Energy Risk Modeling

In O. Henry’s

The Handbook of Hymen

, Mr Pratt is wooing the wealthy Mrs Sampson. Unfortunately for Mr Pratt, he has a rival — a romantic poet. In order to compensate for his romantic disadvantage, the studious Mr Pratt selects quotes from a text on statistical facts in an audacious attempt to dazzle Mrs Sampson into marrying him:

“Let us sit on this log at the roadside,” says I, “and forget the inhumanity and ribaldry of the poets. It is in the glorious columns of ascertained facts and legalized measures that beauty is to be found. In this very log we sit upon, Mrs. Sampson,” says I, “is statistics more wonderful than any poem. The rings show it was sixty years old. At the depth of two thousand feet it would become coal in three thousand years. The deepest coal mine in the world is at Killingworth, near Newcastle. A box four feet long, three feet wide, and two feet eight inched deep will hold one ton of coal. If an artery is cut, compress it above the wound. A man’s leg contains thirty bones. The Tower of London was burned in 1841. “Go on, Mr Pratt,” says Mrs Sampson. “Them ideas is so original and soothing. I think statistics are just lovely as they can be.”

There are many instances where those involved in energy products must make decisions under conditions of uncertainty. An oil producer must decide how much inventory to stock; a risk manger how much economic capital to set aside, and an electricity speculator when to buy or sell. In each of these cases the individuals make their decision on the basis of what they think is likely to occur; their decision is based on the probability that certain events will or will not happen. Most of us have some intuitive understanding of probability. Some people prefer to take the train to their place of work in the knowledge that a serious accident is less likely than if they drive. Others participate in high risk sports such as boxing or sailing, knowing that they are likely to face serious injury or death, but then again the likelihood of such extreme outcomes is actually quite small. Millions of individuals purchase lottery tickets even though the likelihood of wining a very large pay-out is extremely small. If we say that the probability of snow today is one-half, but tomorrow it is only one quarter, we know that snow is more likely today than tomorrow.

Descriptive statistics are those statistical methods that are used to summarize the characteristics of a sample. The main purpose of descriptive statistics is to reduce the original sample into a handful of more understandable metrics without distorting or losing too much of the valuable information contained in the individual observations. We begin by collecting N observations {r₁, r₂,…,r_N} on the price return random variable R. These measurements are then organized and summarized using techniques of descriptive statistics described in this chapter. In most cases we can compactly describe the characteristics of a sample using three distinctive classes of descriptive statistics. The first class summarizes the center of the distribution and are known as measures of central tendency. The second class summarizes the spread or dispersion of the sample and are commonly known as measures of dispersion. The third class known as shape statistics summarizes important elements of the shape of the underlying probability distribution implied by the sample. Our objective in using descriptive statistics is to describe as compactly as possible the key properties of empirical data. This information will then be used to assist us in selecting an appropriate probability model for modeling price risk.

Unlike descriptive statistics, inferential statistics are procedures for determining whether it is possible to make generalizations based on the data collected from a sample. Such generalizations are about an unobserved population. A population consists of all values (past and future) of the random variable of interest. In most circumstances the exact value of a population parameter such as the mean or variance will be unknown, and we will have to make some conjecture about its true value. In Chapter 3, we used sample estimators such as the mean, median, skew, and kurtosis, to provide estimates of the respective population parameters. When a sample is drawn from a population, the evidence contained within it may bolster our conjecture about population values or it may indicate that the conjecture is untenable. Hypothesis testing is a formal mechanism by which we can make and test inferential statements about the characteristics of a population. It uses the information contained in a sample to assess the validity of a conjecture about a specific population parameter.

Throughout the energy sector, risk managers, and analysts face the challenge of uncovering the price and return distributions of various products. Knowledge about the underlying probability distributions generating returns is used both in pricing models and risk management. The selected probability distribution(s) can have a significant impact on the calculated Value at Risk measure of a company’s exposure from trading floor transactions and in the use of derivative pricing tools. It is imperative that risk management metrics such as Value at Risk are calculated using a statistical distribution tailored to the specific characteristics of the energy product of interest. Fitting probability distributions by carefully analyzing energy price returns is an important, although often neglected, activity. This may be partly because the number and variety of distributions to choose from is very large. For a specific product such as the forward price of Brent Crude, or price return of an Electricity index, which of the dozens of distributions should we use? This chapter outlines the process by which the practicing risk manager can begin to answer this question. It starts by assessing the validity of a simple model based on the normal distribution. When normality fails we can adjust the percentiles of the normal probability distribution. If this does not appear to help we might select an alternative probability distribution or else consider a mixture of normal distributions.

