Energy Economics 101 (2021) 105418
Contents lists available at ScienceDirect
Energy Economics
journal homepage: www.elsevier.com/locate/eneeco
Three-factor commodity forward curve model and its joint P and Q dynamics
Sergiy Ladokhin, Svetlana Borovkova ∗
Vrije Universiteit Amsterdam, Netherlands
ARTICLE
INFO
JEL classification:
Q41
G13
G17
C13
C30
C51
Keywords:
Commodity forward curve
Derivatives pricing
Oil futures
Joint dynamics model
Kalman filter
Brent oil futures
ABSTRACT
In this paper, we propose a new framework for modeling commodity forward curves. The proposed model
describes the dynamics of fundamental driving factors simultaneously under physical (𝑃 ) and risk-neutral (𝑄)
probability measures.
Our model is an extension of the forward curve model by Borovkova and Geman (2007), into several
directions. It is a three-factor model, incorporating the synthetic spot price, based on liquidly traded futures,
stochastic level of mean reversion and an analog of the stochastic convenience yield.
We develop an innovative calibration mechanism based on the Kalman filtering technique and apply it to
a large set of Brent oil futures. Additionally, we investigate properties of the time-dependent market price of
risk in oil markets. We apply the proposed modeling framework to derivatives pricing, risk management and
counterparty credit risk. Finally, we outline a way of adjusting the proposed model to account for negative
oil futures prices observed recently due to coronavirus pandemic.
1. Introduction
1.1. Motivation
Commodities is a popular and continuously growing asset class, interesting not only for commodity producers and consumers, but also for
institutional investors. Commodity derivatives markets exhibit a multibillion yearly trading volumes. Trading or investing in commodities is,
however, a risky business: due to new technological developments in
commodities production and rapidly changing geopolitical landscape,
commodity prices show extreme moves, high volatility and dynamic
correlations with other asset classes. A quest by academics as well as
practitioners for realistic commodity price models is far from over.
The following recent example shows the potential impact of model
risk on financial institutions involved in commodity markets. On
September 10, 2018, a major default had happened on NASDAQ Central
Counterparty (CCP) Clearing (Stafford, 2018; Nasdaq-Clearing, 2018).
The default of energy trader Einar Aas results in exhausting of multiple
capital buffers in the CCP default waterfall, namely variation margin,
initial margin, default fund contribution, dedicated NASDAQ resources,
as well as default fund contribution of other members. In a way,
this resulted in a near default situation for NASDAQ CCP, one of the
major clearing houses in the world. The losses were caused by more
than expected loss in the spread trade in European electricity futures.
Although it is hard to pinpoint exact reasons for the default of Einar
Aas’s firm, it is clear that problems in modeling of energy futures were
partly to blame. This shows how important commodity price models
(and especially futures price models) are for internal risk management
of financial institutions as well as for the stability of financial system
as a whole. It is evident that there is a need for realistic and advanced
models for the forward curve dynamics to be used as a core of the
market-risk and counterparty-credit risk systems used across financial
institutions involved in energy derivatives trading.
1.2. Commodity forward curves
Commodity forward curves – collections of futures prices for a
range of maturities – are fundamental objects that are at the center of
commodity trading and risk management. Futures prices are the result
of liquid trading of a large number of market participants and provide
an excellent mechanism of price discovery. This is particularly true
for crude oil futures — the most liquid futures contracts in the world.
Commodity producers and consumers are exposed to the movements
in futures prices more than to movements in the spot price. Furthermore, futures prices reveal the parameters of the risk-neutral measure,
necessary for pricing commodity derivatives.
Commodity futures prices can demonstrate complex patterns (see
Fig. 1 for examples of different shapes of oil forward curves). Crude
∗ Corresponding author.
E-mail addresses: sladokhin@gmail.com (S. Ladokhin), s.a.borovkova@vu.nl (S. Borovkova).
https://doi.org/10.1016/j.eneco.2021.105418
Received 5 February 2021; Received in revised form 17 June 2021; Accepted 25 June 2021
Available online 3 July 2021
0140-9883/© 2021 The Authors. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
oil forward curves can be in the so-called backwardation, when futures
that expire soon are more expensive than those expiring later —
something not often observed in other markets. The opposite situation
is called contango. For seasonal commodities such as gas, electricity or
agricultural commodities, forward curves can exhibit maturity-related
seasonal patterns. So modeling commodity forward curves requires
different tools and techniques than e.g., simple no-arbitrage (buy-andhold) arguments, commonly used in futures pricing for investment
assets.
Until recently, everyone assumed that commodity futures (and spot)
prices can only be positive. A well-known exception to this are electricity prices, which regularly become negative (if only for a short period
at a time). This happens due to market imbalances and oversupply
of electricity (for example, due to high quantity of wind-generated
electricity). This, in combination with inability to store electricity
efficiently, can lead to producers being prepared to pay for someone
to take over the excess supply of electricity in order not to compromise
the network.
On April 20, 2020, for the first time ever a similar situation has
occurred in oil futures market, when the first nearby WTI futures
contract exhibited negative prices never seen before and closed at −35
USD per barrel (see e.g., Borovkova, 2020 for some explanations of
this phenomenon and its impact on quantitative modeling). This event
highlights the need for specific solutions tailored to complexity of oil
(and other commodity) markets, and in this case, variants of models
that would allow for negative futures prices (even though these might
be short-lived). So in this paper we also outline a possible way of
adjusting our proposed model to deal with this phenomenon.
(2013), who showed that models that allow for slowly varying mean fit
market prices better than the standard mean-reversion. In this paper,
we will further develop this approach.
Commodity markets are influenced by multiple economic forces
that have different impacts on the spot price and forward curves. This
results in a complex stochastic behavior that cannot be fully described
by one-factor models. A popular stochastic two-factor model is that
of Schwartz–Smith (Schwartz and Smith, 2000). This model assumes
the first factor to be a zero-mean Ornstein–Uhlenbeck process and it
represents short term fluctuations of price. The second, long-term factor
is modeled by the arithmetic Brownian Motion. The two factors are
assumed to be correlated. The Schwartz–Smith model provides a closedform expression for futures prices and suggests an effective calibration
method. However, often two factors are still not enough to create a
flexible model that fits market quotes well, and an additional factor(s)
is needed. Here we will add one more factor to a two-factor model and
will simultaneously describe long-term behavior as well as medium and
short term price fluctuations.
Commodity price and forward curve models are often an important input to many downstream models, such as valuation models for
energy projects, real option models and such. In these models, both
real world and risk-neutral dynamics of the commodity prices are
important. The approach toward this application of commodity price
models was developed by Hahn et al. (2018), who have shown that the
risk premia, which relates the real world and risk neutral dynamics of
prices, can be estimated by filtering techniques — this is the approach
we also explore in this paper. Their work originates from their earlier
paper (Warren et al., 2014), where Kalman filtering techniques were
applied to estimate long-term level of oil price, again, for energy project
valuation and planning applications. Theirs is either one- or two-factor
model in the spirit of Schwartz and Smith (2000), while ours is a threefactor model, but the filtering techniques we use are the similar. They
have also shown that the risk premia estimates can significantly depend
on the historical period and, moreover, project valuation methods can
be quite sensitive to these estimates.
A major drawback of many commodity forward curve models is
their reliance on the spot price. In practice, often the spot price is not
directly observable; moreover,it is usually determined in a relatively
illiquid OTC market. Borovkova and Geman (2007) introduced an
alternative approach to overcome the issue of unobservable spot price.
As the main driving factor, they use a synthetic spot price which is
a geometric average of observable futures prices (this has an analogy
with some futures-based oil price benchmarks). By construction, this
synthetic spot price is non-seasonal, not prone to jumps and exhibits
lower volatility than the actual spot price (if such is observed at all).
The Borovkova–Geman model also incorporates the convenience yield
and deterministic seasonal premium. However, the model describes the
dynamics of the forward curves only under the real-world probability
measure. This makes it useful for risk management applications, but
not for derivatives pricing. An extension of this model, presented
here, overcomes calibration difficulties under both real world and risk
neutral probability measures.
To conclude this discussion on commodity price models, we would
like to mention that most of the academic and practitioners’ models
(especially those for oil price) are fundamentally lognormal models,
i.e., where the log-price is driven by the Brownian motion. This is also
the standard assumption of the Black–Scholes framework of pricing
options. In this paper, we also assume such a framework. For those
commodities where price can become negative (such as electricity), a
normal model is often used, where the price itself (and not the logprice) is driven by the Brownian motion. However, as in equity, FX
or interest rate markets, commodity prices can exhibit high kurtosis,
i.e., heavy tails, but also stochastic volatility and jumps. An extreme
case of this is electricity price, which can exhibit jumps of hundreds of
percent in a short period of time — this happens due to inability to store
electricity (such jumps are rarely observed in e.g., oil or natural gas
1.3. Commodity price and forward curve models
Generally, there are three distinct approaches to model commodity
futures prices. The first approach starts with a stochastic dynamics of
the spot price and, from that, a formula for futures prices is derived
as the expectation of the future spot under the risk neutral probability
measure. Such models are either calibrated to observed history of spot
prices (under the physical probability measure) or fitted to the observed
forward curves (under the risk-neutral probability measure).
The second approach assumes certain dynamics directly of the
forward prices (see e.g., Amin, Ng and Pirrong (1995)). The forward
curve is considered a given object that is formed as a result of trading.
This approach is useful to price derivatives on futures contracts (instead
on spot commodity). Finally, the third approach is to assume a functional relationship between forward and spot prices. This functional
form usually depends on the spot price as well as on the so-called
convenience yield - a rate of return of owning the commodity rather
than a futures contract on it (such approach sometimes also referred
to as the cost-of-carry).
It is often assumed that commodity prices exhibit a mean-reverting
behavior. This behavior is observed not only in a short-time deviations
of the spot price, but also in a long-term property of the prices to
revert to the stable means over years or even decades. Such behavior of
commodity markets significantly differentiates them from equity markets. For a broad discussion on mean reverting properties of commodity
prices, see an excellent book by Geman (2005).
The standard mean-reversion process assumes that the stochastic
variable (e.g., a commodity spot price) reverts to the constant mean.
To date however, little attention was given to the stationarity of this
mean. Empirical evidence suggests that the mean level of commodity
prices is not constant but stochastic with a (relatively) low volatility.
This has deeper economic justification, e.g., the relationship of commodities to business cycles, extraction and production technologies and
other varying economic fundamentals. So forcing the mean level to
be constant (while it is not) leads to instability and poor performance
of models. The first step toward stochastically varying mean in mean
reversion models for commodities has been set by Borovkova et al.
2
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Fig. 1. Oil forward curves can be in a different shapes, for example contango or backwardation, and all the different regimes in between.
speaking, it is driven by those economic factors that change the
near-end of the forward curve, such as changes in the inventory
levels.
markets). Models that incorporate these features into account have also
been suggested for commodities. For example, B. Trolle and S. Schwartz
(2009) incorporate stochastic volatility into an oil price model. Works
by Benth (e.g., FE and J, 2004; Barndorff-Nielsen et al., 2013) introduce
a class of different driving processes — the so-called Levy processes into
commodity price models, and successfully manage to model heavy tails
in commodity returns. However, practitioners seem to be very attached
to Black–Scholes framework of lognormal prices, due to its analytical
tractability and ease of calibration and use. So while we acknowledge
that there are sophisticated (e.g., Levy-driven) models that might be
better at modeling leptokurtic features of returns, in this paper we stick
to the Brownian motions as the main stochastic drivers of the price
process, to reflect practitioners’ preferences.
We develop an innovative way to calibrate our model under two
probability measures, by a variant of Kalman filtering technique. We
fit the model parameters to the extensive history of Brent oil futures
curves. The model shows good fit to the market prices across all
maturities, as well as consistent factors’ dynamics.
