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Spot Convenience Yield Models for the Energy Markets

Ren´e Carmona and Michael Ludkovski

Abstract. We review that part of the literature on energy spot price models which

involves convenience yield as a factor, our goal being to document the shortcom-

ings of the most commonly used models. From a mathematical point of view, the

introduction of the convenience yield is usually justiﬁed by the desire to reconciliate

dynamical models for the time evolution of commodity prices with standard arbitrage

theory. Since the convenience yield appears as a factor which cannot be observed di-

rectly, stochastic ﬁltering has been proposed as a strategy of choice for its estimation

from observed market prices. We implement these ideas on the models we review,

and on some natural extensions. We illustrate the inconsistencies of the spot models

on readily available data, paving the way for the empirical analysis of models of the

term structure of convenience yield recently proposed as a viable alternative.

1. Introduction

As the energy markets continue to evolve, valuation of energy-linked ﬁnancial instru-

ments has been one of the focal topics of recent mathematical ﬁnance research. One of the

most popular choices for describing asset movements is a class of so-called ”convenience

yield models”. Such models introduce a new unobserved quantity related to physical own-

ership of the asset. In turn, convenience yield models can be broadly split into ”spot”

models and ”term structure factor” models. To the ﬁrst group belong the classical Gibson

and Schwartz model [GS90], as well as later models of Schwartz [Sch97], Hilliard and

Reis [HR98], and Casassus and Collin-Dufresne [CCD03]. The ”term structure” mod-

els, which have been discussed among others by Miltersen and Schwartz [MS98], and

Bjork and Landen [BL01] are similar to Heath-Jarrow-Morton constructions originally

introduced in the analysis of ﬁxed income markets.

In this paper we review the literature of spot convenience yield models, and we

analyze in detail two new extensions. First, we discuss a variant of the Gibson-Schwartz

model with time-dependent parameters. This was ﬁrst suggested by Miltersen [Mil03],

but we provide the ﬁrst full implementation of the model using empirical data. Second, we

describe a new three-factor aﬃne model with stochastic convenience yield and stochastic

market price of risk. The existence of a third factor allows us to achieve a good ﬁt to

the cross-section of futures prices. The idea of time-dependent risk premia in the context

1991 Mathematics Subject Classiﬁcation. Primary 62P05, 62M20; Secondary 60H10.

Key words and phrases. Spot price, convenience yield, energy markets.

1

2 REN´

E CARMONA AND MICHAEL LUDKOVSKI

of spot convenience yield models has been recently proposed by Casassus and Collin-

Dufresne in [CCD03], However, they only consider the deterministic case. We believe

that a stochastic version is much more natural from a theoretical standpoint, especially

with a ﬁltering application in view. Nevertheless, our results indicate that the standard

assumption of Ornstein-Uhlenbeck process for convenience yield may be mis-speciﬁed.

Overall, we fail to ﬁnd a fully satisfactory model that is both consistent with the spot

and the forward curve. In our opinion, this supports the view that the term-structure

paradigm is more appropriate for energy commodities.

Because a commodity can be consumed, its price is a combination of future asset

and current consumption values. However, unlike ﬁnancial derivatives, storage of energy

products is costly and sometimes practically impossible like in the case of electricity. Con-

sequently, physical ownership of the commodity carries an associated ﬂow of services. On

the one hand, the agent has the option of ﬂexibility with regards to consumption (no

risk of commodity shortage). On the other hand, the decision to postpone consumption

implies storage expenses. The net ﬂow of these services per unit of time is termed the con-

venience yield δ. From now on we assume that δis quoted on a continuously compounded

basis. Intuitively, the convenience yield corresponds to dividend yield for stocks.

δ= Beneﬁt of direct access −Cost of carry.

Commodity pricing models are obtained via various assumptions on the behavior

of δt. The implicit assumption is that Stthe spot price process of the commodity in

fact exists. This is not true for some commodities, such as electricity. Even for mature

markets like crude oil where spot prices are quoted daily, the exact meaning of the spot

is diﬃcult to pin down. Nevertheless, we will maintain the industry-standard assumption

of traded spot asset.

