Energy Economics 40 (2013) 989–1000

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Energy Economics

j ourna l homepage: www.e lsev ie r .com/ locate /eneco

Humps in the volatility structure of the crude oil futures market:New evidenceCarl Chiarella a, Boda Kang a, Christina Sklibosios Nikitopoulos a,⁎, Thuy-Duong Tô b

a University of Technology Sydney, Finance Discipline Group, UTS Business School, PO Box 123, Broadway, NSW 2007, Australiab University of New South Wales, Australian School of Business, Sydney NSW 2052, Australia

⁎ Corresponding author. Tel.: +61 2 9514 7768.E-mail addresses: carl.chiarella@uts.edu.au (C. Chiar

(B. Kang), christina.nikitopoulos@uts.edu.au (C.S. Nikito(T.-D. Tô).

0140-9883/$ – see front matter © 2013 Elsevier B.V. Allhttp://dx.doi.org/10.1016/j.eneco.2013.05.019

a b s t r a c t

a r t i c l e i n f o

Article history:Received 3 July 2012Received in revised form 11 March 2013Accepted 24 May 2013Available online 10 June 2013

JEL classification:G13

Keywords:Commodity derivativesCrude oil derivativesUnspanned stochastic volatilityHump-shaped volatilityPricingHedging

This paper analyses the volatility structure of commodity derivatives markets. The model encompasses hump-shaped, unspanned stochastic volatility, which entails a finite-dimensional affine model for the commodityfutures curve and quasi-analytical prices for options on commodity futures. Using an extensive database ofcrude oil futures and futures options spanning 21 years, we find the presence of hump-shaped, partially spannedstochastic volatility in the crude oil market. The hump shaped feature is more pronounced when the market ismore volatile, and delivers better pricing as well as hedging performance under various dynamic factor hedgingschemes.

© 2013 Elsevier B.V. All rights reserved.

1. Introduction

Commodity derivatives serve the very important role of helping tomanage the volatility of commodity prices. Apart from hedgers, thevolatility of commodity prices is also of keen interest to speculators,who have become more dominant in these markets in recent years,see Barone-Adesi et al. (2010) and Cifarelli and Paladino (2010).However, these derivatives have their own volatility, of which theunderstanding and management is of paramount importance. In thispaper, we will provide a tractable model for this volatility, and carryout empirical analysis for the most liquid commodity derivativemarket, namely the crude oil market.

The model used in this paper focuses directly on the volatility ofderivatives. It is set up under the Heath et al. (1992) frameworkthat treats the entire term structure of futures prices as the primarymodelling element. Due to the standard feature that commodity futuresprices are martingales under the risk-neutral measure, the model iscompletely identified by the volatility of futures prices and the initialforward curve.Wemodel this volatility as a multifactor stochastic vola-tility, which may be partially unspanned by the futures contracts. Spotcommodity prices are uniquely determined without the need to specify

ella), boda.kang@uts.edu.aupoulos), td.to@unsw.edu.au

rights reserved.

the dynamics of the convenience yield. Option prices can be obtainedquasi-analytically1 and complex derivative prices can be determinedvia simulation.

Commodity derivatives have been previously studied under theHeath et al. (1992) framework. However, previous works such as thoseof Miltersen and Schwartz (1998), Clewlow and Strickland (2000) andMiltersen (2003) as well as the convenience yield models of Gibsonand Schwartz (1990) and Korn (2005) were restricted to deterministicvolatility. Schwartz and Trolle (2009a) extended the literature signifi-cantly by providing empirical evidence that stochastic volatility modelsare superior to the deterministic volatility models as the latter are notcapable of accommodating the very important feature of unspanned vol-atility in the commodities markets. In addition, Clewlow and Strickland(1999) have demonstrated that convenience yield models can beconverted to forward models which are a class of models employed inour study. However, there are two differences between this paper andthe Schwartz and Trolle (2009a) paper. First, Schwartz and Trolle(2009a) start by modelling the spot commodity and convenience yield.Convenience yield is unobservable and therefore its modelling addscomplexity to model assumptions and estimation. Moreover, sensitivityanalysis has to rely on applying shocks to this unobserved convenienceyield, which makes it less intuitive. Second, the volatility function inthe Schwartz and Trolle (2009a) paper has an exponential decaying

1 Empirical studies that measure and forecast volatility in commodities markets, seefor instance Agnolucci (2009), suggest that more advanced option pricing modelscould provide better results.

4 The usual conditions satisfied by a filtered complete probability space are: (a) F 0

990 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

form, predicting that long term contractswill always be less volatile thanshort term contracts. Ourmodel, on the other hand, uses a hump-shapedvolatility (which can be reduced to an exponential decaying one), andtherefore allows for increasing volatility at the short end of the curve.

The model in this paper falls under the generic framework providedby Andersen (2010) for the construction of Markovianmodels for com-modity derivatives. As an extension to his work, we provide full resultsformodels that allow for hump-shaped, unspanned stochastic volatility.A hump is an important factor in other markets, such as interest ratemarkets, see for example Litterman et al. (1991), Dai and Singleton(2000) and Bekaert et al. (2001). However, limited evidence existsin the crude oil market. As far as we are aware, this feature has onlybeen studied in the working paper version of Schwartz and Trolle(2009a). It is reported that a hump shaped volatility function hadbeen tried, but resulted in very similar estimates and almost indistin-guishable price performance compared to the exponential volatilityfunction. We will re-examine the volatility structure of the crude oilderivatives market. We use a larger panel dataset of crude oil futuresand options traded on the NYMEX, spanning 21 years from 1 January1990 to 31 December 2010. We find that a three-factor stochastic vola-tility model works well. Two of the volatility functions have a humpshape that cannot be captured by the exponential decaying specifica-tion. We also find that the hump shaped volatility matters a lot morewhen the market is volatile than when the market is relatively stable.The extent to which the volatility can be spanned by futures contractsvaries over time, with the lowest spanning being in the recent periodof 2006–2010.

The fact that volatility in the market cannot be spanned by futurescontracts highlights the importance of options for hedging purposes.We analyse the hedging of straddle contracts, the pricing of whichis highly sensitive to volatility. Given the multifactor nature of themodel, factor hedging is employed. Factor hedging has been usedsuccessfully for deterministic and local volatility models,2 such as inClewlow and Strickland (2000) or Fan et al. (2003). We expand themethod to hedge the random shocks coming from stochastic volatility.We show that the hedging performance increases dramatically whenoptions contracts are added to the hedging instrument set. The hedgingperformance is measured under various different factor hedgingschemes, from delta-neutral to delta-vega and delta-gamma neutral.

An alternative approach to the HJM framework is modelling thespot commodity prices directly. A representative list of relevantliterature would include Gibson and Schwartz (1990), Litzenbergerand Rabinowitz (1995), Schwartz (1997), Hilliard and Reis (1998),Casassus and Collin-Dufresne (2005) and Fusai et al. (2008). Thesemodels have been successful in depicting essential and critical featuresof distinct commodity market prices, for instance, the mean-reversionof the agricultural commodity market, the seasonality of the naturalgas market, the spikes and regime switching of the electricity marketand the inverse leverage in the oil market. The disadvantage of thespot commodity models is the requirement to specify and estimatethe unobservable convenience yield. The futures prices are then deter-mined endogenously. In addition, some spot commoditymodels cannotaccommodate unspanned stochastic volatility,3 a feature that can benaturally embedded in HJM models.

