Tag: Volatility

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Intraday Stock Index Forecasting

In a previous post I discussed modelling stock prices processes as Geometric brownian Motion processes:

Understanding Stock Price Range Forecasts

To recap briefly, we assume a process of the form:

Where S0 is the initial stock price at time t = 0.

The mean of such a process is:

and standard deviation:

In the post I showed how to estimate such a process with daily stock prices, using these to provide a forecast range of prices over a one-month horizon. This is potentially useful, for example, in choosing which strikes to select in an option hedge.

Of course, there is nothing to prevent you from using the same technique over different timescales. Here I use the MATH-TWS package to connect Mathematica to the IB TWS platform via the C++ api, to extract intraday prices for the S&P 500 Index at 1-minute intervals. These are used to estimate a short-term GBM process, which provides forecasts of the mean and variance of the index at the 4 PM close.

We capture the data using:

then create a time series of the intraday prices and plot them:

If we want something a little fancier we can create a trading chart, including technical indicators of our choice, for instance:

The charts can be updated in real time from IB, using MATHTWS.

From there we estimate a GBM process using 1-minute close prices:

and then simulate a number of price paths towards the 4 PM close (the mean price path is shown in black):

This indicates that the expected value of the SPX index at the close will be around 4450, which we could estimate directly from:

Where u is the estimated drift of the GBM process.

Similarly we can look at the projected terminal distribution of the index at 4pm to get a sense of the likely range of closing prices, which may assist a decision to open or close certain option (hedge) positions:

Of course, all this is predicated on the underlying process continuing on its current trajectory, with drift and standard deviation close to those seen in the process in the preceding time interval. But trends change, as do volatilities, which means that our forecasts may be inaccurate. Furthermore, the drift in asset processes tends to be dominated by volatility, especially at short time horizons.

So the best way to think of this is as a conditional expectation, i.e. “If the stock price continues on its current trajectory, then our expectation is that the closing price will be in the following range…”.

For more on MATH-TWS see:

MATH-TWS: Connecting Wolfram Mathematica to IB TWS

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Implied Volatility in Merton’s Jump Diffusion Model

The “implied volatility” corresponding to an option price is the value of the volatility parameter for which the Black-Scholes model gives the same price. A well-known phenomenon in market option prices is the “volatility smile”, in which the implied volatility increases for strike values away from the spot price. The jump diffusion model is a generalization of Black\[Dash]Scholes in which the stock price has randomly occurring jumps in addition to the random walk behavior. One of the interesting properties of this model is that it displays the volatility smile effect. In this Demonstration, we explore the Black-Scholes implied volatility of option prices (equal for both put and call options) in the jump diffusion model. The implied volatility is modeled as a function of the ratio of option strike price to spot price.


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Volatility Forecasting in Emerging Markets

The great majority of empirical studies have focused on asset markets in the US and other developed economies.   The purpose of this research is to determine to what extent the findings of other researchers in relation to the characteristics of asset volatility in developed economies applies also to emerging markets.  The important characteristics observed in asset volatility that we wish to identify and examine in emerging markets include clustering, (the tendency for periodic regimes of high or low volatility) long memory, asymmetry, and correlation with the underlying returns process.  The extent to which such behaviors are present in emerging markets will serve to confirm or refute the conjecture that they are universal and not just the product of some factors specific to the intensely scrutinized, and widely traded developed markets.

The ten emerging markets we consider comprise equity markets in Australia, Hong Kong, Indonesia, Malaysia, New Zealand, Philippines, Singapore, South Korea, Sri Lanka and Taiwan focusing on the major market indices for those markets.   After analyzing the characteristics of index volatility for these indices, the research goes on to develop single- and two-factor REGARCH models in the form by Alizadeh, Brandt and Diebold (2002).

Cluster Analysis of Volatility
Processes for Ten Emerging Market Indices

The research confirms the presence of a number of typical characteristics of volatility processes for emerging markets that have previously been identified in empirical research conducted in developed markets.  These characteristics include volatility clustering, long memory, and asymmetry.   There appears to be strong evidence of a region-wide regime shift in volatility processes during the Asian crises in 1997, and a less prevalent regime shift in September 2001. We find evidence from multivariate analysis that the sample separates into two distinct groups:  a lower volatility group comprising the Australian and New Zealand indices and a higher volatility group comprising the majority of the other indices.

