A pricing measure to explain the risk premium in power markets
aa r X i v : . [ q -f i n . P R ] A ug A PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS
FRED ESPEN BENTH AND SALVADOR ORTIZ-LATORREA
BSTRACT . In electricity markets, it is sensible to use a two-factor model with mean reversion for spotprices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accountsfor the small variations. The other factor is an OU process driven by a pure jump L´evy process and modelsthe characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricingmeasure is given by the Esscher transform that preserves the probabilistic structure of the driving L´evyprocesses, while changing the levels of mean reversion. Using this choice one can generate stochastic riskpremiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introducea pricing change of measure, which is an extension of the Esscher transform. With this new change ofmeasure we also can slow down the speed of mean reversion and generate stochastic risk premiums withstochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles withpositive values in the short end of the forward curve and negative values in the long end. Finally, our pricingmeasure allows us to have a stationary spot dynamics while still having randomly fluctuating forward pricesfor contracts far from maturity.
1. I
NTRODUCTION
In modelling and analysis of forward and futures prices in commodity markets, the risk premium playsan important role. It is defined as the difference between the forward price and the expected commodityspot price at delivery, and the classical theory predicts a negative risk premium. The economical argumentfor this is that producers of the commodity is willing to pay a premium for hedging their production (seeGeman [9] for a discussion, as well as a list of references).Geman and Vasicek [10] argued that in power markets, the consumers may hedge the price risk usingforward contracts which are close to delivery, and thus creating a positive premium. Power is a non-storable commodity, and as such may experience rather large price variations over short time (sometimesreferred to as spikes). One might observe a risk premium which may be positive in the short end of theforward market, and negative in the long end where the producers are hedging their power generation. Atheoretical and empirical foundation for this is provided in, for example, Bessembinder and Lemon [5]and Benth, Cartea and Kiesel [3].When deriving the forward price, one specifies a pricing probability and computes the forward price asthe conditional expected spot at delivery. In the power market, this pricing probability is not necessarily aso-called equivalent martingale measure, or a risk neutral probability (see Bingham and Kiesel [6]), as thespot is not tradeable in the usual sense. Thus, a pricing probability can a priori be any equivalent measure,and in effect is an indirect specification of the risk premium. In this paper we suggest a new class ofpricing measures which gives a stochastically varying risk premium.We will focus our considerations on the power market, where typically a spot price model may take theform as a two-factor mean reversion dynamics. Lucia and Schwartz [20] considered two-factor models forthe electricity spot price dynamics in the Nordic power market NordPool. Both arithmetic and geometricmodels where suggested, that is, either directly modelling the spot price by a two-factor dynamics, orassuming such a model for the logarithmic spot prices. Their models were based on Brownian motionand, as such, not able to capture the extreme variations in the power spot markets. Cartea and Figueroa [7]used a compound Poisson process to model spikes, that is, extreme price jumps which are quickly revertedback to ”normal levels”. Benth, ˇSaltyt˙e Benth and Koekebakker [2] give a general account on multi-factor
Date : June 10, 2018.We are grateful for the financial support from the project ”Energy Markets: Modeling, Optimization and Simulation (EM-MOS)”, funded by the Norwegian Research Council under grant Evita/205328. models based on Ornstein-Uhlenbeck processes driven by both Brownian motion and L´evy processes.Empirical studies suggest a stationary power spot price dynamics after explaining deterministic seasonalvariations (see e.g. Barndorff-Nielsen, Benth and Veraart [1] for a study of spot prices at EEX, the Germanpower exchange). We will in this paper focus on a two-factor model for the spot, where each factor is anOrnstein-Uhlenbeck process, driven by a Brownian motion and a jump process, respectively. The firstfactor models the ”normal variations” of the spot price, whereas the second accounts for sudden jumps(spikes) due to unexpected imbalances in supply and demand.The standard approach in power markets is to specify a pricing measure which is preserving the L´evyproperty. This is called the Esscher transform (see Benth et al. [2]), and works for L´evy processes as theGirsanov transform with a constant parameter for Brownian motion. The effect of doing such a measurechange is to adjust the mean reversion level, and it is known that the risk premium becomes deterministicand typically either positive or negative for all maturities along the forward curve.We propose a class of measure changes which slows down the speed of mean reversion of the twofactors. As it turns out, in conjunction with an Esscher transform as mentioned above, we can produce astochastically varying risk premium, where potential positive premiums in the short end of the market canbe traced back to sudden jumps in the spike factor being slowed down under the pricing measure. Thisresult holds for arithmetic spot models, whereas the geometric ones are much harder to analyse underthis change of probability. The class of probabilities preserves the Ornstein-Uhlenbeck structure of thefactors, and as such may be interpreted as a dynamic structure preserving measure change . For the L´evydriven component, the L´evy property is lost in general, and we obtain a rather complex jump process withstate-dependent (random) compensator measure.We can explicitly describe the density process for our measure change. The theoretical contributionof this paper, besides the new insight on risk premium, is a proof that the density process is a true mar-tingale process, indeed verifying that we have constructed a probability measure . This verification is notstraightforward because the kernels used to define the density process, through stochastic exponentiation,are stochastic and unbounded. Hence, the usual criterion by L´epingle-M´emin [19] is difficult to apply and,furthermore, it does not provide sharp results. We follow the same line of reasoning as in a very recentpaper by Klebaner and Lipster [18]. Although their result is more general than ours in some respects, itdoes not apply directly to our case because we need some additional integrability requirements. The proofis roughly as follows. First, we reduce the problem to show the uniform integrability of the sequence ofrandom variables obtained by evaluating at the end of the trading period the localised density process.This sequence of random variables naturally induces a sequence of measure changes which, combinedwith an easy inequality for the logarithm function, allow us to get rid of the stochastic exponential in theexpression to be bounded. Finally, we can reduce the problem to get an uniform bound for the secondmoment of the factors under these new probability measures.Interestingly, as our pricing probability is reducing the speed of mean reversion, we might in the ex-treme situation ”turn off” the mean reversion completely (by reducing it to zero). For example, if wetake the Brownian factor as the case, we can have a stationary dynamics of the ”normal variations” in themarket, but when looking at the process under the pricing probability the factor can be non-stationary, thatis, a drifted Brownian motion. A purely stationary dynamics for the spot will produce constant forwardprices in the long end of the market, something which is not observed empirically. Hence, the inclusionof non-stationary factors are popular in modelling the spot-forward markets. In many studies of com-modity spot and forward markets, one is considering a two-factor model with one non-stationary and onestationary component. The stationary part explains the short term variations, while the non-stationaryis supposed to account for long-term price fluctuations in the spot (see Gibson and Schwartz [11] andSchwartz and Smith [23] for such models applied to oil markets). Indeed, the power spot models in Luciaand Schwartz [20] are of this type. It is hard to detect the long term factor in spot price data, and oneis usually filtering it out from the forward prices using contracts far from delivery. Theoretically, suchcontracts should have a dynamics being proportional to the long term factor. Contrary to this approach,one may in view of our new results, suggest a stationary spot dynamics and introduce a pricing measurewhich turns one of the factors into a non-stationary dynamics. This would imply that one could directly fit
PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 3 a two-factor stationary spot model to power data, and next calibrate a measure change to account for thelong term variations in the forward prices by turning off (or significantly slow down) the speed of meanreversion.Our results are presented as follows: in the next section we introduce the basic assumptions and prop-erties satisfied by the factors in our model. Then, in Section 3, we define the new change of measure andprove the main results regarding the uniform integrability of its density process. We deal with the Brown-ian and pure jump case separately. Finally, in Section 4, we recall the arithmetic and geometric spot pricemodels. We compute the forward price processes induced by this change of measure and we discuss therisk premium profiles that can be obtained.2. T
HE MATHEMATICAL SET UP
Suppose that (Ω , F , {F t } t ∈ [0 ,T ] , P ) is a complete filtered probability space, where T > is a fixedfinite time horizon. On this probability space there are defined W , a standard Wiener process, and L, apure jump L´evy subordinator with finite expectation, that is a L´evy process with the following L´evy-Itˆorepresentation L ( t ) = R t R ∞ zN L ( ds, dz ) , t ∈ [0 , T ] , where N L ( ds, dz ) is a Poisson random measurewith L´evy measure ℓ satisfying R ∞ zℓ ( dz ) < ∞ . We shall suppose that W and L are independent of eachother. The following assumption is minimal, having in mind, on the one hand, that our change of measureextends the Esscher transform and, on the other hand, that we are going to consider a geometric spot pricemodel. Assumption 1.
We assume that Θ L , sup { θ ∈ R + : E [ e θL (1) ] < ∞} , (2.1) is strictly positive constant, which may be ∞ . Actually, to have the geometric model well defined we will need to assume later that Θ L > . Someremarks are in order.
Remark 2.1. In ( −∞ , Θ L ) the cumulant (or log moment generating) function κ L ( θ ) , log E P [ e θL (1) ] iswell defined and analytic. As ∈ ( −∞ , Θ L ) , L has moments of all orders. Also, κ L ( θ ) is convex, whichyields that κ ′′ L ( θ ) ≥ and, hence, that κ ′ L ( θ ) is non decreasing. Finally, as a consequence of L ≥ , a.s.,we have that κ ′ L ( θ ) is non negative. Remark 2.2.
Thanks to the L´evy-Kintchine representation of L we can express κ L ( θ ) and its derivativesin terms of the L´evy measure ℓ. We have that for θ ∈ ( −∞ , Θ L ) κ L ( θ ) = Z ∞ ( e θz − ℓ ( dz ) < ∞ ,κ ( n ) L ( θ ) = Z ∞ z n e θz ℓ ( dz ) < ∞ , n ∈ N , showing, in fact, that κ ( n ) L ( θ ) > , n ∈ N . Consider the OU processes X ( t ) = X (0) + Z t ( µ X − α X X ( s )) ds + σ X W ( t ) t ∈ [0 , T ] , (2.2) Y ( t ) = Y (0) + Z t ( µ Y − α Y Y ( s )) ds + L ( t ) , t ∈ [0 , T ] , (2.3)with α X , σ X , α Y > , µ X , X (0) ∈ R , µ Y , Y (0) ≥ . Note that, in equation (2 . , X is written as a sumof a finite variation process and a martingale. We may also rewrite equation (2 . as a sum of a finitevariation part and pure jump martingale Y ( t ) = Y (0) + Z t ( µ Y + κ ′ L (0) − α Y Y ( s )) ds + Z t Z ∞ z ˜ N L ( ds, dz ) , t ∈ [0 , T ] , BENTH AND ORTIZ-LATORRE where ˜ N L ( ds, dz ) , N L ( ds, dz ) − ds ℓ ( dz ) is the compensated version of N L ( ds, dz ) . In the notationof Shiryaev [24], page 669, the predictable characteristic triplets (with respect to the pseudo truncationfunction g ( x ) = x ) of X and Y are given by ( B X ( t ) , C X ( t ) , ν X ( dt, dz )) = ( Z t ( µ X − α X X ( s )) ds, σ X t, , t ∈ [0 , T ] , and ( B Y ( t ) , C Y ( t ) , ν Y ( dt, dz )) = ( Z t ( µ Y + κ ′ L (0) − α Y Y ( s )) ds, , ℓ ( dz ) dt ) , t ∈ [0 , T ] , respectively. In addition, applying Itˆo formula to e α X t X ( t ) and e α Y t Y ( t ) , one can find the followingexplicit expressions for X ( t ) and Y ( t ) X ( t ) = X ( s ) e − α X ( t − s ) + µ X α X (1 − e − α X ( t − s ) ) + σ X Z ts e − α X ( t − u ) dW ( u ) , (2.4) Y ( t ) = Y ( s ) e − α Y ( t − s ) + µ Y + κ ′ L (0) α Y (1 − e − α Y ( t − s ) ) + Z ts Z ∞ e − α Y ( t − u ) z ˜ N L ( du, dz ) , (2.5)where ≤ s ≤ t ≤ T. Remark 2.3.
Using that the stochastic integral of a deterministic function is Gaussian, one easily getsthat X is a Gaussian process and X ( t ) ∼ N ( m t , Σ t ) with m t = X (0) e − α X t + µ X α X (1 − e − α X t ) , t ∈ [0 , T ] , Σ t = σ X α X (1 − e − α X t ) , t ∈ [0 , T ] .