The paper is organized as follows. Section 2 presents a generalisedunspanned stochastic volatilitymodel for pricing commodity derivativeswithin the HJM framework. Section 3 describes and analyses the data forcrude oil derivatives and explains the estimation algorithm. Section 4presents the results. Section 5 examines the hedging performance.Section 6 concludes. Technical details are presented in the Appendix.

2 Local volatility models refer to models where there is a dependence between vol-atility and the level of the state variables.

3 See the discussion in Collin-Dufresne and Goldstein (2002) for example.

2. The HJM framework for commodity futures prices

We consider a filtered probability space Ω;AT ;AZ ; P� �

, T ∈ (0,∞)with AZ ¼ Atð Þt∈ 0;T½ �, satisfying the usual conditions.4 We introduceV = {Vt, t ∈ [0,T],} a generic stochastic volatility process modellingthe uncertainty in the commodity market. We denote as F(t,T,Vt),the futures price of the commodity at time t ≥ 0, for delivery attime T, (for all maturities T ≥ t). Consequently, the spot price at timet of the underlying commodity, denoted as S(t,Vt) satisfies S(t,Vt) =F(t,t,Vt), t ∈ [0,T]. The futures price process is equal to the expectedfuture commodity spot price under an equivalent risk-neutral probabil-ity measure Q, see Duffie (2001), namely

F t; T;Vtð Þ ¼ EQ S T;VTð Þ½ jAt �:

This leads to the well-known result that the futures price of a com-modity is amartingale under the risk-neutralmeasure, thus the commod-ity futures price process follows a driftless stochastic differential equation.Let W(t) = {W1(t), …, Wn(t)} be an n-dimensional Wiener processdriving the commodity futures prices and WV(t) = {W1

V(t), …, WnV(t)}

be the n-dimensional Wiener process driving the stochastic volatilityprocess Vt, for all t ∈ [0,T].5

Assumption 1. The commodity futures price process follows a driftlessstochastic differential equation under the risk-neutral measure of theform

dF t; T;Vtð ÞF t; T;Vtð Þ ¼

Xni¼1

σ i t; T;Vtð ÞdWi tð Þ; ð1Þ

where σi(t,T,Vt) are the A-adapted futures price volatility processes, forall T > t. The volatility process Vt = {Vt

1, …,Vtn} is an n-dimensional

well-behaved Markovian process evolving as

dVti ¼ aVi t;Vtð Þdt þ σV

i t;Vtð ÞdWVi tð Þ; ð2Þ

for i = 1, …, n, where aiV(t,Vt), σi

V(t,Vt) are A-adapted stochasticprocesses and

EQ dWi tð Þ⋅dWVj tð Þ

h i¼ ρidt; i ¼ j;

0; i ≠ j:

�ð3Þ

Assume that all the above processes are A-adapted boundedprocesses with drifts and diffusions that are regular and predictable sothat the proposed SDEs admit unique strong solutions. The proposedvolatility specification expresses naturally the feature of unspannedstochastic volatility in the model. The correlation structure of the inno-vations determines the extent to which the stochastic volatility isunspanned. If the Wiener processes Wi(t) are uncorrelated with Wi

V(t)then the volatility risk is unhedgeable by futures contracts. When theWiener processes Wi(t) are correlated with Wi

V(t), then the volatilityrisk can be partially spanned by the futures contracts. Thus the volatilityrisk (and consequently options on futures contracts) cannot becompletely hedged by using only futures contracts.

Conveniently, the system of Eqs. (1) and (2) can be expressedin terms of independent Wiener processes. By considering the

contains all the P-null sets of F and (b) the filtration is right continuous. See Protter(2004) for technical details.

5 We essentially assume that the filtration At includes At ¼ Aft∨AV

t , where

Aft

� �t≥0

¼ σ W sð Þ : 0 ≤ s ≤ tð Þf gt≥0;

AVt

� �t≥0

¼ σ WV sð Þ : 0 ≤ s ≤ t� �n o

t≥0:

991C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

n-dimensional independent Wiener processes W1(t) = W(t) andW2(t), then one possible representation is

dF t; T;Vtð ÞF t; T;Vtð Þ ¼

Xni¼1

σ i t; T ;Vtð ÞdW1i tð Þ; ð4Þ

dVti ¼ aVi t;Vt

i� �

dt þ σVi t;Vt

i� �

ρidW1i tð Þ þ

ffiffiffiffiffiffiffiffiffiffiffiffiffi1−ρ2

i

qdW2

i tð Þ�

: ð5Þ

Clearly, the volatility risk of any volatility factors Vti with ρi = 0

cannot be spanned by futures contracts.Let X(t,T) = lnF(t,T,Vt) be the logarithm of the futures prices process,

then from Eq. (4) and an application of Ito's formula, it follows that

dX t; Tð Þ ¼ −12

Xni¼1

σ2i t; T;Vtð Þdt þ

Xni¼1

σ i t; T ;Vtð ÞdWi tð Þ: ð6Þ

Lemma 1. Under the Assumption 1 for the commodity futures pricedynamics, the commodity spot prices satisfy the SDE

dS t;Vtð ÞS t;Vtð Þ ¼ ζ tð Þdt þ

Xni¼1

σ i t; t;Vtð ÞdWi tð Þ; ð7Þ

with the instantaneous spot cost of carry ζ(t) satisfying the relationship

ζ tð Þ ¼ ∂∂t lnF 0; tð Þ−

Xni¼1

∫t0σ i u; t;Vuð Þ ∂∂t σ i u; t;Vuð Þdu

þXni¼1

∫t0∂∂t σ i u; t;Vuð ÞdWi uð Þ:

ð8Þ

Proof. See Appendix A. ■

The commodity HJM model is Markovian in an infinite dimensionalstate space due to the fact that the futures price curve is an infinitedimensional object (one dimension for each maturity T). In addition,the path dependent nature of the integral terms in the drift (8) of thecommodity spot prices also gives the process an infinite dimensionalnature.

2.1. Finite dimensional realisations for a commodity forward model

We specify functional forms for the futures price volatility functionsσi(t,T,Vt) that will allow the proposed commodity forward model toadmit finite dimensional realisations (FDR).

Assumption 2. The commodity futures price volatility functionsσi(t,T,Vt) are of the form

σ i t; T ;Vtð Þ ¼ αi t;Vtð Þφi T−tð Þ; ð9Þ

whereαi : Rþ→R are A

Z-adapted square-integrable stochastic processes

and φi : R→R are quasi exponential functions. A quasi-exponentialfunction φ : R→R has the general form

φ xð Þ ¼ ∑iemix þ∑

jenjx pj xð Þcos kjx

� �þ qj xð Þsin kjx

� �h i; ð10Þ

wheremi, ni and ki are real numbers and pj and qj are real polynomials.These very general volatility specifications have been proposed in

Björk et al. (2004) and can be adapted for commodity forward models.Björk et al. (2004) have demonstrated, by employing methods of Liealgebra, that this functional form is a necessary condition for a forwardinterest rate model with stochastic volatility to admit FDR. In the spiritof Chiarella and Kwon (2001b) and Björk et al. (2004), αi may also

depend on a finite set of commodity futures prices with fixed tenors.When level dependent (or constant direction) volatility is considered,it becomes very difficult to obtain tractable analytical solutions forfutures option prices. For this reason, even though FDR can be obtainedfor a level dependent stochastic volatility model (clearly with a higherdimensional state space), we consider the dependence of αi only onstochastic volatility.