By a basic no-arbitrage argument it follows that the price of a forward contract

F(t, T ) which has payoﬀ STat future time T must equal

(1.1) F(t, T ) = StEQheRT

t(rs−δs)dsi.

Indeed, we can replicate the payoﬀ by either entering into the forward contract, or by

borrowing Sttoday and holding the commodity itself from now until T. As usual, Q

is an equivalent martingale measure used for risk neutral pricing. Reformulating, (1.1)

implies that the risk-neutral drift of the spot commodity must equal (rt−δt)St. From a

modelling point of view, it remains to specify the stochastic processes for St,rtand δt.

However, the diﬃculty lies in the fact that the convenience yield is unobserved. Notice

that δtis deﬁned indirectly as a ”correction” to the drift of the spot process. Thus, for

any model we specify we must address the issue of estimating or ﬁltering the convenience

yield process δtgiven model observables.

In this paper we will present several stochastic convenience yield models and discuss

the resulting ﬁlters. We implement the ﬁltering problems and compare the features of the

resulting estimations using crude oil data. Section 2 reviews the original Gibson-Schwartz

model and its limitations. In Section 3 we investigate the deterministic shift extension

and its relation to work done in ﬁxed income. Our empirical study of this model is the

ﬁrst of its kind. Section 4 is the main thrust of the paper. We analyze the extended model

with stochastic risk premia which is a new method of consistently estimating convenience

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 3

yield given the entire forward curve. The ﬁltering results from this model are summarized

and compared in Section 5. Finally, Section 6 concludes.

2. The Gibson-Schwartz Model

The basic spot model for convenience yield was introduced by Gibson and Schwartz

in 1990 [GS90]. This is a 2-factor model for which the risk neutral dynamics of the

commodity spot price Stis given by a geometric Brownian motion whose rate of growth

is corrected by a stochastic mean-reverting convenience yield δt. To be more speciﬁc, we

assume that (Ω,F,{Ft},P) is a ﬁltered probability space, and we consider a bivariate

state process comprising the spot commodity asset Stand the spot instantaneous conve-

nience yield δt. According to Gibson and Schwartz, the dynamics of the state are given

under the risk-neutral measure Qby a system of Ito stochastic diﬀerential equations of

the form:

dSt= (rt−δt)Stdt +σStdW 1

t,(2.1a)

dδt=κ(θ−δt)dt +γ dW 2

t,(2.1b)

where W1, and W2are 1-dimensional Wiener processes satisfying dhW1, W 2it=ρ dt.

We assume throughout that {Ft}tis the ﬁltration generated by this bivariate Wiener

process. Depending on market conditions, convenience yields can be either positive or

negative, and so unlike with interest rates, the choice of an Ornstein-Uhlenbeck process

for δtin (2.1b) makes sense. From now on we shall also assume that

Assumption 1.The interest rate rtis deterministic.

In practice, one observes that the volatility of the convenience yield is an order of mag-

nitude higher than the volatility of interest rates. Consequently, letting rtbe stochastic

as in Schwartz [Sch97] does not give much qualitative improvement to the model.

The implications of (A1) are crucial. Indeed, non-stochastic interest rates imply that

futures and forward prices are the same, and both equal the risk-neutral expectation of

the future spot price.

If ρ > 0, then the stochastic convenience yield induces weak mean-reversion in the

risk neutral dynamics of the spot price. Observe that according to (2.1), when Stis

increasing due to increment dW 1, thanks to positive correlation ρ,δtis also likely to

increase. In turn, this reduces the drift of the spot. Note that this is a second-order

eﬀect. At the risk of adding to the confusion, one should add that mean reversion of

the historical dynamics of the spot is commonly accepted as an empirical fact, see for

example [GS90], and values of ρ∼0.3−0.7 are viewed as reasonable. The theory of

storage developed back in the 1950s [Bre58] shows that the endogeneous economic link

is through inventory levels: when inventories are low, shortages are likely, causing high

prices, as well as valuable optionality of holding the physical asset.

