Affine‐Structure Models and the Pricing of Energy Commodity Derivatives

Published date01 November 2016
Date01 November 2016
Afne-Structure Models and the
Pricing of Energy Commodity
Ioannis Kyriakou, Nikos K. Nomikos,
Nikos C. Papapostolou and Panos K. Pouliasis
Cass Business School, City University London, 106 Bunhill Row, London EC1Y 8TZ, UK
We consider a seasonal mean-reverting model for energy commodity prices with
jumps and Heston-type stochastic volatility, and three nested models for
comparison. By exploiting the afne form of the log-spot models, we develop a
general valuation framework for futures and discrete arithmetic Asian options. We
investigate ve major petroleum commodities from Europe (Brent crude oil,
gasoil) and US (light sweet crude oil, gasoline, heating oil) and analyse the effects
of the competing tted spot models in futures pricing, Asian options pricing and
hedging. We nd evidence that price jumps and stochastic volatility are important
features of the petroleum price dynamics.
Keywords: energy prices, affine models, futures, arithmetic Asian options, control
variate Monte Carlo
JEL classification: C13, G15, G13, C63
The authors would like to thank the Managing Editor John A. Doukas and two anonymous
referees for their constructive comments and suggestions. Our thanks are extended to
Gianluca Fusai for fruitful discussions and Russell Gerrard for his valuable contribution
that helped improve the paper. Versions of this paper have been presented at the Cass-
ESCP 51st Meeting of the Euro Working Group on Commodities and Financial Modelling,
Winter 2014 Conference of the Multinational Finance Society, 4th Energy Finance
Christmas Workshop (EFC14), Energy Finance Seminar (University of Duisburg-Essen,
April 2015) and Commodity Markets Workshop (Oslo, May 2015). We thank R
Kiesel, Marcel Prokopczuk and other participants for useful feedback. All remaining errors
are of course our own.
European Financial Management, Vol. 22, No. 5, 2016, 853881
doi: 10.1111/eufm.12071
© 2015 John Wiley & Sons, Ltd.
1. Introduction
Understanding the stochastic process governing the price of energy is essential, owing to
the indispensable role of hydrocarbons in the world economy and the response of
macroeconomic aggregates to oil price shocks. Concerns about the security of energy
supply and the inuence of geopolitical events and global economic activity on
petroleum prices create the need for reliable and efcient tools to price energy-related
securities and projects.
Energy commodities are predominantly different from conventional nancial assets
such as equity and xed-income securities and, due to intricate price formation
mechanisms, traditional modelling techniques are not directly applicable. For example,
seasonal effects arise naturally from periodic supply and demand patterns
and have been
successfully modelled by several authors including Routledge et al. (2000) and
Borovkova and Geman (2006). In addition, commodity prices mean-revert to the
marginal cost of production; relevant theoretical arguments and empirical evidence have
been put forward by Bessembinder et al. (1995), Schwartz and Smith (2000) and
Casassus and Collin-Dufresne (2005), among others. In effect, prices may temporarily be
high or low, but will tend toward an equilibrium level. Furthermore, temporary supply
and demand imbalances, changes in market expectations, or even unanticipated
macroeconomic developments may cause sudden jumps in energy prices (Hilliard and
Reis, 1998; Clewlow and Strickland, 2000). Due to construction lags on the supply side,
even a relatively small change in demand can, at times, cause immediate market
movements of large magnitude. Yet, jumps in returns are transient and a more persistent
component may be required. In fact, compared to other markets, energy price volatility is
both relatively higher and more variable over time (see, e.g., Pindyck, 2004). Trolle and
Schwartz (2009) develop a stochastic volatility model for crude oil and highlight its
importance in commodity derivatives pricing. Larsson and Nossman (2011) nd that, in
addition to stochastic volatility, jumps are essential to capture the time series properties
of oil prices. Accounting for both jumps and stochastic volatility provides a reasonable
characterisation of energy commodity prices in that it explains the skewness and fat-tail
feature of commodity return distributions. Furthermore, it gives rise to realistic implied
volatility patterns for short-term options without also affecting long-term smiles.
The previous discussion encourages the empirical testing of different spot model
specications for further use as inputs in derivatives pricing and risk management,
energy investment evaluation, asset allocation and planning. The aim of this paper is to
conduct a comprehensive analysis of stochastic dynamic modelling of European and US
petroleum commodity prices and enrich existing literature with some new insights in
several applications such as futures pricing, options pricing and hedging. In terms of
scope our work shares similarities with recent contributions in the nance literature
focused on other markets such as, for example, the equity index (see Kaeck and
For example, heating oil prices experience an upward pressure during winter when storage
capacity may not be able to smooth out seasonal demand shocks, particularly in the peak
demand season. Instead, lower prices are anticipated during the summer inventory build-up
period. On the other hand, gasoline prices typically trade at a discount during winter, as
demand falls from the peak levels of the summer driving season. Crude oil demand is derived
from the demand for its rened products and seasonal patterns are less evident.
© 2015 John Wiley & Sons, Ltd.
854 Ioannis Kyriakou, Nikos K. Nomikos, Nikos C. Papapostolou and Panos K. Pouliasis

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