These volatility specifications have the flexibility of generating awide range of shapes for the futures price volatility surface. Sometypical examples of interest rate volatility curves include, the exponen-tially declining stochastic volatility structures of the Ritchken andSankarasubramanian (1995), and the hump-shaped volatility struc-tures discussed in Chiarella and Kwon (2001a) and Schwartz andTrolle (2009b), which are special cases of these general specifications.Furthermore some special examples of commodity volatility curvesinclude the exponentially declining stochastic volatility structures ofSchwartz and Trolle (2009a) and the gas volatility structures followinga regular pattern as discussed in Björk et al. (2006). Note that the latterauthors do not consider a stochastic volatility model.

2.2. Hump-shaped unspanned stochastic volatility

Next we propose certain volatility specifications within the generalfunctional form (10) which are not only multi-factor stochastic volatil-ity of Heston (1993) type but also allow for humps.

Assumption 3. The commodity futures price volatility processesσi(t,T,Vt) are of the form

σ i t; T;Vtð Þ ¼ κ0i þ κ i T−tð Þð Þe−ηi T−tð ÞffiffiffiffiffiffiffiVt

iq

; ð11Þ

where κ0i, κi and ηi are constants.When the commodity futures prices volatilities are expressed in

this functional form then finite dimensional realisations of the statespace are possible.

Proposition 1. Under the volatility specifications of Assumption 3, thelogarithm of the instantaneous futures prices at time t with maturity T,namely lnF(t,T,Vt), is expressed in terms of 6n state variables as

lnF t; T ;Vtð Þ ¼ lnF 0; T;V0ð Þ−Xni¼1

�12ðγi1 T−tð Þxi tð Þ þ γi2 T−tð Þyi tð Þ

þγi3 T−tð Þzi tð ÞÞþ βi1 T−tð Þϕi tð Þþβi2 T−tð Þψi tð Þð Þ�;

ð12Þ

where for i = 1, 2, …, n

βi1 T−tð Þ ¼ κ0i þ κ i T−tð Þð Þe−ηi T−tð Þ; ð13Þ

βi2 T−tð Þ ¼ κ ie−ηi T−tð Þ

; ð14Þ

γi1 T−tð Þ ¼ βi1 T−tð Þ2; ð15Þ

γi2 T−tð Þ ¼ 2βi1 T−tð Þβi2 T−tð Þ; ð16Þ

γi3 T−tð Þ ¼ βi2 T−tð Þ2: ð17Þ

The state variables xi(t), yi(t), zi(t), ϕi(t) and ψi(t), i = 1, 2, …, nevolve according to

dxi tð Þ ¼ −2ηixi tð Þ þ Vti

� �dt;

dyi tð Þ ¼ −2ηiyi tð Þ þ xi tð Þ� �dt;

dzi tð Þ ¼ −2ηizi tð Þ þ 2yi tð Þ� �dt;

dϕi tð Þ ¼ −ηiϕi tð Þdt þffiffiffiffiffiffiffiVt

iq

dWi tð Þ;dψi tð Þ ¼ −ηiψi tð Þ þ ϕi tð Þ� �

dt;

ð18Þ

0 1 2 3 4 5 6 7 Dec. Mar June Sept. Dec. Dec. Dec. Dec.Dec.0

0.5

1

1.5

2

2.5

3

3.5 x 105

Maturity (month)

Ope

n In

tere

st

Distribution of Open Interest

Fig. 1. Liquidity of crude oil futures contracts. The figure depicts the average open interest across contracts with different maturities from January 2000 to December 2005.

992 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

subject to xi(0) = yi(0) = zi(0) = ϕi(0) = ψi(0) = 0. The above-mentioned 5n state variables are associated with the stochastic volatilityprocess Vt ¼ Vt

1;…; ;Vt

nn o

which is assumed to be an n- dimensional ofHeston (1993) type process such that

dVti ¼ μV

i νVi −Vt

i� �

dt þ εViffiffiffiffiffiffiffiVt

iq

dWVi tð Þ; ð19Þ

where μiV, νiV, and εiV are constants (they can also be deterministic functions).

Proof. See Appendix B for technical details. ■

Note that the model admits FDR within the affine class of Duffie andKan (1996). Additionally, the model is consistent, by construction,with the currently observed futures price curve, consequently it is atime-inhomogeneous model. However for estimation purposes, it isnecessary to reduce the model to a time-homogeneous one as presentedin Section 3.3 below. Note that the proposed volatility conditions inAndersen (2010) lead to time-inhomogeneous models, which cannot bedirectly applied for estimation purposes.

The price of options on futures can be obtained in closed form as atractable expression for the characteristic function exists. By employingFourier transforms, call and put options on futures contracts can bepriced. These results are summarised in the following propositionwhich is a natural extensions of existing literature and are quotedhere for completeness.

Proposition 2. Under the stochastic volatility specifications (19) andfor t ≤ To ≤ T, the transform ϕ t; v; To; Tð Þ ¼: Et exp vlnF To; T ;VToð Þf g½ �is expressed as

ϕ t; v; To; Tð Þ ¼ exp M t; v; Toð Þ þXni¼1

Ni t; v; Toð ÞVti þ vlnF t; T;Vtð Þ

( );

ð20Þ

where M(t) = M(t;v,To) and for i = 1, …, n, Ni(t) = Ni(t;v,To) satisfythe Riccati ordinary differential equations

dM tð Þdt

¼ −Xni¼1

μvi v

Vi Ni tð Þ; ð21Þ

dni tð Þdt

¼ − v2−v2

φið Þ2− εVi vρiφi−μVi

� �Ni tð Þ−1

2εVi

2N2i tð Þ; ð22Þ

subject to the terminal conditions M(To) = Ni(To) = 0, whereφi ¼ κ0i þ κ i T−tð Þð Þe−ηi T−tð Þ.

The price at time t of a European put option maturing at To with strikeK on a futures contract maturing at time T, is given by

P t; To; T;Kð Þ ¼ EQt e−∫

Tot rsds K−F To; Tð Þð Þþ

" #

¼ P t; Toð Þ KG0;1 log Kð Þð Þ−G1;1 log Kð Þð Þh i ð23Þ

where P(t,To) is the price at time t of a zero-coupon bond maturing at Toand Ga,b(y) is given by

Ga;b yð Þ ¼ ϕ t; a; To; Tð Þ2

− 1π∫∞0

Im ϕ t; aþ ibu; To; Tð Þe−iuyh i

udu: ð24Þ

Note that i2 = −1.

