Modeling is the most useful work we can do for your business. It's purpose is to represent some element of the real world using mathematics. Then this representation can be examined to predict the future or to optimise some aspect.

Mathematical models can take many forms such as differential equations, dynamical systems, or game theoretic models. Our expertise lies in statistical models which model some probabilistic element of the real world. The challenge of these models is to explain as much of the randomness in the world as possible. If your model is better than your competitors' then you are less subject to the whims of chance and you will make more accurate predictions of the future (ie you will make more money in the stock markets).

Models fall into two types: linear and nonlinear. Linear models are a subset of nonlinear models so are more restricted and may not predict the data as well. But their advantage is relative simplicity over nonlinear models. Therefore they are easier to fit and interpret and to optimise. Often nonlinear models are fitted by approximating them with linear models. Nonlinear models are better able to describe complex systems such as chaotic phenomena (the stock market or weather).

Another classification is static vs dynamic models. A static model does not change over time, a dynamic model does. The advantage of the static model is that more data can be used to estimate the model. For example, you can use 10 years worth of data. So the model is estimated more accurately. A dynamic model changes over time. For example with 10 years worth of data you might fit a different model for each year. But each year only has a tenth of the data available for the static model. The skill is in designing models that are dynamic are make maximum use of all the data. This is the focus of most of our modeling and some of our best models are described below. Some applications of those models are also presented below.

Alpha is some vague term that measures the amount by which you beat your competitors in the stock market It is a measure of your skill. On average the alpha for all market participants is zero so if your alpha is positive then you are better than average. The aim of funds management is to "generate alpha." The term is heavily abused and generates much controversy.

More precisely, alpha is the constant term in a linear regression of your gains vs the market gains. Thereby it adjust your returns for the volatility of them - it is a risk-adjusted measure of your skill. Being a term in a regression its standard error can be measured and thus a confidence interval may be calculated. Fund managers may boast about the alpha they have generated but I have never seen a confidence interval on any published alphas. It takes about three to five years worth of investment for alpha confidence intervals to become small enough to be useful in comparing managers.

The stock market is said to be an "efficient" market. This means that it is difficult to generate alpha significantly different from zero. Of all market participants about half will have positive alpha. Of these only a small number will be significantly positive ie will be positive year after year. Only those with the very best prediction models will qualify.

Perhaps the most famous example of a manager who generates significant alpha from quantitative models is Renaissance Technologies which has produced average annual returns of almost 40 percent since 1989 for investors in their Medallion Fund. This is proof that quantitative models work.

The markets display some phenomena that repeat themselves identically throughout history: we call these phenomena invariants. The quest for prediction models is the quest for market invariants. Once you have found them you model their distribution then project them into the future.

Invariants have a precise mathematical definition that we don't need to repeat here (but note that invariant does not mean constant). I mention them because it shows that the quest for alpha has a mathematical definition that real life data can be measured against. Our alpha quest is a systematic quest and not ad-hoc.

Dynamic Factor Models are our best models to date for predicting markets. Factor Models are models that describe how variables behave in terms of other variables. The dependence is described by multiplying each variable by a coefficient to quantify its effect - this use of coefficients and multiplication is what gives rise to the term "factor." The simplest factor model is the regression model y = a + b*x where x is called a factor and b the coefficient.

To fit that factor model you would get a whole lot of y values and x values and do a regression of y vs x and the estimates of a and b that result would allow you to predict y for any future x values.

But what say the relationship of y vs x changes with time? In the stock market, for example, small companies outperform large at some stages of the economic cycle and not others. The markets are not static. The value of b is changing with time. A Dynamic Factor Models lets you fit time-varying coefficients and to predict future values of those coefficients. A very powerful model. For example it lets you predict when small companies are going to outperform large and so you can change your portfolio proactively rather than reactively.

Dynamic Factor Models have been shown in the literature to be capable of generating alpha and designing and fitting such models is a Double-Digit Numerics specialty.

