Cutting Edge: Pure jump models for energy prices

Commodity markets differ from stock and bond markets in several key properties. Supply is determined by production and inventory, with the presence of a quantity risk. Demand is generally inelastic to prices: this is due to essential nature of considered good. The balance of supply and demand can be smoothed by inventories. Non-storability, which uniquely characterises electricity, involves the real-time balancing of supply and demand.

The continuous rebalancing of supply and demand characterises in a special way energy prices and returns, providing peculiar features in trajectories and statistical properties. Energy prices, in particular natural gas ones, show higher and more variabSave and publishle volatility than in stock or bond markets, while at the same time they are affected by jumps and mean reversion, due to physical constraints in production, which lead to an important role for the higher moments of returns.

With deregulation and the development of derivatives markets, the necessity of financial models able to capture these peculiarities arises as a new challenge in mathematical finance. In this work, we conduct an empirical investigation of the use of pure jump processes as modelling blocks to describe distributions of energy returns under the pricing measure by studying option prices. The proposed option-implied approach is able to circumvent most of estimation difficulties related to other models in the literature by using a framework that is very easy to implement.

Click here for a full version of this article (PDF: 604KB)

Lévy processes

Lévy processes are an important mathematical tool to build jump models. They offer a rich class of distributions for modelling financial returns and have been successfully used to capture skewness and kurtosis seen in both the physical and risk-neutral return densities (Cont, 2001). We focus on the so-called pure jump processes that evolve only by jumps: these are characterised by such a high rate of activity that a diffusion component is not needed. Pure jump processes usually lead to a better fit of financial returns than jump-diffusions, allow for more stable parameters and circumvent several estimation issues.1

A discussion on Lévy processes is proposed in Bertoin (1996). The most common Lévy processes in the financial context are: the variance gamma, VG, (Madan & Seneta, 1990; Madan et al, 1998); the normal-inverse Gaussian, NIG, (Barndorff-Nielsen, 1998); hyperbolic (Eberlein & Keller, 1995); generalised hyperbolic (Eberlein, 2001); CGMY (Carr et al, 2002) and the Meixner process (Schoutens, 2002; Madan & Yor, 2008). For a complete overview about this matter see Bertoin (1996) and Cont & Tankov (2004).

We now recall the definition of Lévy processes. and some of their key properties.

Definition

A cádlág stochastic process, (Xt)t≥0, on a filtered probability space (Ω;F;F;P) with values in Rd such that X0=0 almost surely is called a Lévy process if it has stationary and independent increments and it is stochastically continuous.

Let (Xt)t≥0 be a Lévy process, then for any t, the random vector Xt has infinitely divisible distribution and conversely if Ù is an infinitely divisible distribution then there exists a Lévy process (Xt)t≥0 such that the distribution of X1  is Ù. Roughly speaking, Lévy processes characterise the large family of (independent and identically distributed (i.i.d.) processes and the law of increments also governs the process itself. The characteristic function of a Lévy process admits the following Lévy-Khinchin representation.

Proposition

Let (Xt)t≥0 be a Lévy process on a filtered probability space (Ω;F;F;P) with values in Rd and with characteristic triplet (γ,A,ν), where γ∈Rd, A is a symmetric d×d matrix and ν is a positive random measure on Rd, satisfying ν(0)=0 and

Then it has characteristic function

        (1)

with characteristic exponent

        (2)

Therefore the Lévy triplet, (γ,A,ν), uniquely characterises a Lévy process in distribution. This representation simply says that any Lévy process can be written as the sum of a drift, γ, a Brownian motion, Wt, with coefficient A, and an independent jump term, Jt, whose arrivals are governed by ν:

        (3)

We concentrate on the so-called pure jump models in which the diffusive component is absent (A=0) and the rate of jump activity is infinite

A typical example of pure jump process used in finance is the variance gamma, which has a very simple parametrisation and is defined by the following characteristic function:

        (4)

where the mean and the variance are μ and σ2+μ2κ respectively, while the parameter κ induces fat tails and asymmetry, whose sign is determined by μ. The variance gamma process has two intuitive representations: first, the difference of two gammas respresenting respectively positive and negative jumps; second, a Brownian motion with drift, whose drift and variance are driven by a gamma that induces non-Gaussianity and heteroscedasticity.