It is convenient to introduce a special notation for the logarithm of the spot price

Xt= log St. Indeed, the model (2.1) is linear in the state vector Zt= [Xt, δt].

2.1. Model Estimation. Statistical estimation of a model can only be done from

observations, consequently, and it cannot be done directly from the prescriptions of a risk

neutral model as given by (2.1). We need to work backward and recover the dynamics of

the state variables under the objective or historical probability structure. This requires us

4 REN´

E CARMONA AND MICHAEL LUDKOVSKI

to make assumptions on the market prices of risk, say λfor Stand λδfor the unobserved

convenience yield (each random source must have its own market price of risk). The

simplest possible assumption is to assume that:

Assumption 2.λand λδare constant.

Under this assumption, the historical dynamics of the two factors under the is exactly

of the same form as for the risk neutral measure, and estimation will be plain. Indeed,

the model is still linear (at least for the loagrithm of the spot) and hence conditionally

Gaussian, and we can estimate all the parameters empirically using the standard Kalman

ﬁlter method. The historical dynamics of δtare of the form:

(2.2) dδt= [κ(θ−δt)−λδ]dt +γ df

W2

t

for some bivariate Brownian motion under P, say (f

W1,f

W2), having the same correlation

as (W1, W 2) under Q. So the dynamical equation is the same as before as long as we

replace θby b

θ=θ−λδ/κ. Consequently, after a straightforward discretization of the

time, our measurement and transition equations become:

Xn=Xn−1+µrn−δn−σ2

2+λ¶∆t+ξn, ξn∼ N (0, σ2

S∆t),(2.3a)

δn= e−κ∆tδn−1+ (1 −e−κ∆t)ˆ

θ+ηn, ηn∼ N ¡0,γ2

2κ(1 −e−2κ∆t)¢,(2.3b)

and they are clearly amenable to a classical Kalman ﬁlter analysis. Figure 1 shows the

result of such Kalman ﬁltering on crude oil data (see Section 5.1 for description of our

data set). We estimate the model parameters (κ, θ, γ , ρ, λ) using the prediction error

decomposition of the likelihood function coming from the ﬁlter [Har89]. As is usual in

such models, the estimates of κand λare very imprecise. Overall, we ﬁnd that κ∈

[0.1,0.4], γ ∈[0.4,0.5], θ ∈[−0.3,−0.1], ρ ∈[0.5,0.7], λ ∈[−0.1,0.1]. The spot volatility

σis estimated by the historical volatility of the time series, which is about 45% for our

data set. For interest rates we use the current 3-month LIBOR rate.

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 5

Time

$/bbl crude oil

1994 1995 1996 1997 1998 1999 2000 2001 2002

15 20 25 30 35

Time

Filtered Convenience Yield

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.5 -0.1 0.1 0.3

Figure 1. Filtered convenience yield for crude oil, 1994-2002. κ=

0.2, γ = 0.5, θ =−0.15, ρ = 0.7, λ = 0.

2.2. Lack of Consistency. Unfortunately, the basic model (2.1) is not consistent

with the forward curve. In commodity markets, on any given day t, the forward (sic) curve

deﬁnes the term structure of futures contracts. It is deﬁnes as the graph of the function

T →F(t, T ). In the particular case of the risk neutral model (2.1), the convenience yield

is (at least conditionally) a Gaussian process, and the conditional expectation giving the

value of F(t, T ) in (1.1) can be computed explicitely. A direct calculation gives:

F(t, T ) = F(t, T ;Zt) = SteRT

trsdseB(t,T )δt+A(t,T )where(2.4)

B(t, T ) = e−κT −1

κ,(2.5)

A(t, T ) = κθ +ρσsγ

κ2(1 −e−κ(T−t)−κ(T−t))+(2.6)

+γ2

κ3(2κ(T−t)−3 + 4e−κ(T−t)−e−2κ(T−t)).

