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.
FXMomentum Aug 20 2018

Modeling Asset Volatility

I am planning a series of posts on the subject of asset volatility and option pricing and thought I would begin with a survey of some of the central ideas. The attached presentation on Modeling Asset Volatility sets out the foundation for a number of key concepts and the basis for the research to follow.

Perhaps the most important feature of volatility is that it is stochastic rather than constant, as envisioned in the Black Scholes framework.  The presentation addresses this issue by identifying some of the chief stylized facts about volatility processes and how they can be modelled.  Certain characteristics of volatility are well known to most analysts, such as, for instance, its tendency to “cluster” in periods of higher and lower volatility.  However, there are many other typical features that are less often rehearsed and these too are examined in the presentation.

Long Memory
For example, while it is true that GARCH models do a fine job of modeling the clustering effect  they typically fail to capture one of the most important features of volatility processes – long term serial autocorrelation.  In the typical GARCH model autocorrelations die away approximately exponentially, and historical events are seen to have little influence on the behaviour of the process very far into the future.  In volatility processes that is typically not the case, however:  autocorrelations die away very slowly and historical events may continue to affect the process many weeks, months or even years ahead.

Volatility Direction Prediction Accuracy
Volatility Direction Prediction Accuracy

There are two immediate and very important consequences of this feature.  The first is that volatility processes will tend to trend over long periods – a characteristic of Black Noise or Fractionally Integrated processes, compared to the White Noise behavior that typically characterizes asset return processes.  Secondly, and again in contrast with asset return processes, volatility processes are inherently predictable, being conditioned to a significant degree on past behavior.  The presentation considers the fractional integration frameworks as a basis for modeling and forecasting volatility.

Mean Reversion vs. Momentum
A puzzling feature of much of the literature on volatility is that it tends to stress the mean-reverting behavior of volatility processes.  This appears to contradict the finding that volatility behaves as a reinforcing process, whose long-term serial autocorrelations create a tendency to trend.  This leads to one of the most important findings about asset processes in general, and volatility process in particular: i.e. that the assets processes are simultaneously trending and mean-reverting.  One way to understand this is to think of volatility, not as a single process, but as the superposition of two processes:  a long term process in the mean, which tends to reinforce and trend, around which there operates a second, transient process that has a tendency to produce short term spikes in volatility that decay very quickly.  In other words, a transient, mean reverting processes inter-linked with a momentum process in the mean.  The presentation discusses two-factor modeling concepts along these lines, and about which I will have more to say later.

SSALGOTRADING AD

Cointegration
One of the most striking developments in econometrics over the last thirty years, cointegration is now a principal weapon of choice routinely used by quantitative analysts to address research issues ranging from statistical arbitrage to portfolio construction and asset allocation.  Back in the late 1990’s I and a handful of other researchers realized that volatility processes exhibited very powerful cointegration tendencies that could be harnessed to create long-short volatility strategies, mirroring the approach much beloved by equity hedge fund managers.  In fact, this modeling technique provided the basis for the Caissa Capital volatility fund, which I founded in 2002.  The presentation examines characteristics of multivariate volatility processes and some of the ideas that have been proposed to model them, such as FIGARCH (fractionally-integrated GARCH).

Dispersion Dynamics
Finally, one topic that is not considered in the presentation, but on which I have spent much research effort in recent years, is the behavior of cross-sectional volatility processes, which I like to term dispersion.  It turns out that, like its univariate cousin, dispersion displays certain characteristics that in principle make it highly forecastable.  Given an appropriate model of dispersion dynamics, the question then becomes how to monetize efficiently the insight that such a model offers.  Again, I will have much more to say on this subject, in future.

Identifying Drivers of Trading Strategy Performance

Building a winning strategy, like the one in the e-Mini S&P500 futures described here is only half the challenge:  it remains for the strategy architect to gain an understanding of the sources of strategy alpha, and risk.  This means identifying the factors that drive strategy performance and, ideally, building a model so that their relative importance can be evaluated.  A more advanced step is the construction of a meta-model that will predict strategy performance and provided recommendations as to whether the strategy should be traded over the upcoming period.