Modelling energy prices

The most used approach to modelling reduced-form commodity futures can broadly be summarised as: initially, a spot market model is set up and then futures are derived as expected values under the pricing measure. The best known of these models is the two-factor model by Schwartz & Smith (2000), which uses two Brownian motions to model short-term variations and long-term dynamics of commodity spot prices.

These authors also compute prices for futures and options on futures. However, electricity futures are not explicitly modelled. Thus, the applicability to pricing options on electricity futures is limited.

Several models more specific to energy spot markets have extended such an approach: Geman & Eydeland (1999), use Brownian motions extended by stochastic volatilities and poisson jumps; Kholodnyi (2001, 2004) and Kholodnyi & Varshney (2007) introduce a non-Markovian approach that seems to be particularly adept at modelling spikes; Kellerhals (2001) and Culot et al (2006), use affine jump-diffusion processes; Cartea & Figueroa (2005), use a mean-reverting jump-diffusion; Benth & Saltyte-Benth (2004) apply normal inverse Gaussian processes; and Geman & Roncoroni (2006) developed a jump-reversion model based on a jump diffusion process with a regime-switching behaviour.

All of these models are capable of capturing some of the features of spot price dynamics well and imply certain dynamics for futures prices, but these are usually quite involved and difficult to work with. These models are especially not suitable when it comes to option pricing in futures markets, since the evaluation of option pricing formulae is not straightforward. In particular, all models have been fitted to time-series data. This approach requires the estimation of a market price of risk in order to price derivatives. None of the models have been calibrated to option-price data directly as we propose in this article.

Therefore, we assume a very simple characterisation for the underlying price under the pricing measure:

        (5)

where r and q are the risk-free rate and the convenience yield respectively, and ω is the martingale mean-correction of the pure jump Lévy process Xt.

Option pricing

Prices used in this work have been taken from US indexes PJM Electricity (PJM) and Natural Gas (NG), both traded on Nymex, as provided by Bloomberg. Considered options are American-style calls and puts written on futures with a one-month expiry maximum. We have chosen option prices of one week from February 5 to February 11, 2008: restricting attention on particular maturities, strike prices and liquidity level, we have collected in total 247 options for natural gas and 508 for electricity. Maturities of selected contracts are for all months, from one to 18. We have chosen out-of-the-money options with the highest available rate of liquidity.

The interest rate used was the one-year spot rates on the US market for each maturity, , as selected by Bloomberg – at the time of the calibration these were in the range of 2.25–2.45%. The convenience yield used in calibrations is the implied one in quoted prices: we compute it as the one that ensures the put-call parity between the quoted options. The range of estimated values is about 2.25–2.85%.

The following processes are calibrated for both natural gas and electricity: Black-Scholes (BS), Merton jump diffusion (MJD), variance gamma (VG), normal-inverse Gaussian (NIG), Meixner (MXN), Carr-Madan-Geman-Yor (CGMY) and generalised hyperbolic (GH).

The calibration is performed, as usual, by minimising the pricing errors between the observed and theoretical prices:2 the used pricing procedure is the Fourier space time-stepping (FST) technique3 (which exactly solves the associated PIDE problem in the Fourier space) recently proposed by Jackson et al (2007), which only requires the characteristic function of the process that in the Lévy domain is always available in closed form. Calibrations are conducted in the MATLAB environment using the function fminsearch, adjusted for bounds and constraints over parameters.

The robustness of nonlinear optimisation with respect to the choice of initial guess values for parameters has been checked throughout the VG and the NIG cases.