Proof. Follows along the lines of Duffie et al. (2000) and Collin-Dufresneand Goldstein (2002). Technical details of the characteristic function arealso presented in Appendix C. ■

For the market price of volatility risk, a “complete” affine specifica-tion is assumed, see Doran and Ronn (2008) (where they have shownthat the market price of volatility risk is negative) and in particularDai and Singleton (2000). Accordingly, the market price of risk isspecified as,

dWPi tð Þ ¼ dWi tð Þ−λi

ffiffiffiffiffiffiffiVt

iq

dt;

dWPVi tð Þ ¼ dWV

i tð Þ−λVi

ffiffiffiffiffiffiffiVt

iq

dt;ð25Þ

for i = 1, …, n, where WPi tð Þ and WPV

i tð Þ are Wiener processes underthe physical measure P. Note that under these specifications, themodel parameters are 9n, namely; λi, λi

V, κ0i, κi, ηi, μiV, νiV, εiV, and ρithat we will estimate next by fitting the proposed model to crudeoil derivative prices.

Fig. 2. Crude oil futures prices. The figure plots the prices of selected crude oil futures contracts from January 2, 1990 to December 31, 2010 provided by the CME. The selectedcontracts are: (i) the first seven monthly contracts near to the trade date, that we label m1–m7, with the requirement that the first contract has more than 14 days to maturity;(ii) the next three contracts which have either March, June, September or December expiration months, that we label q1–q3; and (iii) the next five December contracts, that weshall denote as y1–y5. Thus, the number of contracts to be used on a daily basis varies between a maximum of 15 and a minimum of 8 and futures maturities are up to 3 yearsbetween 1990 and 1995 while they reach up to 8 years by 2010.

993C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

3. Data and the estimation method

3.1. Data

We estimate the model using an extended dataset of crude oilfutures and options traded on the NYMEX.6 The database spans the21 years from 1 January 1990 to 31 December 2010. This is one ofthe richest databases available on commodity derivatives. In addition,over this period, noteworthy financial market events with extrememarket movements, for instance the oil price crisis in 1990 and thefinancial crisis in 2008, have occurred.

Throughout the sample period, the number of available futurescontracts with positive open interest per day has increased from 17on the 1st of January 1990 to 67 on the 31st of December 2010. Themaximum maturity of futures contracts with positive open interesthas also increased from 499 (calendar) days to 3128 days. We cansee that the price surfaces change significantly throughout the sampleperiod. The maximum futures price was US$40 per barrel in 1990reachingUS$140 per barrel in 2008.

Given the large number of available futures contracts per day, wemake a selection of contracts for estimation purposes based on theirliquidity. Liquidity has increased across the sample. For instance, theopen interest for the futures contract with 6 months to maturity hasincreased from 13,208 contracts in 1990 to 38,766 contracts in2010. For contracts with less than 14 days to expiration, liquidity isvery low, while for contracts with more than 14 days to expiration,liquidity increases significantly. Fig. 1 shows the average openinterest across contracts with different maturities7 from January 2000

6 The database has been purchased from CME.7 Note that futures contracts expire and new contracts are introduced all the time.

to December 2005. Given the marked differences in liquidity, our sam-ple selection begins with the first seven monthly contracts, near to thetrade date, namely m1, m2, m3, m4, m5, m6, and m7.

Note that the first contract should have more than 14 days tomaturity. After that liquidity is mostly concentrated in the contractsexpiring in March, June, September and December. Thus the firstseven monthly contracts are followed by the three contracts whichhave either March, June, September or December expiration months.We name them q1, q2 and q3. Beyond that, liquidity is concentrated inDecember contracts only, therefore the next five December contracts,namely y1, y2, y3, y4 and y5, are included. As a result, the total numberof futures contracts to be used in our analysis is 70,735, with thenumber of contracts to be used on a daily basis varying between amaximum of 15 and a minimum of 8. Fig. 2 plots the selected futuresprices on Wednesdays during the sample period.

With regard to option data, we consider the options on the firstten futures contracts only, namely the futures contracts m1–m7 andq1–q3. We avoid the use of longer maturities because in the proposedmodel we have not taken into account interest rates that vary consid-erably over the sample period and probably stochastically. Due to thismodel constraint, the option pricing Eq. (23) is not very accurate forlonger maturities. Furthermore, the option pricing Eq. (23) providesthe price for European options, not American options that are the optionsof our database. For the conversion of American prices to European prices,including the approximationof the early exercise premium,we follow thesame approach proposed by Broadie et al. (2007) for equity options andapplied by Schwartz and Trolle (2009a) for commodity options.8

8 European Black (1976) prices are computed by using the lognormal implied volatil-ities recovered by inverting the Barone-Adesi and Whaley (1987) formula for Americanoption prices.

0

1

2

Jan90Jan91Jan93

Jan95

0

50

100

Option Maturity

ATM implied volatility

Date

Per

cent

0

0.5

1

1.5

Jan95 Jan96 Jan97 Jan99

0

50

100

Option Maturity

ATM implied volatility

Date

Per

cent

0

1

2

Jan00 Jan02 Jan04 Jan06

0

50

100

Option Maturity

ATM implied volatility

Date

Per

cent

0

12

Jan06 Jan07 Jan08 Jan09 Jan10 Jan11

0

50

100

Option Maturity

ATM implied volatility

Date

Per

cent

Fig. 3. ATM implied volatilities of options on crude oil futures. The figure plots the ATM lognormal implied volatilities of options on selected crude oil futures contracts provided bythe CME. The selected contracts are: (i) the first seven monthly contracts near to the trade date, namely m1–m7;(ii) the next three contracts which have either March, June,September or December expiration months, namely q1–q3. The option maturity is expressed in years. The first panel (left-hand side) displays implied volatilities from January1990 to December 1994, the second panel (right-hand side) from January 1995 to December 1999, the third panel (left-hand side) from January 2000 to December 2005 andthe last panel from January 2006 to December 2010.

Table 1Descriptive statistics.

Maturity 1 M 4 M 7 M 13 M

Period: 1990 - 1994Mean −0.00022 −0.00017 −0.00013 −0.00010Standard deviation 0.02631 0.01837 0.01616 0.01419Sample variance 0.00069 0.00034 0.00026 0.00020Kurtosis 50.19 50.62 31.53 14.35Skewness −2.96698 −3.26779 −2.22760 −1.03636

Period: 1995–1999Mean 0.00030 0.00023 0.00017 0.00009Standard deviation 0.02232 0.01580 0.01354 0.01167Sample variance 0.00049 0.00025 0.00018 0.00014Kurtosis 6.84 7.12 6.60 5.58Skewness 0.20058 0.14862 0.06334 0.03245

Period: 2000–2005Mean 0.00062 0.00069 0.00075 0.00081Standard deviation 0.02426 0.01952 0.01736 0.01528Sample variance 0.00059 0.00038 0.00030 0.00023Kurtosis 6.03 5.09 5.12 4.75Skewness −0.62528 −0.41719 −0.39037 −0.32096

Period: 2006–2010Mean 0.00029 0.00029 0.00029 0.00028Standard deviation 0.02720 0.02272 0.02115 0.01922Sample variance 0.00074 0.00052 0.00045 0.00037Kurtosis 7.71 5.77 5.69 5.59Skewness 0.15898 −0.15396 −0.12377 −0.11221

The table displays the descriptive statistics for daily log returns of futures pricesbetween January 2, 1990 and December 31, 2010. The sample period of 1990–2010 isdivided into 4 subsamples. The first sample period of 1990–1994 covers the Gulf Warperiod. The second sample period of 1995–1999 ends in 1999 just before thereduction in OPEC spare capacity and the increase in the US and China's oil consump-tion and imports. The Iraq War happened in the third sample period of 2000–2005,whereas the Global Financial Crisis happened in the last sample period of 2006–2010.