A side research question discussed in this paper is a time-dependent
market price of risk. The market price of risk links the dynamics of the
asset under physical and risk-neutral probability measures. Typically it
is assumed to be constant. Inspired by the work of Ahmad and Wilmott
(2013), we propose a way to estimate time-dependent market price of
risk. We apply it to the historical market price of risk for Brent oil
futures.
The proposed model has multiple practical applications, such as
derivatives pricing, market risk management, as well as counterparty
credit risk and credit valuation adjustments. These applications are
extremely important for many financial institutions, so we discuss them
in detail at the end of the paper.
The remainder of the paper is organized as follows. We start with
the short overview of popular forward curve models in Section 2. In
Section 3, we introduce the synthetic spot price and other building
blocks of our model. In addition, this section also contains derivations
for the commodity futures prices and their dynamics. This section is
the key section of the article. In Section 5.1, we describe the statespace representation of the model which allows for the application
of Kalman filter. In Section 5, the calibration set-up and results are
described. Model applications are summarized in Section 6. Finally, we
state possible future extensions of the model in Section 7.
Note that we use the terms ‘futures’ and ‘forwards’ interchangeably,
to mean financial contracts that allow to buy or sell a certain amount
of a spot commodity in the future. Such interchangeability is justified
since we focus on the commodity price dynamics and largely ignore
interest rates, transaction costs and counterparty risk. As a result, the
terms ’futures curve’ and ’forward curve’ are also used interchangeably.
1.4. Model summary and goals of the paper
The main aim of this paper is two-fold: to expand the Borovkova–
Geman commodity forward curve model to include stochastically varying price level and to develop its calibration to the risk-neutral probability measure, which allows for derivatives pricing. This results in the
so-called joint-measure model. Such a model simultaneously describes
the dynamics under real-world and risk-neutral probability measures.
As a side research question, we investigate the dynamics of market
price of risk under assumptions of our model. Our extended model is
a useful practical tool for pricing and risk management of commodity
derivatives.
The general modeling framework presented here has three fundamental factors:
• Synthetic spot price. This factor was originally introduced in
the work of Borovkova and Geman; it corresponds to the level
of the commodity forward curve. This factor is based on the
actual quotes of the (liquid) futures, so it better reflects overall
price level compared to the (often non-transparent) spot price.
Moreover, the synthetic-spot factor is not seasonal and does not
exhibit jumps.
• Long-run stochastic mean. This is the major innovative element
of the model: we assume that the synthetic spot price is meanreverting, but in contrast to the previously proposed models, it
reverts to the slowly varying (and stochastic) mean. This echoes
concerns of practitioners, who believe that over time commodity
prices return to some ‘‘psychological’’ expected mean level, but
that this level is not constant.
• Short-term deviation factor, or convenience yield. This factor
corresponds to the well-known concept of convenience yield. This
factor is modeled by zero-mean mean-reverting process. Loosely
2. Overview of related forward curve models
The topic of energy futures modeling is not new in academic literature. The early models were based on the considerations for equity or
interest rate derivatives. Such approaches did not result in successful
models and the need for specialized energy models was recognized.
3
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
2.1. Mean-reverting model
2.4. Spot-forward relationship
It is often stated that energy (and other commodity) prices exhibit
mean-reverting pattern. The early models incorporate mean-reversion
in the commodity spot price. They were fundamentally one-factor models, so the spot price was the single stochastic factor driving evolution of
forward curves, such as in the Schwartz one-factor model, see Schwartz
(1997) for details. The model describes the dynamics of the underlying
spot price as:
The spot price 𝑆(𝑡) and the forward (or futures) prices 𝐹 (𝑡, 𝑇 ) are
linked by the well-known cost-of-carry relationship:
𝑑𝑆(𝑡) = 𝜅(𝜇 − ln 𝑆(𝑡))𝑆(𝑡)𝑑𝑡 + 𝜎𝑆(𝑡)𝑑𝑊 (𝑡),
𝐹 (𝑡, 𝑇 ) = 𝑆(𝑡)𝑒(𝑟−𝛾)(𝑇 −𝑡) ,
where we assume that the spot price 𝑆(𝑡) is observable, 𝑟 is the constant
interest rate, 𝛾 is the so-called convenience yield, which is often defined
by the above relationship. The convenience yield is the rate of return, or
a ‘‘premium’’ of owning the physical commodity rather than a futures
contract. The above relationship stems from no-arbitrage arguments for
‘‘storable’’ commodities. We refer to Geman (2005) for details.
Although frequently used by practitioners, the spot-forward relationship is more a transformation from the observed forward curve
𝐹 (𝑡, 𝑇 ) (and the interest rate 𝑟) to the unobserved convenience yield
𝛾(𝑡). To use the cost of carry relationship, we need a model for the
convenience yield 𝛾(𝑡). The convenience yield that follows from the
forward-spot relationship is not constant for different maturities, moreover it dynamically changes over time.
Furthermore, in this approach, the forward prices are directly based
on the spot price 𝑆(𝑡), which can be unreliable and often even unobserved. In many energy markets, the spot price 𝑆(𝑡) is determined by
a small group of physical commodity traders in a nontransparent way.
In many cases, the spot price is quoted, instead of being determined as
a result of actual trading. This is different from futures market, where
settlement prices of futures contracts 𝐹 (𝑡, 𝑇 ) are determined based on
actual trading.
(1)
where 𝜇 is a constant mean reversion level, 𝜅 speed of mean-reversion,
𝜎 is volatility of the spot price and 𝑊 is the standard Brownian motion.
This model is very similar to the famous Vasicek interest rate model,
with log-spot replacing the short-rate dynamics.
Recall that, in mathematical terms, the commodity’s futures price is
equal to the risk-neutral expectation of the commodity spot price 𝑆(𝑇 ):
𝐹 (𝑡, 𝑇 ) = 𝐸 [𝑆𝑇 |𝑡 ],
(2)
where 𝑡 is filtration on date 𝑡.
By explicitly computing 𝐸 [𝑆𝑇 |𝑡 ], the mean reverting model is
analytically tractable and provides explicit formulas for forward curves
as well as for European options. However, this model is unrealistic for
several reasons. It has only one stochastic factor which is not sufficient
to explain all variability in the forward curve’s shape. Although the
volatility of futures prices is, as expected, decreasing function of time
to maturity, it goes to zero for long maturities, which is empirically
not observed. Finally, the model relies on constant mean 𝜇. Such mean
is hard to calibrate, as it can change significantly over time. In our
model, the mean reversion level will be stochastic (but slowly varying),
introducing a lot more flexibility in traditional mean reverting models.
2.5. Borovkova-Geman model
One way to overcome problems with an unreliable spot price (or its
absence) was introduced by Borovkova and Geman in Borovkova and
Geman (2006). Their model is build around an observable factor 𝐹̄ (𝑡):
the geometric average of all (liquidly traded) futures prices. This factor,
the ‘‘average’’ futures price, is non-seasonal, directly observable, and is
explicitly computed from liquid futures quotes 𝐹 (𝑡, 𝑇 ). In their model,
the forward prices are given by the following formula:
2.2. Schwartz-Smith two-factor model
Energy prices are impacted by multiple economic events, such as
changes in supply–demand, inventory levels, news, political events,
technological developments, and so on. This suggests a need for multiple factors to describe the forward curve dynamic. A popular choice is
the two-factor forward curve model by Schwartz and Smith, introduced
in Schwartz and Smith (2000). The model describes the spot price as a
sum of two (not directly observable) factors:
ln 𝑆𝑡 = 𝜒𝑡 + 𝜉𝑡 .
𝐹 (𝑡, 𝑇 ) = 𝐹̄ (𝑡)𝑒(𝑠(𝑇 )−𝛾(𝑡,𝑇 −𝑡))(𝑇 −𝑡) ,
𝑑𝜉𝑡 = 𝜇𝑑𝑡 + 𝜎𝜉 𝑑𝑊2 ,
(3)
(4)
where 𝑊1 and 𝑊2 are two correlated Brownian motions. The forward
curve is obtained by introducing market price of risk and finding
the expectation of the spot price under the risk neutral probability
measure. Here we will use a similar technique to go from physical to
risk-neutral probability measure. We extend the ideas behind Schwartz–
Smith model by introducing an observable factor and increasing the
total number of model factors to three.
2.3. Gibson-Schwartz stochastic convenience yield model
3. Model definition and specification
Gibson–Schwartz model (Gibson and Schwartz, 1990) is yet another
example of a two factor model. The model describes dynamics of
underlying spot price 𝑆𝑡 and the stochastic convenience yield 𝛿𝑡 . The
dynamics of the state variables is given by the following processes:
𝑑𝑆𝑡 = (𝑟 − 𝛿𝑡 )𝑆𝑡 𝑑𝑡 + 𝜎𝑆 𝑆𝑡 𝑑𝑊𝑆 ,
𝑑𝛿𝑡 = [𝜅(𝛼 − 𝛿𝑡 ) − 𝜆]𝑑𝑡 + 𝜎𝛿 𝑑𝑊𝛿 ,
(7)
where 𝑠(𝑇 ) is a seasonal premium and 𝛾(𝑡, 𝑇 − 𝑡) is a mean-reverting
dynamic convenience yield, which now also depends on the time to
maturity 𝑇 − 𝑡 and hence, exhibits a term structure.
Such model can be applied for both non-seasonal and seasonal
commodities such as natural gas and electricity. As the model describes
the forward curve dynamics under the physical probability measure, it
is useful for simulations (under that measure) and for risk-management
purposes, but it cannot be directly used to price derivatives. One of the
goals of our work is to extend the concept of synthetic spot factor 𝐹̄ (𝑡)
to be used as a main building block for forward curve dynamics not
only under physical, but also under risk neutral probability measure.
These are just a few models in a long list of forward curve models
suggested in the literature. Other models worth mentioning are: TrolleSchwartz model with stochastic volatility (B. Trolle and S. Schwartz,
2009), Geman–Roncoroni model (Geman and Roncoroni, 2006), Geman
model (Geman, 2000).
The dynamics of the factors is described by the following stochastic
differential equations:
𝑑𝜒𝑡 = −𝜅𝜒𝑡 𝑑𝑡 + 𝜎𝜒 𝑑𝑊1 ,
(6)
3.1. Synthetic spot factor
Unlike Borovkova and Geman, we do not worry whether the spot
price of a commodity is or is not observed — it can be either of the
two. However, just like in their model, we start by introducing the
same observable factor which reflects the actual liquid trading; even
for seasonal commodities, this factor is non-seasonal. This is what we
call the ‘‘level’’ factor (it is, in fact, the level of the forward curve). Such
(5)
where 𝑊𝑆 and 𝑊𝛿 are two correlated Brownian motions. The Gibson–
Schwartz model, as Schwartz–Smith model, results in a closed form
formula for futures prices.
4
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
factor should be not influenced by tilts, or contango–backwardation
transitions of the forward curve.
Denote the futures price of a commodity on date 𝑡 as 𝐹 (𝑡, 𝑇 ), where
𝑇 is the expiry date. Time to expiry of the futures contract is given
by 𝑇 − 𝑡. Denote the commodity spot price as 𝑆𝑡 (if it is observed).
As in Borovkova and Geman (2006), we introduce synthetic spot factor
𝐹̄ (𝑡), which is defined as:
√
√ 𝑛
√∏
𝑛
𝐹̄ (𝑡) = √
𝐹 (𝑡, 𝑇𝑖 ),
(8)
If the spot price is not available (or is unreliable), it can be proxied
by the sum of the synthetic spot 𝑋𝑡 and the correction (convenience
yield) 𝑦𝑡 :
ln 𝑆𝑡 = 𝑋𝑡 + 𝑦𝑡 .
(9)
Many commodities also exhibit seasonal effects. In that case, the
above equation can be extended by adding a deterministic seasonal
component, which can be easily estimated from historical price series.
Since this paper focuses on non-seasonal commodities such as crude oil,
we will leave modeling seasonality effects to a subsequent paper.
