www.ssoar.info A Two-Factor Model for the Electricity Forward Market Böerger, Reik H.; Kiesel, Rüdiger; Schindlmayr, Gero Postprint / Postprint Zeitschriftenartikel / journal article Zur Verfügung gestellt in Kooperation mit / provided in cooperation with: www.peerproject.eu Empfohlene Zitierung / Suggested Citation: Böerger, R. H., Kiesel, R., & Schindlmayr, G. (2009). A Two-Factor Model for the Electricity Forward Market. Quantitative Finance, 9(3), 279-287. https://doi.org/10.1080/14697680802126530 Nutzungsbedingungen: Terms of use: Dieser Text wird unter dem "PEER Licence Agreement zur This document is made available under the "PEER Licence Verfügung" gestellt. Nähere Auskünfte zum PEER-Projekt finden Agreement ". 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Diese Version ist zitierbar unter / This version is citable under: https://nbn-resolving.org/urn:nbn:de:0168-ssoar-221234 Quantitative Finance A Two-Factor Model for the Electricity Forward Market For Peer Review Only Journal: Quantitative Finance Manuscript ID: RQUF-2006-0118.R2 Manuscript Category: Research Paper Date Submitted by the 31-Aug-2007 Author: Complete List of Authors: Böerger, Reik; Ulm University, Financial Mathematics Kiesel, Rüdiger; Ulm University, Financial Mathematics Schindlmayr, Gero; EnBW Trading GmbH Energy Markets, Options, Term Structure, Model Calibration, Keywords: Commodity Markets, Implied Volatilities, Multi-factor Models C51 - Model Construction and Estimation < C5 - Econometric JEL Code: Modeling < C - Mathematical and Quantitative Methods Note: The following files were submitted by the author for peer review, but cannot be converted to PDF. You must view these files (e.g. movies) online. RQUF-2006-0118.R2.zip E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Page 2 of 21 Quantitative Finance 1 2 3 4 5 6 7 8 9 A Two-Factor Model for the Electricity Forward 10 11 Market 12 13 14 For Peer Review Only 15 R¨udiger Kiesel (Ulm University) 16 Gero Schindlmayr (EnBW Trading GmbH) 17 18 Reik H. B¨orger (Ulm University) 19 20 21 August 30, 2007 22 23 24 25 26 Abstract 27 28 29 This paper provides a two-factor model for electricity futures, which captures the main 30 features of the market and fits the term structure of volatility. The approach extends 31 the one-factor-model of Clewlow and Strickland to a two-factor model and modifies it to 32 33 make it applicable to the electricity market. We will especially take care of the existence 34 of delivery periods in the underlying futures. Additionally, the model is calibrated to 35 options on electricity futures and its performance for practical application is discussed. 36 37 Keywords: Energy derivatives, Futures, Option, Two-Factor Model, Volatility Term 38 Structure 39 40 41 42 43 1 Introduction 44 45 46 Since the deregulation of electricity markets in the late 1990s, power can be traded on 47 spot and futures markets at exchanges such as the Nordpool or the European Energy 48 Exchange (EEX). Power exchanges established the trade of forwards and futures early 49 on and by now large volumes are traded motivated by risk management and speculation 50 51 purposes. 52 53 Spot electricity is not a tradable asset which is due to the fact that it is non-storable. 54 Thus, the spot trading and prices in electricity markets are not defined in the classical 55 sense. Similarly, electricity futures show contract specifications, that are different to 56 57 many other futures markets. In addition to the characterisation by a fixed delivery 58 59 1 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Quantitative Finance Page 3 of 21 1 2 3 4 5 price per MWh and a total amount of energy to be delivered, power forward and 6 futures contracts specify a delivery period, which will directly influence the price of the 7 contract. 8 9 While much of the research in electricity markets focuses on the spot market (cp. Geman 10 and Eydeland [1999] for an introduction, Ventosa et al. [2005] for a survey of modelling 11 12 approaches, the monographs Eydeland and Wolyniec [2003], Geman [2005] for detailed 13 overviews, Weron [2006] for a discussion of time-series characteristics and distributional 14 properties ofFor electricity Peer prices), only Review few results are available Only for electricity futures and 15 options (on such futures). 