Modeling and Valuation of Energy Structures

Although it is more than ten years old at the time of this writing, Eydeland and Wolyniec (2003, hereafter denoted by EW) remains unparalleled in its presentation of both practical and theoretical techniques for commodity modeling, as well as its coverage of the core structured products in energy markets.¹ We will defer much discussion of the specifics of these markets to EW, as our focus here is on modeling techniques. However, it will still be useful to highlight some central features of energy markets, to provide the proper context for the subsequent analysis.²

Let us start with an example that is very simple, yet illustrates well both the kind of econometric problems faced in energy markets as well as the essential features the econometric analysis must address. Invariably, we are not interested in estimation as such, but only within the context of valuing some structured product/deal. Suppose we are to bid on a deal that entails taking exposure to some (non-traded) entity y, which we believe (for whatever reason) to stand in a relation to some (forward traded) entity x. We thus anticipate putting on some kind of hedge with this entity x. A critical aspect of this deal is that the exposure is realized at a specific time in the future, denoted by T. Examples of proxy hedging include:

We now turn to fundamental questions of valuation and hedging, and their close (indeed, inseparable) connection; we will see that they are really two sides of the same coin. We have already introduced a series of (spread) option structures in Chapter 1 in the context of energy market structures, and we will later focus the discussion on such products. However, it should be stressed that the ideas to be presented here are quite general, and can (in principle, at least) be applied to any nonlinear product.

We have already described the basics of storage deals in Section 1.2.3, and introduced spot-based control problem formulations in Section 3.3.2. (Recall that we also considered storage in Section 3.1.4 in contrasting dynamic hedging strategies [delta-hedging vs. rolling intrinsic].) In fact, in the discussion of control-based approaches, we pointed out that essentially forward-based valuations in terms of baskets-of-spread options provide lower bounds to this generally intractable optimization problem. This connection was not accidental; in Section 3. 1.1 we stressed the basic unity of the two valuation paradigms (spot vs. forward). We will now provide a concrete example in support of these claims.

As we endeavored to emphasize throughout Chapter 2, the reality of nonstationarity (in light of relevant time scales) demands that great care be exercised in analyzing energy-market data (challenging in its own right in light of the hammer and anvil of data sparsity/high volatility). In fact, one may reasonably wonder if there is any extent to which the well-established tools developed for stationary time series have relevance to real-world data. However, it turns out that there is an important concept, known as cointegration, which allows for the investigation of stationary relationships between entities that are individually non-stationary. In some sense it is probably not surprising that such a concept has viability, especially in energy markets. After all, there are a number of fundamental drivers (e.g., weather-driven demand, stack structure, etc.) that place certain physical constraints on (joint) price formation. Obviously, there are different time scales over which these effects occur, but the relevant effect is to constrain the extent to which certain price relationships (e.g., price-gas ratios) can vary. This is the essence ofa cointegrating relationship, and we will now discuss some of the specific techniques that can be brought to bear on such problems. In particular, we point out how they relate to certain, more robust, methods we have already presented.

Spread options are pervasive in energy markets. While we must refer the reader to EW for a complete treatment of the various structures that entail such optionality, for context we will outline a few examples here, stressing that other, more mathematical, issues are the central focus here. To recap, examples include:

As should be quite clear to this stage, the need to understand the joint dependence between multiple stochastic entities is of critical importance in energy markets. We have of course considered numerous examples in the context of spread-option structures, where the relevant measure of dependence in Gaussian scenarios is correlation.¹ We have also considered some fairly rich classes of canonical processes that extend the standard Gaussian framework (affine jump diffusions and Lévy processes). In addition, we examined the interplay between short-term co-movements and long-term stationarity through cointegration analysis. We now examine another concept useful for modeling joint structure, namely, copulas. As will be seen, an interesting facet of copulas is the ability to construct joint dependency in terms of specified marginal distributions. To the extent that marginal distributions may often be extracted from market information (through, say, option prices), the flexibility offered by copula-based models can be quite enticing. Not surprisingly, there is a voluminous literature available, including book-length treatments by Nelsen (1999) and detailed survey articles in Embrechts et al. (2002, 2003). Our objective here is to provide an overview and highlight the most promising directions as they pertain to energy modeling.