We can rewrite the model in the following, perhaps more intuitive
way. Assume that the long-term mean of the (log) oil price (i.e., of
synthetic log spot 𝑋𝑡 ) is 𝐿𝑡 - this is the mean reversion mean of 𝑋𝑡 ,
but which is not constant (instead, we can model it as e.g., a stochastic
process with a very low volatility). Define a new factor
𝑖=1
where 𝑛 = 𝑛𝑡 is the number of all (liquid) available futures traded
on day 𝑡. Synthetic spot price is a non-seasonal characteristic of the
forward curve, corresponding to its level. Due to averaging, this factor
evolves quite smoothly over time — it typically does not exhibit jumps
and has lower volatility than e.g., spot price or first nearby futures
price. Since synthetic spot is a function of (liquid) futures prices, it is
based on active trading (in contrast to the spot price 𝑆𝑡 , which is quoted
by a small pool of commodity producers). In a way, replacing 𝑆𝑡 by 𝐹̄ (𝑡)
as the main building block of the model prevents the same problems as
in interest rate markets with LIBOR-style rates.1
We will work with the logarithmic representation of the synthetic
spot factor:
𝜓𝑡 ≡ 𝑋𝑡 − 𝐿𝑡 .
This factor corresponds to the deviations of the synthetic log-spot 𝑋𝑡
from its long-term mean 𝐿𝑡 . Such deviations are usually triggered by
longer term changes in supply–demand balance or political events.
Reversing the above equation, we get that
𝑋𝑡 = 𝜓𝑡 + 𝐿𝑡 .
𝑋𝑡 ≡ ln 𝐹̄ (𝑡).
In other words, we can say that the ‘‘level’’ of the forward/futures
market is measured by the geometric average of the futures prices, but
it (or, rather, its logarithm 𝑋𝑡 ) is modeled by the sum of 𝜓𝑡 and the
long-term log-price 𝐿𝑡 .
With this new 𝜓-factor at hand, the spot representation (9) can be
re-written into the following three factor model:
A comparison between the synthetic spot factor and the actual spot (if
such is available) is given in Fig. 2. As can be seen from that figure,
the synthetic spot factor corresponds to the level of the curve (given
by the geometric average of futures prices). Note that this average
will, in general, deviate from the actual spot price 𝑆𝑡 . This deviation is
determined by the slope of the forward curve: if the curve is in contango
(as shown in Fig. 2), the synthetic spot is above the actual spot, and
in backwardation this is reversed. This motivates us to introduce the
second fundamental factor, which will be similar to the convenience
yield (as it is that factor that determines whether the curve is in
contango or backwardation) and which we denote by 𝑦𝑡 . This factor
will reflect deviations between the actual spot price 𝑆𝑡 and its synthetic
version 𝐹̄ (𝑡).
ln 𝑆𝑡 = 𝜓𝑡 + 𝑦𝑡 + 𝐿𝑡 ,
(10)
(since 𝜓𝑡 = 𝑋𝑡 − 𝐿𝑡 ). Such spot representation is similar to the one in
Schwartz–Smith model (Schwartz and Smith, 2000), but has a different
interpretation and dynamics of factors.
To summarize: 𝐿𝑡 is the long-term factor, influenced by long term
market developments; 𝜓𝑡 is the ‘‘medium-term’’ factor, it changes faster
than 𝐿𝑡 and it is influenced by medium-term events, and 𝑦𝑡 is the ‘‘shortterm’’ factor, influenced by short term changes in supply and demand
or fluctuations in inventories.
3.2. Three factor model
3.3. Dynamics of factors
The three fundamental factors in our model are:
Recall that both factors 𝜓𝑡 and 𝑦𝑡 are essentially deviations of something from something else, so it is reasonable to assume that both follow
the ordinary mean-zero Ornstein–Uhlenbeck process. The long-term
mean 𝐿𝑡 is fundamentally a macroeconomic variable, indicating longterm stance of the oil price. So in agreement with academic literature,
the dynamic of 𝐿𝑡 is often assumed to be a random walk. This is what
we will assume here also.
Let  be the physical probability measure. We postulate the following factor dynamics under the physical probability measure :
• Synthetic log-spot 𝑋𝑡 , introduced above. This factor corresponds
to the level of the forward curve, it is non-seasonal and it is based
on actual trading in liquid futures.
• The difference between the actual (if available) and the synthetic
spot prices, which we denote 𝑦𝑡 . The need for this factor comes
from the fact that synthetic spot 𝐹̄ (𝑡) is not equal to the actual spot
price 𝑆𝑡 . This factor is related to the ‘‘traditional’’ convenience
yield: it changes sign between backwardation and contango markets (so here we will call it ‘‘convenience yield’’). In economic
terms, this factor is influenced by changes in supply–demand
balance, inventory level or news.
• Long-term average commodity price 𝐿𝑡 . This fundamental factor
captures the long-term price development. This factor is influenced by long term supply–demand expectations as well as by the
speed of technological advances in energy production. Arguably,
this factor is slowly varying, with the volatility much smaller
compared to the volatility of synthetic spot factor 𝑋𝑡 . This factor
is the key innovative element of the proposed model.
𝑑𝜓𝑡 = −𝛼𝜓𝑡 𝑑𝑡 + 𝜎𝑑𝑊1 ,
𝑑𝑦𝑡 = −𝑎𝑦𝑡 𝑑𝑡 + 𝜂𝑑𝑊2 ,
(11)
𝑑𝐿𝑡 = 𝜎𝐿 𝑑𝑊3 ,
where 𝑊𝑖 are (possibly correlated) Brownian motions. In the above
dynamics specification, the parameters 𝛼 and 𝑎 are the mean reversion
speeds of respectively medium- and short-term stochastic factors 𝜓 and
𝑦, and 𝜎 and 𝜂 are their volatilities (and 𝜎𝐿 is the volatility of the
long-term price level 𝐿). We assume these Brownian motions to have
the following correlation structure: 𝑑𝑊1 𝑑𝑊2 = 𝜌, 𝑑𝑊1 𝑑𝑊3 = 0 and
𝑑𝑊2 𝑑𝑊3 = 0. In principle, we could let all correlations to be nonzero, but we restrict two of them to be zero for simplicity and due to
some economic intuition: the moves at the short end of the forward
1
In interest rate markets, many derivatives were quoted based on unreliable and nontransparent LIBOR rates that did not reflect actual market trading
— something that impending LIBOR reform aims to rectify.
5
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Fig. 2. A schematic representation of the synthetic spot compared to the actual spot level.
curve (i.e., of short maturities’ futures prices) are typically in sync
with the synthetic spot price. The long-term price level is impacted by
quite different market forces than the short- or mid-end of the forward
curve, hence the corresponding correlations are assumed to be zero. In
addition, we define 𝜎𝐿 = 𝑞𝜎, with 0 < 𝑞 < 𝑞max < 1. The idea behind
such representation is to ensure that the volatility of long-term price is
smaller than the volatility of the synthetic log-spot factor 𝑋𝑡 . Similar to
the signal processing literature, we call parameter 𝑞 the signal-to-noise
ratio.
In the above model, 𝐿𝑡 is the long-run mean of the stochastic logspot price 𝑋𝑡 but it is itself also stochastic. The idea behind this factor
is to model long-term shared view, or expectation of market participant
about the ‘‘level of crude oil prices’’. Such level incorporates political developments, economic cycles, ongoing technological advances
(e.g., shale oil, under ice exploration, tar sands) as well as global
changes in the demand for oil (e.g., electrification, wider availability
of alternative energy). Stochastic mean is a much slower changing
quantity compared to the other model factors; this property is enforced
by setting 𝑞 < 1 (and typically 𝑞 ≪ 1). The slowly varying long term
mean model for commodity products was also introduced by Borovkova
et al. in Borovkova et al. (2013).
In contrast to 𝐿𝑡 , the factor 𝑦𝑡 corresponds to the short term deviations of the stochastic spot 𝑋𝑡 from the actual log-spot price ln 𝑆𝑡 .
Movements in 𝑦𝑡 result from short term supply and demand changes,
weather impact, news and such factors. As with the traditional convenience yield, it is naturally to assume that 𝑦𝑡 is a mean-reverting process
with zero mean: short term changes in supply or demand will have an
influence on short-term (i.e., spot) commodity price, but once resolved,
it is expected that the spot price will go back to the ‘‘overall’’ level.
Finally, 𝜓𝑡 is a mean-reverting spread between the synthetic spot price
𝑋𝑡 and its long-term mean 𝐿𝑡 .
In general, oil prices are more sensitive to the supply–demand
changes and less to the interest rate or inflation. Moreover, usually
inflation is considered as a result of an increase in oil price and not
other way around (see for example LeBlanc and Chinn, 2004). That is
the reason for not including interest rate or inflation-specific factors
into the model.
clearly not a sustainable solution in the long term. Here we would like
to suggest such a solution, within the framework of our model.
First of all, in calculation of our first fundamental factor 𝐹̄ (𝑡),
we include only those futures prices that are positive. This is not a
significant restriction, since negative prices in oil markets will be shortlived and are not representative of the overall state of the oil market
(this is in contrast to negative interest rates, which can remain negative
for a very long period of time, truly depicting the state of the money
markets, as we observe at the moment of writing of this paper in
Eurozone). The definition of 𝑋(𝑡) remains the same, as the logarithm
of 𝐹̄ (𝑡).
Another modification of our model allowing it to deal with negative
prices is in Eq. (10), where we replace logarithm of spot price by the
spot price itself:
3.4. Model variant for negative futures prices
̃1 ,
𝑑𝜓𝑡 = −𝛼𝜓𝑡 𝑑𝑡 − 𝜆𝜎𝑑𝑡 + 𝜎 𝑑𝑊
𝑆𝑡 = 𝜓𝑡 + 𝑦𝑡 + 𝐿𝑡 .
(12)
The dynamics of all fundamental factors remains the same. So we are
basically going from logarithmic to arithmetic representation of the
model. In this representation, spot as well as nearby futures prices can
become negative.
Other model variants to deal with negative prices are possible. In
our subsequent paper we will further develop this and other model
variants and apply them to WTI futures. But here we proceed with
the logarithmic form of the model and its application to Brent futures
prices.
3.5. Risk neutral dynamics
To derive the formula for futures prices, we need to compute the
expectations of the spot price under the risk neutral measure. Before
that, we present the dynamics of the factors under this risk neutral
measure. Overall, our approach is similar to the one used by Schwartz
and Smith in Schwartz and Smith (2000), where the dynamics under
physical measure is adjusted by the market price of risk.
Let us introduce the equivalent risk-neutral pricing probability measure . This measure is connected with the physical probability measure  by means of the market price of risk 𝜆. The model dynamic under
the risk-neutral measure is then:
̃2 ,
𝑑𝑦𝑡 = −𝑎𝑦𝑡 𝑑𝑡 − 𝜆𝜂𝑑𝑡 + 𝜂 𝑑𝑊
When, in April 2020, WTI futures prices went negative, many financial institutions had big problems with their models and software. A
typical ‘‘duct tape’’ solution was to simply ignore the offending futures
contract (the first nearby one) and base all the calculations of the
remaining futures prices, which remained positive. However, this is
(13)
̃3 ,
𝑑𝐿𝑡 = −𝜆𝜎𝐿 𝑑𝑡 + 𝜎𝐿 𝑑𝑊
̃𝑖 are -Brownian motions with the same correlation structure
where 𝑊
as 𝑊𝑖 . By the above representation we explicitly assume that there exist
a single market price of risk parameter that is used in the dynamics of
6
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
all of the factors. We choose to subtract 𝜆 while some authors add it
in risk-neutral dynamics. Note, that conceptually this is equivalent and
the only difference is that the sign of the calibrated parameter 𝜆. The
quantities 𝜆𝜎, 𝜆𝜂, and 𝜆𝜎𝐿 are also called risk premia. Our representation
explicitly assumes that the risk premia are proportional to the volatility
of each of the factors. An alternative approach would be to assume
a different market price of risk parameter (say 𝜆1 , 𝜆2 , 𝜆3 ) for each
source of risk. However, we will not consider such approach in this
paper. Under our assumption, investors consider risk premium to be
proportional to the volatility of the risk source, and the proportionality
constant is the same of all risk sources.
