Statistical Arbitrage: Part I

Statistical arbitrage is a type algorithmic trading technique that relies on hedging all exposure to risk factors in order to profit from small, mean-reverting, predictable movements in security/currency/commodity prices. Put differently, statistical arbitrage is a quantitative system of trading one instrument against a smaller amount of many others, which allow for macroeconomic variables typically affecting prices of the first instrument to be hedged by similar exposure in the opposite direction by the others.

To do statistical arbitrage, we use a multivariate regression equation, which assumes the instantaneous rate of change in an instrument's price, divided by its price, is equal to its alpha times the change in time, plus sigma for all i of a sensitivity 'beta sub i' times the current value of factor i, and finally, plus an error.
 
In some cases, if the alpha is low, it may be safely ignored, and the value of the instrument then depends exclusively on the betas, the risk factors, and the cointegration residual. By buying 'beta sub i' portfolios of the risk factor i (which can be described by a portfolio that is revealed by principal components analysis), for each risk factor i, we get a complete portfolio whose value is determined only by the cointegration residual of the single instrument and its alpha, which may or may not be zero. (If the cointegration residual has a nonzero drift, this will be our alpha).

This is an extremely simple procedure. Now that we have a portfolio whose value is to be a synthetic time series that we must model. And that's where approaches diverge. Ornstein-Uhlenbeck processes are the most simple, robust, and widely used model that can be used to describe the cointegration residual, but they may not be the best, since they make many assumptions about the time series without any data to support them. For instance, an Ornstein-Uhlenbeck process assumes that the synthetic time series is stationary, has Gaussian distribution with fixed mean, and has a constant, linear mean-reversion speed dependent on distance from the mean. Problems with these assumptions are permanent changes in the alpha value which means that the cointegration residual becomes non-stationary until the problem is corrected and the new drift is grouped as alpha (hence the need to regularly use regression to separate out this drift, while still not having a large sample size to perform the regression accurately, can be a problem.) Next, if the distribution is non-Gaussian, it would be more efficient to capitalize on than to theorize the issue away. Lastly, reversion speed can be better described by something other than distance. It may actually not be proportional to the distance from the mean, if less participants in the market are willing to risk doing Stat Arb in an Extremistan (and not just out-of-equilibrium) environment, and more might be willing stat arbitrageurs when the the residual is small, pushing it too far out of equilibrium on the opposite side. Lastly, if, to avoid this, market participants avoid holding the instrument once it gets past a certain point--say a certain z-score--we would experience a situation where instruments bounced back (at least occasionally) when they get too close to equilibrium.

Of course I don't know what would happen and how it would happen, but Bayesian Networks have more promise in this regard than their rivals, linear stochastic differential equations. Nonlinear Fokker-Planck Equations seem to have some potential for replacing black box predictions like those of Artificial Neural Networks or Hidden Markov Models, but the issue is that there is no great way to derive the 'correct' formula. Either way, we must make assumptions as to the structure of such an equation before we can try to calibrate it.

That all being said, and knowing I'm a fan of hidden Markov models, I would highly recommend those the most for a stat arb prediction engine.

Hidden Markov Models: Part II

Since hidden Markov models help researchers to find what sorts of observations tend to come after other types of observations, we have a situation where we can forecast stock behavior once we know what hidden state the security/commodity/currency/synthetic time series was most recently in. (By a synthetic time series, I mean some type of hedged position with a single value being bet on; cointegration residuals in statistical arbitrage, implied volatilities after delta hedging, and PCA-derived risk factor values.)

To successfully deploy capital using an HMM-driven technique, one needs to avoid overfitting the model to the data. I struggled with this. Closing prices from each day turned out to be too unpredictable to work, because the more time passed, the less the older patterns mattered; consequently, I was forced to limit the size of the training sequence I was using, so that the hidden Markov model would only bother trying to learn from the relevant data. Unfortunately, a training set of only 40 days is too small to work well. But 50 days is pushing it, and more than that is quite outdated any daily trading model.

The way around the problem was to use higher frequency data--because it would be relevant while still providing a wealth of information and hidden patterns. Besides all that, higher frequency data allows for more predictions to be made in any given day, and thus limits volatility in portfolio returns. (Real time prices are available through professional brokerage or subscriptions to specific Reuters or Yahoo Finance services. Delayed, but regularly updated prices are available on Google Finance, and no subscription is necessary.)

In order for a hidden Markov model--or any statistical strategy--to work, the trading techniques must be used many, many times. As the number of times the strategy is used increases, the variability in strategy's overall success decreases, and the strategy has more potential for a clean statistical edge to shine through. Conversely, if only a few instruments are held as a portfolio, the portfolio's return is less certain. Trading a few instruments with a prediction algorithm is like going spearfishing with a toothpick. It really is that impractical.

