Spot convenience yield models for the energy markets

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.

Full text

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