Similar formulae hold in the slightly more general setting of aﬃne processes. The fact

that these exponential aﬃne models, admit analytical futures and options prices in closed

form is emphasized by Bjork and Landen in [BL01]. In any case, one sees that under Q,

the forward contract follows the dynamics given by the stochastic diﬀerential equation:

(2.7) dF (t, T ) = F(t, T )£rtdt +σ dW 1

t+γe−κT −1

κdW 2

t¤.

Also, recall that the convenience yield follows an Ornstein-Uhlenbeck (OU) process. Con-

ditional on Fs,δtis Gaussian with

δt|Fs∼ N ³(1 −e−κ(t−s))δs+ e−κ(t−s)θ, γ2

2κ(1 −e−2κ(t−s))´.

Using the values of the estimated parameters in (2.2) together with the price of a traded

futures contract, say F(t, Ti), we can solve for the convenience yield δt. The values of δt

6 REN´

E CARMONA AND MICHAEL LUDKOVSKI

extracted from the observed futures prices F(t, Ti) are said to be implied by the futures

contract with maturity Ti. Figure 2 gives the plot of the value of the convenience yield

implied by a three month futures oil contract. Comparing this plot with the results in

Figure 1, it is clear that the predictive power of the Gibson-Schwartz model is very lim-

ited in practice. Furthermore, each futures contract seems to carry its own source of risk

as evidenced by the sharp spikes in Figure 2 which result due to sudden uncorrelated

movements in the futures and the spot. As Figure 3 shows, there is a further inconsis-

tency between the implied δt’s from futures contracts of diﬀerent maturities. In the next

sections we will attempt to resolve this problem in two ways – either by considering

time-dependent parameters or by taking into account the forward curve and enlarging

the state space.

Time

Implied Convenience Yield

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.8

Figure 2 : Implied convenience yield using a

3-month futures and same parameters as in

Figure 1.

Time

Difference in implied conv. yield

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.4 -0.2 0.0 0.2 0.4

Figure 3 : Diﬀerence in implied convenience

yields between 3- and 12-month futures.

3. Miltersen Extension

Miltersen [Mil03] proposed to extend (2.1) by allowing some of the parameters to

be a funtion of time. This is a direct analogue of the extension of the Vasicek model for

interest rates done by Hull and White [HW93].

The simplest choice is to make the mean-reversion level θtime-dependent, θ=ϑ(t).

This allows to ﬁt the initial futures prices directly, in the same way that the Hull-

White model can ﬁt the initial term structure of bond prices [BM01]. The calibration

is performed by letting the spot yield be ”dragged around” its changing mean ϑ(t).

Integrating the equation for δtin (2.1) we obtain

(3.1) δt=δse−κ(t−s)+Zt

s

e−κ(t−u)ϑ(u)du +γZt

s

e−κ(t−u)dW 2

u.

If we deﬁne

(3.2) αt:= Zt

.

e−κ(t−u)ϑ(u)du,

it follows that we have the deterministic shift decomposition δt=at+αt, where atfollows

the mean-zero OU process

dat=−κatdt +γ dW 2

t.

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 7

To complete the calibration we must take ϑ(t) to match a chosen set of observed futures

prices F(0, Ti), i = 1,2, . . . , n. For this purpose deﬁne (0, t) via

(3.3) F(0, Ti) = S0eRTi

0(rs−²(0,s))ds.

In HJM-type models [MS98], (0, t) is called the initial term structure of futures conve-

nience yields. Then it can be shown that

(3.4) ϑ(t) = T(0, t)

κ+(0, t) + γ2

2κ2(1 −e−2κt)−ρσγ

κ

or alternatively αt=T(0, t) + γ2

κ2(1 −e−κt)2.

Solving for (0, t) in (3.3) we obtain

(3.5) (0, t) = rt−∂log(F(0, t))

∂t .