Sitting on the very tip of my chair, feigning interest in the mumbled string of motivational buzz words spouting out of the mouth of an unusually dull director of global risk, it occurred to me that if I looked hard enough, through the gray mist of the incoherent mutterings, there would emerge some shape, some form to their ideas, which as yet my colleagues and I could not perceive. I mused on this thought, toyed with the idea of developing a statistical algorithm that would filter out the noise, revealing the underlying structure. My jocose thoughts were shattered by what was supposed to be the motivational crescendo — we all rose to our feet and clapped our hands somewhat like well-fed seals at feeding time at the local zoo — that is, with not much enthusiasm. Unfortunately, for that individual, there was no form to his ideas, no shape to his plan. Needless to say the listless, MBA-clad, mumbo-jumbo speaking “hot shot” was not head of global risk for very long. However, the experience stuck in my mind and re-stimulated my interest in nonparametric statistical methods, a subset of which, non parametric density estimation, is the subject of this chapter. It introduces nonparametric density estimation as a complementary statistical mechanism for describing energy price returns. It begins by discussing, the simplest nonparametric density estimator — the histogram, how to construct it, and its properties and limitations.

To this point, we have dealt almost exclusively with problems of estimation and statistical inference about a parameter of a probability distribution or characteristic of a sample. Another important element of applied statistical modeling of energy risks concerns the relationship between two or more price or other variables. Generally, a risk manager will be interested in whether above (below) average values of one variable tend to be associated with above (below) average values of the other variable. Take for example, a risk manger working for a petroleum refinery, who for hedging purposes, is interested in knowing the relationship between the spot price of Brent Crude and the future price of diesel fuel. If the risk manager simply assumes crude oil and diesel fuel prices always move in tandem, the company will be exposed to the price risk if this relationship breaks down. If on the other hand, the closeness of the two indices is defined in terms of a correlation coefficient, then the manager at least has some rudimentary way of assessing whether or not the relationship exists and its strength.

Regression modeling lies at the heart of modern statistical analysis. It also occupies a key role in much of the analysis carried out in quantitative finance. Given its importance and frequency of use, this chapter provides a hands-on introduction to applied regression modeling. The emphasis is on using R to analyze some simple data sets contained in the R package. The objective is to give you a feel for regression techniques using simple (nonenergy) examples before we move onto discuss more energy-specific applications of regression in the remaining chapters of this book. The emphasis of this chapter is therefore on you the reader becoming comfortable with the ideas surrounding regression and replicating for yourself the examples given in R.

In this chapter we extend the simple linear regression model into the multiple linear regression model. Multiple regression is useful when we can expect more than one independent variable to influence the dependent variable. It allows us to explore the relationship between several independent and a single dependent variable. We also discuss multivariate regression which arises when we have several dependent variables dependent on the same (or some subset) independent variables.

Arguably, one of the most important issues in regression modeling applied to risk management is the correct specification of the regression equation. How can we assess the validity of the pre-specified regression model, which will provide the basis of statistical inference and practical decisions? It turns out that statistical inference concerning the linear regression model depends crucially on the “validity” of the underlying statistical assumptions of the model. If the assumed underlying statistical assumptions are invalid the inference based on it will be unreliable. The primary objective of this chapter is to outline the key assumptions of the linear regression model and provide some elementary techniques for validating or refuting these assumptions given a specific data set.

There is no guarantee that the relationship between the dependent and independent variables will be linear. On many occasions we may find the relationship to have considerable non-linearity. In such circumstances, we might attempt to use polynomial regression, logarithmic regression, exponential regression, or a more general non-linear model. This chapter introduces these models. It also discusses the use of limited dependent regression models.

Say the words “Energy price volatility” and many people will think of the OPEC oil price hikes of the 1970s or perhaps the more recent 2004 sharp upswing in the price of crude oil. Yet price volatility is a characteristic of capitalism and freely operating energy markets are no exception. Accurate estimates of the variation in energy asset values over time are important for the valuation of financial contracts, retail obligations, physical assets, and in solving portfolio allocation problems. As a consequence, modeling and forecasting of price volatility has acquired an unprecedented significance in the industry. In response, various attempts have been made to develop statistical tools to help characterize and predict price volatility. In general these models fall into three categories, Exponentially Weighted Moving Average models, Generalized Autoregressive Conditional Hetroscedasticity models, and Stochastic Volatility Differential Equations. In this chapter we introduce the first two of these modeling approaches.

The main objective of this chapter is twofold. First we introduce a number common stochastic processes used in the valuation of derivative contracts and financial simulations. Second, we consider their relevance to energy risk modeling.