The term structure gives the relationship between the futures prices and the re-spective time to maturity. It provides useful information for hedging or investment decisions because it synthesizes the information available in the market and the operators’ expectations concerning the future. The information is very useful for management purposes: it can be used to hedge exposure on the physical market and to adjust the stock level or the production rate. It can also be used to un-dertake arbitrage transactions, to evaluate derivatives instruments based on futures contracts, and so on. Therefore, stochastic term structure models aim to reproduce the futures prices observed in the market as accurately as possible aiming e.g. to discover futures prices for horizons exceeding exchange traded maturities, to forecast futures price developments under different economic scenarios, to price structured products based on futures contracts with minimized errors or to see the interactions of futures prices with other asset’s price movements.
Over time, different models were introduced ranging from the simplest one factor models to more sophisticated versions of three factor models. Depending on the amount of factors, the following factors are modeled stochastically: the spot price, the convenience yield and the interest rate. Starting in Section 3.4.1 we will intro-duce two examples of one factor models. The first, called Brownian Motion Model, will generate the spot price with a stochastic dynamic coming from a Brownian Motion and a deterministic convenience yield. The second, called Mean Rever-sion Model, will model the spot price over a mean reverting dynamic structure.
In Section 3.4.2 we will introduce the two most accepted two factor models: the Convenience Yield and the Long - Short Term Model. Although, the two models were developed based on different fundamental ideas, the two models are equivalent.
74We will further denote the futures price of a commodity with F (t, T ) = FC(t, T ) and its spot price with P (t) = PC(t) at t ∈ [0, T ].
Closing this section we will give a brief example of a three factor model. It is similar to the Convenience Yield Model of Section 3.4.2, but extended with a stochastic interest rate component.
To evaluate futures prices based on the three input factors commodity spot price, convenience yield and interest rate, given either deterministic or stochastic, the models borrow from the contingent claim analysis developed for stock and interest rate models.75 Therefore, the different models of commodity futures pricing share the following general assumptions: the market for assets is free of frictions, taxes or transaction costs, trading takes place continuously and lending and borrowing rates are equal and there are no short sale constrains.
3.4.1 One Factor Models
One factor models are based on the concept that futures prices are determined as the expectation of the future spot price, conditionally to the available information at time t. Therefore, the spot price is the main determinant of futures prices.
Thus, following [Lautier 2005], most one factor models rely on the spot price. Two general approaches are chosen: either to model the stochastic dynamic of the spot price with a Brownian Motion or with a Mean Reversion Process. The Brownian Motion Model is more excepted in practice than the Mean Reversion Model because it allows for a deterministic convenience yield and therefore, covers the consumption good characteristic of commodities. On the other hand, the Brownian Motion Model does not cover observed mean reversion pattern in commodity futures prices. We will introduce both approaches to give a general overview of common market models, starting in Definition 3.1 with the Brownian Motion Model.
Definition 3.1 Brownian Motion Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], µ ∈ R the drift of the spot price, σP > 0 the spot price volatility and WP(t) a standard Brownian Motion as defined in Definition C.28. Then the dynamic of the spot price in the Geometric Brownian Motion Model is
dP (t) = µP (t)dt + σPP (t)dWP(t), t ∈ [0, T ]. (3.17)
Equation (3.17) stats that the commodity spot price is driven by a stochastic that can be modeled with a simple Brownian Motion. Based on this stochastic process,
75An illustrative introduction can be found in [Zagst 2002].
we will further derive the futures price represented by F (P, t) at time t for delivery of one unit of the commodity at time T . Furthermore, denote:
∂F (P,t)
Using Itˆo’s lemma as of Definition C.1, the instantaneous change in the futures price is given as:
The difference between commodities as simple financial asset and consumption good is captured in the net convenience yield76 assumed to be a proportional to the spot price: c(P, t) = CP (t), with C ∈ R. Following [Brennan Schwartz 1985] the convenience yield is the flow of services that accrues to an owner of the physical commodity. He is able to choose where the commodity will be stored and when to liquidate the inventory. Recognizing the costs for transportation, storage and insurance, the convenience yield ”may be thought of as the value of being able to profit from temporary local shortages of the commodity through ownership of the physical commodity. The profit may arise either from local price variations or from the ability to maintain a production process as a result of ownership of an inventory of raw material.” Therefore, the financial spot price process dP (t) has to be amended with the convenience yield process CP (t)dt yielding to the actual commodity spot
76Compare Remark 3.1 of Section 3.2.