994 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

For each option maturity, we consider six moneyness intervals,0.86 − 0.90, 0.91 − 0.95, 0.96 − 1.00, 1.01 − 1.05, 1.06 − 1.10,1.11 − 1.15. Note that moneyness is defined as option strike dividedby the price of the underlying futures contract. In each moneynessinterval, we use only the out-of-the-money (OTM) and at-the-money(ATM) options that are closest to the interval mean. OTM options aregenerally more liquid and we also benefit by a reduction in the errorsthat occurred in the early exercise approximation.

Based on these selection criteria, we consider 433,137 optioncontracts over the 21 years, with the daily range varying between29 and 100 contracts (per trading day). Note that the total numberof trading days where both futures and options data are available is5272. Fig. 3 displays the ATM lognormal implied volatilities of optionson the first ten oil futures contracts (namely m1–m7 and q1–q3) onWednesdays. The ATM implied volatilities were computed by usingthe Barone-Adesi and Whaley (1987) option pricing formula.

3.2. Sample selection

Fig. 2 shows that the prices of futures contracts change significantlyduring the 21-year period from 1990 to 2010. In their study of crude oilfutures from October 1991 to October 2007, Bekiros and Diks (2008)show that the two periods before and after 1999 differ considerably intheir statistical features. They argue that there are economic reasonsbehind the change, namely the reduction in OPEC spare capacity andthe increase in the US and China's oil consumption and imports.

Our data coverage is longer than that of Bekiros and Diks (2008),namely from January 1990 until December 2010. It covers two moreimportant events related to crude oil prices, namely the Gulf Warand the Global Financial Crisis. It can be seen in Fig. 2 that not onlythe futures prices surged up during the two periods but also there is

Table 2Accumulated percentage of factor contribution.

Time period One factor Two factors Three factors

1990–1994 0.9056 0.9805 0.99611995–1999 0.8913 0.9667 0.99512000–2005 0.8229 0.9059 0.95492006–2010 0.9275 0.9715 0.9887

The table displays the accumulated percentage of PCA factor contribution towardscrude oil futures return variation. We found that three factors are able to explainmost of the variations of the futures returns during each of the subperiods.

Table 3Parameter estimates.

1990–1994 1995–1999

i = 1 i = 2 i = 3 i = 1 i = 2 i = 3

κ0i 0.0954 0.5418 0.3698 0.4173 1.0789 0.2701(0.0011) (0.0087) (0.0056) (0.0062) (0.0098) (0.0024)

κi 1.8975 0.4884 0.0010 1.2634 0.4559 0.0010(0.0101) (0.0069) (0.0000) (0.0095) (0.0057) (0.0000)

ηi 0.3495 0.0010 0.5720 1.3273 0.2104 0.9548(0.0036) (0.0001) (0.0076) (0.0105) (0.0025) (0.0099)

μiV 0.0010 0.0010 1.5480 0.0010 0.0010 2.0000(0.0001) (0.0001) (0.0112) (0.0001) (0.0001) (0.0129)

εiV 0.5212 1.2462 0.4184 0.4483 1.5386 0.0011(0.0074) (0.0113) (0.0046) (0.0041) (0.0132) (0.0001)

ρi −0.0078 −0.0800 0.2363 0.1076 −0.5136 0.2154(0.0005) (0.0010) (0.0039) (0.0021) (0.0042) (0.0018)

λiV −1.0068 1.0279 −1.0058 −0.8542 0.9683 −0.9559

(0.0270) (0.0162) (0.0183) (0.0187) (0.0224) (0.0198)λi 0.7370 1.0312 −0.7494 0.9585 1.0204 −0.7919

(0.0127) (0.0218) (0.0115) (0.0227) (0.0321) (0.0175)F 1.9250 1.9029

(0.0127) (0.0109)σf 0.0010 0.0010

(0.0000) (0.0001)σo 0.0180 0.0100

(0.0005) (0.0004)log L -43294.92 -80252.78

The table displays the quasi maximum-likelihood estimates for the three-factor modelspecifications and the standard errors in parenthesis for two subsamples; January 2,1990 to December 31, 1994 and January 2, 1995 to December 31, 1999. F is the homog-enous futures price at time 0, namely f(0,t) = F, ∀ t. σf and σo are the standard devia-tions of the log futures prices measurements errors and the option price measurementerrors, respectively. To attain identification, we normalized the long run mean of thevolatility process, νi

V, to one.

Table 4Parameter estimates.

2000–2005 2006–2010

i = 1 i = 2 i = 3 i = 1 i = 2 i = 3

κ0i 0.4394 0.1447 0.0010 0.1047 0.4782 0.5435(0.0068) (0.0011) (0.0000) (0.0009) (0.0071) (0.0059)

κi 0.0033 1.1032 0.0940 1.9057 0.4443 0.0010(0.0002) (0.0111) (0.0008) (0.0186) (0.0054) (0.0001)

ηi 1.3300 0.9989 0.0010 0.2685 0.0053 0.5524(0.0122) (0.0101) (0.0001) (0.0021) (0.0004) (0.0045)

μiV 0.0010 0.0010 7.9991 0.0010 0.0010 1.5728(0.0001) (0.0001) (0.0358) (0.0001) (0.0001) (0.0112)

εiV 2.3831 3.0000 3.0000 0.5186 1.2218 0.3085(0.0173) (0.0214) (0.0225) (0.0046) (0.0099) (0.0024)

ρi −0.3803 −0.1123 0.7199 −0.0156 −0.0852 0.2346(0.0033) (0.0008) (0.0040) (0.0007) (0.0012) (0.0022)

λiV −4.0000 −3.9995 0.3563 −1.0017 1.0357 −1.0003

(0.0211) (0.0225) (0.0030) (0.0102) (0.0098) (0.0101)λi 2.2278 1.7151 4.0000 0.7387 1.0238 −0.7623

(0.0210) (0.0164) (0.0267) (0.0043) (0.0099) (0.0065)F 3.1410 1.9324

(0.0087) (0.0015)σf 0.0010 0.0010

(0.0000) (0.0000)σO 0.0487 0.1321

(0.0002) (0.0008)log L -48460.89 -36962.34

The table displays the quasi maximum-likelihood estimates for the three-factor model

995C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

a lot of variation. We therefore break the data further down intosmaller subsamples and analyse their statistical features in Table 1.

Over the last 20 years, the futures returns have increased consistentlyuntil the last period, when the financial crisis hit. However, unlike therelatively simple movement in futures prices (Fig. 2) and returns(Table 1), the behaviour of futures volatility is much more complex(Fig. 3). The variance of returns started high in the first period surround-ing the Gulf War, reduced in the second period 1995–1999, bouncedback in 2000–2005 and finally reached the highest level in 2006–2010(see both Table 1 and Fig. 3). The variances were also mostly driven byextreme values (kurtosis), especially in the Gulf War period. Not onlythis, but also there is a great deal of stochastic variation in the volatilitylevel, even during the relatively quiet period of 1995–2000. Giventhese changes, we will later empirically estimate the model for eachperiod separately.