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Models developed within the single- and two-factor REGARCH framework of Alizadeh, Brandt and Diebold (2002) provide a good fit for many of the volatility series and in many cases have performance characteristics that compare favorably with other classes of models with high R-squares, low MAPE and direction prediction accuracy of 70% or more.   On the debit side, many of the models demonstrate considerable variation in explanatory power over time, often associated with regime shifts or major market events, and this is typically accompanied by some model parameter drift and/or instability.

Single equation ARFIMA-GARCH models appear to be a robust and reliable framework for modeling asset volatility processes, as they are capable of capturing both the short- and long-memory effects in the volatility processes, as well as GARCH effects in the kurtosis process.   The available procedures for estimating the degree of fractional integration in the volatility processes produce estimates that appear to vary widely for processes which include both short- and long- memory effects, but the overall conclusion is that long memory effects are at least as important as they are for volatility processes in developed markets.  Simple extensions to the single-equation models, which include regressor lags of related volatility series, add significant explanatory power to the models and suggest the existence of Granger-causality relationships between processes.

Extending the modeling procedures into the realm of models which incorporate systems of equations provides evidence of two-way Granger causality between certain of the volatility processes and suggests that are fractionally cointegrated, a finding shared with parallel studies of volatility processes in developed markets.

Download paper here.

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The Hedged Volatility Strategy

Being short regular Volatility ETFs or long Inverse Volatility ETFs are winning strategies…most of the time. The challenge is that when the VIX spikes or when the VIX futures curve is downward sloping instead of upward sloping, very significant losses can occur. Many people have built and back-tested models that attempt to move from long to short to neutral positions in the various Volatility ETFs, but almost all of them have one or both of these very significant flaws: 1) Failure to use “out of sample” back-testing and 2) Failure to protect against “black swan” events.

In this strategy a position and weighting in the appropriate Volatility ETFs are established based on a multi-factor model which always uses out of sample back-testing to determine effectiveness. Volatility Options are always used to protect against significant short-term moves which left unchecked could result in the total loss of one’s portfolio value; these options will usually lose money, but that is a small price to pay for the protection they provide. (Strategies should be scaled at a minimum of 20% to ensure options protection.)

This is a good strategy for IRA accounts in which short selling is not allowed. Long positions in Inverse Volatility ETFs are typically held. Suggested minimum capital: $26,000 (using 20% scaling).

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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.

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(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.

 

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Long Memory and Regime Shifts in Asset Volatility

This post covers quite a wide range of concepts in volatility modeling relating to long memory and regime shifts and is based on an article that was published in Wilmott magazine and republished in The Best of Wilmott Vol 1 in 2005.  A copy of the article can be downloaded here.

One of the defining characteristics of volatility processes in general (not just financial assets) is the tendency for the serial autocorrelations to decline very slowly.  This effect is illustrated quite clearly in the chart below, which maps the autocorrelations in the volatility processes of several financial assets.

Thus we can say that events in the volatility process for IBM, for instance, continue to exert influence on the process almost two years later.

This feature in one that is typical of a black noise process – not some kind of rap music variant, but rather:

“a process with a 1/fβ spectrum, where β > 2 (Manfred Schroeder, “Fractals, chaos, power laws“). Used in modeling various environmental processes. Is said to be a characteristic of “natural and unnatural catastrophes like floods, droughts, bear markets, and various outrageous outages, such as those of electrical power.” Further, “because of their black spectra, such disasters often come in clusters.”” [Wikipedia].

Because of these autocorrelations, black noise processes tend to reinforce or trend, and hence (to some degree) may be forecastable.  This contrasts with a white noise process, such as an asset return process, which has a uniform power spectrum, insignificant serial autocorrelations and no discernable trending behavior:

White Noise Power Spectrum
White Noise Power Spectrum

An econometrician might describe this situation by saying that a  black noise process is fractionally integrated order d, where d = H/2, H being the Hurst Exponent.  A way to appreciate the difference in the behavior of a black noise process vs. a white process is by comparing two fractionally integrated random walks generated using the same set of quasi random numbers by Feder’s (1988) algorithm (see p 32 of the presentation on Modeling Asset Volatility).

Fractal Random Walk - White Noise
Fractal Random Walk – White Noise
Fractal Random Walk - Black Noise Process
Fractal Random Walk – Black Noise Process

As you can see. both random walks follow a similar pattern, but the black noise random walk is much smoother, and the downward trend is more clearly discernible.  You can play around with the Feder algorithm, which is coded in the accompanying Excel Workbook on Volatility and Nonlinear Dynamics .  Changing the Hurst Exponent parameter H in the worksheet will rerun the algorithm and illustrate a fractal random walk for a black noise (H > 0.5), white noise (H=0.5) and mean-reverting, pink noise (H<0.5) process.