price process:77
dP (t) + CP (t)dt = µP (t)dt + σPP (t)dWP(t), t ∈ [0, T ]
⇒ dP (t) = (µ − C)P (t)dt + σPP (t)dWP(t) (3.19) Following the no arbitrage pricing methodology, the futures contract delivering one unit of the underlying in T has the same value as the commodity in T . To avoid arbitrage, the two assets must have the same value before T , as well. Therefore, we can construct the following portfolio, called risk free hedge portfolio:
V (t) = P (t) + c(P, t) − δF (P, t)
Thus, the futures price in the Brownian Motion Model is given as the solution of the following partial differential equation with the boundary condition F (P (t), T ) = P (T ):
Ft(P, t) + 1
2σ2PP (t)2FP P(P, t) + FP(P, t)P (t)(rf − C) = 0 (3.20)
77Compare Equation (3.9).
Theorem 3.3 Futures Price in the Brownian Motion Model
Let the notations be as in Definition 3.1, c(P, t) = CP (t) with C ∈ R be the deter-ministic convenience yield and rf be the constant risk free interest rate. Then the futures price F (P, t) is a function of the spot price and the time to maturity:
F (P, t) = P (t)e(rf−C)(T −t). (3.21)
Proof: It has been shown in [Zagst 2002], that the futures price F (t) of an asset is the conditional expectation as of Definition C.24, whereby conditional is regarding the available information of today embodied in σ-Algebra Ft as of Definition C.5, of its future spot price P (T ) under the equivalent martingale measure ˜Q as of Definition C.32:78
F (P, t) = EQ˜[P (T )|Ft], t ∈ [0, T ] (3.22) Using the Feynman-Kac representation of Theorem C.5, there is an indirect way to get (3.22). Someone can solve the Cauchy-Problem as given in Definition C.36 to get the solution of the stochastic differential equation underlying the futures price.
The Feynman-Kac representation then stats that if there exists a solution that it is equal to conditional expectation of (3.22).79 To get F (P, t) = v(P, t) as requested in Equation (C.25), we have to define: x := P , r(P, t) ≡ 0 and D(P ) := P (T ).
Therewith, we have to show that F solves the Cauchy-Problem as defined in C.36.
For it, we first have to transfer the spot price P in the world of the equivalent mar-tingale measure ˜Q which exists because of the Girsanov-Theorem as of Theorem C.3.
Denote with d ˜W the increments of the Brownian motion as of Definition C.28 under Q. Using the Girsanov-Theorem as of Theorem C.3, we have:˜
d ˜W (t) = λ(t)dt + dW (t), t ∈ [0, T ] (3.23) where λ : R 7→ R is called the market price of risk. It results
dP (t) = [µ − σPλ(t)]P (t)dt + σPP (t)d ˜WP(t), t ∈ [0, T ] (3.24) where it has to be µ − σPλ(t) = rf because the discounted spot price process has to be a martingale as of Definition C.29. It is
dP (t) = rfP (t)dt + σPP (t)d ˜WP(t), t ∈ [0, T ]. (3.25)
78Also compare Equation (3.2).
79Attention: The opposite direction is not always true. See [Zagst 2002].
Again, we have to amend the financial spot price of commodities with the conve-nience yield process as in Equation (3.19). It follows
dP (t) = (rf − C)P (t)dt + σPP (t)d ˜WP(t), t ∈ [0, T ] (3.26) Then, the adapted Cauchy-Problem as of Definition C.36 is given as:
Ft(P, t) + 1
2σP2P (t)2FP P(P, t) + FP(P, t)P (t)(rf − C) = 0, t ∈ [0, T ] (3.27) with the terminal boundary condition F (P, T ) = P (T ). Recall, this is equal to Equation (3.20) and therefore shows, that solving the Cauchy Problem is in line with solving the differential equation developed over the no arbitrage approach.
Under the assumption of Equation (3.21), F (P, t) = P (t)e(rf−C)(T −t), with t ∈ [0, T ], we get:
FP(P, t) = e(rf−C)(T −t), FP,P(P, t) = 0,
Ft(P, t) = −(rf − C)F (P, t), t ∈ [0, T ].
Putting this into the Cauchy-Problem, Equation (3.27), it follows 0 + (rf − C)F (P, t) − (rf − C)F (P, t) = 0, ∀t ∈ [0, T ] which shows, that the futures price is indeed F (P, t) = P (t)e(rf−C)(T −t).