For now we assume that the market price of risk 𝜆 is constant,
although later on we will relax this assumption. We will introduce a
time dependent market price of risk 𝜆𝑡 and suggest a way to estimate
it from the data. Although our estimation approach will differ from the
one proposed by Wilmott and Ahmad in Ahmad and Wilmott (2013),
we obtain a similar interpretation of results.
Note that the dynamics of 𝑦𝑡 can be rewritten in terms of uncorrelated Brownian motions:
√
̃4 ,
̃1 + 𝜂 1 − 𝜌2 𝑑𝑊
(14)
𝑑𝑦𝑡 = −𝑎𝑦𝑡 𝑑𝑡 − 𝜆𝜂𝑑𝑡 + 𝜂𝜌𝑑𝑊
where
[𝜎
]
𝜂
(1 − 𝑒−𝛼(𝑇 −𝑡) ) + (1 − 𝑒−𝑎(𝑇 −𝑡) ) + 𝜎𝐿 (𝑇 − 𝑡) +
𝛼
𝑎
𝜎𝜂𝜌
𝜎2
(1 − 𝑒−2𝛼(𝑇 −𝑡) ) +
(1 − 𝑒−(𝛼+𝑎)(𝑇 −𝑡) )+
4𝛼
(𝛼 + 𝑎)
𝜎
𝜌2 𝜂 2
𝜂 2 (1 − 𝜌2 )
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) +
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) + 𝐿 (𝑇 − 𝑡).
4𝑎
4𝑎
2
𝐴(𝑡, 𝑇 ) = −𝜆
The above formula provides the futures price at time 𝑡 (so also at
𝑡 = 0) for maturity 𝑇 . This price depends on the current value of the
three state variables 𝜓𝑡 , 𝑦𝑡 and 𝐿𝑡 as well as on the parameters of their
dynamics.
A natural requirement for a forward price is the condition that, for
a short time-to-expiry, forward price would converge to the spot price.
It is easy to see that, when 𝑇 = 𝑡, the forward price formula above will
indeed simplify to the spot price and will be 𝑆𝑇 ≡ 𝐹 (𝑇 , 𝑇 ) = 𝑒𝜓𝑇 +𝑦𝑇 +𝐿𝑇 .
We will illustrate this later on empirical data.
If the arithmetic version the model is used, the derivations in this
and following paragraphs simplify significantly, as we only need to
deal with a simple sum of Brownian Motions. So here we will proceed
with a more involved, geometric version of the model, and leave the
corresponding derivations of futures prices and their dynamics to a
future research.
̃1 , 𝑑𝑊
̃2 and 𝑑𝑊
̃4 are pair-wise uncorrelated, and 𝜌 is the
where 𝑑𝑊
correlation between Brownian motions driving short- and medium
stochastic factors. We will use this representation in further steps of
the model building.
4.2. Futures price dynamics
For many applications, such as derivatives pricing, we need the dynamics (and especially the volatility) of the futures price. Moreover, we
must impose a natural condition that the futures prices are martingales
under .
Here we derive, by Ito’s lemma, the dynamics of the futures price
𝑑𝐹 (𝑡, 𝑇 ), which depends on three stochastic variables 𝜓𝑡 , 𝑦𝑡 , 𝐿𝑡 .
4. Forward curve and its dynamics
4.1. Forward price
We can now proceed to deriving formula for forward prices. In line
with literature, the forward price is the expectation of the spot price
under the risk-neutral probability measure . The model dynamics
allows us to solve for this expectation. Processes 𝜓 and 𝑦𝑡 are described
by Ornstein–Uhlenbeck process and 𝐿𝑡 is described by arithmetic Brownian motion. Such representation allows us obtain log-spot price at time
𝑇 , given 𝐹𝑡 :
𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 = 𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) −
𝑇
√
𝜂 1 − 𝜌2 𝑒−𝑎(𝑇 −𝑡)
∫𝑡
̃1 (𝑠) + 𝜂𝜌𝑒
𝑒 𝑑𝑊
𝑎𝑠
∫𝑡
̃1 (𝑠)+
𝑒 𝑑𝑊
𝑇
̃4 (𝑠) + 𝜎𝐿
𝑒𝑎𝑠 𝑑 𝑊
∫𝑡
(16)
The resulting process has zero drift, so is a martingale. This is
in line with our expectation about the form of the forward price
dynamics. From this dynamics we can easily obtain the time- and
maturity-dependent Black’s volatility:
√
(17)
𝜎𝐵 (𝑡, 𝑇 ) = (𝜎𝑒−𝛼(𝑇 −𝑡) + 𝜂𝜌𝑒−𝑎(𝑇 −𝑡) )2 + 𝜂 2 (1 − 𝜌2 )𝑒−2𝑎(𝑇 −𝑡) + 𝜎𝐿2 .
𝑇
−𝑎(𝑇 −𝑡)
𝛼𝑠
𝑇
∫𝑡
𝑑𝐹 (𝑡, 𝑇 )
̃1 +
= (𝜎𝑒−𝛼(𝑇 −𝑡) + 𝜂𝜌𝑒−𝑎(𝑇 −𝑡) )𝑑𝑊
𝐹 (𝑡, 𝑇 )
√
̃4 + 𝜎𝐿 𝑑𝑊
̃3 .
𝜂 1 − 𝜌2 𝑒−𝑎(𝑇 −𝑡) 𝑑𝑊
𝜆𝜂
𝜆𝜎
(1 − 𝑒−𝛼(𝑇 −𝑡) ) + 𝑦𝑡 𝑒−𝑎𝑡 −
(1 − 𝑒−𝑎(𝑇 −𝑡) )
𝛼
𝑎
−𝛼(𝑇 −𝑡)
+𝐿𝑡 − 𝜆𝜎𝐿 (𝑇 − 𝑡) + 𝜎𝑒
Proposition 2. The dynamics of the forward price of commodity, whose
spot price follows the 3 factor model with dynamics (13) is given by:
̃3 (𝑠).
𝑒𝛼𝑠 𝑑 𝑊
The above solution allows us to explicitly compute the expected value
𝐸 [𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 |𝑡 ] and the variance of 𝑉 𝑎𝑟 [𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 |𝑡 ] of the
log-spot price:
This volatility can be directly used in Black-76 formula to price options
on futures (see Black, 1976). An example of the volatility term structure
is given in Fig. 3. As can be seen in the picture, the volatility decreases
with the time to maturity; for long maturities it will converge to nonzero 𝜎𝐿 . This property overcomes existing problems with many simple
mean-reverting models, where volatility decreases to zero for long
maturities.
Inspired by yield curve modeling, commodity forward curve modeling can follow two main approaches. First one assumes, as a starting
point, the dynamics of the spot (or short rate), while another one
assumes directly the dynamics of the forward prices (or instantaneous
interest rate forwards). The second approach is also referred to as
Heath–Jarrow–Morton (HJM) framework (Heath et al., 1990). In our
paper, we have chosen for the first approach, with the goal to obtain
a closed form formula for futures prices. Formula (16) is an important
link between our approach and the HJM-style framework in commodity
forward curve modeling — this is the approach taken by e.g., Amin
et al. (1995). Equipped with this formula, we can extend the modeling
considerations outlined in this paper to pricing of exotic derivatives on
both commodity spot and futures contracts.
𝐸[ln 𝑆𝑇 |𝑡 ] = 𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) + 𝑦𝑡 𝑒−𝑎(𝑇 −𝑡) + 𝐿𝑡 −
[𝜂
]
𝜎
𝜆 (1 − 𝑒−𝑎(𝑇 −𝑡) ) + 𝜎𝐿 (𝑇 − 𝑡) + (1 − 𝑒−𝛼(𝑇 −𝑡) ) ,
𝑎
𝛼
2𝜎𝜂𝜌
𝜎2
𝑉 𝑎𝑟[ln 𝑆𝑇 |𝑡 ] =
(1 − 𝑒−2𝛼(𝑇 −𝑡) ) +
(1 − 𝑒−(𝛼+𝑎)(𝑇 −𝑡) )+
2𝛼
(𝛼 + 𝑎)
𝜌2 𝜂 2
𝜂 2 (1 − 𝜌2 )
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) +
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) + 𝜎𝐿 (𝑇 − 𝑡).
2𝑎
2𝑎
Since ln 𝑆𝑇 is normally distributed, the spot price is log-normally
distributed with mean:
[
]
]
[
])
( [
1
𝐸 𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 |𝑡 = 𝑒𝑥𝑝 𝐸 𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 |𝑡 + 𝑉 𝑎𝑟 𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 |𝑡 .
2
The above results allow us to formulate the following proposition.
Proposition 1. Forward price of a commodity, where spot price is represented by the 3 factors following the dynamics (13), is given by:
(
)
𝐹 (𝑡, 𝑇 ) = 𝐸 [𝑆𝑇 | ] = 𝑒𝑥𝑝 𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) + 𝑦𝑡 𝑒−𝑎(𝑇 −𝑡) + 𝐿𝑡 + 𝐴(𝑡, 𝑇 ) ,
(15)
7
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
The explicit representation of the problem in terms of measurement
and transition equations allows us to apply Kalman filter machinery,
including explicit formula for the log-likelihood of residuals. In this
work we will not give further overview of the approach and will refer
to Durbin and Koopman (2012) instead.
The Kalman filtering approach allows us to calibrate the model
simultaneously under the physical  and the risk-neutral  probability
measures. From a practical perspective, it means that the single model
can be used for risk management applications (under ) as well as
for pricing applications (under ). Using the single model for these
two purposes can significantly reduce model risk as well as costs (of
development and validation) in a financial institution. Moreover, such
an approach can result in a more consistent modeling practices between
departments within one financial institution.
5.2. Data
Fig. 3. Volatility term structure. The shape is consistent with patterns observed in
implied volatility of futures options on futures, the so-called Samuelson effect.
We apply our model to an extensive dataset of ICE Brent futures
prices. Brent oil is the major oil benchmark and one of the most heavily
traded oil grades in the world with many derivatives using it as the
underlying. We use Brent futures for a period from 7-02-2005 till 2407-2018. Note that Brent futures have not become negative in 2020 as
WTI futures did. So in our next paper, we will apply the model to WTI
futures, including most recent negative prices.
There are 3134 daily observations of the Brent oil forward curves
in our dataset. Brent contracts have a monthly expiration schedule, but
for each day we use 1-st, 2-nd, 3-rd, 4-th, 5-th, 6-th, 9-th, 12-th, 18-th,
24-th, 30-th and 36-th to expiry futures contract. That results in a daily
observations of 12 points which form the forward curve. Such choice
is guided by higher trading volumes of the front futures compared to
the low liquid far end of the curve. In addition, to reflect liquidity of
the contracts, we have used the first 6 contracts on a daily calculation
of 𝐹̄ (𝑡).
5. Calibration
5.1. State-space representation
By design, our model is a joint-dynamics model. This term was
introduced by Hull, White, Sokol in Hull et al. (2014) in the context
of interest rate models. The parameters of such a model describe
dynamics under both physical and risk-neutral probability measures.