One way of modeling the kind of behavior demonstrated by volatility process is by using long memory models such as ARFIMA and FIGARCH (see pp 47-62 of the Modeling Asset Volatility presentation for a discussion and comparison of various long memory models).  The article reviews research into long memory behavior and various techniques for estimating long memory models and the coefficient of fractional integration d for a process.

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But long memory is not the only possible cause of long term serial correlation.  The same effect can result from structural breaks in the process, which can produce spurious autocorrelations.  The article goes on to review some of the statistical procedures that have been developed to detect regime shifts, due to Bai (1997), Bai and Perron (1998) and the Iterative Cumulative Sums of Squares methodology due to Aggarwal, Inclan and Leal (1999).  The article illustrates how the ICSS technique accurately identifies two changes of regimes in a synthetic GBM process.

In general, I have found the ICSS test to be a simple and highly informative means of gaining insight about a process representing an individual asset, or indeed an entire market.  For example, ICSS detects regime shifts in the process for IBM around 1984 (the time of the introduction of the IBM PC), the automotive industry in the early 1980’s (Chrysler bailout), the banking sector in the late 1980’s (Latin American debt crisis), Asian sector indices in Q3 1997, the S&P 500 index in April 2000 and just about every market imaginable during the 2008 credit crisis.  By splitting a series into pre- and post-regime shift sub-series and examining each segment for long memory effects, one can determine the cause of autocorrelations in the process.  In some cases, Asian equity indices being one example, long memory effects disappear from the series, indicating that spurious autocorrelations were induced by a major regime shift during the 1997 Asian crisis. In most cases, however, long memory effects persist.

Excel Workbook on Volatility and Nonlinear Dynamics 

There are several other topics from chaos theory and nonlinear dynamics covered in the workbook, including:

More on these issues in due course.

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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.

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Market Timing in the S&P 500 Index Using Volatility Forecasts

There has been a good deal of interest in the market timing ideas discussed in my earlier blog post Using Volatility to Predict Market Direction, which discusses the research of Diebold and Christoffersen into the sign predictability induced by volatility dynamics.  The ideas are thoroughly explored in a QuantNotes article from 2006, which you can download here.

There is a follow-up article from 2006 in which Christoffersen, Diebold, Mariano and Tay develop the ideas further to consider the impact of higher moments of the asset return distribution on sign predictability and the potential for market timing in international markets (download here).

Trading Strategy
To illustrate some of the possibilities of this approach, we constructed a simple market timing strategy in which a position was taken in the S&P 500 index or in 90-Day T-Bills, depending on an ex-ante forecast of positive returns from the logit regression model (and using an expanding window to estimate the drift coefficient).  We assume that the position is held for 30 days and rebalanced at the end of each period.  In this test we make no allowance for market impact, or transaction costs.

Results
Annual returns for the strategy and for the benchmark S&P 500 Index are shown in the figure below.  The strategy performs exceptionally well in 1987, 1989 and 1995, when the ratio between expected returns and volatility remains close to optimum levels and the direction of the S&P 500 Index is highly predictable,  Of equal interest is that the strategy largely avoids the market downturn of 2000-2002 altogether, a period in which sign probabilities were exceptionally low.

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In terms of overall performance, the model enters the market in 113 out of a total of 241 months (47%) and is profitable in 78 of them (69%).  The average gain is 7.5% vs. an average loss of –4.11% (ratio 1.83).  The compound annual return is 22.63%, with an annual volatility of 17.68%, alpha of 14.9% and Sharpe ratio of 1.10.

The under-performance of the strategy in 2003 is explained by the fact that direction-of-change probabilities were rising from a very low base in Q4 2002 and do not reach trigger levels until the end of the year.  Even though the strategy out-performed the Index by a substantial margin of 6% , the performance in 2005 is of concern as market volatility was very low and probabilities overall were on a par with those seen in 1995.  Further tests are required to determine whether the failure of the strategy to produce an exceptional performance on par with 1995 was the result of normal statistical variation or due to changes in the underlying structure of the process requiring model recalibration.

Future Research & Development
The obvious next step is to develop the approach described above to formulate trading strategies based on sign forecasting in a universe of several assets, possibly trading binary options.  The approach also has potential for asset allocation, portfolio theory and risk management applications.

Market Timing in the S&amp;P500 Index
Market Timing in the S&P500 Index
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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.