Also, if you know how to do something with a synthetic time series, do it. There tends to be much less variability in outcomes when unwanted risk factors are hedged, and thus much less uncertainty regarding the hidden Markov model's predictive ability.

A New Kind of Global Macro

This is going to be kind of esoteric because I'm holding my cards close to the vest.

Imagine:
Since the entire basis of global macro is risk management, what if we could put portfolios together one risk factor at a time? Wouldn't that be interesting? Risk exposure is the beginning and the end of all global macro strategy. So why wait until the end to integrate it? It would be much better to bet on the risk factors right off the bat? Why focus on certain commodities, currencies, and corporation's securities (or the respective derivatives)? That was never the point of global macro. We care about capitalizing on macroeconomic changes anyway. Why get exposure to those changes from a few instruments when you could get right to the source, and just buy the factor itself?

It's simpler. It's more efficient. It's more potent. It's better diversified.

So the answer is to construct portfolios consisting of nothing but a few uncorrelated risk factors selected in whatever relative portion desired. The method driving the prediction of risk factors is not the issue. That is for another time. What we're concerned with is portfolio allocation once the decisions have been made.

And best of all? Portfolio optimization is easy. The risk factors are approximated by baskets of many instruments, and the instruments each have a covariance with each other. But that has already addressed by principal components analysis. We can merely treat each factor as an instrument to be bought. (This is reasonable--because it is like buying a stock, which is a basket of risk factors, only these ones have with non-zero sensitivity to only one risk factor.) Once we have bought these baby, de-facto 'stocks,' we realize that they have no correlation at all. Thus, the terms inside the first sigma in the Markowitz Mean-Variance model are eliminated, and the second one is easily maximized by investing the most money where the highest return is expected. This is similar to setting our risk adversity parameter to zero, (though we have done nothing of the kind,) since it is now trivial.

This may seem dangerous, because the allocation weights are not constrained, but a rule can be added separately, allowing a maximum position size in any given risk factor, as well as a maximum position and/or maximum portfolio variance. On top of that, we can use VaR systems and stress testing like any global macro fund would.

The exact method for predicting factors varies considerably. There are several approaches that could work, including the John Paulson approach, the Soros approach, and the Robert Frey approach.

High Frequency Portfolios

An investment management company must keep track of risk across all positions and strategies used in each of its funds. At high frequency, however, this matters less because the positions are held only a short time, and the strategies can be made market neutral or even componentless (as measured by Principal Components Analysis). Instead, the focus is on transaction costs. How much money does the strategy lose from market impact when entering and exiting the trade? How much of those remaining profits are eaten away at by broker fees? These costs will determine how much of a position may be built in the time available for the strategy to work. Thus, the costs and the time series forecast determine how much may be invested, and the investment is entirely time based. Enter here; wait for the time series' next value; exit here if prediction is unfavorable or if a better post-transaction cost profit is predicted elsewhere--and hold position if it is favorable; repeat.

However, the size of orders should also be limited to a certain percentage of portfolio value, the leverage should remain fixed, and a Value-at-Risk system should also be used to determine whether a trade's marginal VaR is too high to make it a worthwhile investment.

One theoretical (gasp!) price impact model is written as an integral, (from the initial price to the final acceptable transaction price) where the integrand is the price multiplied by the liquidity function, which is a function of the price, and can be derived empirically through experimentation and the use of some statistical regression. Of course, we integrate with respect to the price.

This is intuitive because we have an infinitely small amount we can buy without moving the price. The price moves as we buy, so we have the price times the amount we can buy at that price, for every price between the initial and the final one. (And this works in reverse if selling--the top term of the integral will just be lower than the bottom.)

If there is a certain largest absolute price impact that the investor is willing to allow, he can simply set that impact equal to the integral described above, and solve for the upper term. Then, he can see how many shares he will be buying by integrating the liquidity function with respect to the price, and using the earlier boundaries for the new boundaries of integration. Conversely, if an investor wants to see how much of an impact will be made by a trade of a certain size, the later expression can be set equal to a certain number of shares, and the upper bound of the integral can be solved for. In that case, we plug the upper bound into the former expression, and integrate to find the market impact of the trade.

From there, the investor can construct a search heuristic (particle swarm optimization, cuckoo search, etc.) to search for the optimal trade size. From there, we can use a sort of execution algorithm that allows for transactions across time--something that minimizes (if buying) or maximizes (if selling) the VWAP or TWAP.