3. T
HE CHANGE OF MEASURE
We will consider a parametrized family of measure changes which will allow us to simultaneouslymodify the speed and the level of mean reversion in equations (2 . and (2 . . The density processes ofthese measure changes will be determined by the stochastic exponential of certain martingales. To thisend consider the following families of kernels G θ ,β ( t ) , σ − X ( θ + α X β X ( t )) , t ∈ [0 , T ] , (3.1) H θ ,β ( t, z ) , e θ z (cid:18) α Y β κ ′′ L ( θ ) zY ( t − ) (cid:19) , t ∈ [0 , T ] , z ∈ R . (3.2)The parameters ¯ β , ( β , β ) and ¯ θ , ( θ , θ ) will take values on the following sets ¯ β ∈ [0 , , ¯ θ ∈ ¯ D L , R × D L , where D L , ( −∞ , Θ L / and Θ L is given by equation (2 . . By Assumption (1) andRemarks 2.1 and 2.2 these kernels are well defined.
Remark 3.1.
Under the assumption R ∞ z e Θ L z ℓ ( dz ) < ∞ , which is stronger than R ∞ e Θ L z ℓ ( dz ) < ∞ , one can consider the set cl( D L ) = ( −∞ , Θ L / and our results still hold by changing κ ′ L ( θ ) , κ ′′ L ( θ ) and κ (3) L ( θ ) by its left derivatives at the rigth end of D L . Example 3.2.
Typical examples of ℓ, Θ L and D L are the following: (1) Bounded support: L has a jump of size a, i.e. ℓ = δ a . In this case Θ L = ∞ and D L = R . (2) Finite activity: L is a compound Poisson process with exponential jumps, i.e., ℓ ( dz ) = ce − λz (0 , ∞ ) dz, for some c > and λ > . In this case Θ L = λ and D L = ( −∞ , λ/ . (3) Infinite activity: L is a tempered stable subordinator, i.e., ℓ ( dz ) = cz − (1+ α ) e − λz (0 , ∞ ) dz, forsome c > , λ > and α ∈ [0 , . In this case also Θ L = λ and D L = ( −∞ , λ/ . PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 5
Next, for ¯ β ∈ [0 , , ¯ θ ∈ ¯ D L , define the following family of Wiener and Poisson integrals ˜ G θ ,β ( t ) , Z t G θ ,β ( s ) dW ( s ) , t ∈ [0 , T ] , (3.3) ˜ H θ ,β ( t ) , Z t Z ∞ ( H θ ,β ( s, z ) −
1) ˜ N L ( ds, dz ) , t ∈ [0 , T ] , (3.4)associated to the kernels G θ ,β and H θ ,β , respectively. Remark 3.3.
Let M be a semimartingale on (Ω , F , {F t } t ∈ [0 ,T ] , P ) and denote by E ( M ) the stochasticexponential of M, that is, the unique strong solution of d E ( M )( t ) = E ( M )( t − ) dM ( t ) , t ∈ [0 , T ] , E ( M )( t ) = 1 . When M is a local martingale, E ( M ) is also a local martingale. If E ( M ) is positive, then E ( M ) is also asupermartingale and E P [ E ( M )( t )] ≤ , t ∈ [0 , T ] . In that case, one has that E ( M ) is a true martingaleif and only E P [ E ( M )( T )] = 1 . If E ( M ) is a positive true martingale, it can be used as a density processto define a new probability measure Q, equivalent to P, that is, dQdP (cid:12)(cid:12)(cid:12) F t = E ( M )( t ) , t ∈ [0 , T ] . The desired family of measure changes is given by Q ¯ θ, ¯ β ∼ P, ¯ β ∈ [0 , , ¯ θ ∈ ¯ D L , with dQ ¯ θ, ¯ β dP (cid:12)(cid:12)(cid:12)(cid:12) F t , E ( ˜ G θ ,β + ˜ H θ ,β )( t ) , t ∈ [0 , T ] , (3.5)where we are implicitly assuming that E ( ˜ G θ ,β + ˜ H θ ,β ) is a strictly positive true martingale. Then, byGirsanov’s theorem for semimartingales (Thm. 1 and 3, p. 702 and 703 in Shiryaev [24]), the process X ( t ) and Y ( t ) become X ( t ) = X (0) + B XQ ¯ θ, ¯ β ( t ) + σ X W Q ¯ θ, ¯ β ( t ) , t ∈ [0 , T ] ,Y ( t ) = Y (0) + B YQ ¯ θ, ¯ β ( t ) + Z t Z ∞ z ˜ N LQ ¯ θ, ¯ β ( ds, dz ) , t ∈ [0 , T ] , (3.6)with B XQ ¯ θ, ¯ β ( t ) = Z t ( µ X + θ − α X (1 − β ) X ( s )) ds, t ∈ [0 , T ] , (3.7) B YQ ¯ θ, ¯ β ( t ) = Z t ( µ Y + κ ′ L (0) − α Y Y ( s )) ds + Z t Z ∞ z ( H θ ,β ( s, z ) − ℓ ( dz ) ds (3.8) = Z t { ( µ Y + κ ′ L (0) − α Y Y ( s )) + Z ∞ z ( e θ z − ℓ ( dz )+ α Y β κ ′′ L ( θ ) Z ∞ z e θ z ℓ ( dz ) Y ( s − ) } ds = Z t (cid:0) µ Y + κ ′ L ( θ ) − α Y (1 − β ) Y ( s ) (cid:1) ds, t ∈ [0 , T ] , where W Q ¯ θ, ¯ β is a Q ¯ θ, ¯ β -standard Wiener process and the Q ¯ θ, ¯ β -compensator measure of Y (and L ) is v YQ ¯ θ, ¯ β ( dt, dz ) = v LQ ¯ θ, ¯ β ( dt, dz ) = H θ ,β ( t, z ) ℓ ( dz ) dt. In conclusion, the semimartingale triplet for X and Y under Q ¯ θ, ¯ β are given by ( B XQ ¯ θ, ¯ β , σ X t, and ( B YQ ¯ θ, ¯ β , , v YQ ¯ θ, ¯ β ) , respectively. BENTH AND ORTIZ-LATORRE
Remark 3.4.
Under Q ¯ θ, ¯ β , X and Y still satisfy Langevin equations with different parameters, that is, themeasure change preserves the structure of the equations. The process L is not a L´evy process under Q ¯ θ, ¯ β ,but it remains a semimartingale. Therefore, one can use Itˆo formula again to obtain the following explicitexpressions for X and YX ( t ) = X ( s ) e − α X (1 − β )( t − s ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( t − s ) ) (3.9) + σ X Z ts e − α X (1 − β )( t − u ) dW Q ¯ θ, ¯ β ( u ) ,Y ( t ) = Y ( s ) e − α Y (1 − β )( t − s ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( t − s ) ) (3.10) + Z ts Z ∞ e − α Y (1 − β )( t − u ) z ˜ N LQ ¯ θ, ¯ β ( du, dz ) , where ≤ s ≤ t ≤ T. Remark 3.5.
Looking at equations (3 . and (3 . , one can see how the values of the parameters ¯ θ and ¯ β change the drift. Setting ¯ θ = (0 , we keep fixed the level to which the process reverts and changethe speed of mean reversion by changing ¯ β . If ¯ β = (0 , we fix the speed of mean reversion and changethe level by changing ¯ θ. By choosing β = 1 , say, we observe that X ( t ) in (3.9) becomes (using a limitconsideration in the second term) X ( t ) = X ( s ) + ( µ X + θ )( t − s ) + σ X ( W Q ¯ θ, ¯ β ( t ) − W Q ¯ θ, ¯ β ( s )) . (3.11) Hence, X is a drifted Brownian motion and we have a non-stationary dynamics under the pricing measurewith this choice of β . Obviously, we can choose β = 1 and obtain similarly a non-stationary dynamicsfor the jump component as well, however, this will not be driven by a L´evy process under Q ¯ θ, ¯ β . The previous reasonings rely crucially on the assumption that Q ¯ θ, ¯ β is a probability measure. Hence, wehave to find sufficient conditions on the L´evy process L and the possible values of the parameters ¯ θ and ¯ β that ensure E ( ˜ G θ ,β + ˜ H θ ,β ) to be a true martingale with strictly positive values. As [ ˜ G θ ,β , ˜ H θ ,β ] , thequadratic co-variation between ˜ G θ ,β and ˜ H θ ,β , is identically zero, by Yor’s formula (equation II.8.19in [14]) we can write E ( ˜ G θ ,β + ˜ H θ ,β )( t ) = E ( ˜ G θ ,β )( t ) E ( ˜ H θ ,β )( t ) , t ∈ [0 , T ] , (3.12)and, as the stochastic exponential of a continuous process is always positive, we just need to ensure thepositivity of E ( ˜ H θ ,β )( t ) . Assume that E ( ˜ H θ ,β ) is positive, then remark 3.3 yields that E ( ˜ G θ ,β +˜ H θ ,β ) is a true martingale if and only if E P [ E ( ˜ G θ ,β + ˜ H θ ,β )( T )] = 1 . Using the independence of ˜ G θ ,β and ˜ H θ ,β and the identity (3 . , we get E P [ E ( ˜ G θ ,β + ˜ H θ ,β )( T )] = E P [ E ( ˜ G θ ,β )( T )] E P [ E ( ˜ H θ ,β )( T )] , showing that E ( ˜ G θ ,β + ˜ H θ ,β ) is a martingale if and only if E ( ˜ G θ ,β ) and E ( ˜ H θ ,β ) are also martin-gales. Hence, we can write dQ ¯ θ, ¯ β dP (cid:12)(cid:12)(cid:12)(cid:12) F t = dQ θ ,β dP (cid:12)(cid:12)(cid:12)(cid:12) F t × dQ θ ,β dP (cid:12)(cid:12)(cid:12)(cid:12) F t , t ∈ [0 , T ] , where dQ θ ,β dP (cid:12)(cid:12)(cid:12) F t , E ( ˜ G θ ,β )( t ) and dQ θ ,β dP (cid:12)(cid:12)(cid:12) F t , E ( ˜ H θ ,β )( t ) , t ∈ [0 , T ] . The previous reasonings allow us to reduce the proof that Q ¯ θ, ¯ β is a probability measure equivalent to P, Q ¯ θ, ¯ β ∼ P , to prove that E ( ˜ G θ ,β ) is martingale (or Q θ ,β ∼ P ) and E ( ˜ H θ ,β ) is a martingale withstrictly positive values (or Q θ ,β ∼ P ). The literature on this topic is huge, see for instance Kazamaki[17], Novikov [21], L´epingle and M´emin [19] and Kallsen and Shiryaev [16]. The main difficulty whentrying to use the classical criteria is that our kernels depend on the processes X and Y, which are un-bounded. To prove that E ( ˜ G θ ,β ) is a martingale one could use a localized version of Novikov’s criterion. PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 7
However, this approach would entail to show that the expectation of the exponential of the integral of astochastic iterated integral of order two is finite. Although these computations seem feasible, they aredefinitely very stodgy. On the other hand, the most widely used sufficient criterion for martingales withjumps is the L´epingle-M´emin criterion. This criterion is very general but the conditions obtained are farfrom optimal. Using this criterion we are only able to prove the result by requiring the L´evy process L tohave bounded jumps.In a very recent paper, assuming some structure on the processes, Klebaner and Lipster [18] give a fairlygeneral criterion which seems easier to apply than those of Novikov and L´epingle-M´emin. Although wecan not apply directly their criteria, at least not in the pure jump case, we can reason similarly to prove thedesired result for E ( ˜ G θ ,β ) and E ( ˜ H θ ,β ) . Finally, note that these results can be extended, in a straightforward manner, to any finite number ofLangevin equations driven by Brownian motions and L´evy processes, independent of each other. In thefollowing two subsections, we will drop the subindices in the parameters θ and β. Brownian driven OU-process.
We first show that the process ˜ G θ,β is a martingale under P . Proposition 3.6.