Such approach poses some challenges to the calibration of the model:
the calibrated model should fit observed market prices of derivatives
(futures) and at the same time produce parameters that evolve from one
day to another according to the specified dynamics. A suitable approach
to solve such problems is filtering approach, known from the field of
signal processing. We estimate the model with a dynamic panel data
set of futures prices via Kalman filter combined with the method of
maximum likelihood. In order to apply Kalman filtering technique, the
state-space representation of the problem has to be written first. The
state-space representation consists of the measurement and transition
equations. First, we formulate the measurement equation for the model
(10) (geometric version of the model):
⎡ ln 𝐹 (𝑡, 𝑇1 ) ⎤ ⎡ 𝑒−𝛼(𝑇1 −𝑡)
⎢ ln 𝐹 (𝑡, 𝑇 ) ⎥ ⎢ 𝑒−𝛼(𝑇2 −𝑡)
2 ⎥
⎢
⎢
...
...
⎢
⎥=⎢
⎢ln 𝐹 (𝑡, 𝑇𝑁 )⎥ ⎢𝑒−𝛼(𝑇𝑁 −𝑡)
⎢
⎥ ⎢
1
⎣ ln(𝐹̄ (𝑡)) ⎦ ⎣
𝑒−𝑎(𝑇1 −𝑡)
𝑒−𝑎(𝑇2 −𝑡)
...
𝑒−𝑎(𝑇𝑁 −𝑡)
0
1⎤
⎡ 𝐴(𝑡, 𝑇1 ) ⎤
1 ⎥ ⎡𝜓𝑡 ⎤ ⎢ 𝐴(𝑡, 𝑇2 ) ⎥
⎥⎢ ⎥ ⎢
⎥
...⎥ 𝑦𝑡 + ⎢ ... ⎥ + 𝜖𝑡 ,
⎢
⎥
1 ⎥ ⎣𝐿𝑡 ⎦ ⎢𝐴(𝑡, 𝑇𝑁 )⎥
⎥
⎢
⎥
1⎦
⎣ 0 ⎦
5.3. Calibration approach and results
The model is calibrated by minimizing the negative log-likelihood
function of the Kalman filter. The optimization is performed on changed
(unbounded) variables, which are then converted back to the original
(bounded) variables. We do this conversion to ensure a number of
conditions (such as positive volatility and mean-reversion speed). The
optimization is performed in two steps. On the first step, the global
solver is applied (we have used Simulated Annealing algorithm) to
determine a region with the global minimum. This algorithm is applied
with arbitrary chosen values of the input parameters. On the second
step, a local solver is used with the starting values of parameters
obtained on the first step. We use a Quasi-Newton method with numerical estimation of derivatives as a local solver. Such a two-stage
approach allows avoiding local minima associated with parameters that
are unrealistic close to their bounds.
The calibration procedure takes as an input prices of futures 𝐹 (𝑡, 𝑇1 ),
𝐹 (𝑡, 𝑇2 ), . . . , 𝐹 (𝑡, 𝑇𝑁 ), history of synthetic spot factor 𝐹̄ (𝑡), value of
𝑞Max as well as a set of initial values for model parameters 𝜃0 , where
𝜃0 =< 𝜆0 , 𝛼0 , 𝜎0 , 𝜌0 , 𝜂, 𝑞, 𝜓0 , 𝑦0 , 𝐿0 >. We design the calibration in
such way that the following constrains are satisfied: 𝛼 > 0, 𝑎 > 0, 𝜎 > 0,
𝜂 > 0, 0 < 𝑞 < 𝑞Max , −1 < 𝜌 < 1, ln 30 < 𝐿0 < ln 100.
In addition, we also calculate two error measures to compute the
goodness of fit of theoretical forward curves vs observed ones:
√
√
𝑁 ∑
𝐾
√
∑
(
)2
√ 1
𝐹
(𝑡 , 𝑇 ) − 𝐹Model (𝑡𝑗 , 𝑇𝑖 ) ,
𝐴𝑏𝑠𝑜𝑙𝑢𝑡𝑒𝑅𝑀𝑆𝐸 = √
𝑁 + 𝐾 𝑖=1 𝑗=1 Market 𝑗 𝑖
√
√
)
𝑁 ∑
𝐾 (
√
∑
𝐹Market (𝑡𝑗 , 𝑇𝑖 ) − 𝐹Model (𝑡𝑗 , 𝑇𝑖 ) 2
√ 1
𝑅𝑒𝑙𝑎𝑡𝑖𝑣𝑒𝑅𝑀𝑆𝐸 = √
,
𝑁 + 𝐾 𝑖=1 𝑗=1
𝐹Model (𝑡𝑗 , 𝑇𝑖 )
(18)
with [𝜓𝑡 𝑦𝑡 𝐿𝑡 ]𝑇 being a state vector. 𝜖𝑡 is an 𝑁 +1 vector of disturbances
with 𝐸[𝜖𝑡 ] = 0 and 𝑉 𝑎𝑟[𝜖𝑡 ] = 𝐻𝑡 , where matrix 𝐻𝑡 is a diagonal matrix
with elements equal to ℎ𝑡 . The last row of the measurement equation is
a way to enforce the condition that the filtered value of the synthetic
spot is equal to the observed value: 𝑋𝑡 = ln 𝐹̄ (𝑡).
Transition equation describes the dynamics of the state vector and
is given by:
⎡𝜓𝑡 ⎤ ⎡1 − 𝛼𝛥𝑡
⎢𝑦 ⎥ = ⎢ 0
⎢ 𝑡⎥ ⎢
⎣𝐿𝑡 ⎦ ⎣ 0
0
1 − 𝑎𝛥𝑡
0
0⎤ ⎡𝜓t-1 ⎤
0⎥ ⎢ 𝑦t-1 ⎥ + 𝑤𝑡
⎥⎢
⎥
1⎦ ⎣𝐿t-1 ⎦
(19)
In the above equation, 𝑤𝑡 is a residual vector with zero expected value
𝐸[𝑤𝑡 ] = 0 and the variance given by:
⎡ 𝜎 2 𝛥𝑡
𝑉 𝑎𝑟[𝑤𝑡 ] = ⎢𝜎𝜂𝜌𝛥𝑡
⎢
⎣ 0
𝜎𝜂𝜌𝛥𝑡
𝜂 2 𝛥𝑡
0
0 ⎤
0 ⎥.
⎥
2
𝜎𝐿 𝛥𝑡⎦
Note that, for our application, 𝛥𝑡 corresponds to daily changes and is
equal to 1∕365 or 1∕250, depending on whether calendar or trading days
are considered.
8
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Fig. 4. Filtered time series of model factors. Signal-to-noise ratio of 𝑞 = 0.1.
Table 1
Calibration results with 𝑞Max = 0.9. This results are obtained after 10 000
iterations of the first step of the solver, 56 iterations of the second step
of the solver and result in log-likelihood of 159 280.
Parameter
Value (Standard error)
𝜆
𝛼
𝜎
𝑎
𝜂
𝜌
𝑞
0.1443254∗∗∗ (0.009361403)
0.2430296∗∗∗ (0.001635522)
0.4000135∗∗∗ (0.003455691)
3.7857089∗∗∗ (0.024889426)
0.003885854∗∗∗ (0.003885854)
0.5457884∗∗∗ (0.027449226)
0.5246947∗∗∗ (0.010475912)
Table 2
Sensitivity of model calibration results to the choice of 𝑞Max parameter.
Parameter
Test 1
𝑞Max
0.9
𝜆
0.1443254
𝛼
0.2430296
𝜎
0.4000135
𝑎
3.7857089
𝜂
0.1187412
𝜌
0.5457884
𝑞
0.5246947
Log-likelihood
159280.0
Num. Iterations (step 1) 10000
Num. Iterations (step 2) 56
Absolute RMSE
0.3085206
Relative RMSE
0.004492449
Test 2
Test 3
Test 4
0.5
0.1152516
0.2427340
0.4000178
3.7472506
0.1183404
0.5413630
0.4866871
159263.1
10000
64
0.409216
0.005627727
0.25
0.0952823
0.2406266
0.4512142
3.8200394
0.1036434
0.2817274
0.2499994
158507.9
10000
120
0.3377983
0.004748221
0.1
0.13578620
0.23215921
0.49656735
3.82370828
0.10015373
0.11572452
0.09999984
154705.2
10000
144
0.4753287
0.006187431
Fig. 5. Differences between filtered time series of 𝐿𝑡 + 𝜓𝑡 and observed synthetic spot
time series of 𝑋𝑡 . Signal-to-noise ratio of 𝑞 = 0.1.
with our model specification. Moreover, we see that stochastic log
mean 𝐿𝑡 is varying, but stays in a relatively smaller range compared
to the changes of synthetic spot factor 𝑋𝑡 . This is also in line with
empirical observations, that energy prices mean-revert, but not to the
constant, but to some kind of floating level. In addition, in Fig. 5 we
show that the sum of the filtered time series 𝐿𝑡 + 𝜓𝑡 is very close
to the observed synthetic spot time series 𝑋𝑡 . This is an additional
confirmation of correctness of our model. Finally, an example of the
fit of the theoretical model (𝑞Max = 0.9) to the market data is given in
Fig. 6. As can be seen from this figure, the model fits observed market
prices of futures very well.
where 𝐾 = 3469 is the total number of daily observations in the dataset.
We have used R statistical software for all of the computations. The
optimal parameters calibrated with fixing 𝑞Max = 0.9 are presented
in Table 1. The calibration procedure depends on 𝑞Max as an input
parameter. Natural question could be a sensitivity of the model to the
choice of 𝑞Max . The sensitivity of the model calibration to the choice
of 𝑞Max parameter is summarized in Table 2. The run with 𝑞Max = 0.9
results in the best market fit as well as in the highest value of the
likelihood function. The run with 𝑞Max = 0.1 gives attractable model
features, such as low volatility of the long-term factor 𝐿𝑡 ; as well as
still acceptable quality of the model fit. We choose to use these two
settings, the particular value of the 𝑞Max depends on the potential model
use. For each application considered in this paper we will specify the
value of 𝑞Max used. In addition, Table 2 contains the values of Absolute
and Relative RMSE assuming constant market price of risk (MPR).
The filtered time series of the state variables are presented in Fig. 4.
As can be seen in these figures, volatility of the short-term deviation
factor 𝑦𝑡 is smaller compared to the volatility of 𝜓𝑡 - this is inline
5.4. Model-implied spot price
One of the main distinctive features of our model is the synthetic
spot factor 𝑋𝑡 . This factor is determined using only futures prices
and not the actual spot price. However, our model can be used to
determine the model-implied spot price 𝑆𝑡 = 𝑒𝜓𝑡 +𝑦𝑡 +𝐿𝑡 . Such a spot
price depends on the liquid derivatives products and reflects significant
trading volumes, rather than a small panel of large market players. The
time series of differences between the model-implied spot price and
the actual Brent spot price is given in Fig. 7. This figure shows that
9
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Fig. 6. Examples of the model fit to Brent futures market data. The boundary of signal-to-noise ratio is 𝑞Max = 0.9.
Fig. 7. Differences between model-implied spot 𝑆𝑡 = 𝑒𝜓𝑡 +𝑦𝑡 +𝐿𝑡 , observed Brent spot price and synthetic spot 𝐹̄ . Signal-to-noise ratio of 𝑞 = 0.1.
the model-implied spot follows the actual spot price remarkably well,
as the differences are generally less than 5 cents per barrel (with the
exception of one observation).
alternative (to the first nearby futures price or spot price) commodity
‘‘reference price’’.
When financial press talks about ‘‘the oil price’’, they can mean
different things. As the ‘‘state’’ of oil market is given by the combination
of the current spot price and the today’s forward curve, different
reference prices can be chosen as ‘‘the oil price’’. Typically, it is the
price of the first-to-expire futures contract, but sometimes (but rarely)
it is the oil spot price. Our model suggests two more candidates for such
a reference price: the synthetic spot price 𝐹̄𝑡 and the model-implied spot
price 𝑆𝑡 = 𝑒𝜓𝑡 +𝑦𝑡 +𝐿𝑡 .