Of course our data consists of just {F(0, Ti)}so we must interpolate those smoothly

and then take partial derivatives to infer the implied (0, t)’s. We recommend ﬁtting the

forward curve with B-splines using 5-8 degrees of freedom. B-splines provide accurate ﬁt

to market data and also generate smooth ﬁrst and second derivatives w.r.t. T (note that

for (3.4) we also need ∂²(t,T )

∂T ). Figures 4 and 6 provide examples of such B-spline ﬁtting

with six degrees of freedom. As our input we use various 28-month crude oil forward

curves from the last few years. The parameter values are the same as in Figure 1.

Months to Maturity

$/bbl

0 5 10 15 20 25

24 26 28 30 32 34 36

Months to maturity

Conv. yield

0 5 10 15 20 25

0.15 0.25 0.35

Figure 4 : Crude oil forward curve and the

interpolated term structure of convenience

yields. Backwardation of 11/21/2000.

Months to maturity

Theta

0 5 10 15 20 25

-0.2 0.0 0.2 0.4 0.6

Figure 5 : Term structure of mean reversion

level ϑt, 11/21/2000.

8 REN´

E CARMONA AND MICHAEL LUDKOVSKI

Months to Maturity

$/bbl

0 5 10 15 20 25 30

17 18 19 20 21 22

Months to maturity

Conv. yield

0 5 10 15 20 25

-0.10 -0.05 0.0 0.05

Figure 6 : Crude oil forward curve and the

interpolated term structure of convenience

yields. Contango observed on 1/18/2002.

Months to maturity

Theta

0 5 10 15 20 25

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6

Figure 7 : Term structure of mean reversion

level ϑt, 1/18/2002.

From a more general calibration point of view, we now ﬁt exactly a set of benchmark

futures prices and instead concentrate on modelling the term structure of volatility. The

empirical work in this direction is complicated by the need to estimate volatility and

the numerical diﬃculty of estimating the rest of the parameters given time-dependent

models. Also, it is rather unnatural to ﬁt the initial term-structure, while remaining in

a spot paradigm since this leaves the model exposed to inconsistencies in time (i.e. the

need to constant re-calibration).

4. Enlarging the Observation Equation

The inconsistency alluded to in the end of Section 2 is serious. We have an explicit

formula (2.2) for the value of the futures price as a function of the spot and the conve-

nience yield, and this formula is invertible. Hence, at least in principle, we can recover

the exact value of δtfrom futures prices. Unfortunately, there are many futures traded,

and if we compare the implied δtfrom a 3-month futures it does not agree with the

implied δtimplied from a 6-month contract (cf. Figure 3). The obvious solution to this

dilemma is to assign to each contract, its own source of idiosyncratic risk due to tempo-

rary miss-pricing, bid-ask spread and liquidity concerns. Such an assumption is made by

Schwartz [Sch97]. Considering (2.2), we see that ﬁxing St, the entire forward curve is

just 1-dimensional, which is certainly not what one observes in practice. A more elaborate

idea is to assume a whole term structure of convenience yields, one for each maturity.

This leads to HJM-type constructions [MS98].

The natural solution to model consistency is to have the futures (or any other vanilla

derivatives that are liquidly traded) as part of our observation. After including the futures

Fi

s=F(s, Ti), i = 1,2, . . . n, the observable ﬁltration is now FO

t=σ{Ss, F i

s: 0 ≤s≤t}.

By enlarging the set of observable instruments we are also able to address other

limitations of the model. Recall that the behavior of the market price of risk λfor the

spot was so far ”swept under the rug”. By (A2) λis constant. Ignoring for a second

the validity of such assumption, we are still faced with the problem of estimating its

value. While the market price of risk can be in principle inferred from prices of traded

assets, this method is cumbersome and imprecise. An alternative suggestion elaborated

by Runggaldier et al. in a series of papers [BCR01, GJR02, GR01, Run03] is to

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 9

make λtalso a stochastic process. Because a ﬁlter is robust to model speciﬁcation, this

approach is valid even if λtis deterministic or just constant.

Runggaldier [Run03] suggests using another OU process for λtsince mean reversion

and the resulting stationarity is a desirable feature:

(4.1) dλt=κλ(¯

λ−λt)dt +σλdW 3

t.

