Tag: Direction Prediction

Posted on

Forecasting Financial Markets – Part 1: Time Series Analysis

The presentation in this post covers a number of important topics in forecasting, including:

  • Stationary processes and random walks
  • Unit roots and autocorrelation
  • ARMA models
  • Seasonality
  • Model testing
  • Forecasting
  • Dickey-Fuller and Phillips-Perron tests for unit roots

Also included are a number of detailed worked examples, including:

  1. ARMA Modeling
  2. Box Jenkins methodology
  3. Modeling the US Wholesale Price Index
  4. Pesaran & Timmermann study of excess equity returns
  5. Purchasing Power Parity

 

Forecasting 2011 - Time Series

 

Posted on

Can Machine Learning Techniques Be Used To Predict Market Direction? The 1,000,000 Model Test.

During the 1990’s the advent of Neural Networks unleashed a torrent of research on their applications in financial markets, accompanied by some rather extravagant claims about their predicative abilities.  Sadly, much of the research proved to be sub-standard and the results illusionary, following which the topic was largely relegated to the bleachers, at least in the field of financial market research.

With the advent of new machine learning techniques such as Random Forests, Support Vector Machines and Nearest Neighbor Classification, there has been a resurgence of interest in non-linear modeling techniques and a flood of new research, a fair amount of it supportive of their potential for forecasting financial markets.  Once again, however, doubts about the quality of some of the research bring the results into question.

SSALGOTRADING AD

Against this background I and my co-researcher Dan Rico set out to address the question of whether these new techniques really do have predicative power, more specifically the ability to forecast market direction.  Using some excellent MatLab toolboxes and a new software package, an Excel Addin called 11Ants, that makes large scale testing of multiple models a snap, we examined over 1,000,000 models and model-ensembles, covering just about every available non-linear technique.  The data set for our study comprised daily prices for a selection of US equity securities, together with a large selection of technical indicators for which some other researchers have claimed explanatory power.

In-Sample Equity Curve for Best Performing Nonlinear Model
In-Sample Equity Curve for Best Performing Nonlinear Model

The answer provided by our research was, without exception, in the negative: not one of the models tested showed any significant ability to predict the direction of any of the securities in our data set.  Furthermore, our study found that the best-performing models favored raw price data over technical indicator variables, suggesting that the latter have little explanatory power.

As with Neural Networks, the principal difficulty with non-linear techniques appears to be curve-fitting and a failure to generalize:  while it is very easy to find models that provide an excellent fit to in-sample data, the forecasting performance out-of-sample is often very poor.

Out-of-Sample Equity Curve for Best Performing Nonlinear Model
Out-of-Sample Equity Curve for Best Performing Nonlinear Model

Some caveats about our own research apply.  First and foremost, it is of course impossible to prove a hypothesis in the negative.  Secondly, it is plausible that some markets are less efficient than others:  some studies have claimed success in developing predictive models due to the (relative) inefficiency of the F/X and futures markets, for example.  Thirdly, the choice of sample period may be criticized:  it could be that the models were over-conditioned on a too- lengthy in-sample data set, which in one case ran from 1993 to 2008, with just two years (2009-2010) of out-of-sample data.  The choice of sample was deliberate, however:  had we omitted the 2008 period from the “learning” data set, it would be very easy to criticize the study for failing to allow the algorithms to learn about the exceptional behavior of the markets during that turbulent year.

Despite these limitations, our research casts doubt on the findings of some less-extensive studies, that may be the result of sample-selection bias.  One characteristic of the most credible studies finding evidence in favor of market predictability, such as those by Pesaran and Timmermann, for instance (see paper for citations), is that the models they employ tend to incorporate independent explanatory variables, such as yield spreads, which do appear to have real explanatory power.  The finding of our study suggest that, absent such explanatory factors, the ability to predict markets using sophisticated non-linear techniques applied to price data alone may prove to be as illusionary as it was in the 1990’s.

