A Structural Risk‐neutral Model for Pricing and Hedging Power Derivatives

Abstract

We develop a structural risk-neutral model for energy market modifying along several directions the approach introduced in Aïd et al. In particular, a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contracts on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closedform formulae for futures prices and semiexplicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.

A Survey of Commodity Markets and Structural Models for Electricity Prices

“…For example, for an n fuel market, one might define n + 1 regimes, one driven by each underlying fuel, and one for spikes. Although this does not incorporate an overlap regime, the simplification to a single marginal fuel is intuitively appealing and similar in spirit to the work of Aïd et al [1,2]. Instead of their strict capacity-driven thresholds, we then use our demand-dependent regime probabilities to ensure that fuels higher in the merit order are more likely to be used when demand is high.…”

“…This provides much more convenient formulas for pricing derivatives, but at the expense of a major oversimplification of spot price dynamics. In an extension of the earlier model, Aïd et al [2] extended this approach to improve spot price dynamics and capture spikes, by multiplying a 'scarcity function' (of margin) by the heat rate and fuel price of the marginal fuel. They choose this function to be a power law of the reserve margin (with a cap), arguing this to be more effective than the common choice of exponential.…”

“…A big obstacle to either approach is that the merit order may change, particularly over medium to long time horizons. Indeed, in [2], to retain mathematical tractability in the n fuel case, the authors assume that the initial merit order is fixed and enforce this by modeling the spreads between neighbouring fuels as Geometric Brownian Motions, a departure from commonly-used models for fuel prices. In this section, we present another variation in order to retain the chance of future merit order changes, following more closely the original setup of [1].…”

A Model for Hedging Load and Price Risk in the Texas Electricity Market

“…• We jointly estimate all parameters which appear in (2) given observed values of load and gas price, 1 and under the assumption that X t is a standard Gaussian random variable at every t. This assumption is reasonable since we expect X to be a very fast mean-reverting process, reflecting short-term noise. Parameters γ 1 and γ 2 provide the appropriate scaling Table 2.…”

“…In such a model, power price is written as a function of several underlying supply and demand factors, and its dynamics are therefore not specified directly through an SDE (stochastic differential equation), but produced indirectly as a result of the dynamics chosen for the factors. Early work by Barlow [3] treated demand as the only driving factor, before various authors extended this branch of the literature to include factors such as fuel prices [20,7], capacity changes [5,10], or both [11,2].…”