A Two-Factor Model for Capturing Momentum and Mean Reversion in Stock Returns


Financial modeling has long sought to develop frameworks that accurately capture the complex dynamics of asset prices. Traditional models often focus on either momentum or mean reversion effects, struggling to incorporate both simultaneously. In this blog post, we introduce a two-factor model that aims to address this issue by integrating both momentum and mean reversion effects within the stochastic processes governing stock prices.

The development of the two-factor model is motivated by the empirical observation that financial markets exhibit periods of persistent trends (momentum) and reversion to historical means or intrinsic values (mean reversion). Capturing both effects within a single framework has been a challenge in financial econometrics. The proposed model seeks to tackle this challenge by incorporating momentum and mean reversion effects within a unified framework.

The two-factor model consists of two main components: a drift factor and a mean-reverting factor. The drift factor, denoted as d μ(t), represents the long-term trend or momentum of a stock’s price. It incorporates a constant drift parameter θ, reflecting the underlying direction driven by broader market forces or fundamental changes. The mean-reverting factor, denoted as d θt, captures the short-term deviations from the drift. It is characterized by a mean-reversion speed κ, which determines the rate at which prices revert to their long-term equilibrium following temporary fluctuations. These factors are influenced by their respective volatilities (σμ, σθ) and driven by correlated Wiener processes, allowing the model to reflect the interaction between momentum and mean reversion observed in markets

To demonstrate the model’s application, the research applies the two-factor framework to daily returns data of Coca-Cola (KO) and PepsiCo (PEP) over a twenty-year period. This empirical analysis explores the model’s potential for informing pairs trading strategies. The parameter estimation process employs a maximum likelihood estimation (MLE) technique, adapted to handle the specifics of fitting a two-factor model to real-world data. This approach aims to ensure accuracy and adaptability, enabling the model to capture the evolving dynamics of the market.

The introduction of the two-factor model contributes to the field of quantitative finance by providing a framework that incorporates both momentum and mean reversion effects. This approach can lead to a more comprehensive understanding of asset price dynamics, potentially benefiting risk management, asset allocation, and the development of trading strategies. The model’s insights may be particularly relevant for pairs trading, where identifying relative mispricings between related assets is important.

The two-factor model presented in this blog post offers a new approach to financial modeling by integrating momentum and mean reversion effects. The model’s empirical application to Coca-Cola and PepsiCo demonstrates its potential for informing trading strategies. As quantitative finance continues to evolve, the two-factor model may prove to be a useful tool for researchers, practitioners, and investors seeking to understand the dynamics of financial markets.

Two-Factor-Model-of-Stock-Returns-ver_1_1

Range-Based EGARCH Option Pricing Models (REGARCH)

The research in this post and the related paper on Range Based EGARCH Option pricing Models is focused on the innovative range-based volatility models introduced in Alizadeh, Brandt, and Diebold (2002) (hereafter ABD).  We develop new option pricing models using multi-factor diffusion approximations couched within this theoretical framework and examine their properties in comparison with the traditional Black-Scholes model.

The two-factor version of the model, which I have applied successfully in various option arbitrage strategies, encapsulates the intuively appealing idea of a trending long term mean volatility process, around which oscillates a mean-reverting, transient volatility process.  The option pricing model also incorporates asymmetry/leverage effects and well as correlation effects between the asset return and volatility processes, which results in a volatility skew.

The core concept behind Range-Based Exponential GARCH model is Log-Range estimator discussed in an earlier post on volatility metrics, which contains a lengthy exposition of various volatility estimators and their properties. (Incidentally, for those of you who requested a copy of my paper on Estimating Historical Volatility, I have updated the post to include a link to the pdf).

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We assume that the log stock price s follows a drift-less Brownian motion ds = sdW. The volatility of daily log returns, denoted h= s/sqrt(252), is assumed constant within each day, at ht from the beginning to the end of day t, but is allowed to change from one day to the next, from ht at the end of day t to ht+1 at the beginning of day t+1.  Under these assumptions, ABD show that the log range, defined as:

is to a very good approximation distributed as

where N[m; v] denotes a Gaussian distribution with mean m and variance v. The above equation demonstrates that the log range is a noisy linear proxy of log volatility ln ht.  By contrast, according to the results of Alizadeh, Brandt,and Diebold (2002), the log absolute return has a mean of 0.64 + ln ht and a variance of 1.11. However, the distribution of the log absolute return is far from Gaussian.  The fact that both the log range and the log absolute return are linear log volatility proxies (with the same loading of one), but that the standard deviation of the log range is about one-quarter of the standard deviation of the log absolute return, makes clear that the range is a much more informative volatility proxy. It also makes sense of the finding of Andersen and Bollerslev (1998) that the daily range has approximately the same informational content as sampling intra-daily returns every four hours.