Let θ ∈ R and β ∈ [0 , . Then, ˜ G θ,β = { ˜ G θ,β ( t ) } t ∈ [0 ,T ] , defined by (3 . , is a squareintegrable martingale under P .Proof. We have to show that G θ,β ∈ L (Ω × [0 , T ]; P ⊗ Leb) . We get E P [ Z T G θ,β ( t ) dt ] ≤ σ − X { θ T + α X E P [ Z T X ( t ) dt ] } . By remark 2.3 and the properties of the Gaussian distribution, one has E P [ Z T X ( t ) dt ] = Z T ( m t + Σ t ) dt ≤ T sup t ∈ [0 ,T ] ( m t + Σ t ) < ∞ , because m t and Σ t are continuous functions on [0 , T ] . (cid:3) Theorem 3.7.
Let θ ∈ R and β ∈ [0 , . Then E ( ˜ G θ,β ) = {E ( ˜ G θ,β )( t ) } t ∈ [0 ,T ] is a martingale under P. Proof. As ˜ G θ,β is a martingale with continuous paths, we have that E ( ˜ G θ,β ) is a positive local martingale.By remark 3.3, it suffices to prove that E P [ E ( ˜ G θ,β )( T )] = 1 . Note that the sequence of stopping times τ n = inf { t : E ( ˜ G θ,β ) > n } ∧ T, n ≥ is a reducing sequence for E ( ˜ G θ,β ) . That is, τ n converges a.s. to T and, for every n ≥ fixed, the stopped process E ( ˜ G θ,β ) τ n ( t ) , E ( ˜ G θ,β )( t ∧ τ n ) is a (bounded) martingaleon [0 , T ] . Therefore, E P [ E ( ˜ G θ,β ) τ n ( T )] = E P [ E ( ˜ G θ,β ) τ n (0)] = 1 , n ≥ , and if we show that lim n →∞ E P [ E ( ˜ G θ,β ) τ n ( T )] = E P [ E ( ˜ G θ,β )( T )] (3.13)we will have finished. To show (3 . is equivalent to show the uniform integrability of the sequence ofrandom variable {E ( ˜ G θ,β ) τ n ( T ) } n ≥ , that is, to show lim M →∞ sup n ≥ E P [ E ( ˜ G θ,β ) τ n ( T ) {E ( ˜ G θ,β ) τn ( T ) >M } ] = 0 . It is not difficult to prove that if Λ( t ) is a non-negative function such that lim t →∞ Λ( t ) /t = ∞ and sup n ≥ E P [Λ( E ( ˜ G θ,β ) τ n ( T ))] < ∞ , then {E ( ˜ G θ,β ) τ n ( T ) } n ≥ is uniformly integrable. We consider the test function Λ( t ) = 1 + t log( t ) . Hence, it suffices to prove that sup n ≥ E P [ E ( ˜ G θ,β ) τ n ( T ) log( E ( ˜ G θ,β ) τ n ( T ))] < ∞ . (3.14) BENTH AND ORTIZ-LATORRE
Note that we can use the sequence of martingales on [0 , T ] given by {E ( ˜ G θ,β ) τ n } n ≥ to define a sequenceof probability measures { Q nθ,β } n ≥ with Radon-Nykodim densities given by dQ nθ,β dP (cid:12)(cid:12)(cid:12) F t , E ( ˜ G θ,β ) τ n ( t ) , t ∈ [0 , T ] , n ≥ . In addition, one has that E ( ˜ G θ,β ) τ n ( t ) = exp (cid:18)Z t ∧ τ n G θ,β ( s ) dW ( s ) − Z t ∧ τ n G θ,β ( s ) ds (cid:19) (3.15) = exp (cid:18)Z t [0 ,τ n ] ( s ) G θ,β ( s ) dW ( s ) − Z t ( [0 ,τ n ] ( s ) G θ,β ( s )) ds (cid:19) = E ( ˜ G nθ,β )( t ) , t ∈ [0 , T ] , n ≥ , where ˜ G nθ,β ( t ) , R t [0 ,τ n ] ( s ) G θ,β ( s ) dW ( s ) , t ∈ [0 , T ] , n ≥ . On the other hand, from (3 . , we havethe trivial bound log( E ( ˜ G θ,β ) τ n ( T )) ≤ ˜ G τ n θ,β ( T ) . Combining the last bound with the change of measuregiven by { Q nθ,β } n ≥ we get that sup n ≥ E Q nθ,β [ ˜ G τ n θ,β ( T )] < ∞ , (3.16)implies that (3 . holds. Applying Girsanov’s Theorem, we can write ˜ G τ n θ,β ( T ) = Z T [0 ,τ n ] ( t )( G θ,β ( t )) dt + Z T [0 ,τ n ] ( t ) G θ,β ( t ) dW Q nθ,β ( t ) , where W Q nθ,β is a Q nθ,β -Brownian motion. Therefore, it suffices to prove that sup n ≥ E Q nθ,β [ Z T [0 ,τ n ] ( t )( G θ,β ( t )) dt ] < ∞ , (3.17)because this imply that R T ∧ τ n G θ,β ( t ) dW Q nθ,β ( t ) is a Q nθ,β -martingale with zero expectation and, in pass-ing, that (3 . holds. Now we proceed as in the proof of Proposition 3.6. We have that E Q nθ,β [ Z T [0 ,τ n ] ( t )( G θ,β ( t )) dt ] ≤ σ − X { θ T + α X E Q nθ,β [ Z T [0 ,τ n ] ( t ) X ( t ) dt ] } , but now the term with X ( t ) is more delicate to treat. Using Remark 2.3, we know that X ( t ) condi-tioned to τ n is Gaussian, but we do not know the distribution of τ n and, hence, a direct computation of E Q nθ,β [ [0 ,τ n ] ( t ) X ( t ) ] is not possible. However, we have that E Q nθ,β [ Z T [0 ,τ n ] ( t ) X ( t ) dt ] ≤ { E Q nθ,β [ Z T [0 ,τ n ] ( t ) (cid:18) X (0) e − α X (1 − β ) t + µ X + θα X (1 − β ) (cid:16) − e − α X (1 − β ) t (cid:17)(cid:19) dt ]+ σ X E Q nθ,β [ Z T [0 ,τ n ] ( t ) (cid:18)Z t e − α X (1 − β )( t − u ) dW Q nθ,β ( u ) (cid:19) dt ] }≤ T { ( | X (0) | + ( | µ X | + | θ | ) T ) + σ X T } < ∞ , where we have used that the function η ( x ) , (1 − e − xa ) /x ≤ a for x, a ≥ , and that E Q nθ,β "(cid:18)Z t e − α X (1 − β )( t − u ) dW Q nθ,β ( u ) (cid:19) = Z t e − α X (1 − β )( t − u ) du ≤ T. Hence, we have shown (3 . and the result follows. (cid:3) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 9
L´evy driven OU-processes.
First we will prove that ˜ H θ,β is a square integrable martingale. Proposition 3.8.
Let θ ∈ D L , β ∈ [0 , . Then ˜ H θ,β = { ˜ H θ,β ( t ) } t ∈ [0 ,T ] , defined by (3 . , is a squareintegrable martingale under P. Proof.
According to Ikeda-Watanabe [13], p. 59-63, we have to check that E P [ R T R ∞ | H θ,β ( s, z ) − | ℓ ( dz ) dt ] < ∞ . We can write E P [ Z T Z ∞ | H θ,β ( s, z ) − | ℓ ( dz ) dt ] ≤ T Z ∞ | e θz − | ℓ ( dz )+ α Y (cid:0) κ ′′ L ( θ ) (cid:1) Z ∞ e θz z ℓ ( dz ) Z T E P [ | Y ( t ) | ] dt. By the mean value theorem in integral form we have that | e θz − | = | θz R e λθz dλ | ≤ θ z e (2 θ ∨ z . Hence, as θ ∈ D L , Z ∞ | e θz − | ℓ ( dz ) ≤ θ Z ∞ z e θz ℓ ( dz ) = θ κ ′′ L (2 θ ∨ < ∞ . Therefore, the result follows by showing that sup t ∈ [0 ,T ] E P [ | Y ( t ) | ] < ∞ . We have that sup t ∈ [0 ,T ] E P [ | Y ( t ) | ] ≤ t ∈ [0 ,T ] { (cid:18) Y (0) e − α Y t + µ Y + κ ′ L (0) α Y (1 − e − α Y t ) (cid:19) + E P [ (cid:18)Z t Z ∞ ze − α Y ( t − s ) ˜ N L ( ds, dz ) (cid:19) ] }≤ (cid:18) Y (0) + µ Y + κ ′ L (0) α Y (cid:19) + sup t ∈ [0 ,T ] Z t Z ∞ z e − α Y ( t − s ) ℓ ( dz ) ds ≤ (cid:18) Y (0) + µ Y + κ ′ L (0) α Y (cid:19) + T κ ′′ L (0) < ∞ . (cid:3) Note that the stochastic exponential E ( ˜ H θ,β ) satisfies the following SDE E ( ˜ H θ,β )( t ) = 1 + Z t E ( ˜ H θ,β )( s − ) d ˜ H θ,β ( s ) = 1 + Z t Z ∞ E ( ˜ H θ,β )( s − ) (cid:16) ˜ H θ,β ( s, z ) − (cid:17) ˜ N L ( ds, dz ) , and it can be represented explicitly as E ( ˜ H θ,β )( t ) = e ˜ H θ,β ( t ) Y − , up toan evanescent set. Moreover, by the definition of ˜ H θ,β ( t ) and H θ,β ( t, z ) we have that ∆ ˜ H θ,β ( t ) = H θ,β ( t, ∆ L ( t )) − e θ ∆ L ( t ) −
1) + α Y βκ ′′ L ( θ ) ∆ L ( t ) e θ ∆ L ( s ) Y ( t − ) , t ∈ [0 , T ] , (3.19)which yields the condition P ( α Y βκ ′′ L ( θ ) (∆ L ( t )) Y ( t − ) > − , t ∈ [0 , T ]) = 1 . (3.20) Remark 3.9.
As we assume that L is a subordinator and Y (0) ≥ and µ ≥ , we have that P ( Y ( t ) ≥ , t ∈ [0 , T ]) = 1 , condition (3 . is automatically satisfied and E ( ˜ H θ,β ) , is strictly positive. Theorem 3.10.