2 Although, the Brownian Motion Model is probably the most simple and therewith the most known one, it has the drawback of not covering mean reversion occurring in commodity spot prices caused by the consumption good characteristic of commodi-ties reflecting producers and consumers actions in the physical market.80 When the spot price is low, industrials expect prices to rise and fill their inventories. Producers react with a reduction of output providing only low benefits. The increased demand and the simultaneous reduction of supply have a rising influence on the spot price.
Conversely, when the spot price is higher than its long run average, industrials will serve their demand with inventories that were build up at low commodity price times and producers increase their production rate expecting higher margins for the same
80See the latest work [Markert 2005].
output. Both movements will push the spot price to lower levels. [Schwartz 1997]
published a one factor model that directly incorporates the mean reversion effect into the spot price.
Definition 3.2 Mean Reversion Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], µ ∈ R the long-run mean, κ > 0 the speed of adjustment of the spot price, σP > 0 the spot price volatility and WP(t) a standard Brownian motion as defined in Definition C.28.
Then the dynamic of the spot price in the Mean-Reverting Model is
dP (t) = P (t)κ[µ − ln P (t)]dt + σPP (t)dWP(t), t ∈ [0, T ]. (3.28)
The model covers two characteristics of mean reversion: the spot price has the prosperity to return to its long-term mean, but simultaneously, random shocks can move it away in the short-run allowing for sudden price peaks.
Based on the spot price movements we can calculate the futures price in the Mean Reversion Model:
Theorem 3.4 Futures Price in the Mean Reversion Model
Let the notations be as in Definition 3.2, λ : R 7→ R be the market price of risk introduced in the Girsanov-Theorem as of Theorem C.3 and rf be the constant risk free interest rate. Then the futures price F (P, t) is a function of the spot price and the time to maturity and is expressed by
F (P, t) = exp[e−κ(T −t)ln P (t) + (1 − e−κ(T −t))(µ − σ2P/2κ − λ) + σP2
4κ(1 − e−2κ(T −t))], (3.29)
with t ∈ [0, T ].
Proof: As shown in Proof 3.4.1, the futures price F must solve the Cauchy-Problem as defined in Definition C.36 with x := P , r(P, t) ≡ 0 and D(P ) := P (T ) for all P (T ) ∈ R and t ∈ [0, T ]. Following the methodology of Proof 3.4.1 we have to transfer the stochastic process for the factors into the world of the equivalent martingale measure ˜Q. Using the Girsanov-Theorem as of Theorem C.3, we have:
dP (t) = P (t)κ[µ − ln P (t) − λ]dt + σPP (t)d ˜WP(t), t ∈ [0, T ].
d ˜WP(t) is the increment of a Brownian motions under the equivalent martingale measure. Based on this equations the adapted Cauchy-Problem as of Definition C.36
is given as:
1
2σP2P (t)2FP,P(P, t) + P (t)κ[µ − ln P (t) − λ]FP(P, t) + Ft(P, t) = 0 (3.30) with t ∈ [0, T ] and the terminal boundary condition F (P, T ) = P (T ).
Under the assumption of Equation (3.34), that
F (P, t) = exp
e−κ(T −t)ln P (t) + (1 − e−κ(T −t))(µ − σ2P/2κ − λ) + σP2
4κ(1 − e−2κ(T −t))
with t ∈ [0, T ], we can calculate the respective derivatives:
FP(P, t) = e−κ(T −t) F (P,t)
Putting this into the Cauchy-Problem of Equation (3.30) it follows:
1
which shows that F (P, t) solves the Cauchy-Problem as of Equation (3.30).
Finally, we have to prove that our assumption solves the terminal boundary condi-tion F (P, T ) = P (T ). Under our assumpcondi-tion it holds
The model has the major drawback that it treats positive and negative mean rever-sion in the same way. Following [Lautier 2005] contango is limited to the storage costs until a certain maturity resulting in an upper boundary for price spreads be-tween two maturity following futures contracts, while backwardation is not. To cover this phenomena, more complex models are needed.
3.4.2 Two Factor Models
Two factor models determine the uncertainty in the commodity spot price over two random processes. Two approaches are excepted in literature: the convenience yield and the long-short term approach. Although the models look different on the first view, they are equivalent what we will show later. Starting this section we introduce the Convenience Yield Model allowing for a stochastic spot price implicitly driven by a stochastic convenience yield. Recall, convenience yield determines why and how commodity spot prices deviate from classical asset prices. The following sto-chastic model specifies the spot price implicitly driven by the convenience yield that is modeled exogenously as a mean revering process and determine the futures price as the risk neutral expectation of future spot prices. The model was first introduced in [Schwartz 1997] and is given in Definition 3.3.