Except for the model of Chou (2001), GARCH-type volatility models rely on squared or absolute returns (which have the same information content) to capture variation in the conditional volatility ht. Since the range is a more informative volatility proxy, it makes sense to consider range-based GARCH models, in which the range is used in place of squared or absolute returns to capture variation in the conditional volatility. This is particularly true for the EGARCH framework of Nelson (1990), which describes the dynamics of log volatility (of which the log range is a linear proxy).

ABD consider variants of the EGARCH framework introduced by Nelson (1990). In general, an EGARCH(1,1) model performs comparably to the GARCH(1,1) model of Bollerslev (1987).  However, for stock indices the in-sample evidence reported by Hentschel (1995) and the forecasting performance presented by Pagan and Schwert (1990) show a slight superiority of the EGARCH specification. One reason for this superiority is that EGARCH models can accommodate asymmetric volatility (often called the “leverage effect,” which refers to one of the explanations of asymmetric volatility), where increases in volatility are associated more often with large negative returns than with equally large positive returns.

The one-factor range-based model (REGARCH 1)  takes the form:

where the returns process Rt is conditionally Gaussian: Rt ~ N[0, ht2]

and the process innovation is defined as the standardized deviation of the log range from its expected value:

Following Engle and Lee (1999), ABD also consider multi-factor volatility models.  In particular, for a two-factor range-based EGARCH model (REGARCH2), the conditional volatility dynamics) are as follows:

and

where ln qt can be interpreted as a slowly-moving stochastic mean around which log volatility  ln ht makes large but transient deviations (with a process determined by the parameters kh, fh and dh).

The parameters q, kq, fq and dq determine the long-run mean, sensitivity of the long run mean to lagged absolute returns, and the asymmetry of absolute return sensitivity respectively.

The intuition is that when the lagged absolute return is large (small) relative to the lagged level of volatility, volatility is likely to have experienced a positive (negative) innovation. Unfortunately, as we explained above, the absolute return is a rather noisy proxy of volatility, suggesting that a substantial part of the volatility variation in GARCH-type models is driven by proxy noise as opposed to true information about volatility. In other words, the noise in the volatility proxy introduces noise in the implied volatility process. In a volatility forecasting context, this noise in the implied volatility process deteriorates the quality of the forecasts through less precise parameter estimates and, more importantly, through less precise estimates of the current level of volatility to which the forecasts are anchored.

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2-Factor REGARCH Model for the S&P500 Index

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.

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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.

Crash-Proof Investing

As markets continue to make new highs against a backdrop of ever diminishing participation and trading volume, investors have legitimate reasons for being concerned about prospects for the remainder of 2016 and beyond, even without consideration to the myriad of economic and geopolitical risks that now confront the US and global economies. Against that backdrop, remaining fully invested is a test of nerves for those whose instinct is that they may be picking up pennies in front an oncoming steamroller.  On the other hand, there is a sense of frustration in cashing out, only to watch markets surge another several hundred points to new highs.

In this article I am going to outline some steps investors can take to match their investment portfolios to suit current market conditions in a way that allows them to remain fully invested, while safeguarding against downside risk.  In what follows I will be using our own Strategic Volatility Strategy, which invests in volatility ETFs such as the iPath S&P 500 VIX ST Futures ETN (NYSEArca:VXX) and the VelocityShares Daily Inverse VIX ST ETN (NYSEArca:XIV), as an illustrative example, although the principles are no less valid for portfolios comprising other ETFs or equities.

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Risk and Volatility

Risk may be defined as the uncertainty of outcome and the most common way of assessing it in the context of investment theory is by means of the standard deviation of returns.  One difficulty here is that one may never ascertain the true rate of volatility – the second moment – of a returns process; one can only estimate it.  Hence, while one can be certain what the closing price of a stock was at yesterday’s market close, one cannot say what the volatility of the stock was over the preceding week – it cannot be observed the way that a stock price can, only estimated.  The most common estimator of asset volatility is, of course, the sample standard deviation.  But there are many others that are arguably superior:  Log-Range, Parkinson, Garman-Klass to name but a few (a starting point for those interested in such theoretical matters is a research paper entitled Estimating Historical Volatility, Brandt & Kinlay, 2005).