Let θ ∈ D L and β ∈ [0 , . Then E ( ˜ H θ,β ) = {E ( ˜ H θ,β )( t ) } t ∈ [0 ,T ] is a martingale under P .Proof. As ˜ H θ,β is a martingale on [0 , T ] , we have that E ( ˜ H θ,β ) , is a local martingale on [0 , T ] . Hence,there exists a sequence of increasing stopping times such that τ n ↑ T, P - a.s. and the stopped processes E ( ˜ H θ,β ) τ n , n ≥ are martingales on [0 , T ] . By Remark 3.3 and the same reasonings as in the proof ofTheorem 3.7, to show that E ( ˜ H θ,β ) is a martingale is equivalent to show that E [ E ( ˜ H θ,β )( T )] = 1 andthis is equivalent to prove that the sequence {E ( ˜ H θ,β ) τ n ( T ) } n ≥ is uniformly integrable. A sufficientcondition for the uniform integrability of {E ( ˜ H θ,β ) τ n ( T ) } n ≥ is given by sup n ≥ E P [ E ( ˜ H θ,β ) τ n ( T ) log( E ( ˜ H θ,β ) τ n ( T ))] < ∞ . (3.21)By equation (3 . , we get log( E ( ˜ H θ,β ) τ n ( T )) ≤ ˜ H τ n θ,β ( T ) − X ≤ t ≤ τ n ∧ T ∆ ˜ H θ,β ( t ) − log(1 + ∆ ˜ H θ,β ( t )) ≤ ˜ H τ n θ,β ( T ) , because the function x − log(1 + x ) ≥ for x > − . Hence, we can write E P [ E ( ˜ H θ,β ) τ n ( T ) ˜ H τ n θ,β ( T )]= E P [ (cid:18) Z T ∧ τ n E ( ˜ H θ,β )( t − ) d ˜ H θ,β ( t ) (cid:19) ˜ H τ n θ,β ( T )]= E P [ (cid:18) Z T E ( ˜ H θ,β ) τ n ( t − ) d ˜ H τ n θ,β ( t ) (cid:19) ˜ H τ n θ,β ( T )]= E P [ ˜ H τ n θ,β ( T )] + E P [ (cid:18)Z T E ( ˜ H θ,β ) τ n ( t − ) d ˜ H τ n θ,β ( t ) (cid:19) (cid:18)Z T [0 ,τ n ] ( t ) d ˜ H τ n θ,β ( t ) (cid:19) ]= Z T Z ∞ E P [ [0 ,τ n ] ( t ) E ( ˜ H θ,β ) τ n ( t ) (cid:18) e θz − α Y βκ ′′ L ( θ ) e θz zY ( t ) (cid:19) ] ℓ ( dz ) dt = E P [ E ( ˜ H θ,β ) τ n ( T ) Z T Z ∞ [0 ,τ n ] ( t ) (cid:18) e θz − α Y βκ ′′ L ( θ ) e θz zY ( t ) (cid:19) ℓ ( dz ) dt ] ≤ T Z ∞ (cid:12)(cid:12)(cid:12) e θz − (cid:12)(cid:12)(cid:12) ℓ ( dz ) + 2 α Y κ ′′ L (2 θ )( κ ′′ L ( θ )) E P [ E ( ˜ H θ,β ) τ n ( T ) Z T ∧ τ n Y ( t ) dt ] , (3.22)where we have used that for any stopping time τ ≤ T the process ˜ H τθ,β ( T ) is a P -martingale with zeroexpectation. In addition, we have used that ∀ n ≥ fixed, E P [ E ( ˜ H θ,β ) τ n ( T )] = 1 and E P [ E P [ E ( ˜ H θ,β ) τ n ( T ) [0 ,τ n ] ( t ) (cid:18) e θz − α Y βκ ′′ L ( θ ) e θz zY ( t ) (cid:19) |F t ]]= E P [ [0 ,τ n ] ( t ) E P [ E ( ˜ H θ,β ) τ n ( T ) |F t ] (cid:18) e θz − α Y βκ ′′ L ( θ ) e θz zY ( t ) (cid:19) ]= E P [ [0 ,τ n ] ( t ) E ( ˜ H θ,β ) τ n ( t ) (cid:18) e θz − α Y βκ ′′ L ( θ ) e θz zY ( t ) (cid:19) ] , because τ n is a reducing sequence for the local martingale E ( ˜ H θ,β ) . One can reason as in the proof ofProposition 3.8 to show that the terms R ∞ (cid:12)(cid:12) e θz − (cid:12)(cid:12) ℓ ( dz ) and κ ′′ L (2 θ ) , in equation (3 . , are finite. PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 11
Note that R T ∧ τ n Y ( t ) dt = R T ∧ τ n Y ( t ∧ τ n ) dt ≤ R T Y τ n ( t ) dt, thus, it just remains to prove that sup n ≥ E P [ E ( ˜ H θ,β ) τ n ( T ) Z T Y τ n ( t ) dt ] < ∞ , to finish the proof. As E ( ˜ H θ,β ) τ n is a strictly positive martingale, by Remark 3.9, we can define theprobability measure Q nθ,β ∼ P by setting dQ nθ,β dP (cid:12)(cid:12)(cid:12) F t , E ( ˜ H θ,β ) τ n ( t ) , t ∈ [0 , T ] , and, hence, it sufficesto prove that sup n ≥ E Q nθ,β [ R T Y τ n ( t ) dt ] < ∞ . Using Girsanov’s Theorem with Q nθ,β ∼ P, n ≥ , theprocess Y τ n can be written as Y τ n ( t ) = Y (0) + ˜ B τ n ( t ) + Z t Z ∞ [0 ,τ n ] ( s ) z ˜ N LQ nθ,β ( ds, dz ) t ∈ [0 , T ] , where ˜ B τ n ( t ) = Z t [0 ,τ n ] ( s )( µ Y + κ ′ L (0) − α Y Y ( s )) ds + Z t Z R z [0 ,τ n ] ( s )( H θ,β ( s, z ) − ℓ ( dz ) ds = Z t [0 ,τ n ] ( s ) { ( µ Y + κ ′ L (0) − α Y Y ( s )) + Z R z ( e θz − ℓ ( dz ) + α Y βκ ′′ L ( θ ) Z R z e θz ℓ ( dz ) Y ( s ) } ds = Z t [0 ,τ n ] ( s ) (cid:0) µ Y + κ ′ L ( θ ) − α Y (1 − β ) Y ( s ) (cid:1) ds, t ∈ [0 , T ] , and ˜ N LQ nθ,β ( ds, dz ) is the compensated version of the random measure N LQ nθ,β ( ds, dz ) with Q nθ,β -compensatorgiven by ˜ ν LQ nθ,β ( ds, dz ) = { [0 ,τ n ] ( s )( H θ,β ( s, z ) −
1) + 1 } ℓ ( dz ) ds. Hence, E Q nθ,β [( Y τ n ( t )) ] ≤ { Y (0) + E Q nθ,β [ (cid:18)Z t [0 ,τ n ] ( s )( µ Y + κ ′ L ( θ ) + α Y (1 − β ) Y ( s )) ds (cid:19) ]+ E Q nθ,β [ (cid:18)Z t Z ∞ [0 ,τ n ] ( s ) z ˜ N LQ nθ,β ( ds, dz ) (cid:19) ] }≤ { Y (0) + T E Q nθ,β [ Z t [0 ,τ n ] ( s )( µ Y + κ ′ L ( θ ) + α Y (1 − β ) Y τ n ( s )) ds ]+ E Q nθ,β [ Z t Z ∞ [0 ,τ n ] ( s ) z { [0 ,τ n ] ( s )( H θ,β ( s, z ) −
1) + 1 } ℓ ( dz ) ds ] } . On the one hand, E Q nθ,β [ Z t [0 ,τ n ] ( s )( µ Y + κ ′ L ( θ ) + α Y (1 − β ) Y τ n ( s )) ds ] ≤ T ( µ Y + κ ′ L ( θ )) + 2 α Y Z t E Q nθ,β [( Y τ n ( s )) ] ds. On the other hand, E Q nθ,β [ Z t Z ∞ [0 ,τ n ] ( s ) z { [0 ,τ n ] ( s )( H θ,β ( s, z ) −
1) + 1 } ℓ ( dz ) ds ]= E Q nθ,β [ Z t Z ∞ [0 ,τ n ] ( s ) z H θ,β ( s, z ) ℓ ( dz ) ds ]= E Q nθ,β [ Z t Z ∞ [0 ,τ n ] ( s ) z (cid:18) e θz + α Y βκ ′′ L ( θ ) e θz zY ( s − ) (cid:19) ℓ ( dz ) ds ] ≤ T Z ∞ z e θz ℓ ( dz ) + E Q nθ,β [ Z t Z ∞ [0 ,τ n ] ( s ) α Y βκ ′′ L ( θ ) e θz z Y τ n ( s ) ℓ ( dz ) ds ] ≤ T κ ′′ L ( θ ) + α Y βκ ′′ L ( θ ) Z ∞ z e θz ℓ ( dz ) Z t E Q nθ,β [ Y τ n ( s )] ds ≤ T κ ′′ L ( θ ) + α Y κ (3) L ( θ ) κ ′′ L ( θ ) Z t E Q nθ,β [( Y τ n ( s )) ] ds. To sum up, E Q nθ,β [( Y τ n ( t )) ] ≤ C + C R t E Q nθ,β [( Y τ n ( s )) ] ds, where C = C ( Y (0) , µ Y , θ, T ) , Y (0) + 8 T ( µ Y + κ ′ L ( θ )) + 4 T κ ′′ L ( θ ) ,C = C ( α Y , T ) , T α Y + 4 α Y κ (3) L ( θ ) κ ′′ L ( θ ) , and applying Gronwall’s lemma to the function E Q nθ,β [ Y τ n ( t ) ] , we get that E Q nθ,β [ Y τ n ( t ) ] ≤ C e C T . (3.23)Finally, using Fubini-Tonelli and inequality (3 . we obtain sup n ≥ E Q nθ,β [ Z T Y τ n ( t ) dt ] ≤ sup n ≥ Z T E Q nθ,β [ Y τ n ( t ) ] dt ≤ T C e C T < ∞ , and the proof is finished. (cid:3) Remark 3.11. If L has finite activity, that is ℓ ((0 , ∞ )) < ∞ , then one can use the kernel M θ,β ( t, z ) , e θz (cid:18) α Y βκ ′ L ( θ ) Y ( t − ) (cid:19) , t ∈ [0 , T ] , z ∈ R , and the Poisson integral ˜ M θ,β ( t ) , Z t Z ∞ ( M θ,β ( s, z ) −
1) ˜ N L ( ds, dz ) , to define the change of measure. The results in Proposition 3.8 and Theorem 3.10, below, also hold. Notethat the change of measure with ˜ M θ,β does not work for the infinite activity case. This is because, inthe analogous proofs of the statements in Proposition 3.8 and Theorem 3.10 using the change of measureinduced by ˜ M θ,β , it appears the integral R ∞ e θz ℓ ( dz ) , which is divergent if ℓ ((0 , ∞ )) = ∞ .
4. S
TUDY OF THE RISK PREMIUM
We are interested in applying the previous probability measure change to study the risk premium inelectricity markets. As we discussed in the Introduction, there are two reasonable models for the spotprice S in this market: the arithmetic and the exponential model. We define the arithmetic spot pricemodel by S ( t ) = Λ a ( t ) + X ( t ) + Y ( t ) , t ∈ [0 , T ∗ ] , (4.1)and the geometric spot price model by S ( t ) = Λ g ( t ) exp( X ( t ) + Y ( t )) , t ∈ [0 , T ∗ ] , (4.2)where T ∗ > is a fixed time horizon. The processes Λ a and Λ g are assumed to be deterministic and theyaccount for the seasonalities observed in the spot prices.One of the particularities of electricity markets is that power is a non storable asset and for that reasonis not a directly tradeable asset. This entails that one can not derive the forward price of electricity fromthe classical buy-and-hold hedging arguments. Using a risk-neutral pricing argument (see Benth, ˇSaltyt˙eBenth and Koekebakker [2]), under the assumption of deterministic interest rates, the forward price, withtime of delivery < T < T ∗ , at time < t < T is given by F Q ( t, T ) , E Q [ S ( T ) |F t ] , where Q is anyprobability measure equivalent to the historical measure P and F t is the market information up to time t . In what follows we will use the probability measure Q discussed in the previous sections. However,in electricity markets, the delivery of the underlying takes place over a period of time [ T , T ] , where PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 13 < T < T < T ∗ . We call such contracts swap contracts and we will denote their price at time t < T by F Q ( t, T , T ) , E Q [ 1 T − T Z T T S ( T ) dT |F t ] . We can use the stochastic Fubini theorem to relate the price of forwards and swaps F Q ( t, T , T ) , T − T Z T T F Q ( t, T ) dT. The risk premium for forward prices is defined by the following expression R FQ ( t, T ) , E Q [ S ( T ) |F t ] − E P [ S ( T ) |F t ] , and for swap prices by R SQ ( t, T , T ) , F Q ( t, T , T ) − E Q [ 1 T − T Z T T S ( T ) dT |F t ] = 1 T − T Z T T R FQ ( t, T ) dT. (4.3)In order to compute the previous quantities we need to know the dynamics of S (that is, of X and Y )under P and under Q. Explicit expressions for X and Y under P are given in equations (2 . and (2 . , respectively. In the rest of the paper, Q = Q ¯ θ, ¯ β , ¯ θ ∈ ¯ D L , ¯ β ∈ [0 , defined in (3 . , and the explicitexpressions for X and Y under Q are given in Remark . , equations (3 . and (3 . , respectively. Remark 4.1.
We will use the subindices a and g to denote the arithmetic and the geometric spot models,respectively. That is, we will use the notation R Fa,Q ( t, T ) , R Fg,Q ( t, T ) , R Sa,Q ( t, T , T ) and R Sg,Q ( t, T , T ) . Remark 4.2.
In the discussion to follow, we are interested in finding values of the parameters ¯ θ, ¯ β suchthat some empirical features of the observed risk premium profiles are reproduced by our pricing measure.In particular, we show that is possible to have the sign of the risk premium changing stochastically frompositive values on the short end of the market to negative values on the long end. This is proved for forwardcontracts in, both, the arithmetic and geometric model. Equation (4 . just tell us that the risk premiumfor swaps becomes the average of the risk premium for forwards with fixed-delivery. Hence, we can obtainstochastic sign change also for these, depending on the length of delivery. Worth noticing is that contractsin the short end have short delivery (a day, or a week), while in the long end have month/quarter/yeardelivery. Average for negative is negative, for the long end, and average over short period, dominantlypositive, gives positive, in the short end. Arithmetic spot price model.