Definition 3.3 Convenience Yield Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], c(t) the convenience yield at time t, µ ∈ R the drift of the spot price, α ∈ R is the long-run level to which the convenience yield reverts, κ > 0 is the speed of adjustment of the convenience yield, σP > 0 the spot price volatility, σc > 0 the convenience yield volatility and dWP(t) and dWc(t) the increments of two Brownian Motions as defined in Definition C.28 with a correlation
dWP(t)dWc(t) = ρdt, t ∈ [0, T ], ρ ∈ [−1, 1]. (3.31) The spot price and the instantaneous convenience yield process are assumed to have the following form:
dP (t) = P (t)[µ − c(t)]dt + σPP (t)dWP(t), (3.32) dc(t) = κ[α − c(t)]dt + σcdWc(t), t ∈ [0, T ] (3.33)
The spot price P (t) of Equation (3.32) follows a geometric Brownian Motion as of Definition C.28 with a stochastic convenience yield defined in Equation (3.33).
The stochastic convenience yield c(t) as of Equation (3.33) is assumed to be mean reverting and follows a Mean Reversion process. The inclusion of this process into Equation (3.32) introduces an implicit mean reversion effect on the commodity spot price process, when the respective Brownian Motions are positively correlated: An increase in P (t) from a positive dWP(t) is typically associated with a positive dWc(t) and an increase of c(t) entering negative the drift rate of P (t) and decreasing the spot price.81 Based on the spot price movements we can calculate the futures price in the Convenience Yield Model:
Theorem 3.5 Futures Price in the Convenience Yield Model
Let the notations be as in Definition (3.3), λ : R 7→ R be the market price of risk introduced in the Girsanov-Theorem as of Theorem C.3 and rf be the constant risk free interest rate. Then the futures price F (P, c, t) is a function of the spot price, the convenience yield and the time to maturity and is expressed by
F (P, c, t) = P (t) exp
Proof: As shown in Proof 3.4.1, the futures price F must solve the Cauchy-Problem as defined in Definition C.36 with x := P , r(P, t) ≡ 0 and D(P ) := P (T ) for all P (T ) ∈ R and t ∈ [0, T ]. Following the methodology of Proof 3.4.1, we have to transfer the stochastic process for the factors into the world of the equivalent martingale measure ˜Q as defined in Definition C.32. Using the Girsanov-Theorem as of Theorem C.3, we have:
dP (t) = P (t)[rf − c(t)]dt + σPP (t)d ˜WP(t), dc(t) = (κ[α − c(t)] − λ)dt + σcd ˜Wc(t), d ˜WP(t)d ˜Wc(t) = ρdt, t ∈ [0, T ]
d ˜WP(t) and d ˜Wc(t) are the increments of two Brownian motions under the equiva-lent martingale measure. Based on these equations we can formulated the specific
81See [Markert 2005] for empirical evidence.
Cauchy-Problem:
Under the assumption of Equation (3.34) that
F (P, c, t) = P (t) exp we can calculate the respective derivatives:
FP(P, c, t) = F (P,c,t)P (t) ,
Putting this into the Cauchy-Problem Equation (3.36) it follows
which shows that indeed F (P, c, t) as of Equation (3.34) solves the Cauchy-Problem Equation (3.36). Still we have to prove that our assumption solves the terminal boundary condition F (P, c, T ) = P (T ). Under our assumption it holds
F (P, c, T ) = P (T ) exp Thinking about real options under the purpose to find the optimal exercise moment for exploration ventures, brought up the thought of long term trends and short term fluctuations in commodity markets. [Schwartz Smith 2000] used the idea and
pub-lished their Long - Short Term Model that models mean reversion in short term prices and uncertainty in the equilibrium level to which prices revert. Although, these variables are not directly observable in the market, the authors used the fol-lowing intuition to estimate the parameters of the model from market data: move-ments in prices for long maturing futures contracts provide information about the equilibrium price level, and differences between the prices for the short and long term contracts provide information about short term variations. The mathematical formulation of the model is given in Definition 3.4.