Leaving questions of estimation to one side, one issue with using standard deviation as a measure of risk is that it treats upside and downside risk equally – the “risk” that you might double your money in an investment is regarded no differently than the risk that you might see your investment capital cut in half.  This is not, of course, how investors tend to look at things: they typically allocate a far higher cost to downside risk, compared to upside risk.

One way to address the issue is by using a measure of risk known as the semi-deviation.  This is estimated in exactly the same way as the standard deviation, except that it is applied only to negative returns.  In other words, it seeks to isolate the downside risk alone.

This leads directly to a measure of performance known as the Sortino Ratio.  Like the more traditional Sharpe Ratio, the Sortino Ratio is a measure of risk-adjusted performance – the average return produced by an investment per unit of risk.  But, whereas the Sharpe Ratio uses the standard deviation as the measure of risk, for the Sortino Ratio we use the semi-deviation. In other words, we are measuring the expected return per unit of downside risk.

There may be a great deal of variation in the upside returns of a strategy that would penalize the risk-adjusted returns, as measured by its Sharpe Ratio. But using the Sortino Ratio, we ignore the upside volatility entirely and focus exclusively on the volatility of negative returns (technically, the returns falling below a given threshold, such as the risk-free rate.  Here we are using zero as our benchmark).  This is, arguably, closer to the way most investors tend to think about their investment risk and return preferences.

In a scenario where, as an investor, you are particularly concerned about downside risk, it makes sense to focus on downside risk.  It follows that, rather than aiming to maximize the Sharpe Ratio of your investment portfolio, you might do better to focus on the Sortino Ratio.

 

Factor Risk and Correlation Risk

Another type of market risk that is often present in an investment portfolio is correlation risk.  This is the risk that your investment portfolio correlates to some other asset or investment index.  Such risks are often occluded – hidden from view – only to emerge when least wanted.  For example, it might be supposed that a “dollar-neutral” portfolio, i.e. a portfolio comprising equity long and short positions of equal dollar value, might be uncorrelated with the broad equity market indices.  It might well be.  On the other hand, the portfolio might become correlated with such indices during times of market turbulence; or it might correlate positively with some sector indices and negatively with others; or with market volatility, as measured by the CBOE VIX index, for instance.

Where such dependencies are included by design, they are not a problem;  but when they are unintended and latent in the investment portfolio, they often create difficulties.  The key here is to test for such dependencies against a variety of risk factors that are likely to be of concern.  These might include currency and interest rate risk factors, for example;  sector indices; or commodity risk factors such as oil or gold (in a situation where, for example, you are investing a a portfolio of mining stocks).  Once an unwanted correlation is identified, the next step is to adjust the portfolio holdings to try to eliminate it.  Typically, this can often only be done in the average, meaning that, while there is no correlation bias over the long term, there may be periods of positive, negative, or alternating correlation over shorter time horizons.  Either way, it’s important to know.

Using the Strategic Volatility Strategy as an example, we target to maximize the Sortino Ratio, subject also to maintaining very lows levels of correlation to the principal risk factors of concern to us, the S&P 500 and VIX indices. Our aim is to create a portfolio that is broadly impervious to changes in the level of the overall market, or in the level of market volatility.

 

One method of quantifying such dependencies is with linear regression analysis.  By way of illustration, in the table below are shown the results of regressing the daily returns from the Strategic Volatility Strategy against the returns in the VIX and S&P 500 indices.  Both factor coefficients are statistically indistinguishable from zero, i.e. there is significant no (linear) dependency.  However, the constant coefficient, referred to as the strategy alpha, is both positive and statistically significant.  In simple terms, the strategy produces a return that is consistently positive, on average, and which is not dependent on changes in the level of the broad market, or its volatility.  By contrast, for example, a commonplace volatility strategy that entails capturing the VIX futures roll would show a negative correlation to the VIX index and a positive dependency on the S&P500 index.

Regression

 

Tail Risk

Ever since the publication of Nassim Taleb’s “The Black Swan”, investors have taken a much greater interest in the risk of extreme events.  If the bursting of the tech bubble in 2000 was not painful enough, investors surely appear to have learned the lesson thoroughly after the financial crisis of 2008.  But even if investors understand the concept, the question remains: what can one do about it?