We assume in this section that the spot price S ( t ) is given by thedynamics (4.1) for ≤ t ≤ T ∗ , T ∗ > , with the maturity time of the forward contract T satisfying < T < T ∗ . Using equations (2 . and (2 . and the basic properties of the conditional expectation weget E P [ S ( T ) |F t ] = Λ a ( T ) + E P [ X ( t ) e − α X ( T − t ) + µ X α X (1 − e − α X ( T − t ) ) |F t ]+ E P [ Y ( t ) e − α Y ( T − t ) + µ Y + κ ′ L (0) α Y (1 − e − α Y ( T − t ) ) |F t ]+ E P [ σ X Z Tt e − α X ( T − s ) dW ( s ) + Z Tt Z ∞ e − α Y ( T − s ) z ˜ N L ( ds, dz ) |F t ]= Λ a ( T ) + X ( t ) e − α X ( T − t ) + Y ( t ) e − α Y ( T − t ) + µ X α X (1 − e − α X ( T − t ) ) + µ Y + κ ′ L (0) α Y (1 − e − α Y ( T − t ) )+ E P [ σ x Z Tt e − α X ( T − s ) dW ( s )] + E P [ Z Tt Z ∞ e − α Y ( T − u ) z ˜ N L ( ds, dz )]= Λ a ( T ) + X ( t ) e − α X ( T − t ) + Y ( t ) e − α Y ( T − t ) + µ X α X (1 − e − α X ( T − t ) ) + µ Y + κ ′ L (0) α Y (1 − e − α Y ( T − t ) ) . Note that we have also used that W and ˜ N L have independent increments under P to write conditionalexpectations as expectations. If we assume that α , α X = α Y , then E P [ S ( T ) |F t ] = Λ a ( T ) + ( S ( t ) − Λ( t )) e − α ( T − t ) + µ X + µ Y + κ ′ L (0) α (1 − e − α ( T − t ) ) . This last expression for E P [ S ( T ) |F t ] is considerably simpler and depends explicitly on S ( t ) , the spotprice at time t, which is directly observable in the market.To find a similar expression for E Q [ S ( T ) |F t ] we need the following lemma. Lemma 4.3.
We have that R t R ∞ e α Y (1 − β ) s z ˜ N LQ ( ds, dz ) is a Q -martingale on [0 , T ] , T > . Proof.
We have to prove that E Q [ R t R ∞ e α Y (1 − β ) s zv LQ ( ds, dz )] < ∞ . One has that E Q [ Z t Z ∞ e α Y (1 − β ) s zv LQ ( ds, dz )] = E Q [ Z t Z ∞ e α Y (1 − β ) s zH θ ,β ( s, z ) ℓ ( dz ) ds ]= E Q [ Z t Z ∞ e α Y (1 − β ) s z (cid:18) e θ z + α Y β κ ′′ L ( θ ) e θ z zY ( s ) (cid:19) ℓ ( dz ) ds ] ≤ e α Y T { T κ ′ L ( θ ) + α Y T sup ≤ t ≤ T E Q [ Y ( t )] } , and κ ′ L ( θ ) < ∞ because θ ∈ D L . The proof that sup ≤ s ≤ T E Q [ Y ( s )] is finite follows the same linesas the last part of Theorem 3.10. Using the semimartingale representation of Y, equation (3 . , we obtainthat there exist constants C and C such that E Q [ Y ( t )] ≤ C + C R t E Q [ Y ( s )] ds. Applying Gronwall’sLemma we get that E Q [ Y ( t )] ≤ C e C T and the result follows. (cid:3) Remark 4.4.
We need the previous lemma because Girsanov’s Theorem just ensures that Z t Z ∞ e α Y (1 − β ) s z ˜ N LQ ( ds, dz ) (4.4) is a Q -local martingale. We want (4 . to be a Q -martingale because then it follows trivially that E Q [ Z Tt Z ∞ e α Y (1 − β ) s z ˜ N LQ ( ds, dz ) |F t ] = 0 . Note that we can not reduce the previous conditional expectation (unless β = 0 , which coincides withthe Esscher change of measure) to an expectation because the compensator of N LQ depends on Y and,therefore, ˜ N LQ does not has independent increments. Using the basic properties of the conditional expectation, Remark 4.4 and equations (3 . and (3 . we get E Q [ S ( T ) |F t ] = Λ a ( T ) + E Q [ X ( t ) e − α X (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) |F t ]+ E Q [ Y ( t ) e − α Y (1 − β )( T − t ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) ) |F t ]+ E Q [ σ X Z Tt e − α X (1 − β )( T − s ) dW Q ( s ) |F t ]+ E Q [ Z Tt Z ∞ e − α Y (1 − β )( T − s ) z ˜ N LQ ( ds, dz ) |F t ]= Λ a ( T ) + X ( t ) e − α X (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 15 + Y ( t ) e − α Y (1 − β )( T − t ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) )+ E Q [ σ X Z Tt e − α X (1 − β )( T − s ) dW Q ( s )]+ e − α Y (1 − β ) T E Q [ Z Tt Z ∞ e α Y (1 − β ) s z ˜ N LQ ( ds, dz ) |F t ]= Λ a ( T ) + X ( t ) e − α X (1 − β )( T − t ) + Y ( t ) e − α Y (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) ) . Therefore, we have proved the following result.
Proposition 4.5.
The forward price F Q ( t, T ) in the arithmetic spot model (4.1) is given by F Q ( t, T ) = Λ a ( T ) + X ( t ) e − α X (1 − β )( T − t ) + Y ( t ) e − α Y (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) ) . In Lucia and Schwartz [20] a two-factor model (among others) is proposed as the dynamics for powerspot prices in the Nordic electricity market NordPool. Following the model of Schwartz and Smith [23],they consider a non-stationary long term variation factor together with a stationary short term variationfactor. In our context, one could let the mean reversion in X be zero, to obtain a non-stationary factor as adrifted Brownian motion under the pricing measure Q . After doing a measure transform with β = 1 , wecan price forwards as in Proposition 4.5 to find F Q ( t, T ) = Λ a ( T ) + X ( t ) + Y ( t ) e − α Y (1 − β )( T − t ) + ( µ X + θ )( T − t )+ µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) ) . When T − t becomes large, i.e. when we are far out on the forward curve, we see that F Q ( t, T ) ∼ Λ a ( T ) + X ( t ) + ( µ X + θ )( T − t ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) . (4.5)Thus, the forward curve moves stochastically as the non-stationary factor X . If one, on the other hand, let X be stationary, we find that the forward price in Proposition 4.5 will behave for large time to maturities T − t as F Q ( t, T ) ∼ Λ a ( T ) + µ X + θ α X (1 − β ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) . The forward prices becomes constant after subtracting the seasonal function, with no stochastic move-ments. This is not what is observed for forward data in the market. However, following the empiricalstudy in Barndorff-Nielsen, Benth and Veraart [1], electricity spot prices on the German power exchangeEEX are stationary. One way to have a stationary spot dynamics, and still maintain forward prices whichmoves randomly in the long end, is to apply our measure change to slow down the mean reversion in oneor more factors of the (stationary) spot. In the extreme case, we can let β = 1 , and obtain a non-stationaryfactor X under the pricing measure, in which case we obtain the same long term asymptotic behaviour asin the generalization of the Lucia and Schwartz model (4.5). In conclusion, our pricing measure allowsfor a stationary spot dynamics and a forward price dynamics which is not constant in the long end.Let us return back to the risk premium, which in view of Prop. 4.5 becomes: Proposition 4.6.
The risk premium R Fa,Q ( t, T ) for the forward price in the arithmetic spot model (4 . isgiven by R Fa,Q ( t, T ) = X ( t ) e − α X ( T − t ) ( e α X β ( T − t ) −
1) + Y ( t ) e − α Y ( T − t ) ( e α Y β ( T − t ) − + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + µ Y + κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β )( T − t ) ) − µ X α X (1 − e − α X ( T − t ) ) − µ Y + κ ′ L (0) α Y (1 − e − α Y ( T − t ) ) . We analyse different cases for the risk premium in the next subsection.4.1.1.
Discussion on the risk premium.
The first remarkable property of this measure change is that, aslong as the parameter ¯ β = (0 , , the risk premium is stochastic. This might be a desirable feature inview of the discussion in the Introduction where we referred to the economical and empirical evidencein Geman and Vasicek [10], Bessembinder and Lemon [5] and Benth, Cartea and Kiesel [3]. Note thatwhen ¯ β = (0 , , our measure change coincides with the Esscher transform (see Benth, ˇSaltyt˙e Benth andKoekebakker [2]). In the Esscher case, the risk premium has a deterministic evolution given by R Fa,Q ( t, T ) = θ α X (1 − e − α X ( T − t ) ) + κ ′ L ( θ ) − κ ′ L (0) α Y (1 − e − α Y ( T − t ) ) , (4.6)an already known result, see Benth and Sgarra [4].Another interesting feature of the empirical risk premium is that its sign might change from positiveto negative when the time to maturity τ , T − t increases. Hence, we are interested in theoreticalmodels that allow to reproduce such empirical property. From now on we shall rewrite the expressionsfor the risk premium in terms of the time to maturity τ and, slightly abusing the notation, we will write R Fa,Q ( t, τ ) instead of R Fa,Q ( t, t + τ ) . We fix the parameters of the model under the historical measure P, i.e., µ X , α X , σ X , µ Y , and α Y , and study the possible sign of R Fa,Q ( t, τ ) in terms of the change of measureparameters, i.e., ¯ β = ( β , β ) and ¯ θ = ( θ , θ ) and the time to maturity τ. Note that the present time justenters into the picture through the stochastic components X and Y. We are going to assume µ X = µ Y = 0 . This assumption is justified, from a modeling point of view, because we want the processes X and Y torevert toward zero. In this way, the seasonality function Λ a accounts completely for the mean pricelevel. On the other hand it is also reasonable to expect that α X < α Y , which means that the componentaccounting for the jumps reverts the fastest (e.g., being the factor modelling the spikes). The factor X isreferred to as the base component, modelling the normal price variations when the market is not underparticular stress. The expression for R Fa,Q ( t, τ ) given in Proposition 4.6 allows for a quite rich behaviour.We are going to study the cases ¯ θ = (0 , , ¯ β = (0 , and the general case separately. Moreover, inorder to graphically illustrate the discussion we plot the risk premium profiles obtained assuming that thesubordinator L is a compound Poisson process with jump intensity c/λ > and exponential jump sizeswith mean λ. That is, L will have the L´evy measure given in Example 3.2. We shall measure the timeto maturity τ in days and plot R Fa,Q ( t, τ ) for τ ∈ [0 , , roughly one year. We fix the values of thefollowing parameters α X = 0 . , α Y = 0 . , c = 0 . , λ = 2 . The speed of mean reversion for the base component α X yields a half-life of seven days, while the one forthe spikes α Y yields a half-life of two days (see e.g., Benth, Saltyte Benth and Koekebakker [2] for theconcept of half-life). The values for c and λ give jumps with mean . and frequency of spikes a month.The following lemma will help us in the discussion to follow. Lemma 4.7. If µ X = µ Y = 0 and α X < α Y , we have that the risk premium R Fa,Q ( t, τ ) satisfies R Fa,Q ( t, τ ) = X ( t ) e − α X τ ( e α X β τ −
1) + Y ( t ) e − α Y τ ( e α Y β τ − (4.7) + θ α X (1 − β ) (1 − e − α X (1 − β ) τ ) + κ ′ L ( θ ) − κ ′ L (0) α Y (1 − β ) (1 − e − α Y (1 − β ) τ )+ κ ′ L (0) α Y Λ( α Y τ, − β ) , PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 17
50 100 150 200 250 300 350 Τ- - H Τ L (a) θ = 0 . , θ = 0
50 100 150 200 250 300 350 Τ- - H Τ L (b) θ = − . , θ = 0
50 100 150 200 250 300 350 Τ- - H Τ L (c) θ = 0 , θ = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (d) θ = 0 , θ = − . F IGURE
1. Risk premium profiles when L is a compound Poisson process with expo-nentially distributed jumps. Esscher transform: case ¯ β = (0 , . Arithmetic spot pricemodel where Λ( x, y ) = 1 − e − xy y − (1 − e − x ) , x ∈ R + , y ∈ [0 , , lim x →∞ Λ( x, y ) = 1 − yy , lim x → ∂∂x Λ( x, y ) = 0 , is a non-negative function. Moreover, lim τ →∞ R Fa,Q ( t, τ ) = θ α X (1 − β ) + κ ′ L ( θ ) − κ ′ L (0) α Y (1 − β ) + κ ′ L (0) α Y β − β , (4.8) lim τ → ∂∂τ R Fa,Q ( t, τ ) = X ( t ) α X β + Y ( t ) α Y β + θ + κ ′ L ( θ ) − κ ′ L (0) . (4.9) Proof.