16 17 For commodity futures modelling approaches can broadly be divided into two cate- 18 19 gories. The first approach is to set up a spot-market model and derive the futures as 20 expected values under a pricing measure. The best-known representative is the two- 21 factor model by Schwartz and Smith [2000], which uses two Brownian motions to model 22 short-term variations and long-term dynamics of commodity spot prices. The authors 23 24 also compute futures prices and prices for options on futures. However, electricity fu- 25 tures are not explicitly modelled. In particular, the fact that electricity futures have 26 a delivery over a certain period (instead of delivery at a certain point in time) is not 27 taken into account. Thus, the applicability to pricing options on electricity futures is 28 29 limited. 30 Several models more specific to electricity spot markets have extended the approach 31 by Schwartz and Smith [2000] during the last few years. The typical model ingre- 32 dients are a deterministic seasonality function plus some stochastic factors modelled 33 34 by L´evyprocesses. Typical representatives are Geman and Eydeland [1999], who use 35 Brownian motions extended by stochastic volatilities and poisson jumps, Kellerhals 36 [2001] and Culot et al. [2006], who use affine jump-diffusion processes, Cartea and 37 Figueroa [2005], in which a mean-reverting jump-diffusion is suggested, Benth and 38 39 Saltyte-Benth [2004], who apply Normal Inverse Gaussian processes, which is extended 40 to non-Gaussian Ornstein-Uhlenbeck processes in Benth et al. [2005]. Furthermore, a 41 regime-switching factor process has been used in Huisman and Mahieu [2003]. All of 42 these models are capable of capturing some of the features of the spot price dynamics 43 well and imply certain dynamics for futures prices, but these are usually quite involved 44 45 and difficult to work with. Especially, these models are not suitable when it comes to 46 option pricing in futures markets, since the evaluation of option pricing formulae is not 47 straightforward. In particular, none of the models has been calibrated to option price 48 data. 49 50 51 The other line of research is to model futures markets directly, without considering 52 spot prices, using Heath-Jarrow-Morton-type of models (HJM). Here Chapter 5 of the 53 monograph by Eydeland and Wolyniec [2003] provides a general summary of the mod- 54 elling approaches for forward curves, but it does not apply a fully specified model to 55 electricity futures data. They rather point out that the modelling philosophy coming 56 57 from the interest rate world can be applied to commodities markets in a refined ver- 58 59 2 60 E-mail: [email protected] URL://http.manuscriptcentral.com/tandf/rquf Page 4 of 21 Quantitative Finance 1 2 3 4 5 sion as well. The paper by Hinz et al. [2005] follows this line of research and provides 6 arguments to justify this intuitive approach. 7 8 More direct approaches start with the well-known one-factor model by Clewlow and 9 Strickland [1999] for general commodity futures. In principle, they can capture roughly 10 term-structure features, which are present in many commodities markets, but their 11 12 emphasis is on the evaluation of options such as caps and floors, the derivation of spot 13 dynamics within the model and building of trees, which are consistent with market 14 prices and allowFor for efficient Peer pricing routinesReview for derivatives Only on spot prices. Again, they 15 do not apply their model to the products of electricity markets and do not discuss 16 17 an efficient way of estimating parameters. A general discussion of HJM-type models 18 in the context of power futures is given in Benth and Koekebakker [2005] (which can 19 be viewed as an extension of Koekebakker and Ollmar [2005]). They devote a large 20 part of their analysis to the relation of spot-, forward and swap-price dynamics and 21 22 derive no-arbitrage conditions in power future markets and conduct a statistical study 23 comparing a one-factor model with several volatility specifications using data from Nord 24 Pool market. Among other things, they conclude that a strong volatility term-structure 25 is present in the market. The main empirical focus of both papers is to assess the fit of 26 27 the proposed models to futures prices.