, the combination portfolio performs worse than a 50% allocation to Value and a 50% allocation to Momentum.
A More Sophisticated Model
Yangru Wu of Rutgers has been doing interesting work in this area over the last 15 years, or more. His 2005 paper (with Ronald Balvers), Momentum and mean reversion across national equity markets, considers joint momentum and mean-reversion effects and allows for complex interactions between them. Their model is of the form 
where the excess return for country i (relative to the global equity portfolio) is represented by a combination of mean-reversion and autoregressive (momentum) terms. Balvers and Wu find that combination momentum-contrarian strategies, used to select from among 18 developed equity markets at a monthly frequency, outperform both pure momentum and pure mean-reversion strategies. The results continue to hold after corrections for factor sensitivities and transaction costs. The researchers confirm that momentum and mean reversion occur in the same assets. So in establishing the strength and duration of the momentum and mean reversion effects it becomes important to control for each factor’s effect on the other. The momentum and mean reversion effects exhibit a strong negative correlation of 35%. Accordingly, controlling for momentum accelerates the mean reversion process, and controlling for mean reversion may extend the momentum effect.
Momentum, Mean Reversion and Volatility
The presence of strong momentum and mean reversion in volatility processes provides a rationale for the kind of volatility strategy that we trade at Systematic Strategies. One sophisticated model is the Range Based EGARCH model of Alizadeh, Brandt, and Diebold (2002) . The model posits a two-factor volatility process in which a short term, transient volatility process mean-reverts to a stochastic long term mean process, which may exhibit momentum, or long memory effects (details here).
In our volatility strategy we model mean reversion and momentum effects derived from the level of short and long term volatility-of-volatility, as well as the forward volatility curve. These are applied to volatility ETFs, including levered ETF products, where convexity effects are also important. Mean reversion is a well understood phenomenon in volatility, as, too, is the yield roll in volatility futures (which also impacts ETF products like VXX and XIV).
Momentum effects are perhaps less well researched in this context, but our research shows them to be extremely important. By way of illustration, in the chart below I have isolated the (gross) returns generated by one of the momentum factors in our model.
Quant Strategies in 2018
Quant Strategies – Performance Summary Sept. 2018
The end of Q3 seems like an appropriate time for an across-the-piste review of how systematic strategies are performing in 2018. I’m using the dozen or more strategies running on the Systematic Algotrading Platform as the basis for the performance review, although results will obviously vary according to the specifics of the strategy. All of the strategies are traded live and performance results are net of subscription fees, as well as slippage and brokerage commissions.
Volatility Strategies
Those waiting for the hammer to fall on option premium collecting strategies will have been disappointed with the way things have turned out so far in 2018. Yes, February saw a long-awaited and rather spectacular explosion in volatility which completely destroyed several major volatility funds, including the VelocityShares Daily Inverse VIX Short-Term ETN (XIV) as well as Chicago-based hedged fund LJM Partners (“our goal is to preserve as much capital as possible”), that got caught on the wrong side of the popular VIX carry trade. But the lack of follow-through has given many volatility strategies time to recover. Indeed, some are positively thriving now that elevated levels in the VIX have finally lifted option premiums from the bargain basement levels they were languishing at prior to February’s carnage. The Option Trader strategy is a stand-out in this regard: not only did the strategy produce exceptional returns during the February melt-down (+27.1%), the strategy has continued to outperform as the year has progressed and YTD returns now total a little over 69%. Nor is the strategy itself exceptionally volatility: the Sharpe ratio has remained consistently above 2 over several years.
Hedged Volatility Trading
Investors’ chief concern with strategies that rely on collecting option premiums is that eventually they may blow up. For those looking for a more nuanced approach to managing tail risk the Hedged Volatility strategy may be the way to go. Like many strategies in the volatility space the strategy looks to generate alpha by trading VIX ETF products; but unlike the great majority of competitor offerings, this strategy also uses ETF options to hedge tail risk exposure. While hedging costs certainly acts as a performance drag, the results over the last few years have been compelling: a CAGR of 52% with a Sharpe Ratio close to 2.
F/X Strategies
One of the common concerns for investors is how to diversify their investment portfolios, especially since the great majority of assets (and strategies) tend to exhibit significant positive correlation to equity indices these days. One of the characteristics we most appreciate about F/X strategies in general and the F/X Momentum strategy in particular is that its correlation to the equity markets over the last several years has been negligible. Other attractive features of the strategy include the exceptionally high win rate – over 90% – and the profit factor of 5.4, which makes life very comfortable for investors. After a moderate performance in 2017, the strategy has rebounded this year and is up 56% YTD, with a CAGR of 64.5% and Sharpe Ratio of 1.89.
Equity Long/Short
Thanks to the Fed’s accommodative stance, equity markets have been generally benign over the last decade to the benefit of most equity long-only and long-short strategies, including our equity long/short Turtle Trader strategy , which is up 31% YTD. This follows a spectacular 2017 (+66%) , and is in line with the 5-year CAGR of 39%. Notably, the correlation with the benchmark S&P500 Index is relatively low (0.16), while the Sharpe Ratio is a respectable 1.47.
Equity ETFs – Market Timing/Swing Trading
One alternative to the traditional equity long/short products is the Tech Momentum strategy. This is a swing trading strategy that exploits short term momentum signals to trade the ProShares UltraPro QQQ (TQQQ) and ProShares UltraPro Short QQQ (SQQQ) leveraged ETFs. The strategy is enjoying a banner year, up 57% YTD, with a four-year CAGR of 47.7% and Sharpe Ratio of 1.77. A standout feature of this equity strategy is its almost zero correlation with the S&P 500 Index. It is worth noting that this strategy also performed very well during the market decline in Feb, recording a gain of over 11% for the month.
Futures Strategies
It’s a little early to assess the performance of the various futures strategies in the Systematic Strategies portfolio, which were launched on the platform only a few months ago (despite being traded live for far longer). For what it is worth, both of the S&P 500 E-Mini strategies, the Daytrader and the Swing Trader, are now firmly in positive territory for 2018. Obviously we are keeping a watchful eye to see if the performance going forward remains in line with past results, but our experience of trading these strategies gives us cause for optimism.
Conclusion: Quant Strategies in 2018
There appear to be ample opportunities for investors in the quant sector across a wide range of asset classes. For investors with equity market exposure, we particularly like strategies with low market correlation that offer significant diversification benefits, such as the F/X Momentum and F/X Momentum strategies. For those investors seeking the highest risk adjusted return, option selling strategies like the Option Trader strategy are the best choice, while for more cautious investors concerned about tail risk the Hedged Volatility strategy offers the security of downside protection. Finally, there are several new strategies in equities and futures coming down the pike, several of which are already showing considerable promise. We will review the performance of these newer strategies at the end of the year.
Go here for more information about the Systematic Algotrading Platform.