It is reasonable to assume that the market price of risk carries its own Brownian motion

and that W3is correlated with W1, but not with W2the noise of the convenience yield.

Intuitively, the market risk premia are independent from storage costs of the commodity.

We also expect that there is negative correlation between the spot and the market price

of risk ρSλ <0. This is to strengthen the empirical mean-reversion in the spot [CCD03].

Working in an incomplete market we pick the minimal martingale measure Qso that

the Girsanov transformation corresponding to λtaﬀects only W1(f

W1under P):

dW 1

t=df

W1

t−λtdt.

Our extended state is now Zt= [Xt, δt, λt] which is again Markovian, and in fact

conditionally Gaussian. Thus we preserve the exponential aﬃne model which allows for

easy pricing of futures and options.

Following the standard martingale method for pricing derivatives, we assume that

the price of a futures contract is a function of the state vector

(4.2) Fi

t=Fi(t, Zt) = EQ[STi| Ft].

Then applying Itˆo’s formula and using the fact that discounted traded asset prices are

Q-martingales we must have

(4.3) dF i

t=rtFi

tdt +σSt

∂F i

t

∂S dW 1

t+γ∂F i

t

∂δ dW 2

t+σλ

∂F i

t

∂λ dW 3

t.

In our case we already have the explicit expression (2.2) which held for the Gibson-

Schwartz model. This was derived by a replication argument. Since λonly aﬀects the

distribution under P, the replication argument still goes through in the extended model.

In other words, ∂F i

t

∂λ = 0 and (4.3) simpliﬁes to

(4.4) dFt

Ft

=rtdt +µσ+ργ e−κT −1

κ¶λtdt +σdf

W1

t+γe−κT −1

κdf

W2

t+α dW F

t.

The last term α dW F

tis the idiosyncratic risk associated with Ftand used to smooth

out the data. We expect αto be an order smaller than the other volatilities in (4.4).

Summarizing, the complete ﬁltering model under the real world probability Pis given by

10 REN´

E CARMONA AND MICHAEL LUDKOVSKI

dλt=κλ(¯

λ−λt)dt +σλdW 3

t

(4.5a)

dδt=κ(ˆ

θ−δt)dt +γ d ˜

W2

t

(4.5b)

dSt= (rt−δt+σλt)Stdt +σStd˜

W1

t

(4.5c)

dF i

t=µrt+σλt+ργ e−κTi−1

κλt¶Fi

tdt +σF i

td˜

W1

t+γF i

t

e−κTi−1

κd˜

W2

t+α dW Fi

t.

(4.5d)

According to the setup, δtand λtare not observable, Stis fully observed and Fi

tis

imperfectly observed in the market.

Notice that after taking logarithms of observed prices, the entire system is still linear

and hence amenable to Kalman ﬁltering. We note that this linearity is more an artifact

of the model rather than its goal. Our choice of OU processes for λand δhas been

motivated by heuristic arguments, not by modelling convenience.

Figure 8 shows the result of applying (4.5) to crude oil data. Once again we estimate

the parameters (κ, ˆ

θ, γ, ρ, κλ,¯

λ, σλ, ρλ) using the prediction error decomposition from

the ﬁlter. We ﬁnd that the ﬁlter is relatively insensitive to the parameters of the λ

process. The range of λappears to be much smaller (on the order of 0.05) and that the

correlation between λand the spot is essentially zero. The presence of λalso reduces

the estimated correlation between spot and δtto around 0.5. The empirical correlation

between λtand δtis 0.258. The idiosyncratic futures noise αcorresponds to a small error

of about 0.05. Empirically, this is suﬃcient to remove the undesirable spikes of Figure 2.

However, the empirical results are unsatisfactory. Even with the noise term α dW F

t, the

ﬁltered convenience yield exhibits severe spikes which contradict the initial assumption

of δtfollowing an OU process. Furthermore, in Figure 8, δtdoes not exhibit any sort

of persistence, contrary to economic intuition. In short, while the theoretical model is

superior, the empirical results fall short of the simpler Gibson-Schwartz case.