 

ONE MILLION MODELS

Posted on

On Testing Direction Prediction Accuracy


As regards the question of forecasting accuracy discussed in the paper on Forecasting Volatility in the S&P 500 Index, there are two possible misunderstandings here that need to be cleared up.  These arise from remarks by one commentator  as follows:

“An above 50% vol direction forecast looks good,.. but “direction” is biased when working with highly skewed distributions!   ..so it would be nice if you could benchmark it against a simple naive predictors to get a feel for significance, -or- benchmark it with a trading strategy and see how the risk/return performs.”

(i) The first point is simple, but needs saying: the phrase “skewed distributions” in the context of volatility modeling could easily be misconstrued as referring to the volatility skew. This, of course, is used to describe to the higher implied vols seen in the Black-Scholes prices of OTM options. But in the Black-Scholes framework volatility is constant, not stochastic, and the “skew” referred to arises in the distribution of the asset return process, which has heavier tails than the Normal distribution (excess Kurtosis and/or skewness). I realize that this is probably not what the commentator meant, but nonetheless it’s worth heading that possible misunderstanding off at the pass, before we go on.

SSALGOTRADING AD

(ii) I assume that the commentator was referring to the skewness in the volatility process, which is characterized by the LogNormal distribution. But the forecasting tests referenced in the paper are tests of the ability of the model to predict the direction of volatility, i.e. the sign of the change in the level of volatility from the current period to the next period. Thus we are looking at, not a LogNormal distribution, but the difference in two LogNormal distributions with equal mean – and this, of course, has an expectation of zero. In other words, the expected level of volatility for the next period is the same as the current period and the expected change in the level of volatility is zero. You can test this very easily for yourself by generating a large number of observations from a LogNormal process, taking the difference and counting the number of positive and negative changes in the level of volatility from one period to the next. You will find, on average, half the time the change of direction is positive and half the time it is negative.

For instance, the following chart shows the distribution of the number of positive changes in the level of a LogNormally distributed random variable with mean and standard deviation of 0.5, for a sample of 1,000 simulations, each of 10,000 observations.  The sample mean (5,000.4) is very close to the expected value of 5,000.

Distribution Number of Positive Direction Changes

So, a naive predictor will forecast volatility to remain unchanged for the next period and by random chance approximately half the time volatility will turn out to be higher and half the time it will turn out to be lower than in the current period. Hence the default probability estimate for a positive change of direction is 50% and you would expect to be right approximately half of the time. In other words, the direction prediction accuracy of the naive predictor is 50%. This, then, is one of the key benchmarks you use to assess the ability of the model to predict market direction. That is what test statistics like Theil’s-U does – measures the performance relative to the naive predictor. The other benchmark we use is the change of direction predicted by the implied volatility of ATM options.
In this context, the model’s 61% or higher direction prediction accuracy is very significant (at the 4% level in fact) and this is reflected in the Theil’s-U statistic of 0.82 (lower is better). By contrast, Theil’s-U for the Implied Volatility forecast is 1.46, meaning that IV is a much worse predictor of 1-period-ahead changes in volatility than the naive predictor.

On its face, it is because of this exceptional direction prediction accuracy that a simple strategy is able to generate what appear to be abnormal returns using the change of direction forecasts generated by the model, as described in the paper. In fact, the situation is more complicated than that, once you introduce the concept of a market price of volatility risk.

 

Posted on

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.

Posted on

Forecasting Volatility in the S&P500 Index

Several people have asked me for copies of this research article, which develops a new theoretical framework, the ARFIMA-GARCH model as a basis for forecasting volatility in the S&P 500 Index.  I am in the process of updating the research, but in the meantime a copy of the original paper is available here