The place to start is by looking at the fundamental characteristics of the portfolio returns.  Here we are not such much concerned with risk, as measured by the second moment, the standard deviation. Instead, we now want to consider the third and forth moments of the distribution, the skewness and kurtosis.

Comparing the two distributions below, we can see that the distribution on the left, with negative skew, has nonzero probability associated with events in the extreme left of the distribution, which in this context, we would associate with negative returns.  The distribution on the right, with positive skew, is likewise “heavy-tailed”; but in this case the tail “risk” is associated with large, positive returns.  That’s the kind of risk most investors can live with.

 

skewness

 

Source: Wikipedia

 

 

A more direct measure of tail risk is kurtosis, literally, “heavy tailed-ness”, indicating a propensity for extreme events to occur.  Again, the shape of the distribution matters:  a heavy tail in the right hand portion of the distribution is fine;  a heavy tail on the left (indicating the likelihood of large, negative returns) is a no-no.

Let’s take a look at the distribution of returns for the Strategic Volatility Strategy.  As you can see, the distribution is very positively skewed, with a very heavy right hand tail.  In other words, the strategy has a tendency to produce extremely positive returns. That’s the kind of tail risk investors prefer.

SVS

 

Another way to evaluate tail risk is to examine directly the performance of the strategy during extreme market conditions, when the market makes a major move up or down. Since we are using a volatility strategy as an example, let’s take a look at how it performs on days when the VIX index moves up or down by more than 5%.  As you can see from the chart below, by and large the strategy returns on such days tend to be positive and, furthermore, occasionally the strategy produces exceptionally high returns.

 

Convexity

 

The property of producing higher returns to the upside and lower losses to the downside (or, in this case, a tendency to produce positive returns in major market moves in either direction) is known as positive convexity.

 

Positive convexity, more typically found in fixed income portfolios, is a highly desirable feature, of course.  How can it be achieved?    Those familiar with options will recognize the convexity feature as being similar to the concept of option Gamma and indeed, one way to produce such a payoff is buy adding options to the investment mix:  put options to give positive convexity to the downside, call options to provide positive convexity to the upside (or using a combination of both, i.e. a straddle).

 

In this case we achieve positive convexity, not by incorporating options, but through a judicious choice of leveraged ETFs, both equity and volatility, for example, the ProShares UltraPro S&P500 ETF (NYSEArca:UPRO) and the ProShares Ultra VIX Short-Term Futures ETN (NYSEArca:UVXY).

 

Putting It All Together

While we have talked through the various concepts in creating a risk-protected portfolio one-at-a-time, in practice we use nonlinear optimization techniques to construct a portfolio that incorporates all of the desired characteristics simultaneously. This can be a lengthy and tedious procedure, involving lots of trial and error.  And it cannot be emphasized enough how important the choice of the investment universe is from the outset.  In this case, for instance, it would likely be pointless to target an overall positively convex portfolio without including one or more leveraged ETFs in the investment mix.

Let’s see how it turned out in the case of the Strategic Volatility Strategy.

 

SVS Perf

 

 

Note that, while the portfolio Information Ratio is moderate (just above 3), the Sortino Ratio is consistently very high, averaging in excess of 7.  In large part that is due to the exceptionally low downside risk, which at 1.36% is less than half the standard deviation (which is itself quite low at 3.3%).  It is no surprise that the maximum drawdown over the period from 2012 amounts to less than 1%.

A critic might argue that a CAGR of only 10% is rather modest, especially since market conditions have generally been so benign.  I would answer that criticism in two ways.  Firstly, this is an investment that has the risk characteristics of a low-duration government bond; and yet it produces a yield many times that of a typical bond in the current low interest rate environment.

Secondly, I would point out that these results are based on use of standard 2:1 Reg-T leverage. In practice it is entirely feasible to increase the leverage up to 4:1, which would produce a CAGR of around 20%.  Investors can choose where on the spectrum of risk-return they wish to locate the portfolio and the strategy leverage can be adjusted accordingly.

 

Conclusion

The current investment environment, characterized by low yields and growing downside risk, poses difficult challenges for investors.  A way to address these concerns is to focus on metrics of downside risk in the construction of the investment portfolio, aiming for high Sortino Ratios, low correlation with market risk factors, and positive skewness and convexity in the portfolio returns process.

Such desirable characteristics can be achieved with modern portfolio construction techniques providing the investment universe is chosen carefully and need not include anything more exotic than a collection of commonplace ETF products.