κ= 0.2κλ= 0.4

γ= 0.5σλ= 0.35

ˆ

θ=−0.1¯

λ= 0.02

ρ= 0.5ρλ=−0.1

Table 1. Estimated parameter values for Model (4.5).

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 11

Time

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.2 -0.1 0.0 0.1 0.2

Convenience Yield

Market Price of Risk

Figure 8. Filtered convenience yield for crude oil using the spot and

3-month futures. Parameter values are given in Table 1.

5. Empirical Implementation

5.1. Description of Oil Data. For our data set we choose the West Texas Inter-

mediate oil contracts during the period Jan 1994 - July 2002. Crude oil is one of the most

mature energy markets with a well-developed forward curve spanning several years.

The WTI futures contract trades on NYMEX since 1993. Monthly futures contracts

are traded with maturities of 1,2,3, ...23,24,30,36,48,and 60 months. The longer matu-

rities usually trade as a set of strips (for example a January-June strip of 6 months) and

consequently exhibit a high correlation. A single futures contract is for 1,000bbl to be

delivered anytime during the delivery month at Cushing, Oklahoma. Trading terminates

on the third business day prior to the 25th calendar day of the month preceding the

delivery month. The spot price is deﬁned as the Balance-of-the-Month (BOM) contract

and has varying liquidity.

12 REN´

E CARMONA AND MICHAEL LUDKOVSKI

Time

Crude Oil $/bbl

1994 1995 1996 1997 1998 1999 2000 2001 2002

15 20 25 30 35

Spot

6-month Futures

Figure 9. Time series for crude oil spot and 6-month futures, 1994-2002.

As opposed to many other energy assets, crude oil exhibits little seasonality and

the prices tend to be somewhat stable. Consequently the value of having the physical

commodity at hand is small and the net convenience yield is dominated by storage

costs. As a result, the forward curve is usually in backwardation, i.e. T→F(t, T ) is

a decreasing function. Occasionally we have a ﬂip and the forward curve appears as a

contango. Litzenberger and Rabinowitz [LR95] report that 80%-90% of the time the oil

forward curve is in backwardation. Historically, periods of contango when futures prices

are above the spot price are highly correlated with periods of low prices.

Oil futures also exhibit the well-known Samuelson eﬀect. When the contracts are ﬁrst

started and are far away from maturity (2+ years), they are thinly traded and exhibit

low volatility. As the maturity nears both trading volume and volatility increase. As a

rule of thumb, the nearest month contract is both the most liquid and the most volatile.

The spot contract is rather distinct, since it is used for diﬀerent purposes. Speciﬁcally,

spot contracts are usually used for balancing day-to-day needs and consequently exhibit

high volatility and medium volume. Thus, the term structure of oil futures volatility

T→σ(t, T ) is usually decreasing.

5.2. Results. First, we compare the consistency of the three-factor model (4.5) with

respect to the forward curve. We ﬁnd that our estimate of convenience yield is stable when

using futures contracts with diﬀerent maturities as inputs. Figure 10 shows the estimated

δtusing three diﬀerent sets of two futures contracts as inputs. All three estimates are

quite close to each other. Thus our model succeeds in removing the inconsistency of

Figure 3.

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 13

Time

Conv Yield

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.10 -0.05 0.00 0.05 0.10 0.15 0.20

3 & 6 months

4 & 9 months

2 & 12 months

Figure 10. Estimates of δtusing three diﬀerent pairs of futures contracts.

Secondly, we test the notion that the convenience yield is the ﬂow of services accruing

to the holder of the physical asset but not to the holder of a futures contract. This

interpretation implies that when the convenience yield is positive δt>0, the forward

curve should be in backwardation, and conversely when the convenience yield is negative

we should be in contango. Thus, if our model ﬁts well, the estimated δtcan be used

as an indicator to predict the switches from contango and backwardation. Such market

transitions are very important for actual trading. In this vein we compare the performance

of our δt-indicator versus two other indicators proposed recently by Borovkova [Bor03b,

Bor03a].

