Affine‐Structure Models and the Pricing of Energy Commodity Derivatives

• Author: Nikos K. Nomikos,Ioannis Kyriakou,Nikos C. Papapostolou,Panos K. Pouliasis
• Published date: 01 November 2016
• DOI: http://doi.org/10.1111/eufm.12071
• Date: 01 November 2016

Affine-Structure Models and the

Pricing of Energy Commodity

Derivatives

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

E-mails: ioannis.kyriakou@city.ac.uk; n.nomikos@city.ac.uk; n.papapostolou@city.ac.uk;

p_pouliasis@city.ac.uk

Abstract

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 affine form of the log-spot models, we develop a

general valuation framework for futures and discrete arithmetic Asian options. We

investigate five 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 fitted spot models in futures pricing, Asian options pricing and

hedging. We find 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€

udiger

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, 853–881

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 influence of geopolitical events and global economic activity on

petroleum prices create the need for reliable and efficient tools to price energy-related

securities and projects.

Energy commodities are predominantly different from conventional financial assets

such as equity and fixed-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

1

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) find 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

specifications 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 finance literature

focused on other markets such as, for example, the equity index (see Kaeck and

1

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 refined 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