5.5. Stochastic market price of risk
The model specified in (13) contains a constant parameter for the
market price of risk 𝜆 (MPR). As was argued in Ahmad and Wilmott
(2013), MPR parameter is actually time-dependent, moreover, it is most
likely stochastic. We can expand our model to accommodate for the
time-dependent market price of risk. This is done by changing the specification of the Kalman filter and introducing one more unobservable
variable 𝜆𝑡 . This is possible because the pricing formula for futures can
be re-written as follows:
The problem with the first nearby futures contract is that it does
not represent the ‘‘constant maturity’’ object, because its time to expiry
changes every day (i.e., it decreases by one day). Both synthetic spot
𝐹̄𝑡 and the model-implied spot price are synthetic objects without an
expiry date. This is a convenient feature for the modeling purposes.
−𝛼(𝑇 −𝑡) +𝑦 𝑒−𝑎(𝑇 −𝑡) +𝐿 +𝜆 𝐵(𝑡,𝑇 )+𝐶(𝑡,𝑇 )
𝑡
𝑡
𝑡
𝐹 (𝑡, 𝑇 ) = 𝑒𝜓𝑡 𝑒
,
(20)
where in case of a constant MPR 𝐴(𝑡, 𝑇 ) = 𝜆𝐵(𝑡, 𝑇 ) + 𝐶(𝑡, 𝑇 ). The
specification of the measurement equation of the Kalman filter changes
too:
While the implied spot price is model-dependent (it depends not
only on the underlying processes, but also on the choice of calibration
method), the synthetic spot 𝐹̄𝑡 is model-independent. It also has an
advantage that it reflects liquid and traded contracts. Finally, the
synthetic spot 𝑋𝑡 has lower volatility than the nearby futures price,
observed and model-implied spot prices. In all, it provides a great
−𝛼(𝑇 −𝑡)
⎡ ln 𝐹 (𝑡, 𝑇1 ) ⎤ ⎡ 𝑒 1
⎢ ln 𝐹 (𝑡, 𝑇2 ) ⎥ ⎢ 𝑒−𝛼(𝑇2 −𝑡)
⎢
⎥ ⎢
...
⎢
⎥ = ⎢ ...
⎢ln 𝐹 (𝑡, 𝑇 )⎥ ⎢𝑒−𝛼(𝑇𝑁 −𝑡)
𝑁
⎢
⎥ ⎢
⎣ ln(𝐹̄ (𝑡)) ⎦ ⎣
1
10
𝑒−𝑎(𝑇1 −𝑡)
𝑒−𝑎(𝑇2 −𝑡)
...
𝑒−𝑎(𝑇𝑁 −𝑡)
0
1
1
...
1
1
𝐵(𝑡, 𝑇1 ) ⎤
𝐶(𝑡, 𝑇1 ) ⎤
⎡𝜓 ⎤ ⎡
𝐵(𝑡, 𝑇2 ) ⎥ ⎢ 𝑡 ⎥ ⎢ 𝐶(𝑡, 𝑇2 ) ⎥
⎥ ⎢ 𝑦𝑡 ⎥ ⎢
⎥
+ ⎢ ... ⎥ + 𝜖𝑡 ,
... ⎥
⎢
⎥
𝐿
𝐵(𝑡, 𝑇𝑁 )⎥ ⎢ 𝑡 ⎥ ⎢𝐶(𝑡, 𝑇𝑁 )⎥
⎥ ⎣𝜆 ⎦ ⎢
⎥
⎣ 0 ⎦
0 ⎦ 𝑡
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Fig. 8. Time-dependent market price of risk 𝜆𝑡 for Brent oil futures.
Fig. 9. An example of the model fit with time-dependent MPR 𝜆𝑡 to Brent futures
market data on 29-12-2008.
Table 3
Quality of fit of theoretical forward curve to market quotes with time dependent MPR
𝜆𝑡 and the boundary of signal-to-noise ratio 𝑞𝑀𝑎𝑥 = 0.9.
Table 4
Comparison of quality of fit of the proposed model to Gibson–Schwartz model.
Test type
Absolute RMSE
Relative RMSE
Test type
Model
Absolute RMSE
Relative RMSE
Constant MPR
Time dependent MPR
0.3085206
0.2113817
0.004492449
0.003041120
In-sample
In-sample
Out-of-sample
Out-of-sample
Gibson–Schwartz
Proposed model
Gibson–Schwartz
Proposed model
0.4477562
0.3085206
0.2521213
0.3369156
0.006629341
0.004492449
0.005555177
0.005278424
The transition equation then becomes:
⎡𝜓𝑡 ⎤ ⎡1 − 𝛼𝛥𝑡
⎢ ⎥ ⎢
⎢ 𝑦𝑡 ⎥ = ⎢ 0
⎢ 𝐿𝑡 ⎥ ⎢ 0
⎢𝜆 ⎥ ⎢ 0
⎣ 𝑡⎦ ⎣
0
1 − 𝑎𝛥𝑡
0
0
0
0
1
0
5.6. Comparison to gibson–schwartz model and out-of-sample performance
0⎤ ⎡𝜓t-1 ⎤
⎥⎢
⎥
0⎥ ⎢ 𝑦t-1 ⎥
+ 𝑤𝑡 ,
0⎥ ⎢𝐿t-1 ⎥
⎥
⎢
⎥
1⎦ ⎣ 𝜆t-1 ⎦
In this section we describe model benchmarking tests, where we
compare the performance of our 3-factor model to the performance
of Gibson–Schwartz model. Gibson–Schwartz model is a very popular
choice among practitioners, so it is natural to choose it as the comparison. We perform two performance comparisons tests: in-sample and
out-of-sample. For the in-sample test, we fit both models on the whole
Brent futures data set described in Section 5.2 and calculate errors
within the same data set. For the out-of sample comparison, we first
calibrate the models on the data from 07-02-2005 to 29-12-2017, then
we filter unobserved variables for dates from 02-01-2018 to 24-072018. Finally, we calculate the errors on trading dates from 02-01-2018
to 24-07-2018. In such a way, the differences between market and
model prices are evaluated on the data different from the one used for
model calibration.
where 𝑤𝑡 is a residual vector with zero expected value 𝐸[𝑤𝑡 ] = 0 and
the variance given by:
⎡ 𝜎 2 𝛥𝑡
⎢𝜎𝜂𝜌𝛥𝑡
𝑉 𝑎𝑟[𝑤𝑡 ] = ⎢
⎢ 0
⎢ 0
⎣
𝜎𝜂𝜌𝛥𝑡
𝜂 2 𝛥𝑡
0
0
0
0
𝜎𝐿2 𝛥𝑡
0
0 ⎤
0 ⎥⎥
.
0 ⎥
⎥
2
𝜎𝜆 𝛥𝑡⎦
Given the new specification of the Kalman filter, we explicitly assume
MPR to follow a simple, driftless stochastic process. We make such
an assumption to impose as few constraints on the structure of the
MPR process as possible. We apply a similar two-steps optimization
procedure to determine optimal parameters of the new filter. For this
test, we keep the boundary of signal-to-noise ratio at 𝑞𝑀𝑎𝑥 = 0.9. The
resulting filtered time series of 𝜆𝑡 for Brent oil is presented in Fig. 8.
As can be seen from the picture, MPR increases during the times of
price increase, and drops during the periods of the market stress. These
findings are similar to the ones observed in the interest rate markets
in Ahmad and Wilmott (2013). Willmott defines the periods of low (or
even negative) market price of risk as periods of ‘‘fear’’, while high
positive values of MPR would correspond to the periods of ‘‘greed’’.
Finally, the model with time dependent MPR 𝜆𝑡 results in a lower
RMSE and ARMSE measures (see Table 3 for details). An example of
the model fit with the time-dependent MPR is presented in Fig. 9. Additionally, from Fig. 8, we observe that the filtered MPR resembles the
Ornstein–Uhlenbeck process path. Potentially, the proposed model can
be extended by introducing the fourth stochastic factor that describes
the MPR.
For the benchmarking tests, we have used R implementation of
the Gibson–Schwartz model provided by package Schwartz97 (refer to
package documentation (Luthi et al., 2014a) and (Luthi et al., 2014b)
for more details). Gibson–Schwartz model was calibrated using functionality of Schwartz97 package; calibrated parameters are described
in Appendix A.5.
For the comparisons tests we have used a calibration version of
our 3-factor model with 𝑞𝑀𝑎𝑥 = 0.9. This setting results in the highest
value of the likelihood function and lowest RMSE. The results of model
benchmarking for in-sample and out-of-sample performance are given
in Table 4.
Our 3-factor model performs superior to Gibson–Schwartz model for
in-sample comparison test. It also performs with a comparable error for
out-of-sample test. The proposed model has an advantage of a number
of pre-defined features (such as synthetic spot price, long-run mean)
that give our model a more intuitive interpretation without the loss of
a good fit to the observed futures prices.
11
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
6. Model applications
6.1. Scenario generation
Our three-factor model has multiple practical applications. In this
section, we discuss some of them. One natural application of the model
is forward curve scenario generation. For example, for counterparty
credit risk (CCR) computations, one would consider long-horizon simulations under the risk neutral probability measure. Such computations
can be used as a part of the credit valuation adjustment (CVA) engine,
where one computes potential negative and positive counterparty exposures of a commodity derivatives book. The critical steps of the CVA
calculations include generating a large set of scenarios of risk factors
(in our case, energy forward curves), re-pricing all the OTC derivatives linked to these factors and calculating potential positive/negative
exposures for the selected time horizons. Finally, calculate the CVA
adjustment based on the exposures and probabilities of default assigned
to each counterparty. A realistic model for generating the forward
curves scenarios is paramount for the model performance. Such a model
should capture all main empirical properties of the underlying energy
forward curves (e.g., contango–backwardation switch, mean-reversion,
etc.). The general approach for implementing the exposure model is:
Fig. 10. Example of Monte-Carlo simulations using the three factor synthetic spot
model. Simulation start on 2018-07-11 and the simulation end date is 2019-07-11.
Such simulations allow practitioners to compute exposures on 2019-07-11.
• Calibrate model parameters (including the market price of risk
𝜆), based on historical observations of forward curves. Typically
a few years of historical data is used, possibly including a period
of market stress if required by the regulation.
• Using model dynamics under risk-neutral probability , generate
scenarios of the state variables 𝜓𝑡 , 𝑦𝑡 and 𝐿𝑡 . This is done by
generating a number of paths up to some required time horizon.
Typically, 5000 to 10000 paths are generated.
• Based on the generated scenarios of the state variables, calculate
scenarios of commodity forward curves using formula (15).
• For each curve scenario, re-price all the derivatives associated
with this curve. This will result in the risk-neutral distribution of
portfolio values for a given counterparty at a given future time.
• Based on the scenarios of portfolio values, calculate statistics of
interest such as Expected Positive Exposure, Expected Negative
Exposure or other.
Calendar spread options can be viewed as a particular case of a
more general construction — basket or spread options. Calendar spread
consists of the futures from the same forward curve, while general
basket and spread options can contain different commodity futures. A
typical example is 3:2:1 crack spread which is a difference between
different quantities of crude oil, heating oil and unleaded gasoline.
The main challenge in pricing of basket options lies in the fact that
a linear combination of log-normally distributed random variables is
not log-normally distributed. Moreover, it does not follow any known
probability distribution. As the result, the Black–Scholes framework for
pricing options cannot be applied. Several methods have been proposed
for such options, starting with the seminal paper by Margrabe (1978),
where zero strike is assumed. If the strike of a spread option is not
zero, then Kirk method can be applied (E, 1995) or the Wakeman
method (Turnbull and Wakeman, 1991).
A visual example of Monte-Carlo simulations under the risk-neutral
probability measure is shown in Fig. 10.
Another possible application of the model is for market risk management purposes. Here one would generate forward curve scenarios with
a short horizon (typically 2–10 days) and the simulations should be
based on the physical probability measure. This is done in a similar way
as above, but using Eq. (11) and by computing different risk metrics
(such as Value at Risk or Expected Shortfall), based on the generated
set of scenarios. Such applications can be especially valuable for initial
margin modeling for clearing houses and brokers.