In this analysis we are concerned with the issue of whether market forecasts of volatility, as expressed in the Black-Scholes implied volatilities of at-the-money European options on the S&P500 Index, are superior to those produced by a new forecasting model in the GARCH framework which incorporates long-memory effects.  The ARFIMA-GARCH model, which uses high frequency data comprising 5-minute returns, makes volatility the subject process of interest, to which innovations are introduced via a volatility-of-volatility (kurtosis) process.  Despite performing robustly in- and out-of-sample, an encompassing regression indicates that the model is unable to add to the information already contained in market forecasts.  However, unlike model forecasts, implied volatility forecasts show evidence of a consistent and substantial bias.  Furthermore, the model is able to correctly predict the direction of volatility approximately 62% of the time whereas market forecasts have very poor direction prediction ability.  This suggests that either option markets may be inefficient, or that the option pricing model is mis-specified.  To examine this hypothesis, an empirical test is carried out in which at-the-money straddles are bought or sold (and delta-hedged) depending on whether the model forecasts exceed or fall below implied volatility forecasts.  This simple strategy generates an annual compound return of 18.64% over a four year out-of-sample period, during which the annual return on the S&P index itself was -7.24%.  Our findings suggest that, over the period of analysis, investors required an additional risk premium of 88 basis points of incremental return for each unit of volatility risk.

Posted on

Using Volatility to Predict Market Direction

Decomposing Asset Returns

 

We can decompose the returns process Rt as follows:

While the left hand side of the equation is essentially unforecastable, both of the right-hand-side components of returns display persistent dynamics and hence are forecastable. Both the signs of returns and magnitude of returns are conditional mean dependent and hence forecastable, but their product is conditional mean independent and hence unforecastable. This is an example of a nonlinear “common feature” in the sense of Engle and Kozicki (1993).

Although asset returns are essentially unforecastable, the same is not true for asset return signs (i.e. the direction-of-change). As long as expected returns are nonzero, one should expect sign dependence, given the overwhelming evidence of volatility dependence. Even in assets where expected returns are zero, sign dependence may be induced by skewness in the asset returns process.  Hence market timing ability is a very real possibility, depending on the relationship between the mean of the asset returns process and its higher moments. The highly nonlinear nature of the relationship means that conditional sign dependence is not likely to be found by traditional measures such as signs autocorrelations, runs tests or traditional market timing tests. Sign dependence is likely to be strongest at intermediate horizons of 1-3 months, and unlikely to be important at very low or high frequencies. Empirical tests demonstrate that sign dependence is very much present in actual US equity returns, with probabilities of positive returns rising to 65% or higher at various points over the last 20 years. A simple logit regression model captures the essentials of the relationship very successfully.

Now consider the implications of dependence and hence forecastability in the sign of asset returns, or, equivalently, the direction-of-change. It may be possible to develop profitable trading strategies if one can successfully time the market, regardless of whether or not one is able to forecast the returns themselves.  

There is substantial evidence that sign forecasting can often be done successfully. Relevant research on this topic includes Breen, Glosten and Jaganathan (1989), Leitch and Tanner (1991), Wagner, Shellans and Paul (1992), Pesaran and Timmerman (1995), Kuan and Liu (1995), Larsen and Wozniak (10050, Womack (1996), Gencay (1998), Leung Daouk and Chen (1999), Elliott and Ito (1999) White (2000), Pesaran and Timmerman (2000), and Cheung, Chinn and Pascual (2003).

There is also a huge body of empirical research pointing to the conditional dependence and forecastability of asset volatility. Bollerslev, Chou and Kramer (1992) review evidence in the GARCH framework, Ghysels, Harvey and Renault (1996) survey results from stochastic volatility modeling, while Andersen, Bollerslev and Diebold (2003) survey results from realized volatility modeling.

Sign Dynamics Driven By Volatility Dynamics

Let the returns process Rt be Normally distributed with mean m and conditional volatility st.

The probability of a positive return Pr[Rt+1 >0] is given by the Normal CDF F=1-Prob[0,f]


 

 

For a given mean return, m, the probability of a positive return is a function of conditional volatility st. As the conditional volatility increases, the probability of a positive return falls, as illustrated in Figure 1 below with m = 10% and st = 5% and 15%.

In the former case, the probability of a positive return is greater because more of the probability mass lies to the right of the origin. Despite having the same, constant expected return of 10%, the process has a greater chance of generating a positive return in the first case than in the second. Thus volatility dynamics drive sign dynamics.  

 Figure 1

Email me at jkinlay@investment-analytics.com.com for a copy of the complete article.