3.3. Estimation method

The estimation approach is quasi-maximum likelihood in com-bination with the extended Kalman filter. The model is cast into astate-space form, which consists of the system equations and theobservation equations.

For estimation purposes, a time-homogeneous version of themodel (13) is considered, by assuming for all T, F(0,T) = fo, wherefo is a constant representing the long-term futures price (at infinitematurity). This constant is an additional parameter that is also to beestimated. In the estimation we normalized the long run mean ofthe volatility process, νiV, to one to achieve identification. 9

The system equations describe the evolution of the underlying statevariables. In our case, the state vector is Xt = {Xti, i = 1, 2,…, n} whereXti consists of the six state variables xi(t), yi(t), zi(t), ϕi(t), ψi(t) and Vt

i.The continuous time dynamics (under the physical probability mea-sure) of these state variables are defined by Eqs. (18), (19) and (25).The corresponding discrete evolution is

Xtþ1 ¼ Φ0 þΦXXt þwtþ1;wtþ1 ∼ iid N 0;Qtð Þ; ð26Þ

where Φ0, ΦX and Qt can be computed in closed form. Details can befound in Appendix D.

The observation equation describes how observed options andfutures prices are related to the state variables, namely

zt ¼ h Xtð Þ þ ut ;ut ∼ iid N 0;Ωð Þ: ð27Þ

In particular, log futures prices are linear functions of the state vari-ables (as described in Eq. (12)) and the options prices are nonlinearfunctions of the state variables (as described in Eqs. (23) and (24)) sothe function h will have to vary accordingly.

9 For details see for example the discussion on invariant transformations in (Dai andSingleton, 2000).

specifications and the standard errors in parenthesis for two subsamples; January 2,2000 to December 31, 2004 and January 2, 2005 to December 31, 2010. F is the homog-enous futures price at time 0, namely f(0,t) = F, ∀ t. σf and σo denote the standard de-viations of the log futures prices measurements errors and the option pricemeasurement errors, respectively. To attain identification, we normalized the longrun mean of the volatility process, νiV, to one.

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3Volatility functions

Time to maturity τ

σ(τ)

Factor1Factor2Factor3

Jan90 Jan91 Jan92 Jan93 Jan94 Jan950

1

2

3

4

5

6

7

8

t

Vti

Vti

Factor1Factor2Factor3

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3Volatility functions

Time to maturity τ

σ(τ)

Factor1Factor2Factor3

Jan94 Jan95 Jan96 Jan97 Jan98 Jan99 Jan000

1

2

3

4

5

6

7

8

t

Vt

i

Vti

Factor1Factor2Factor3

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3Volatility functions

Time to maturity τ

σ(τ)

Factor1Factor2Factor3

Jan99 Jan00 Jan01 Jan02 Jan03 Jan04 Jan05 Jan060

1

2

3

4

5

6

7

8

t

Vti

Vti

Factor1Factor2Factor3

0 5 10 15 20 25 30 35 40 45 500

0.5

1

1.5

2

2.5

3Volatility functions

Time to maturity τ

σ(τ)

Factor1Factor2Factor3

Jan05 Jan06 Jan07 Jan08 Jan09 Jan10 Jan110

1

2

3

4

5

6

7

8

t

Vti

Vti

Factor1Factor2Factor3

Fig. 4. Estimated futures price volatility processes. The panels on the left-hand side show the futures price volatility function φi(T − t) of each futures price volatility process. Thepanels on the right-hand side show the estimated time-series of each volatility state factor f Vt

i for the three-factor model. Top panels: January 1990 to December 1994; secondpanels: January 1995 to December 1999; third panels: January 2000 to December 2005; bottom panels: January 2006 to December 2010.

996 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

Table 5Contribution and shape of volatility factors.

1990–1994 1995–1999 2000–2005 2006–2010

σ1 28.80% hump 1.18% hump 0.75% exp 26.12% humpσ2 57.27% hump(e) 77.55% hump 76.04% hump 52.64% hump(e)σ3 13.93% exp 21.27% exp 23.21% hump(e) 21.24% exp

The table reports the contribution and the shape of each volatility factor to the total variance for the three factor model. “Hump" means hump-shaped volatility and “exp" meansexponential decaying volatility. “Hump(e)" refers to a hump-shaped volatility the hump of which has not appeared within the relevant maturity range (less than five years). Thetotal contribution of the two hump-shaped volatility factors is at least 78% of the total futures return variation.

997C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

3.4. Other considerations

3.4.1. Number of stochastic factorsThe number of driving stochastic factors affecting the evolution of the

futures curve can be determined by performing a principal componentanalysis (PCA) of futures price returns. Table 2 shows that we do notneed more than three factors to capture the variations in the futuresreturns. We therefore will estimate a 3-factor model for all sampleperiods in our empirical analysis and check their performance againstcorresponding 2-factor models.

3.4.2. The discount functionThe discount function P(t,T) is obtained by fitting aNelson and Siegel

(1987) curve each trading day to LIBOR and swap data consisting of1-, 3-, 6-, 9- and 12-month LIBOR rates and the 2-year swap rate,similar to Schwartz and Trolle (2009a).

Let f(t,T) denote the time − t instantaneous forward interest rate totime T. Nelson and Siegel (1987) parameterize the forward interestrate curve as

f t; Tð Þ ¼ β0 þ β1e−θ T−tð Þ þ β2θ T−tð Þe−θ T−tð Þ ð28Þ

from which we can price LIBOR and swap rates. This also yields forzero-coupon bond prices the expression

P t; Tð Þ ¼ exp β0 T−tð Þ þ β1 þ β2ð Þ1θ

1−e−θ T−tð Þ� �þ β2 T−tð Þe−θ T−tð Þ

� :

ð29ÞOn each observation date, we recalibrate the parameters β0, β1, β2

and θ, by minimizing the mean squared percentage differences be-tween the model implied forward rates (as described in Eq. (28))and the observed LIBOR and swap curve consisting of the 1-, 3-, 6-,9- and 12-month LIBOR rates and the 2-year swap rate on that date.

3.4.3. Computational detailsThe loglikehood function is maximised by using the constrained

optimization routine “e04jy" in the NAG library. We begin with severaldifferent initial hypothetical parameter values, firstly on monthly data,then onweekly data and finally on daily data, aimed at obtaining globaloptima.

The ODE's (21) and (22) are solved by a standard fourth-orderRunge–Kutta algorithm using complex arithmetic. The integral inEq. (24), is approximated by the Gauss–Legendre quadrature formulawith 30 integration points and truncating the integral at 400.

4. Empirical results

4.1. Parameter estimation

Parameter estimates for our three-factor unspanned stochasticvolatility model can be found in Tables 3 and 4. Estimation is carriedout for four different subsamples due to the marked difference intheir price behaviour, as can be seen in Fig. 3 and Table 1.10

10 Estimation for two-factor models was also carried out. Results are published in theworking paper series of the authors' institution.