Definition 3.4 Long - Short Term Model
Let P (t) be the spot price of a commodity at time t ∈ [0, T ], χ(t) the short-term deviation in prices at time t, ξ(t) the equilibrium price level at time t, µ ∈ R the drift of the equilibrium price level, κ > 0 the speed of adjustment of the short-term deviation, σχ > 0 the short-term prices volatility, σξ > 0 the equilibrium price level volatility and dWχ and dWξ the increments of two standard Brownian Motions as defined in Definition C.28 with a correlation
dWχ(t)dWξ(t) = ρdt, t ∈ [0, T ], ρ ∈ [−1, 1]. (3.37) Then the dynamic of this model is
ln P (t) = χ(t) + ξ(t) (3.38)
dχ(t) = −κχ(t)dt + σχdWχ(t) (3.39)
dξ(t) = µξdt + σξdWξ(t), t ∈ [0, T ] (3.40)
Temporary price changes, caused e.g. by abrupt weather alteration or supply inter-ruptions, are embodied in the short term component χ(t). They are not expected to persist because market participants will switch to inventories to adjust changing market conditions. Following [Gabillon 1995], production, consumption, stock level and the fear of inventory disruptions are the most important explanatory factors in the short run. Information of these factors are mainly needed for hedging purposes.
Changes in the long term level represent fundamental modifications of the market conditions and are therefore, are expected to persist. Latter can be caused e.g. by a change in the number of producers in the industry or the availability of a commodity.
It is also determined by expectations of exhausting supply, improving technology for the production and macroeconomic influences like inflation, politics and regulatory effects. Following [Gabillon 1995], the information is used for investment purposes.
The derivation of the price of a futures contract with the underlying stochastic processes as of Definition 3.4 can be found in [Schwartz Smith 2000]. Conceptually, its derivation runs as the methodology of Proof 3.4.1 and Proof 3.4.2. To avoid redundance, we will focus on another interesting fact. Although, the model does not explicitly consider changes in the convenience yield, it is equivalent to the Con-venience Yield Model of Definition 3.3. The following theorem gives the explanation how the variables of the one model can be expressed as linear combination of the variables of the other model:
Theorem 3.6 Equivalence of the Convenience Yield and the Long - Short Term Model
The Convenience Yield Model as of Definition 3.3 and the Long - Short Model as of Definition 3.4 are equivalent with the following parameters:
Long - Short Model Convenience Yield Model
κ κ
Proof: Following Definition 3.3, the price dynamics in the two factor convenience yield model are given as of Equation (3.32) and Equation (3.33):
dP (t) = P (t)[µ − c(t)]dt + σPP (t)dWP(t), dc(t) = κ[α − c(t)]dt + σcdWc(t), t ∈ [0, T ]
With Itˆo as of Lemma C.1 the log spot price dynamic of (3.32) are given as:
dln(P (t)) =
Then, the variables in the long - short model can be written in terms of the variables of the stochastic convenience yield model as follows:
χ(t) = short term deviation = 1
κ(c(t) − α) (3.42) Therewith, it follows
dχ(t) = 1
κdc(t)
(3.33)
z}|{= 1
κ(κ[α − c(t)]dt + σcdWc(t))
= [α − c(t)]dt +σc
κdWc(t)
(3.42)
z}|{= −κχ(t)dt + σc κ
|{z}≡σχ
dWc(t)
| {z }
≡dWχ(t)
Moreover
ξ(t) = equilibrium price level
= ln(P (t)) − χ(t)
= ln(P (t)) − 1
κ(c(t) − α) (3.43)
Therewith, it follows
dξ(t) = dln(P (t)) − 1 κdc(t)
= [µ − c(t) − 1
2σP2]dt + σPdWP(t) − 1
κ(κ[α − c(t)]dt + σcdWc(t))
= [µ − α − 1 2σP2
| {z }
≡µξ
]dt + σPdWP(t) − σc
κdWc(t)
| {z }
≡σξ≡dWξ(t)
Finally, showing the last equation of Table 3.1.
2 [Schwartz Smith 2000] showed that the model works best for mid term maturities.
Moreover, the model includes the two one factor models Brownian Motion and Mean Reversion. The first one is generated by setting σχ equal to zero, i.e. assuming that there is uncertainty in equilibrium prices, only. A Mean Reversion Model is given by assuming a constant equilibrium price, i.e. setting σξ equal to zero. Statistical comparison of the three models by the authors showed significant advantages in cap-turing the characteristics of commodity futures prices through the two factor model.
But as [Lautier 2005] stats, there is still one question remaining: is it interesting to represent a stable equilibrium with a stochastic variable? On the other hand, some pricing perspectives, especially in the real options environment, focus on long term prices and do not care about short term fluctuation.82
3.4.3 Three Factor Models
Not until 1997, the first three factor model was introduced: [Schwartz 1997] pro-posed his three factor model with the extension of stochastic interest rates because the hypothesis of constant interest rates as in the one and two factor models amounts to saying that the term structure of interest rates is flat, which is far from reality.
Moreover, under this assumption forward and futures prices are equivalent, which
Moreover, under this assumption forward and futures prices are equivalent, which