Borovkova et al. (2007) proposed a robust way to price basket
options based on displaced log-normal distribution. The approach approximates the distribution of the basket value based on the moment
matching technique. The paper describes close-form pricing formulas
for options on very general baskets and so also on calendar spreads.
The framework in Borovkova et al. (2007) depends on the following
assumptions about futures dynamics:
𝑑𝐹𝑖 (𝑡, 𝑇𝑖 )
= 𝜎𝑖 𝑑 𝑊̄ 𝑖 (𝑡), 𝑖 = 1, 2, … , 𝑁
𝐹𝑖 (𝑡, 𝑇𝑖 )
6.2. Pricing options on calendar spread futures
(21)
where 𝜎𝑖2 is a variance of the futures 𝑖, 𝑁 is a number of assets in the
basket, 𝑊̄ 𝑖 (𝑡) and 𝑊̄ 𝑗 (𝑡) are Brownian motions driving futures 𝑖 and 𝑗
with correlation 𝜌̄𝑖,𝑗 .
Our model can be used also for pricing commodity derivatives. The
model naturally yields closed-form expression for futures and forwards
contracts. In this section we discuss possible application of the model
to pricing of a more complex derivative products, namely options on
calendar spreads.
The payoff of a calendar spread is the difference between two futures with different expiry dates 𝑅(𝑡, 𝑇1 , 𝑇2 ) = 𝐹 (𝑡, 𝑇2 )−𝐹 (𝑡, 𝑇1 ). A typical
example is the calendar spread between the second and the first nearby
futures. Calendar spreads can be used, for example, to hedge against
the move of the commodity curve from contango to backwardation and
vice verse. A European option on a calendar spread is an option whose
payoff depends on the value of the spread 𝑅(𝑡𝑒𝑥𝑝 , 𝑇1 , 𝑇2 ) at the option’s
expiry date 𝑡𝑒𝑥𝑝 ≤ 𝑇1 .
The same framework (as well as methods of Magrabe, Kirk or
Wakeman) can be applied to the problem of pricing European option on calendar spreads. All these methods mentioned above require
the volatilities and correlation between assets underlying the basket or the spread option. Our 3-factor model provides such modelimplied volatilities and correlations, without having to estimate them
separately.
To do that, we need to determine the variances of the futures in the
spread and the correlation coefficient. This can be done by re-writing
the dynamics of the futures 𝐹 (𝑡, 𝑇1 ) and 𝐹 (𝑡, 𝑇2 ) from the representation
(16). The correlation between two futures is naturally given as the ratio
12
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
∙ It specifies the model not only under the physical, but also under
the risk-neutral probability measure.
∙ It adds a stochastic but slowly varying long-term mean to the mean
reverting dynamics of the commodity price.
∙ We develop an innovative calibration approach based on the
Kalman filtering methodology.
Furthermore, we outlined possible model adjustments for dealing
with recently observed negative futures prices.
We also demonstrated that the market price of risk plays an important role in transition from 𝑃 to 𝑄 measure, and that the market price of
risk is not constant, but stochastic and is related to the price level. This
is in line with findings of Willmot and Ahmad in Ahmad and Wilmott
(2013).
We described practical applications of the model, ranging from scenario generation (under physical or risk-neutral probability measures)
to pricing options on calendar spreads.
Future research will be focused on including stochastic volatility
and seasonal effects into the model, as well as applying the arithmetic
version of the model to WTI futures which recently exhibited negative
values.
Fig. 11. Correlation between two legs of calendar spread 𝐹1 and 𝐹2 represented as a
function of difference between corresponding time to expiry of the futures 𝑇2 − 𝑇1 . The
time to expiry of the first future is fixed at 10 days 𝑇1 = 10∕365.
Appendix A. Appendices
A.1. Derivation of forward pricing formula
of covariance to the variances of the corresponding legs of the spread:
𝜌̄1,2 = √
𝐶𝑜𝑣1,2
.
In this section we will derive forward contracts pricing formula (15).
As was mentioned before the log-spot is assumed to depend on three
stochastic factors ln 𝑆𝑇 = 𝜓𝑇 + 𝑦𝑇 + 𝐿𝑇 . The dynamics of these factors
under the risk-neutral probability measure is given by Eq. (13).
The dynamics of 𝜓𝑇 is described by Ornstein–Uhlenbeck process,
which results in the following solution is:
(22)
𝑉 𝑎𝑟1 𝑉 𝑎𝑟2
The covariance can be calculated from the dynamics of the forward
prices (16) and is given by the following formula:
𝐶𝑜𝑣1,2 = 𝜎 2 𝑒−𝛼(𝑇1 +𝑇2 −2𝑡) + 𝜂𝜌𝜎𝑒−𝛼(𝑇1 −𝑡)−𝑎(𝑇2 −𝑡) + 𝜂𝜌𝜎𝑒−𝑎(𝑇1 −𝑡)−𝛼(𝑇2 −𝑡) +
𝜓𝑇 = 𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) −
𝜂 2 𝜌2 𝑒−𝑎(𝑇1 +𝑇2 −2𝑡) + 𝜂 2 (1 − 𝜌2 )𝑒−𝑎(𝑇1 +𝑇2 −2𝑡) + 𝜎𝐿2 .
𝑇
̃1 (𝑠).
𝑒𝛼𝑠 𝑑 𝑊
The dynamics of 𝑦𝑇 can be solved in a similar way leading to:
(23)
𝑦𝑇 = 𝑦𝑡 𝑒−𝑎(𝑇 −𝑡) −
The variance for futures 𝐹 (𝑡, 𝑇𝑖 ) is:
𝑉 𝑎𝑟𝑖 ≡ 𝜎𝑖2 = (𝜎 2 𝑒−𝛼(𝑇𝑖 −𝑡) + 𝜂𝜌𝑒−𝑎(𝑇𝑖 −𝑡) )2 + 𝜂 2 (1 − 𝜌2 )𝑒−2𝑎(𝑇𝑖 −𝑡) + 𝜎𝐿2 .
𝜆𝜎
(1 − 𝑒−𝛼(𝑇 −𝑡) ) + 𝜎𝑒−𝛼(𝑇 −𝑡)
∫𝑡
𝛼
𝜆𝜂
(1 − 𝑒−𝑎(𝑇 −𝑡) ) + 𝜂𝜌𝑒−𝑎𝑡
∫𝑡
𝑎
√
𝜂 1 − 𝜌2 𝑒−𝑎(𝑇 −𝑡)
(24)
The above formulas depend on the calibrated model parameters. The
correlation coefficient (22) depends on the time difference between
the legs of the spread 𝑇2 − 𝑇1 . The dependency of the correlation to
the value of 𝑇2 − 𝑇1 is presented in Fig. 11. As can be seen from that
figure, the correlation decreases when time spread increases. Overall,
correlation stays high which is consistent with empirical observation
that two futures prices with nearby time to expiry are highly correlated.
Plugging in Eqs. (23) and (24), which result from the dynamics of
our 3-factor model into your favorite spread option valuation method
(Magrabe, Kirk, Wakeman or the framework proposed in Borovkova
et al., 2007), we are able to price calendar spread options in a fast and
efficient way.
𝑇
̃1 (𝑠)+
𝑒𝑎𝑠 𝑑 𝑊
𝑇
∫𝑡
̃4 (𝑠).
𝑒𝑎𝑠 𝑑 𝑊
Finally, the dynamics of the long-run mean 𝐿𝑇 is described by arithmetic Brownian motion, which also enables us to easily obtain the
solution:
𝑇
𝐿𝑇 = 𝐿𝑡 − 𝜆𝜎𝐿 (𝑇 − 𝑡) + 𝜎𝐿
̃3 (𝑠).
𝑒𝛼𝑠 𝑑 𝑊
∫𝑡
The next step is to compute expected value and variance of log-spot
process ln 𝑆𝑇 . These calculations are based on the fact that all three
stochastic factors are normally distributed. The expected part is easily
obtained from the above solutions of SDE:
𝐸[ln 𝑆𝑇 |𝑡 ] = 𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) + 𝑦𝑡 𝑒−𝑎(𝑇 −𝑡) + 𝐿𝑡
[𝜂
]
𝜎
− 𝜆 (1 − 𝑒−𝑎(𝑇 − 𝑡)) + 𝜎𝐿 (𝑇 − 𝑡) + (1 − 𝑒−𝛼(𝑇 −𝑡) ) .
𝑎
𝛼
The variance of log-spot is given by the sum of variances with respect
to individual uncorrelated Brownian motions:
7. Concluding remarks and further research
In this paper we proposed a new three-factor model to describe the
dynamics of the commodity (energy) forward curves. The new model
is an example of the so-called ‘‘joint dynamics’’ model: i.e., the model
that describes dynamics of the state variables under both risk-neutral
and physical probability measures.
The model is based on the three factors with the synthetic spot
factor being the main one. The synthetic spot factor was originally
introduced by Borovkova and Geman in Borovkova and Geman (2006)
and has a number of useful properties. Our work extends the work
in Borovkova and Geman (2006) in three dimensions:
𝑉 𝑎𝑟[ln 𝑆𝑇 |𝑡 ] = 𝑉1 + 𝑉2 + 𝑉3 ,
where
𝑉1 =
𝑇(
∫𝑡
𝜎2
2𝛼
13
𝑇(
∫𝑡
𝜎𝑒−𝛼(𝑇 −𝑡) 𝑒𝛼𝑠 + 𝜂𝜌𝑒−𝑎(𝑇 −𝑡) 𝑒𝑎𝑠
𝜎 2 𝑒−2𝛼(𝑠−(𝑇 −𝑡)) + 2𝜂𝜌𝜎𝑒(𝑎+𝛼)(𝑠−(𝑇 −𝑡)) + 𝜂 2 𝜌2 𝑒2𝑎(𝑠−(𝑇 −𝑡))
(1 − 𝑒−2𝛼(𝑇 −𝑡) ) +
𝜂 2 𝜌2
)2
)2
𝑑𝑠 =
𝑑𝑠 =
2𝜎𝜂𝜌
(1 − 𝑒−(𝑎+𝛼)(𝑇 −𝑡) ) +
(1 − 𝑒−2𝑎(𝑇 −𝑡) ),
𝛼+𝑎
2𝑎
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
𝑉2 = 𝜎𝐿2 (𝑇 − 𝑡)
Table A.5
Calibration results for time-dependent MPR test with
𝑞Max = 0.9. This results are obtained after 10 000
iterations of the first step of the solver, 116 iterations of the second step of the solver and result in
log-likelihood of 175684.7.
𝜂 2 (1 − 𝜌2 )
(1 − 𝑒−2𝑎(𝑇 −𝑡) ).
𝑉3 =
2𝑎
Finally, since all the stochastic factors are normally distributed, the logspot price is also normally distributed. As the consequence, spot price
is log-normally distributed. The forward price is equal to the expected
value of the spot price, which yields the following pricing formula:
−𝛼(𝑇 −𝑡) +𝑦 𝑒−𝑎(𝑇 −𝑡) +𝐿 +𝐴(𝑡,𝑇 )
𝑡
𝑡
𝐹 (𝑡, 𝑇 ) = 𝐸 [𝑆𝑇 | ] = 𝑒𝜓𝑡 𝑒
,
where
[𝜎
]
𝜂
𝐴(𝑡, 𝑇 ) = −𝜆 (1 − 𝑒−𝛼(𝑇 −𝑡) ) + (1 − 𝑒−𝑎(𝑇 −𝑡) ) + 𝜎𝐿 (𝑇 − 𝑡)+ +
𝛼
𝑎
𝜎𝜂𝜌
𝜎2
(1 − 𝑒−2𝛼(𝑇 −𝑡) ) +
(1 − 𝑒−(𝛼+𝑎)(𝑇 −𝑡) )+
4𝛼
(𝛼 + 𝑎)
𝜎
𝜌2 𝜂 2
𝜂 2 (1 − 𝜌2 )
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) +
(1 − 𝑒−2𝑎(𝑇 −𝑡) ) + 𝐿 (𝑇 − 𝑡).