State Space Models are time series models that let you include parameters that you cannot directly observe. They differ from factor models in that factor models can only use observable factors (for example in y = a + b*x you must have measured values of y and x, a State Space Model lets you include and estimate an unobservable factor z). The unobserved variable is called the "state" of the system (the term came from physics).

Unobservable variables occur frequently in finance. For example, the alpha and beta of a stock is unobservable. All that you can observe are the stock prices. So State Space Models can be used to estimate stock betas, especially when that beta is time varying.

A restriction of State Space Models is that the unobservable variable must be a number or real quantity such as 1.43. It cannot be a state such as "bear market" or "bull market." Hidden Variable Models allow for discrete states. (You would think that the two model should swap names).

The most famous hidden model is the Hidden Markov Model (HMM) which is used in temporal pattern recognition in fields such as speech processing as well as markets. The HMM is used widely in statistical learning and Bayesian Analysis. This is a hot area for research and offers plenty of opportunities for you to have better models than your competitors.

Arbitrage is where you buy and sell securities to take advantages of price discrepancies in two different markets. Doing this can guarantee a profit - buy at one price, sell at another. Such opportunities are not frequently available to most market participants because they represent a free lunch. Statistical Arbitrage is where the arbitrage is not free but is only available in probability. Models are used to calculate the probabilities and so able to determine whether trades are likely to be profitable or not.

The most famous Statistical Arbitrage strategy and once with which DDNUM has experience, is pairs trading. Using that strategy you buy one stock while simultaneously selling short another stock in the hope that the two stock prices will converge or diverge according to predictions.

Because you are both long and short in the market you are closer to market neutral ie you are not concerned with whether the market goes up or down, only with what the two stocks do relative to each other.

This is an application of market modeling where you have eliminated some variables that, although observable, are out of your control. So you juggle with the model until you get a combination (in this case of two stocks) that cancels out the variables you don't want.

We always want to, all other things being equal, reduce the volatility of our investments. Some securities may produce the same expected returns as others but have more volatility than the others. So optimally we want to reduce their weight in the portfolio.

This means that volatility modeling is important in addition to return modeling. Volatility modeling lets us both measure past volatility and predict future volatility. You might think that measuring past volatility does not require a model - simply calculate the stand deviation of past prices. But to do this you need a mean. What mean do you use? A static mean? A (more realistic) dynamic mean? And if you use a dynamic mean how do you calculate it? You require a model of some sort.

The above models can be used for volatility modeling. And many volatility models come from Energy Markets since volatility behaves more like electricity prices than stock prices (technically we say that volatility exhibits clustering).

VaR stands for Value at Risk and is concerned with the probability of extreme events (such as your portfolio dropping in value by more than 5% in a month). This is important statistically because the modeling of extreme events is different from the modeling of common events. Extreme events are those that occur infrequently (and may cause you to lose a whole year's gains in a single week) whereas common events are those that occur frequently (such as daily profits that you portfolio makes).

A lot of alpha is what we at DDNUM call "faux alpha." This is alpha that is positive for, say, most years and occasionally negative, say one year in 10. In the long run the net alpha is zero. But for a whole 10 year period it may be positive and thus delude the managers into thinking that they are generating positive alpha. So the calculation of VaR and extreme events requires careful modeling because it has far less data to work with than for modeling returns.

Statistically, returns may be represented quite well by the mean and variance of those returns. But VaR needs to look at the skewness and kurtosis of those returns since these give greater emphasis to the tails of the distribution. Extreme value theory can be used to model the tails specifically.

Derivatives are any security that derives it price from another and so to accurately price derivatives you need to be able to accurately model the original prices and the relationship between the two. The mathematics is usually intractable but this is overcome by using simulation methods to do the calculation.

For decades this has been a hot area for research. It is now so specialised and advanced that it might not be easy to get better results than your competitors unless you are involved in a new field such as weather derivatives.