Strategy Performance – Case Study

Let’s take a look at how this works in practice.  Our case study makes use of the following daytrading strategy in e-Mini futures.

Fig1

The overall performance of the strategy is quite good.  Average monthly PNL over the period from April to Oct 2015 is almost $8,000 per contract, after fees, with a standard deviation of only $5,500. That equates to an annual Sharpe Ratio in the region of 5.0.  On a decent execution platform the strategy should scale to around 10-15 contracts, with an annual PNL of around $1.0 to $1.5 million.

Looking into the performance more closely we find that the win rate (56%) and profit factor (1.43) are typical for a profitable strategy of medium frequency, trading around 20 times per session (in this case from 9:30AM to 4PM EST).

fig2

Another attractive feature of the strategy risk profile is the Max Adverse Execution, the drawdown experienced in individual trades (rather than the realized drawdown). In the chart below we see that the MAE increases steadily, without major outliers, to a maximum of only around $1,000 per contract.

Fig3

One concern is that the average trade PL is rather small – $20, just over 1.5 ticks. Strategies that enter and exit with limit orders and have small average trade are generally highly dependent on the fill rate – i.e. the proportion of limit orders that are filled.  If the fill rate is too low, the strategy will be left with too many missed trades on entry or exit, or both.  This is likely to damage strategy performance, perhaps to a significant degree – see, for example my post on High Frequency Trading Strategies.

The fill rate is dependent on the number of limit orders posted at the extreme high or low of the bar, known as the extreme hit rate.  In this case the strategy has been designed specifically to operate at an extreme hit rate of only around 10%, which means that, on average, only around one trade in ten occurs at the high or low of the bar.  Consequently, the strategy is not highly fill-rate dependent and should execute satisfactorily even on a retail platform like Tradestation or Interactive Brokers.

Drivers of Strategy Performance

So far so good.  But before we put the strategy into production, let’s try to understand some of the key factors that determine its performance.  Hopefully that way we will be better placed to judge how profitable the strategy is likely to be as market conditions evolve.

In fact, we have already identified one potential key performance driver: the extreme hit rate (required fill rate) and determined that it is not a major concern in this case. However, in cases where the extreme hit rate rises to perhaps 20%, or more, the fill ratio is likely to become a major factor in determining the success of the strategy.  It would be highly inadvisable to attempt implementation of such a strategy on a retail platform.

SSALGOTRADING AD

What other factors might affect strategy performance?  The correct approach here is to apply the scientific method:  develop some theories about the drivers of performance and see if we can find evidence to support them.

For this case study we might conjecture that, since the strategy enters and exits using limit orders, it should exhibit characteristics of a mean reversion strategy, which will tend to do better when the market moves sideways and rather worse in a strongly trending market.

Another hypothesis is that, in common with most day-trading and high frequency strategies, this strategy will produce better results during periods of higher market volatility.  Empirically, HFT firms have always produced higher profits during volatile market conditions  – 2008 was a banner year for many of them, for example.  In broad terms, times when the market is whipsawing around create additional opportunities for strategies that seek to exploit temporary mis-pricings.  We shall attempt to qualify this general understanding shortly.  For now let’s try to gather some evidence that might support the hypotheses we have formulated.

I am going to take a very simple approach to this, using linear regression analysis.  It’s possible to do much more sophisticated analysis using nonlinear methods, including machine learning techniques. In our regression model the dependent variable will be the daily strategy returns.  In the first iteration, let’s use measures of market returns, trading volume and market volatility as the independent variables.

Fig4

The first surprise is the size of the (adjusted) R Square – at 28%, this far exceeds the typical 5% to 10% level achieved in most such regression models, when applied to trading systems.  In other words, this model does a very good job of account for a large proportion of the variation in strategy returns.

