Choices, choices and more choices
Choosing the appropriate process for modelling energy prices is essential to calculate value, risk and hedging for energy derivatives and assets. The best choice of which price process to is especially vexing in the energy sector since energy spot prices are particularly difficult to model when compared to other asset classes. This stems from the underlying physical nature of the market(s) and their fluctuations.
Starting off is the seasonality in price and model parameters that feed in from the term structure of forward prices and volatilities associated with differences in supply/demand throughout the year. On a shorter term horizon, spot prices react immediately as a result of short-term supply and demand shocks. In most markets, demand tends to be inelastic in the short term and thus supply shocks (the loss of a power station, delay to an LNG cargo or pipeline outage for example) can cause prices to rise sharply but there is a strong tendency for mean reversion once the friction has been removed. In fact, many energy spot prices exhibit large changes in price and these price jumps can be dramatic and extremely punchy. Modelling these price spikes is the Achilles’ heel of many price process, but jumps can be incorporated based on information regarding the expected jump frequency and, the expected jump magnitude and the duration.
Power, Gas and Oil Spot prices
The differences between the prices in energy and financial markets can often be traced to issues related to the physical delivery of the commodity. Power markets tend to be extremely volatile, highly mean reverting and have frequent spikes driven by the lack of widespread storage to hedge short term disruptions and the high price of replacement generation. A grid operator, when faced with short term loss of large base-load unit, will tend replace this power with expensive pumped or oil burning alternatives whilst a replacement is found. Contrasting this the natural gas market which, despite high volatility during injection season, mean reversion over long horizons and prices may spike during high-demand periods when storage levels are low. Finally, crude oil, products and coal show lower levels of volatility – supply options a more varied – mean reversion is apparent over long horizons and spikes in oil markets are the result of geopolitical tensions.
Stochastic spot price processes
Spot price processes are used to simulate price paths and probability distributions at different future horizons. The choice of spot process has a large impact of the implied valuation, hedging parameters and risk calculation.
Brownian motion, or random walk, is one of the most simple stochastic processes and dates back to 1827 when the Scottish botanist Robert Brown observed the random motion of pollen grains on the surface of water. The movement of many financial processes appear, at first glance, to bear a striking similarity to these chaotic movements. In fiance, the most common continuous time process is the Geometric Brownian Motion (GBM) which is one of the key assumptions behind the Black-Scholes derivatives pricing model. Two of the main assumptions behind GBM is that price changes are ‘memory-less’ and that they have constant drift and volatility. In short, any future price movement is not influenced by a historic price movement plus the scale and direction of this movement is constant.
In a GBM process, prices evolve according to random shocks that follow a lognormal distribution, but do not allow of mean reversions of jumps. A mean reverting process incorporates a reversion level as well consideration as to how rapid the return to that level is. Building upon this, mean reversion jump diffusion (MRJD) models also factor a stochastic jump component too.
Assessing different price models
The choice of the price process to be used should be a function of the market being modelled. Knowledge of the physical drivers as well as historical analysis of the empirical behavior of spot, forward prices and spreads can provide valuable insights into the dynamics present. For example, power and gas spot prices often exhibit mean reversions and jumps, while oil and coal markets do not have such pronounced features over the short term.
As far as the calibration of model parameters is concerned, it is important to consider that historical price movements are poor guides. Whilst statistical models to exist that will map historical price movements, they tend to fair poorly when attempting to forecast movements. This stems from the fact that past physical price drivers – the main underlying support of much of the spot market – is not applicable (or is significantly different) in the future. Recent examples of this include structural changes in the US Gas market due to Shale Gas and well as the price decoupling of oil and gas.
Finally, model risk is prevalent and wide spread in energy derivatives, long term contracts and physical asset valuation, but it can be mitigated by understanding the assumptions and limitations as well as using models that fit the specific needs of the valuation and risk problem.
Extensions to mean reversion jump diffusion
There are many extensions of the mean reversion jump diffusion (MRJD) model, some of which include introducing information related to jump duration times, price caps, modelling the dependence structure between jumps, as well as relaxing the assumption of deterministic volatility and other parameters. Other attempts include the use of different mean-reversion levels and mean reversion rates for jumps versus non-jump shocks.
A common choice by quantitative analysts is to decouple the non-jump diffusion from the jump component to allow for greater modelling flexibility. In terms of modelling hourly prices, due to the complexity of the hourly price dynamics, some models perform the calibration and simulation for on-peak and off-peak daily prices and then impose hourly shapes using econometric techniques
such as bootstrapping or principal component analysis. Finally, hybrid models represent a promising intermediate step between purely stochastic processes and full-blown fundamental models. The idea is that it is possible to summarise the supply and demand curves for power and produce reasonable simulated levels of power prices driven by underlying physical market conditions.