Fig. 4 depicts the estimated deterministic part φi of each volatilityfactor σi (σi = φiVi) and the estimated volatility state factor Vi. Thedeterministic part of the volatility functions (i.e. φi) determines theshape of each volatility factor of the futures price returns and hasbeen classified into three types. One type is the exponential volatility,which dies out as the time to maturity increases. Another type is thehump volatility, where volatility increases with time to maturity to apeak level, then decreases as time to maturity gets longer. The lasttype is what we call the “hump(e)" volatility, which is a hump-shapedvolatility exhibiting a peak beyond 5 years to maturity. Therefore, ingraphs featuring a time tomaturity under 5 years, the hump(e) volatil-ity appears as an increasing volatility.

From the parameter estimates in Tables 3 and 4, the significance of κiin all subsamples indicates the existence of hump-shaped volatilityfactors.11 In all except the second subsample, all of the three types ofvolatility shapes (exponential, hump and hump(e)) coexist. In thesecond subsample, there is no hump(e) volatility, but rather one expo-nential and two hump-shaped volatility factors.12 Table 5 shows thecontribution of each volatility factor σi to the total variance. We con-clude that, in general, the total contribution of the two hump-shapedvolatility (hump and hump(e)) factors accounts for at least 78% of thetotal futures return variation.

Given that hump(e) volatility is the type of volatility that affectsthe market in the long term, and only dies out for the very distantfutures maturities, we can conjecture that hump(e) is the type ofvolatility caused by strong market events that create long term mar-ket uncertainty. This is confirmed by historical observation. Unlikethe quiet period of 1995–1999, during the other three periods,there were three major events that affected respectively the volatil-ity of the crude oil market, namely the Gulf War 1990–1991, the IraqWar 2003 and the Global Financial Crisis in 2008. The implied volatil-ity especially for short-dated options increased by more than 100%over the 1991 and 2003 crises while implied volatilities for bothshort-dated and long-dated options increased by 90% and 50%,respectively, during the 2008 crisis. However, the effect of theshock to the implied volatility was more persistent during the 2008crisis.

The quietness of the 1995–1999 period is also reflected by thelow volatility factor estimates as well as the limited stochasticbehaviour signalled by the almost flat state factors Vt

i displayed inthe right hand side of Panel 2, Fig. 4. The period 2000–2005 is themost volatile period, whereby volatility factors fluctuated consider-ably in the few year periods leading towards the Iraq War as well asthe few years after that. In all of the periods, all of the volatility factorsare highly persistent (evidenced by the very low value of μiV),suggesting that they are important for the pricing of futures and op-tions of all maturities. For each of the subsamples, the innovation toat least one of the volatility factors has a very low correlation (abso-lute values from 0.7% to 11%) with the innovations to the futuresprices, implying the large extent to which the volatility is unspannedby the futures contracts.

11 For each subsample estimation, the likelihood ratio tests strongly reject the two-factor models in favour of the three-factor models.12 Note that the order of the volatility functions is not important.

Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul120

1

2

3

4

5

6

7

8

Time t

RM

SE

(P

erce

ntag

e)

RMSE of the futures prices

Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul120

2

4

6

8

10

12

14

16

18

20

Time t

RM

SE

(P

erce

ntag

e)

RMSE of the implied volatilities

Fig. 5. Model goodness of fit. The figure shows the RMSEs of the percentage differences between actual and fitted futures prices (the left-hand panel) as well as of the differencebetween actual and fitted implied option volatilities (the right-hand panel) for the three-factor model with hump-shaped volatility. The model is estimated separately for eachsample in the period January 1990 to December 2010, and then combined into one graph.

Table 6Model fit.

Sample Hump-shaped Improvement compared toexponential decaying

Futures Option Futures Options ln L0–ln L1

1990–1994 1.45 3.39 1.69 4.15 20401.561995–1999 1.28 3.54 2.85 0.76 8861.782000–2005 1.59 1.73 1.54 1.79 9756.182006–2010 1.55 1.48 3.57 10.86 14264.89

The table reports the average of the RMSEs for futures and options for four periods.These periods are January 2, 1990 to December 31, 1994; January 2, 1995 toDecember 31, 1999; January 2, 2000 to December 31, 2005 and January 2, 2006 toDecember 31, 2010. The futures pricing errors is computed as the differences

998 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

4.2. Pricing performance

Fig. 5 graphs the RMSEs of the percentage differences between actualand fitted futures prices as well as of the difference between actual andfitted implied option volatilities, whereas Table 6 gives the averagevalues. The overall goodness of fit is quite good, except during the specialevents of 1991, 2003 and 2008. Table 6 also compares the goodness ofthe fit of the hump-shaped volatility specification to the exponentialdecaying specification.13 The log likelihood ratio tests clearly favour thehump volatility specification. The improvement for the fit of futuresprices averages at 2.4%. The improvement for thefit of option implied vol-atility is not much for the period 1995–2005, but very significant duringthe periods 1990–1994 and 2006–2010 at 4.15% and 10.86% respectively.

Fig. 6 displays time series of implied volatilities and the fit to thethree-factor model. There were a lot of fluctuations in the implied vol-atilities over the last 21 years. The model does well in capturing thesechanges, as well as the special periods of the Gulf War 1990–1991, theIraq War 2003 and the Global Financial Crisis 2008.

5. Hedging performance

To gauge the impact of the hump-shaped volatility specificationcompared to exponential decaying only volatility specification, weassess the hedging performance of option portfolios on crude oil fu-tures by using the hedge ratios implied by the corresponding models.The various factors of the model manifested by the empirical analysisrepresent different dimensions of risk to which a portfolio of oil de-rivatives is exposed. In our stochastic volatility modelling framework,the variation in the crude oil forward curve is instigated by randomchanges of these forward curve volatility factors as well as randomchanges in a general stochastic volatility factor. By extending thetraditional factor hedging method to accommodate the stochastic vola-tility specification, a set of futures and futures options are used to hedgethe risk associated to the forward curve variation. The technical detailsof the extended factor hedging are presented in Appendix F.

The portfolio that we choose to hedge is a straddle, which is a typicaloption portfolio that is traded in thesemarkets and is sensitive to volatil-ity. A long straddle consisting of a call and a put with the same strike of130 and the same maturity of February 2009 is constructed and hedgedby using weights implied by the three-factor models in Section 4.14 The

13 Appendix E presents the model specifications that allow for exponential decayingvolatility structures. The estimation results for the model with exponential volatilityare presented in the working paper series of the authors' academic institutions.14 Seeking a representative example of a period in which the market was very vola-tile, the hedging performance of an option during the financial crisis in 2008 has beenselected. The hedging result is not sensitive to the particular straddle chosen.

hedging period is from August 1, 2008 to the straddle maturity ofFebruary 17, 2009.

A large number of derivative contracts are available to serve as thehedging instruments. Motivated by the presence of unspanned stochas-tic volatility, we will start by using futures contracts alone, then using amixture of futures and option contracts. The three futures contracts thatwill be used have maturities of six-months, nine-months and one-year(i.e. February 2009, May 2009 and August 2009), chosen due to theirliquidity. The three option contracts used as hedging instruments havethe same maturities as the (three) futures contracts but different strikesfrom the target option. Their strikes are 133,128 and 132.5 respectively.