Note that the returns in the underlying S&P50o index play no part (the coefficient is not statistically significant). We might expect this: ours is is a trading strategy that is not specifically designed to be directional and has approximately equivalent performance characteristics on both the long and short side, as you can see from the performance report.

Now for the next surprise: the sign of the volatility coefficient.  Our ex-ante hypothesis is that the strategy would benefit from higher levels of market volatility.  In fact, the reverse appears to be true (due to the  negative coefficient).  How can this be?  On further reflection, the reason why most HFT strategies tend to benefit from higher market volatility is that they are momentum strategies.  A momentum strategy typically enters and exits using market orders and hence requires  a major market move to overcome the drag of the bid-offer spread (assuming it calls the market direction correctly!).  This strategy, by contrast, is a mean-reversion strategy, since entry/exits are effected using limit orders.  The strategy wants the S&P500 index to revert to the mean – a large move that continues in the same direction is going to hurt, not help, this strategy.

Note, by contrast, that the coefficient for the volume factor is positive and statistically significant.  Again this makes sense:  as anyone who has traded the e-mini futures overnight can tell you, the market tends to make major moves when volume is light – simply because it is easier to push around.  Conversely, during a heavy trading day there is likely to be significant opposition to a move in any direction.  In other words, the market is more likely to trade sideways on days when trading volume is high, and this is beneficial for our strategy.

The final surprise and perhaps the greatest of all, is that the strategy alpha appears to be negative (and statistically significant)!  How can this be?  What the regression analysis  appears to be telling us is that the strategy’s performance is largely determined by two underlying factors, volume and volatility.

Let’s dig into this a little more deeply with another regression, this time relating the current day’s strategy return to the prior day’s volume, volatility and market return.

Fig5

In this regression model the strategy alpha is effectively zero and statistically insignificant, as is the case for lagged volume.  The strategy returns relate inversely to the prior day’s market return, which again appears to make sense for a mean reversion strategy:  our model anticipates that, in the mean, the market will reverse the prior day’s gain or loss.  The coefficient for the lagged volatility factor is once again negative and statistically significant.  This, too, makes sense:  volatility tends to be highly autocorrelated, so if the strategy performance is dependent on market volatility during the current session, it is likely to show dependency on volatility in the prior day’s session also.

So, in summary, we can provisionally conclude that:

This strategy has no market directional predictive power: rather it is a pure, mean-reversal strategy that looks to make money by betting on a reversal in the prior session’s market direction.  It will do better during periods when trading volume is high, and when market volatility is low.

Conclusion

Now that we have some understanding of where the strategy performance comes from, where do we go from here?  The next steps might include some, or all, of the following:

(i) A more sophisticated econometric model bringing in additional lags of the explanatory variables and allowing for interaction effects between them.

(ii) Introducing additional exogenous variables that may have predictive power. Depending on the nature of the strategy, likely candidates might include related equity indices and futures contracts.

(iii) Constructing a predictive model and meta-strategy that would enable us assess the likely future performance of the strategy, and which could then be used to determine position size.  Machine learning techniques can often be helpful in this content.

I will give an example of the latter approach in my next post.

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: Fig 1
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.

SSALGOTRADING AD

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:

  1. Momentum = Total return over the past twelve months (ignoring the last month)
  2. Value = EBIT/(Total Enterprise Value)

They form the simple Value and Momentum portfolios as follows:

  1. EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
  2. MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
  3. Universe VW = Value-weight returns to the universe of firms.
  4. 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. Fig 2   The researchers then form the following four portfolios:

  1. EBIT VW = Highest decile of firms ranked on Value (EBIT/TEV). Portfolio is value-weighted.
  2. MOM VW = Highest decile of firms ranked on Momentum. Portfolio is value-weighted.
  3. 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.
  4. 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:

Fig 3 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 Fig 4 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.

Fig 6