It follows trivially from Proposition 4.6 and the assumptions on the coefficients µ X , µ Y , α X and α Y . (cid:3) Remark 4.8.
The previous Lemma shows that the risk premium R Fa,Q ( t, τ ) vanishes with rate given byequation (4 . at the short end of the forward curve, when τ converges to zero, and approaches the valuegiven in equation (4 . at long end of the forward curve, when τ tends to infinity. It follows that the signof R Fa,Q ( t, τ ) in the short end of the forward curve will be positive if (4 . is positive and negative if (4 . is negative. Hence, a sufficient condition to obtain the empirically observed risk premium profiles (with
50 100 150 200 250 300 350 Τ- - H Τ L (a) θ = − . , θ = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (b) θ = 0 . , θ = − .
50 100 150 200 250 300 350 Τ- - H Τ L (c) θ = − . , θ = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (d) θ = − . , θ = 0 . F IGURE
2. Risk premium profiles when L is a compound Poisson process with expo-nentially distributed jumps. Esscher transform: case ¯ β = (0 , . Arithmetic spot pricemodel positive values in the short end and negative values in the long end of the forward curve) is to choose thevalues of the parameters ¯ θ ∈ ¯ D L and ¯ β ∈ [0 , such that the following two conditions are simultaneouslysatisfied θ α X (1 − β ) + κ ′ L ( θ ) − κ ′ L (0) α Y (1 − β ) + κ ′ L (0) α Y β − β < ,X ( t ) α X β + Y ( t ) α Y β + θ + κ ′ L ( θ ) − κ ′ L (0) > . We also recall here that, according to Remark 2.2, κ ′ ( θ ) is positive, increasing function, so the sign of κ ′ L ( θ ) − κ ′ L (0) is equal to the sign of θ . Moreover, it is easy to see that − κ ′ L (0) < κ ′ L ( θ ) − κ ′ L (0) < κ ′ L (Θ L / − κ ′ L (0) < ∞ . • Changing the level of mean reversion (Esscher transform) , ¯ β = (0 ,
0) :
Setting ¯ β = (0 , , the probability measure Q only changes the level of mean reversion (which is assumed to be zerounder the historical measure P ). On the other hand, the risk premium is deterministic and cannotchange with changing market conditions. From equation (4 . , we get that if we set θ = 0 , whichmeans that we just change the level of the regular factor X, the sign of R Fa,Q ( t, τ ) is the same forany time to maturity τ and it is equal to the sign of θ , see Figures 1(a) and 1(b). The situation issimilar if we set θ = 0 , then the sign of R Fa,Q ( t, τ ) is constant over the time to maturity τ endequal to the sign of κ ′ L ( θ ) − κ ′ L (0) , that is to the sign of θ , see Figures 1(c) and 1(d).When both θ and θ are different from zero the situation is more interesting, the sign of R Fa,Q ( t, τ ) may change depending on the time to maturity. By Remark 4.8 it suffices to choose PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 19 θ < and θ > satisfying θ α X + κ ′ L ( θ ) − κ ′ L (0) α Y < , (4.10) θ + κ ′ L ( θ ) − κ ′ L (0) > , (4.11)(these exist because α X < α Y and κ ′ L ( θ ) is increasing) to get that R Fa,Q ( t, τ ) > for τ close tozero and R Fa,Q ( t, τ ) < for τ large enough, see Figure 2(a). This corresponds to the situationof a premium induced from consumers’ hedging pressure on short-term contracts and long termhedging of producers. We can also choose values for θ > and θ < such that equations 4.10and 4.11 are satisfied but with inverted inequalities. In this way, we can get that R Fa,Q ( t, τ ) < for τ close to zero and R Fa,Q ( t, τ ) > for τ large enough, see Figure 2(b). Risk premium profileswith constant sign can also be generated, see Figures 2(c) and 2(d). • Changing the speed of mean reversion, ¯ θ = (0 ,
0) :
Setting ¯ θ = (0 , , the probability measure Q only changes speed of mean reversion. Note that in this case the risk premium is stochastic andit changes with market conditions. By Lemma 4.7 we have that the risk premium is given by R Fa,Q ( t, τ ) = X ( t ) e − α X τ ( e α X β τ −
1) + Y ( t ) e − α Y τ ( e α Y β τ − κ ′ L (0) α Y Λ( α Y τ, − β ) , and lim τ →∞ R Fa,Q ( t, τ ) = κ ′ L (0) α Y β − β ≥ , lim τ → ∂∂τ R Fa,Q ( t, τ ) = X ( t ) α X β + Y ( t ) α Y β . Hence the risk premium will approach to a non negative value in the long end of the market. In theshort end, it can be both positive or negative and stochastically varying with X ( t ) and Y ( t ) , but Y ( t ) will always contribute to a positive sign. Actually, as the function Λ( x, y ) is non-negativeand κ ′ L (0) is strictly positive, the only negative contribution to R Fa,Q ( t, τ ) comes from the termdue to the base component X . Hence, if β = 0 or X ( t ) ≥ , then R Fa,Q ( t, τ ) will be positive forall times to maturity. Some of the possible risk profiles that can be obtained are plotted in Figure3. • Changing the level and speed of mean reversion simultaneously : The general case is quitecomplex to analyse. As we are more interested in how the change of measure Q influence thecomponent Y ( t ) , responsible for the spikes in the prices, we are going to assume that β = 0 . This means that Q may change the level of mean reversion of the regular component X ( t ) , but notthe speed at which this component reverts to that level. The first implication of this assumptionis that the possible stochastic component in R Fa,Q ( t, τ ) due to X ( t ) vanish. This simplifies theanalysis as this term could be positive or negative. By Lemma 4.7 we get that R Fa,Q ( t, τ ) = Y ( t ) e − α Y τ ( e α Y β τ −
1) + θ α X (1 − e − α X τ )+ κ ′ L ( θ ) α Y (1 − β ) (1 − e − α Y (1 − β ) τ ) − κ ′ L (0) α Y (1 − e − α Y τ ) . and lim τ →∞ R Fa,Q ( t, τ ) = θ α X + κ ′ L ( θ ) − κ ′ L (0) α Y (1 − β ) + κ ′ L (0) α Y β − β , (4.12) lim τ → ∂∂τ R Fa,Q ( t, τ ) = Y ( t ) α Y β + θ + κ ′ L ( θ ) − κ ′ L (0) . (4.13)
50 100 150 200 250 300 350 Τ- - H Τ L (a) β = 0 . , β = 0 . , X ( t ) = 2 . , Y ( t ) = 2 .
50 100 150 200 250 300 350 Τ- - H Τ L (b) β = 0 . , β = 0 , X ( t ) = − . , Y ( t ) = 2 .
50 100 150 200 250 300 350 Τ- - H Τ L (c) β = 0 . , β = 0 . , X ( t ) = − . , Y ( t ) = 0
50 100 150 200 250 300 350 Τ- - H Τ L (d) β = 0 . , β = 0 . , X ( t ) = − . , Y ( t ) = 2 . F IGURE
3. Risk premium profiles when L is a compound Poisson process with exponen-tially distributed jumps. Case ¯ θ = (0 , . Arithmetic spot price model
50 100 150 200 250 300 350 Τ- - H Τ L (a) β = 0 , β = 0 . , θ = − . , θ = 0 . , X ( t ) = R , Y ( t ) = 5 F IGURE
4. Risk premium profiles when L is a compound Poisson process with exponen-tially distributed jumps. Arithmetic spot price modelNote that we can make equation (4 . negative by simply choosing θ θ < − α X α Y (1 − β ) (cid:0) κ ′ L ( θ ) − κ ′ L (0) + β κ ′ L (0) (cid:1) . (4.14) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 21
On the other hand, to make equation (4 . positive, we have to choose θ satisfying θ > − ( κ ′ L ( θ ) − κ ′ L (0)) − Y ( t ) α Y β . (4.15)Equations (4 . and (4 . are compatible if the following inequality is satisfied κ ′ L ( θ ) − κ ′ L (0) + Y ( t ) α Y β > α X α Y (1 − β ) (cid:0) κ ′ L ( θ ) − κ ′ L (0) + β κ ′ L (0) (cid:1) . (4.16)For any θ > , which yields κ ′ L ( θ ) − κ ′ L (0) > (and θ < ), we have that there exists β ∗ ∈ (0 , such that if β < β ∗ equation (4 . is satisfied. Actually, the larger the value of Y ( t ) , the larger the value of β ∗ . If Y ( t ) is close to κ ′ L (0) /α Y , then β ∗ is close to ( α Y − α X ) /α Y . This just says that if the speed of mean reversion of the spikes component is large (in absolutevalue and relatively to the speed of mean reversion of the base component) one can choose β close to one. Even in the case that Y ( t ) = 0 , equation (4 . is satisfied by choosing β smallenough. To sum up, we can create a measure Q that can have a positive premium in the short endof the forward market due to sudden positive spikes in the price (that is, Y increases), whereas inthe long end of the market these spikes are not influential and we have a negative premium, seeFigure 4.4.2. Geometric spot price model.