We apply a small shock to the system, including both the shock tothe stochastic volatility component and the shock directly to the futurecurve. We then calculate the hedging portfolio weight so that theresulting portfolio is delta neutral, delta-veganeutral, or delta-gammaneutral. We adopt a dynamic hedging scheme with fortnightly re-balancing. To update the hedging portfolio, we re-estimate the modelbased on the past one year data and then use the new estimates tore-calculate/update the hedge ratios. The daily P&L of the hedged andunhedged positions are computed, by using the root mean squarederror (RMSE) to assess the hedging performance. The daily P&L of a per-fect hedge should be 0. The RMSE of our hedged position is computed as

RMSEhedge ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi∑day

P&Lð Þ2dayr

:

between actual and fitted futures prices divided by the actual prices and reported inpercentages. The option pricing errors are computed as the difference between actualand fitted implied option volatilities and reported in percentages. Columns 2–3 showthe average RMSEs for the three-factor model with hump-shaped volatilities. Columns4–5 compare that with the RMSEs from the three-factor model with exponentialdecaying volatilities and report the percentage of improvement for futures and options.The difference in the log likelihood of the two estimations is shown in column 6.

Jan90 Jul92 Jan95 Jul97 Jan00 Jul02 Jan05 Jul07 Jan10 Jul120

0.2

0.4

0.6

0.8

1

1.2

1.4Implied Volatility of ATM options on the 4th monthly contract

Time t

Impl

ied

Vol

atili

ty

Model Fit

Market

Fig. 6. Fit to implied volatilities. The graph compares the time series of market andmodel implied volatilities of ATM options on the fourth monthly contract. The marketimplied volatilities are denoted by ‘—’ and the model implied volatilities are denotedby ‘…’. The model does well in capturing variation in volatility, even in the periods ofthe Gulf War 1990–1991, the Iraq War 2003 and the Global Financial Crisis 2008.

Table 8Hedging Stability.

Standard deviation

hump exp

Delta hedge (3 futures) 0.6019 3.9512Delta-gamma hedge (3 futures + 3 options) 0.9978 25.1324Delta-vega hedge (3 futures + 3 options) 0.1342 8.6253

The table shows the standard deviation of the hedging errors when we apply 10,000different shocks to the system. “Hump" indicates the model with hump-shaped volatil-ities, whereas “exp" indicates the model with exponential decaying volatilities.

999C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

We repeat the procedure 10,000 times for each of the two modelspecifications (hump-shaped volatility and exponential volatility),under each of the three different hedging schemes, with differentcombinations of hedging instruments. Table 7 shows the hedgingerrors of the best hedged portfolios. The reported R-squared is thepercentage of the variation accounted for in the residuals of theunhedged positions.

Regarding the best hedged positions, three observations stand out.First, delta hedging is not as effective as delta-gamma and delta-vegahedging, confirming the existence of stochastic volatility. Moreover, thesignificant improvement from delta hedging to delta-vega hedginghighlights the relative importance of the volatility shocks. Second,hedging performance improves when we replace futures with optionsas the hedging instruments, accentuating the feature of unspannedfutures volatility. Third, hedging performance is always better underthe hump-shaped volatility specification compared to the exponentialvolatility specification. Under the simple delta hedging scheme, thehedge under the hump-shaped volatility specification can explain33.47% of the variation of the unhedged residuals, whereas the hedgeunder the exponential specification can explain only 1.56% of thevariation. The R-squared for the hump-shaped volatility specification

Table 7Hedging straddles.

RMSE R2 (%)

hump exp hump exp

Unhedged 2.6170 2.6170Delta hedge (3 futures) 2.1346 2.5965 33.4688 1.5605Delta hedge (2 futures + 1 option) 1.9118 2.4832 46.6324 9.9640Delta-gamma hedge (3 futures + 3 options) 1.5534 2.2648 64.7662 25.1051Delta-vega hedge (3 futures + 3 options) 1.3401 1.9167 73.7779 46.3585

The table compares the RMSEs and R2 of various hedging schemes; unhedged, delta hedge,delta-gamma hedge and delta-vega hedge. The portfolio to be hedged is a straddleconsisting of a call and a putwith the same strike of 130 and the samematurity of February2009. The hedging period is from August 1, 2008 to the straddle maturity of February 17,2009. The hedging instruments are the three futures contracts with maturities February2009, May 2009 and August 2009 and the three options contracts with the samematuritiesas the (three) futures contracts and strikes 133,128 and 132.5 respectively. Hedging isdynamic with fortnightly re-balancing. The reported R2 measures the reduction in thevariance of the hedged residuals compared to the variance of the unhedged residuals.

increases to 73.8% with the more sophisticated delta-vega hedgingscheme.

To understand whether this best hedging performance is represen-tative of the hedging performance in general, we investigate the stabil-ity of the hedging performance. Table 8 shows the standard deviation ofthe hedging errors when we apply 10,000 different shocks to the sys-tem. The hedging performance is quite stable under the hump shapedvolatility specification. On the contrary, the exponential volatilityspecification results in a very wide range of hedging errors. This resultclearly favours the use of the hump-shaped volatility specification.

6. Conclusion

A multi-factor stochastic volatility model for commodity futurescurves within the Heath et al. (1992) framework is proposed. Themodel aims to capture themain characteristics of the volatility structurein commodity futures markets. The model accommodates exogenousstochastic volatility processes that may be partially unspanned byfutures contracts. We specify a hump component for the volatility ofthe futures curves, which can generate a finite dimensional Markovianforward model. The resulting model is highly tractable with quasi-analytical prices for European options on futures contracts.

The model was fitted to an extensive database of crude oil futuresprices and option prices traded in the NYSE over 21 years. We findsupporting evidence for three volatility factors, two of which exhibit ahump. This provides new evidence on the volatility structure in crudeoil futures markets, which has been traditionally modelled with expo-nentially declining volatility functions. Finally, by using hedge ratiosimplied by the proposed unspanned hump-shaped stochastic volatilitymodel, the hedging performance of factor hedging schemes is exam-ined. The results favour the proposed model compared to a modelwith only exponential decaying volatility.

The currentwork suggests newdevelopments in commoditymarketmodelling. Firstly, it will be interesting to verify the existence of humpsin the volatility structure of other commodities. Our methodology isgeneric and can be adapted to any commodity futuresmarket. Addition-ally, the current model can be adjusted to accommodate stochasticconvenience yield and stochastic interest rates. This direction has thepotential to provide useful insights on the features of convenienceyields in commodity markets.

Acknowledgements

We thank seminar participants at the 2012 French Finance Associ-ation, the 2012 Bachelier Finance Society World Congress, the 2012Computing in Economics and Finance and the UTS Finance DisciplineGroup internal research seminars for fruitful discussions and helpfulsuggestions. We would also wish to thank the anonymous refereesfor their valuable comments and the Australian Research Council forfinancial support (DP 1095177, The Modelling and Estimation ofVolatility in Energy Markets).

1000 C. Chiarella et al. / Energy Economics 40 (2013) 989–1000

Appendix: Supplementary data

Supplementary data associated with this article can be found, inthe online version, at http://dx.doi.org/10.1016/j.eneco.2013.05.019.

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