4𝑎
4𝑎
2
Parameter
Value
𝛼
𝜎
𝑎
𝜂
𝜌
𝑞
𝜎𝜆
0.7926642222
0.3999862774
1.8170002163
0.1007189394
0.1552717310
0.8334227542
0.1012842194
Table A.6
Calibrated values of Gibson–Schwartz model that are used for the model bench-marking.
The interest rate 𝑟 is taken as an average USD Fed-Fund rate over the dates within the
data set and is equal to 𝑟 = 0.0134.
A.2. Derivation of forward dynamics
In this section we will derive dynamics of the forward price and
we will do it in two steps. First, we will compute partial derivatives
of log-forward price 𝐺(𝑡, 𝑇 ) ≡ ln 𝐹 (𝑡, 𝑇 ) with respect to the stochastic
factors and time:
𝜕𝐺
𝜕𝐺
= 𝐹 (𝑡, 𝑇 )𝑒−𝛼(𝑇 −𝑡) ,
= 𝐹 (𝑡, 𝑇 )𝑒−𝑎(𝑇 −𝑡) ,
𝜕𝜓
𝜕𝑦
𝜕2 𝐺
𝜕𝐺
= 𝐹 (𝑡, 𝑇 ),
= 𝐹 (𝑡, 𝑇 )𝑒−2𝛼(𝑇 −𝑡) ,
𝜕𝐿
𝜕𝜓 2
𝜕2 𝐺
𝜕2 𝐺
= 𝐹 (𝑡, 𝑇 )𝑒−2𝑎(𝑇 −𝑡) ,
= 𝐹 (𝑡, 𝑇 ),
𝜕𝑦2
𝜕𝐿2
[
𝜆𝜂𝑎 −𝑎(𝑇 −𝑡)
𝜕𝐺
= 𝐹 (𝑡, 𝑇 ) 𝛼𝜓𝑡 𝑒−𝛼(𝑇 −𝑡) + 𝑎𝑦𝑡 𝑒−𝑎(𝑇 −𝑡) +
𝑒
𝜕𝑡
𝑎
2
𝜆𝜎𝛼 −𝛼(𝑇 −𝑡) 2𝛼𝜎 −2𝛼(𝑇 −𝑡)
+𝜎𝐿 𝜆 +
𝑒
−
𝑒
−
𝛼
4𝛼
𝜎𝜂𝜌(𝛼 + 𝑎) −(𝛼+𝑎)(𝑇 −𝑡)
𝑒
−
(𝛼 + 𝑎)
2
𝜂 2 𝜌2 2𝑎 −2𝑎(𝑇 −𝑡) 𝜎𝐿 𝜂 2 (1 − 𝜌2 )2𝑎 −2𝑎(𝑇 −𝑡) ]
𝑒
−
−
𝑒
.
4𝑎
2
4𝑎
After applying Ito’s lemma and canceling some terms, we will obtain:
√
𝑑𝐹 (𝑡, 𝑇 )
̃1 + 𝜂 1 − 𝜌2 𝑒−𝑎(𝑇 −𝑡) 𝑑 𝑊
̃4 + 𝜎𝐿 𝑑 𝑊
̃3 .
= (𝜎𝑒−𝛼(𝑇 −𝑡) + 𝜂𝜌𝑒−𝑎(𝑇 −𝑡) )𝑑 𝑊
𝐹 (𝑡, 𝑇 )
Parameter
All data set
From 07-02-2005 till 29-12-2017
𝛼
𝜎𝑆
𝜅
𝜎𝛿
𝜌
𝜆
0.4177644
0.537943
0.5190225
0.2471361
0.8656749
0.2623438
0.595662
0.6050017
0.5142756
0.2530181
0.9071371
0.3732789
coefficients 𝑎𝑖𝑗 . Volatilities 𝑠𝑖𝑔𝑚𝑎 and 𝑒𝑡𝑎 as well as correlation 𝜌
can be easily derived from covariance matrix 𝑅.
• For −𝑐Min ≤ 𝜃𝑖 ≤ 𝑐Max the following transformation is used:
𝑐
+ 𝑐Min
𝑐Max − 𝑐Min
arctan(𝜃̃𝑖 ) + Max
,
𝜋
2
((
)
𝑐Max + 𝑐Min )
𝜋
𝜃̃𝑖 = tan 𝜃𝑖 −
,
2
(𝑐Max − 𝑐Min )
𝜃𝑖 =
where 𝑐Max and 𝑐Min are two constants that bound the variable 𝜃𝑖 .
This transformation is used for bounded variables, such as 𝐿0 , 𝑞,
as well as for 𝑎11 , 𝑎12 , 𝑎22 etc.
A.4. Optimal permeates time-dependent MPR test
Optimal parameters for Kalman filter with time-dependent MPR 𝜆𝑡
are presented in Table A.5.
A.3. Change of variables in optimization
A.5. Parameters of Gibson–Schwartz model
The model calibration is performed in terms of changed variables
𝜃̃𝑖 , this is done to ensure realistic constrains on the optimal parameters
values (such as positive volatility). There are few types of variable
change used in the model. The model is defined in terms of bounded
variables 𝜃𝑖 , but the optimization routines (both global and local) are
defined in terms of unbounded variables 𝜃̃𝑖 . We will list the types of
variable constrains and the corresponding transformation functions:
Optimal parameters for Gibson–Schwartz model used for model
bench-marking are presented in A.6.
Appendix B. Supplementary data
Supplementary material related to this article can be found online
at https://doi.org/10.1016/j.eneco.2021.105418.
• For 𝜃𝑖 > 0 the following transformation is used:
̃
𝜃𝑖 = 𝑒𝜃𝑖 + 𝑐,
References
𝜃̃𝑖 = ln(𝜃𝑖 − 𝑐),
Ahmad, R., Wilmott, P., 2013. The market price of interest-rate risk: Measuring and
modelling fear and greed in the fixed-income markets. WILMOTT Mag. 64–70.
Amin, K., Ng, V., Pirrong, S., 1995. Valuing energy derivatives. Risk Publ. Enron Capital
Trade Resour. 57, 57–70.
B. Trolle, A., S. Schwartz, E., 2009. Unspanned stochastic volatility and the pricing of
commodity derivatives. Rev. Financ. Stud. 22, 4423–4461.
Barndorff-Nielsen, O.E., Benth, F.E., Veraart, A.E.D., 2013. Modelling energy spot prices
by volatility modulated Lévy-driven Volterra processes. Bernoulli 19 (3), 803–845.
http://dx.doi.org/10.3150/12-BEJ476.
Black, F., 1976. The pricing of commodity contracts. J. Financ. Econ. 3, 167–179.
Borovkova, S., 2020. Negative oil price: what does it mean for you?. URL https:
//www.financialinvestigator.nl/nl/nieuws-detailpagina/2020/04/23/Negative-oilprice-what-does-it-mean-for-you.
where 𝑐 = 1𝑒 − 08 is a small positive constant. This transformation
is used for mean-reversion coefficients and volatilities that are not
part of covariance matrix (such as 𝜎𝐿 ).
• For correlation/covariance matrix a special transformation is applied. This is done to ensure three things: that volatilities are positive, that correlations are within [−1, 1] and that the covariance
matrix is positive semidefinite. This is achieved by performing
optimization in terms of coefficients of Cholesky matrix decomposition 𝑎11 , 𝑎12 , 𝑎22 , where 𝑐 < 𝑎𝑖𝑗 < 104 . The covariance matrix
𝑅 is obtained as 𝐴𝐴𝑇 , where A is a lower diagonal matrix with
14
Energy Economics 101 (2021) 105418
S. Ladokhin and S. Borovkova
Heath, D., Jarrow, R., Morton, A., 1990. Bond pricing and the term structure of interest
rates: A discrete time approximation. J. Financ. Quant. Anal. 25, 419–440.
Hull, J.C., Sokol, A., White, A., Modeling the short rate: The real and risk-neutral
worlds, Rotman School of Management Working Paper, No. 2403067.
LeBlanc, M., Chinn, M.D., Do high oil prices presage inflation? The evidence from
G-5 countries, UC Santa Cruz Economics Working Paper No. 561, SCCIE Working
Paper No. 04-04, https://ssrn.com/abstract=509262 or http://dx.doi.org/10.2139/
ssrn.509262.
Luthi, D., Erb, P., Otziger, S., 2014a. Using the Schwartz97 Package. Tech. rep..
Luthi, D., Erb, P., Otziger, S., 2014b. Schwartz 1997 Two-Factor Model Technical
Document. Tech. rep..
Margrabe, W., 1978. The value of an option to exchange one asset for another. 33.
Nasdaq-Clearing, 2018. Clearing member default that occurred 10 september,
2018. URL https://business.nasdaq.com/updates-on-the-Nasdaq-Clearing-MemberDefault/index.html.
Schwartz, E., 1997. The stochastic behavior of commodity prices: Implications for
valuation and hedging. J. Finance 52 (3), 923–973.
Schwartz, E., Smith, J.E., 2000. Short-term variations and long-term dynamics in
commodity prices. Manage. Sci. 46 (7), 893–911.
Stafford, P., 2018. How clearing houses aim to avert market disasters. URL https:
//www.ft.com/content/01596fde-b805-11e8-b3ef-799c8613f4a1.
Turnbull, S.M., Wakeman, L.M., 1991. A quick algorithm for pricing European average
options. J. Financ. Quant. Anal. 26 (3), 377–389, URL http://www.jstor.org/stable/
2331213.
Warren, J.H., DiLellio, J.A., Dyer, J.S., 2014. What do market-calibrated stochastic
processes indicate about the long-term price of crude oil?. Energy Econ. 44,
212–221.
Borovkova, S., Bunk, H., Gooeij, W., Mechev, D., Veldhuizen, D., 2013. Collaterized
commodity obligations. J. Altern. Investments 16 (2), 9–29. http://dx.doi.org/10.
3905/jai.2013.16.2.009.
Borovkova, S., Geman, H., 2006. Seasonal and stochastic effects in commodity forward
curves. Rev. Derivatives Res. 9 (2), 167–186. http://dx.doi.org/10.1007/s11147007-9008-4.
Borovkova, S., Permana, F.J., Weide, H.V., 2007. A closed form approach to the
valuation and hedging of basket and spread option. J. Derivatives 14 (4), 8–
24. http://dx.doi.org/10.3905/jod.2007.686420, URL https://jod.iijournals.com/
content/14/4/8.
Durbin, J., Koopman, S.J., 2012. Time Series Analysis by State Space Methods (Oxford
Statistical Science Series Book 38), second ed. OUP Oxford.
E, K., 1995. Correlation in the energy markets. In: Managing Energy Price Risk. Risk
Publications and Enron, pp. 71–78.
FE, B., J, S.-B., 2004. The normal inverse Gaussian distribution and spot price modelling
in energy markets. Int. J. Theor. Appl. Finance 7(02), 177–192.
Geman, H., 2000. Scarcity and price volatility in oil markets. EDF Trading Technical
Report.
Geman, H., 2005. Commodities and Commodity Derivatives. Modelling and Pricing for
Agriculturals, Metals and Energy). Jhon Wiley and Sons Ltd..
Geman, H., Roncoroni, A., 2006. Understanding the fine structure of electricity prices.
J. Bus. 79 (3), 1225–1261.
Gibson, R., Schwartz, E.S., 1990. Stochastic convenience yield and the pricing of oil
contingent claims. J. Finance 45, 959–976.
Hahn, W.J., DiLellio, J.A., Dyer, J.S., 2018. Risk premia in commodity price forecasts
and their impact on valuation. Energy Econ. 72, 393–403.
15