Table 1 presents the calibration results from a subset of prices selected in order to minimise distortions due to liquidity and seasonality.4

The first thing to note is that the BS model heavly underperforms with respect to jump models, with errors that double those of other models. Second, we can see that all jump models perform in a similar way, producing pricing errors quite close in magnitude. Nevertheless, these sampling errors are coherent with those expected in theory: as the VG model is a subcase of the CGMY, and the VG and NIG are subcases of the GH model, these models all produce errors larger then the ones of their parent models. CGMY and GH are four parameters models, so they provide the smallest pricing errors – they therefore seem to be the best candidate models for describing energy prices, given the data set involved.

Moreover, pure jump models seem to improve the goodness of the fit also with respect to the jump-diffusion model, which is the most-used approach in the energy literature.

Thirdly, statistical features of the risk-neutral measures implied in option prices are very similar between the models. Variance and skewness are indeed almost identical for each jump process, only the excess of kurtosis is relatively varying. Both natural gas and electricity measures show a strong distance from Gaussianity. A calibration example with the VG is shown in figure 1 and optimal parameters are reported in table 2.

We have also tested the robustness of results in terms of parameter stationarity and seasonality. In order to check for stationarity of parameters, we have separately selected contracts from each day during the period February 5 to February 11 – for each of these, all maturities were considered together, from one to 18 months. Results show that calibrating together all maturities produces high pricing errors and, through days, variability in paramters, which only marginally affects the second moment of risk-neutral measures but affects their non-Gaussianity more significantly. Table 3 presents the calibration results.

We also provide a second robustness test to analyse seasonal effects in calibrations. The aim of the test is to find a way to account for seasonality in order to be able to compare estimated risk-neutral measures. While time-series of returns can be deseasonalised by using a sinusoidal functional when performing a regression, we do not know of any instances in the literature of a corresponding methodology to account for seasonality under the risk-neutral measure and starting from an option data set. Our idea is then not to eliminate seasonality but to eliminate its distorsive effects in calibration results. We start from the simple but empirically proven assumption that there are two high peaks per year, in winter (January) and summer (July), and other two low peaks in spring (April) and autumn (October). We then divide the option data set in to two subsamples: we select options with maturities expiring in the high periods and separate the ones expiring in low periods.

In figure 2 we show the seasonal cycle through maturities: between the shortest maturity, February 2008, and the longest one, August 2009, we can see three high and three low peaks.

We consider contracts of different days together, while we select maturities in order to obtain two subsets where seasonal effects are homogeneous. Table 4 presents the calibration results.

The first thing to notice is that, on average, pricing errors are lower than the ones obtained with the first test. We can therefore deduce that, even though here we do not eliminate the distorsive effects of variability of parameters in time, the subsampling procedure seems to eliminate distorsive effects, which we can impute to seasonality, providing a global benefit in pricing errors.

Secondly, we can see that jump models overperform the Black-Scholes model in a more significant way than in the stationary test: then we can deduce that the elimination of seasonality effects is beneficial in particular for jump models – in the absence of distorsive effects, jump models can capture features in a better way than the risk-neutral measure. As in the previous cases, the performance of pure jump models is similar, and on average, CGMY and GH seem to be the best processes.5

Conclusion

This article provides an option-implied approach under non-Gaussian distributions to model energy prices. It avoids the difficulties related to the estimation of the risk premium and can be easily implemented using any Lévy process. Calibration results, as expected, show the overperformance of the Lévy models with respect to the Black-Scholes and the jump-diffusion model and the coherence between the jump models where CGMY and GH seem to be the best choices. However, in order to obtain a good fit of quoted prices, the distortive effects of seasonality have to be ruled out. Using our subsampling procedure, all models produce smaller errors, leading to a good fit of the volatility surface.

Roberto Marfè, Swiss Finance Institute – Université de Lausanne

Email: Roberto.Marfe@unil.ch