We assume in this section that the spot price S ( t ) follows the geomet-ric model (4.2) for ≤ t ≤ T ∗ , T ∗ > and with the maturity of the forward contract being < T < T ∗ .In our setting, the geometric model is harder to deal with than the arithmetic one. The results obtained arefair less explicit and some additional integrability conditions on L are required. A first, natural, additionalassumption on L is that the constant Θ L appearing in Assumption 1 to be bigger than . This condition isreasonable to expect because it just states that E [ e L ( t ) ] < ∞ , for all t ∈ R , and if we want E [ e Y ( t ) ] to befinite it seems a minimal assumption. Note, however that this is not entirely obvious because the process Y has a mean reversion structure that L does not have. On the other hand, the complex probabilisticstructure of the spike factor Y under the new probability measure Q, makes the computations much moredifficult. Still, it is possible to compute the risk premium analytically in some cases. In general, one hasto rely on numerical techniques.In what follows, we shall compute the conditional expectations involved under Q (note that Q = P, when θ = θ = β = β = 0 ). First, we show that the problem can be reduced to the study of the spikecomponent Y. Due to the independence of X and Y, we have that E Q [ S ( T )] = Λ g ( T ) E Q [exp( X ( T ) + Y ( T ))]= Λ g ( T ) E Q [exp( X ( T ))] E Q [exp( Y ( T ))] , which is finite if and only E Q [exp( X ( T ))] < ∞ and E Q [exp( Y ( T ))] < ∞ . As X ( T ) is a Gaussianrandom variable it has finite exponential moments. To determine whether E Q [exp( Y ( T ))] is finite or notis not as straightforward. Let us assume, for now, that it is finite. Then, it makes sense to compute thefollowing conditional expectation E Q [ S ( T ) |F t ] = Λ g ( T ) E Q [exp( X ( T ) + Y ( T )) |F t ]= Λ g ( T ) E Q [ E Q [exp( X ( T )) |F t ∨ σ ( { Y ( t ) } ≤ t ≤ T )] exp( Y ( T )) |F t ] . Using (3 . , the fact that X is independent of σ ( { Y ( t ) } ≤ t ≤ T ) and basic properties of the conditionalexpectation we get that E Q [exp( X ( T )) |F t ∨ σ ( { Y ( t ) } ≤ t ≤ T )]= exp (cid:18) X ( t ) e − α X (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) (cid:19) × E Q [exp (cid:18) σ X Z Tt e − α X (1 − β )( T − s ) dW Q ( s ) (cid:19) ]= exp (cid:18) X ( t ) e − α X (1 − β )( T − t ) + µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) (cid:19) × exp (cid:18) σ X α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) (cid:19) . Hence, we have reduced the problem to the study of E Q [exp( Y ( T )) |F t ] . Let us start with the Esscher case Q = Q θ ,β with θ ∈ D L and β = 0 . We have that E Q [exp( Y ( T ))] = exp (cid:26) Y (0) e − α Y T + µ Y + κ ′ L ( θ ) α Y (1 − e − α Y T ) (cid:27) × E Q [exp (cid:18)Z T Z ∞ ze − α Y ( T − s ) ˜ N LQ ( ds, dz ) (cid:19) ]= exp (cid:26) Y (0) e − α Y T + µ Y α Y (1 − e − α Y T ) (cid:27) × E Q [exp (cid:18)Z T Z ∞ ze − α Y ( T − s ) N LQ ( ds, dz ) (cid:19) ]= exp (cid:26) Y (0) e − α Y T + µ Y α Y (1 − e − α Y T ) (cid:27) × exp (cid:26)Z T Z ∞ ( e ze − αY ( T − s ) − e θ z ℓ ( dz ) ds (cid:27) , where we have used that the compensator of L under Q is v LQ ( ds, dz ) = e θ z ℓ ( dz ) ds (note that e θ z ℓ ( dz ) is a L´evy measure) and Proposition 3.6 in Cont and Tankov [8]. Of course, the previous result holds aslong as the integral in the exponential is finite. A sufficient condition for the integrability of exp( Y ( T )) follows from Z T Z ∞ ( e ze − αY ( T − s ) − e θ z ℓ ( dz ) ds = Z T Z ∞ ze − α Y ( T − s ) (cid:18)Z e λze − αY ( T − s ) dλ (cid:19) e θ z ℓ ( dz ) ds ≤ Z T Z ∞ ze − α Y ( T − s ) e z ( θ + e − αY ( T − s ) ) ℓ ( dz ) ds ≤ T κ ′ L ( θ + 1) . As θ ∈ D L , to have κ ′ L ( θ + 1) < ∞ yields the condition θ ∈ D gL , D L ∩ ( −∞ , Θ L −
1) = ( −∞ , (Θ L − ∧ (Θ L / . Note that for θ ∈ D gL to be strictly positive and, therefore, include the case Q = P , we need to have Θ L > . This, of course, is a restriction on the structure of the jumps. For instance, if L is a compoundPoisson process with exponentially distributed jump sizes, Example 3.2 (Case 2), we have that the jumpsizes must have a mean less than one. Note also that, if Θ L > then D gL = D L . Using expression (3 . and repeating the previous arguments we obtain E Q [exp( Y ( T )) |F t ] = exp (cid:26) Y ( t ) e − α Y ( T − t ) + µ Y α Y (1 − e − α Y ( T − t ) ) (cid:27) × exp (cid:26)Z T − t Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds (cid:27) . Hence we have proved the following result:
Proposition 4.9.
In the Esscher case for the spike component Y , i.e., θ ∈ D gL , β = 0 , and assuming Θ L > , the forward price F Q ( t, T ) in the geometric spot model (4 . is given by F Q ( t, T ) = Λ g ( T ) exp (cid:16) X ( t ) e − α X (1 − β )( T − t ) + Y ( t ) e − α Y ( T − t ) (cid:17) × exp (cid:18) µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + µ Y α Y (1 − e − α Y ( T − t ) ) (cid:19) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 23 × exp (cid:18) σ X α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + Z T − t Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds (cid:19) . and the risk premium for the forward price R Fg,Q ( t, T ) is given by R Fg,Q ( t, T ) = E P [ S ( T ) |F t ] { exp( R Fa,Q ( t, T )) × exp (cid:18) σ X α X (1 − β ) (1 − e − α X (1 − β )( T − t ) − σ X α X (1 − e − α X ( T − t ) (cid:19) × exp (cid:18) − κ ′ L ( θ ) − κ ′ L (0) α Y (1 − e − α Y ( T − t ) ) (cid:19) × exp (cid:18)Z T − t Z ∞ ( e ze − αY s − e θ z − ℓ ( dz ) ds (cid:19) − } , where R Fa,Q ( t, T ) is also understood under the assumption β = 0 . Corollary 4.10.
Setting θ = 0 in Proposition 4.9 we get E P [ S ( T ) |F t ] = Λ g ( T ) exp (cid:16) X ( t ) e − α X ( T − t ) + Y ( t ) e − α Y ( T − t ) (cid:17) × exp (cid:18) µ X α X (1 − e − α X ( T − t ) ) + µ Y α Y (1 − e − α Y ( T − t ) ) (cid:19) × exp (cid:18) σ X α X (1 − e − α X (1 − β )( T − t ) ) + Z T − t Z ∞ ( e ze − αY s − ℓ ( dz ) ds (cid:19) . The previous result is as far as one can go using ”basic” martingale techniques. In the general case, inorder to find conditions under which E Q [exp( Y ( T ))] < ∞ , and also to compute E Q [exp( Y ( T )) |F t ] , itis convenient to look at Y as an affine Q -semimartingale process with state space R + . In the sequel wefollow the notation in Kallsen and Muhle-Karbe [15], but taking into account that in our case the L´evycharacteristics do not depend on the time parameter. The L´evy-Kintchine triplets of Y are ( β , γ , ϕ ( dz )) = ( µ Y + κ ′ L ( θ ) , , (0 , ∞ ) e θ z ℓ ( dz ))( β , γ , ϕ ( dz )) = ( − α Y (1 − β ) , , α Y β κ ′′ L ( θ ) (0 , ∞ ) ze θ z ℓ ( dz )) , which, according to Definition 2.4 in Kallsen and Muhle-Karbe [15], are (strongly) admissible. Note that,as the triplets do not depend on t, we can choose any truncation function. Moreover, as Y is a special Q -semimartingale, we choose the (pseudo) truncation function h ( x ) = x. Associated to the previousL´evy-Kintchine triplets we have the following L´evy exponents Λ θ ,β ( u ) = (cid:0) µ Y + κ ′ L ( θ ) (cid:1) u + Z ∞ ( e uz − − uz ) e θ z ℓ ( dz )= µ Y u + Z ∞ ( e uz − e θ z ℓ ( dz )= µ Y u + κ L ( u + θ ) − κ L ( θ ) , Λ θ ,β ( u ) = − α Y (1 − β ) u + α Y β κ ′′ L ( θ ) Z ∞ ( e uz − − uz ) ze θ z ℓ ( dz )= − α Y u + α Y β κ ′′ L ( θ ) Z ∞ ( e uz − ze θ z ℓ ( dz )= − α Y u + α Y β κ ′′ L ( θ ) (cid:0) κ ′ L ( u + θ ) − κ ′ L ( θ ) (cid:1) . We have the following result.
Theorem 4.11.
Let ¯ β ∈ [0 , , ¯ θ ∈ ¯ D gL , R × D gL . Assume Θ L > , that Ψ θ ,β , Ψ θ ,β ∈ C ([0 , T ] , R ) satisfy the ODE ddt Ψ θ ,β ( t ) = Λ θ ,β (Ψ θ ,β ( t )) , Ψ θ ,β (0) = 1 , ddt Ψ θ ,β ( t ) = Λ θ ,β (Ψ θ ,β ( t )) , Ψ θ ,β (0) = 0 , (4.17) and that the integrability condition κ ′′ L ( θ + sup t ∈ [0 ,T ] Ψ θ ,β ( t )) = Z ∞ z exp { ( θ + sup t ∈ [0 ,T ] Ψ θ ,β ( t )) z } ℓ ( dz ) < ∞ , (4.18) holds. Then, we have that the forward price F Q ( t, T ) in the geometric spot model (4 . is given by F Q ( t, T ) = Λ g ( T ) exp (cid:16) X ( t ) e − α X (1 − β )( T − t ) + Y ( t )Ψ θ ,β ( T − t ) + Ψ θ ,β ( T − t ) (cid:17) × exp (cid:18) µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) + σ X α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) (cid:19) , and the risk premium for the forward price R Fg,Q ( t, T ) is given by R Fg,Q ( t, T ) = E P [ S ( T ) |F t ] { exp( X ( t ) e − α X ( T − t ) ( e α X β ( T − t ) − × exp( Y ( t )(Ψ θ ,β ( T − t ) − e − α Y ( T − t ) )) × exp (cid:18) µ X + θ α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) − µ X α X (1 − e − α X ( T − t ) ) (cid:19) × exp (cid:18) σ X α X (1 − β ) (1 − e − α X (1 − β )( T − t ) ) − σ X α X (1 − e − α X ( T − t ) ) (cid:19) × exp (cid:18) Ψ θ ,β ( T − t ) − µ Y α Y (1 − e − α Y ( T − t ) ) − Z T − t Z ∞ ( e ze − αY s − ℓ ( dz ) ds (cid:19) − } . Proof.
We apply Theorem 5.1 in Kallsen and Muhle-Karbe [15]. Note that making the change of variable t → T − t the ODE (4 . is reduced to the one appearing in items 2. and 3. of Theorem 5.1. Theintegrability assumption (4 . implies conditions 1. and 5., in Theorem 5.1, and condition 4. is triviallysatisfied because Y (0) is deterministic. Hence, the conclusion of that theorem, with p = 1 , holds and weget E Q [exp( Y ( T )) |F t ] = exp (cid:0) Y ( t )Ψ θ ,β ( T − t ) + Ψ θ ,β ( T − t ) (cid:1) , t ∈ [0 , T ] . (4.19)The result now follows easily. (cid:3) Remark 4.12.
Equation (4 . is called a generalised Riccati equation in the literature. Note that theequation for Ψ θ ,β ( t ) is trivially solved, once we know Ψ θ ,β ( t ) , by Ψ θ ,β ( t ) = Z t Λ θ ,β (Ψ θ ,β ( s )) ds. Hence, the problem is really reduced to study the equation for Ψ θ ,β ( t ) . Remark 4.13.
The Esscher case can be obtained from Theorem 4.11, as Ψ θ , ( t ) = e − α Y t and Ψ θ , ( t ) = µ Y α Y (1 − e − α Y t ) + Z t Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds, solve ddt Ψ θ , ( t ) = − α Y Ψ θ , ( t ) , Ψ θ , (0) = 1 , ddt Ψ θ , ( t ) = µ Y Ψ θ , ( t ) + κ L (Ψ θ , ( t ) + θ ) − κ L ( θ ) , Ψ θ , (0) = 0 . As sup t ∈ [0 ,T ] Ψ θ , ( t ) = 1 , the integrability condition (4 . is satisfied because θ ∈ D gL . PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 25
In general, one cannot find explicit solutions for the non-linear differential equation (4 . in Theorem4.11 and has to rely on numerical techniques. However, the main problem that we find is that the maximaldomain of definition of Ψ θ ,β and Ψ θ ,β may be a proper subset of [0 , ∞ ) , in particular when β isclose to . As we are particularly interested in the solution of (4 . for large T , we shall give a generalsufficient criterion for global (defined for any t > ) existence and uniqueness of the solution of (4 . .The next theorem classifies the behaviour of the solutions of (4 . . Theorem 4.14.