Borovkova proposes to use the moving-average inter-month spread indicator:

(5.1) I1(t) = 1

M

M

X

k=0

n

X

i=1

γi(Fi+1(t−k∆t)−Fi(t−k∆t))

and the PCA indicator:

(5.2) I2(t) = 1

M

M

X

k=0 X

i

wi¯

Fi(t−k∆t).

Above γis the discount factor used in weighing diﬀerent inter-month spreads. The implicit

assumption is that the closer contracts are more important and so the weights can be

written as wk=γk. Similarly, wiare the factor loads corresponding to the ﬁrst factor

from PCA performed on a de-meaned forward curve. This factor generally corresponds

to ”slope”. In [Bor03b]I2(t) is anticipative because the PCA is performed on the entire

data sample.

14 REN´

E CARMONA AND MICHAEL LUDKOVSKI

Figure 5.2 compares the performance of these two indicators, as well as the nega-

tive ﬁltered convenience yield −δt, on the oil data from 1994-2002. The areas of con-

tango/backwardation were identiﬁed a priori by looking directly at the forward curves.

As can be seen, all three indicators perform well. The times when the indicators are

near zero generally correspond to periods of market transition or uncertainty. Of the

three the convenience yield one usually gives the clearest signal, but also gave a false

backwardation signal in early 1995.

Time

Indicator

1994 1995 1996 1997 1998 1999 2000 2001 2002

-0.2 0.0 0.2 0.4 0.6

Contango

Flat

Backwardation

Unstable

Contango

Backwardation

Contango

Backwdtn

Figure 11. Three diﬀerent indicators for forward curve state: I1,

I2(gray) and IGS δt(dotted line) using a 3-month futures from Gibson-

Schwartz model. Parameter values are the same as in Figure 1.

We ﬁnd that the forward curve indicator from the extended stochastic-λmodel is not

a good predictor for market transitions. From Figures 8 and 10 we can see the estimated

convenience yield is too close to zero to give a clear signal on the state of the forward

curve.

6. Conclusion

In this paper we investigated ﬁltering of convenience yield models. We discussed the

inadequacies of the classical Gibson-Schwartz model and analyzed two new extensions.

The time-inhomogeneous extension ﬁrst proposed by Miltersen seems to work well but

requires further investigation into the forward curve volatility structure to fully judge its

consistency. This is fraught with complications because of discrepancies between realized

and implied volatilities. The stochastic market price of risk extension was also studied.

This approach provides a ”clean” solution while maintaining the exponential aﬃne fea-

ture of the model crucial for fast implementation. Furthermore, the model is consistent

with the observed forward curve. However, the ﬁltered convenience yield displays ex-

treme spikeness which contradicts the assumption of a driving OU process. As well, the

model is not as good as the basic one when it comes to predicting forward curve state

SPOT CONVENIENCE YIELD MODELS FOR THE ENERGY MARKETS 15

transitions. This fact is unsettling because it goes against the economic underpinnings of

the convenience yield deﬁnition.

Our work provided some clues for future research and exposed the weaknesses of

various spot models. Overall, we believe that this study shows the need for more sophis-

ticated term-structure models in order to explain both the spot and the forward curve.

In turn, this requires a more careful analysis of the term structure of futures volatility.

Further work also needs to be carried on applying the models to natural gas markets

where seasonality is strong. Decoupling of seasonality and unobserved state variables

remains an open theoretical problem.

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16 REN´

E CARMONA AND MICHAEL LUDKOVSKI

Bendheim Center for Finance, Department of Operations Research and Financial Engi-

neering, Princeton University, Princeton, NJ 08544

E-mail address:rcarmona@princeton.edu

Bendheim Center for Finance, Department of Operations Research and Financial Engi-

neering, Princeton University, Princeton, NJ 08544

E-mail address:mludkovski@princeton.edu