Assume that Θ L > . For any δ > , the system of ODEs ( . ) with β ∈ (0 , and θ ∈ D gL ( δ ) , ( −∞ , (Θ L − − δ ) ∧ (Θ L / admits a unique local solution Ψ θ ,β ( t ) and Ψ θ ,β ( t ) . In addition, let u ∗ ( θ , β ) be the unique strictlypositive solution of the following equation u = β κ ′′ L ( θ ) (cid:0) κ ′ L ( u + θ ) − κ ′ L ( θ ) (cid:1) . (4.20) The behaviour of Ψ θ ,β ( t ) and Ψ θ ,β ( t ) is characterised as follows: (1) If u ∗ ( θ , β ) > , then Ψ θ ,β ( t ) and Ψ θ ,β ( t ) are globally defined, satisfy < Ψ θ ,β ( t ) ≤ , ≤ Ψ θ ,β ( t ) ≤ Z ∞ Λ θ ,β (Ψ θ ,β ( s )) ds < ∞ , and lim t →∞ t log(Ψ θ ,β ( t )) = − α Y (1 − β ) , (4.21) lim t →∞ Ψ θ ,β ( t ) = Z ∞ Λ θ ,β (Ψ θ ,β ( s )) ds < ∞ . (4.22)(2) If u ∗ ( θ , β ) = 1 , then Ψ θ ,β ( t ) ≡ and Ψ θ ,β ( t ) = { µ Y + κ L (1 + θ ) − κ L ( θ ) } t. (3) If u ∗ ( θ , β ) < , then the maximal domain of definition of Ψ θ ,β ( t ) and Ψ θ ,β ( t ) is [0 , t ∞ ) , where < t ∞ = Z Θ L − θ (Λ θ ,β ( u )) − du < ∞ . In addition, lim t ↑ t ∞ Ψ θ ,β ( t ) = Θ L − θ , lim t ↑ t ∞ Ψ θ ,β ( t ) = Z t ∞ Λ θ ,β (Ψ θ ,β ( s )) ds, where the previous integral is non negative and may be finite or infinite.Proof. We have to study the vector field Λ θ ,β ( u ) = − α Y u + α Y β κ ′′ L ( θ ) Z ∞ ( e uz − ze θ z ℓ ( dz ) , β ∈ [0 , , θ ∈ D gL . Consider D (Λ θ ,β ) , int( { u ∈ R : Λ θ ,β ( u ) < ∞} ) = int( { u ∈ R : κ ′ L ( u + θ ) < ∞} ) = ( −∞ , Θ L − θ ) , and, for any δ > , define D δ , int( \ β ∈ [0 , ,θ ∈ D gL ( δ ) D (Λ θ ,β )) = ( −∞ , Θ L − ((Θ L − − δ ) ∧ (Θ L / −∞ , (1 + δ ) ∨ (Θ L / . On the other hand, for u, v ∈ D (Λ θ ,β ) , one has that (cid:12)(cid:12)(cid:12) Λ θ ,β ( u ) − Λ θ ,β ( v ) (cid:12)(cid:12)(cid:12) ≤ α Y | u − v | + α Y β κ ′′ L ( θ ) Z ∞ | e uz − e vz | ze θ z ℓ ( dz ) , and Z ∞ | e uz − e vz | ze θ z ℓ ( dz ) ≤ | u − v | Z ∞ e ( u ∨ v + θ ) z z ℓ ( dz ) , Moreover, note that int( { u ∈ R : Z ∞ z e ( u + θ ) z ℓ ( dz ) < ∞} ) = ( −∞ , Θ L − θ ) = D (Λ θ ,β ) . Hence, the vector field Λ θ ,β ( u ) , θ ∈ D gL ( δ ) , β ∈ [0 , is well defined (i.e., finite) and locally Lipschitzin D δ . As the initial condition for Ψ θ ,β ( t ) is Ψ θ ,β (0) = 1 , it is natural to require that ∈ D δ and thisis precisely the role of δ > . Then, by Picard-Lindel¨of Theorem, see Theorem 3.1, pag. 18, in Hale[12], we have local existence and uniqueness for Ψ θ ,β ( t ) and Ψ θ ,β (0) ∈ D δ . In addition, we have that ∈ D δ and, hence, we have local existence and uniqueness for solutions of Ψ θ ,β ( t ) with Ψ θ ,β (0) = 0 . As Λ θ ,β (0) = 0 , we have that Ψ θ ,β ( t ) ≡ is the unique global solution of equation (4 . starting at0. As a consequence, it is sufficient to study the vector field Λ θ ,β ( u ) for u ≥ , because any solutionof equation (4 . with Ψ θ ,β (0) = 1 cannot cross to the negative real line without contradicting theuniqueness result at . The unicity of Ψ θ ,β ( t ) trivially follows from that of Ψ θ ,β ( t ) . The next step is tostudy the zeros of Λ θ ,β ( u ) , u ∈ D δ ∩ [0 , ∞ ) . We have to solve the non-linear equation θ ,β ( u ) = − α Y u + α Y β κ ′′ L ( θ ) (cid:0) κ ′ L ( u + θ ) − κ ′ L ( θ ) (cid:1) . (4.23)Note that equation (4 . has the trivial solution u = 0 . As the first and second derivatives of Λ θ ,β ( u ) are ddu Λ θ ,β ( u ) = − α Y + α Y β κ ′′ L ( θ ) κ ′′ L ( u + θ ) ,d du Λ θ ,β ( u ) = α Y β κ ′′ L ( θ ) κ (3) L ( u + θ ) > , we have that there exists a unique < u ∗ ( θ , β ) < Θ L − θ for θ ∈ D gL ( δ ) and β ∈ (0 , suchthat equation (4 . is satisfied. Moreover Λ θ ,β ( u ) < for u ∈ (0 , u ∗ ( θ , β )) and Λ θ ,β ( u ) > for ( u ∗ ( θ , β ) , Θ L − θ ) . When β ↓ , u ∗ ( θ , β ) converges to Θ L − θ . On the other hand, when β ↑ , u ∗ ( θ , β ) converges to zero. Therefore, we have three possible cases to discuss • Case : If u ∗ ( θ , β ) > , then Ψ θ ,β ( t ) will monotonically converge to and, by uniquenessof solutions, it will take an infinite amount of time to reach . Hence, Ψ θ ,β ( t ) will be a globallydefined bounded solution. The exponential rate of convergence of Ψ θ ,β ( t ) to zero, equation4.21, follows by applying H ˆopital’s rule to lim t →∞ t − log(Ψ θ ,β ( t )) = lim t →∞ ddt Ψ θ ,β ( t )Ψ θ ,β ( t )= lim t →∞ Λ θ ,β (Ψ θ ,β ( t ))Ψ θ ,β ( t )= lim t →∞ − α Y Ψ θ ,β ( t ) + α Y β κ ′′ L ( θ ) { κ ′ L (Ψ θ ,β ( t ) + θ ) − κ ′ L (+ θ ) } Ψ θ ,β ( t )= − α Y + α Y β κ ′′ L ( θ ) lim t →∞ R κ ′′ L ( θ + λ Ψ θ ,β ( t )) dλ Ψ θ ,β ( t )Ψ θ ,β ( t )= − α Y (1 − β ) . PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 27
It follows that Ψ θ ,β ( t ) will be also globally defined and, as Λ θ ,β ( u ) = µ Y u + R κ ′ L ( θ + λu ) dλ > for u ∈ (0 , , by monotone convergence lim t →∞ Ψ θ ,β ( t ) = Z ∞ Λ θ ,β (Ψ θ ,β ( s )) ds. To show that the previous integral is actually finite, it suffices to prove that Λ θ ,β (Ψ θ ,β ( t )) converges to zero faster than t − (1+ ε ) , for some ε > , when t tends to infinity. We have that Λ θ ,β (Ψ θ ,β ( t )) = µ Y Ψ θ ,β ( t ) + κ L (Ψ θ ,β ( t ) + θ ) − κ L ( θ )= { µ Y + Z κ ′ L ( θ + λ Ψ θ ,β ( t )) dλ } Ψ θ ,β ( t ) , and ddt (cid:16) Λ θ ,β (Ψ θ ,β ( t )) (cid:17) = ddu Λ θ ,β ( u ) (cid:12)(cid:12)(cid:12)(cid:12) u =Ψ θ ,β ( t ) ddt Ψ θ ,β ( t )= { µ Y + κ ′ L ( θ + Ψ θ ,β ( t )) }× {− α Y Ψ θ ,β ( t ) + α Y β κ ′′ L ( θ ) (cid:0) κ ′ L (Ψ θ ,β ( t ) + θ ) − κ ′ L ( θ ) (cid:1) } = { µ Y + κ ′ L ( θ + Ψ θ ,β ( t )) }× {− α Y + α Y β κ ′′ L ( θ ) Z κ ′′ L ( θ + λ Ψ θ ,β ( t )) dλ } Ψ θ ,β ( t ) . By H ˆopital’s rule and equation (4 . t →∞ t (1+ ε ) Λ θ ,β (Ψ θ ,β ( t )) = lim t →∞ (1 + ε ) t ε − ddt Λ θ ,β (Ψ θ ,β ( t )) (cid:16) Λ θ ,β (Ψ θ ,β ( t )) (cid:17) − = (1 + ε ) µ Y + κ ′ L ( θ ) α Y (1 − β ) lim t →∞ t ε Ψ θ ,β ( t ) = 0 , and we can conclude that equation (4 . holds. • Case : If u ∗ ( θ , β ) = 1 , then Ψ θ ,β ( t ) ≡ , will be the unique global solution and Ψ θ ,β ( t ) = Z t Λ θ ,β (Ψ θ ,β ( s )) ds = { µ Y + κ L (1 + θ ) − κ L ( θ ) } t. • Case : If u ∗ ( θ , β ) < , then Ψ θ ,β ( t ) will increase monotonically to Θ L − θ , becausethe vector field Λ θ ,β is strictly positive in [1 , Θ L − θ ) . Separating variables an integratingthe equation for Ψ θ ,β ( t ) with Ψ θ ,β (0) = 1 we get that the maximal domain of definition of Ψ θ ,β ( t ) is [0 , t ∞ ) with t ∞ , Z Θ L − θ (Λ θ ,β ( u )) − du. To show that t ∞ is actually finite we have to distinguish between the case Θ L < ∞ and Θ L = ∞ . If Θ L < ∞ , then (Λ θ ,β ( u )) − is bounded in [1 , Θ L − θ ) and the integral is obviously finite.If Θ L = ∞ we have to ensure that (Λ θ ,β ( u )) − converges to zero fast enough when u tends toinfinity. Note that, by monotone convergence, one has that lim θ →∞ κ L ( θ ) = Z ∞ lim θ →∞ ( e θz − ℓ ( dz ) = ∞ , lim θ →∞ κ ( n ) L ( θ ) = Z ∞ lim θ →∞ z n e θz ℓ ( dz ) = ∞ , n ≥ . For any < ε < , we have that lim u →∞ u − (1+ ε ) Λ θ ,β ( u ) = lim u →∞ u − (1+ ε ) {− α Y u + α Y β κ ′′ L ( θ ) ( κ ′ L ( u + θ ) − κ ′ L ( θ )) } = α Y β κ ′′ L ( θ ) lim u →∞ κ ′ L ( u + θ ) u (1+ ε ) = α Y β (1 + ε ) κ ′′ L ( θ ) lim u →∞ κ ′′ L ( u + θ ) u ε = α Y β (1 + ε ) εκ ′′ L ( θ ) lim u →∞ u − ε κ (3) L ( u + θ ) = ∞ , which yields that the integral defining t ∞ is finite. According to Remark 4.12, we have that lim t → t ∞ Ψ θ ,β ( t ) = Z t ∞ Λ θ ,β (Ψ θ ,β ( s )) ds, (4.24)which may be finite or infinite depending, of course, on how fast Λ θ ,β (Ψ θ ,β ( s )) diverges toinfinity when s approaches to t ∞ . (cid:3) As it does not seem possible to give simple conditions for the finiteness (or not) of the integral (4 . and it is not relevant in the discussion to follow, we do not proceed further in the analysis. Remark 4.15. If β = 0 , then Ψ θ , ( t ) = e − α Y t and Ψ θ , ( t ) = Z t µ Y e − α Y s ds + Z t κ L ( e − α Y s + θ ) − κ L ( θ ) ds = µ Y α Y (1 − e − α Y t ) + Z t Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds. Obviously lim t →∞ e α Y t Ψ θ , ( t ) = 1 and lim t →∞ Ψ θ , ( t ) = µ Y α Y + Z ∞ Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds < ∞ . Note that Z ∞ Z ∞ ( e ze − αY s − e θ z ℓ ( dz ) ds = Z ∞ (cid:18)Z ∞ ( e ze − αY s − ds (cid:19) e θ z ℓ ( dz ) ≤ Z ∞ (cid:18)Z ∞ (cid:18)Z e λze − αY s dλ (cid:19) ze − α Y s ds (cid:19) e θ z ℓ ( dz ) ≤ α Y Z ∞ ze (1+ θ ) z ℓ ( dz ) = κ ′ L (1 + θ ) α Y < ∞ . If β = 1 , we have that ddu Λ θ ,β ( u ) = − α Y + α Y κ ′′ L ( θ ) κ ′′ L ( u + θ ) = α Y ( κ ′′ L ( u + θ ) κ ′′ L ( θ ) − > , for u ∈ (0 , Θ L − θ ) , which yields that Ψ θ , ( t ) > and monotonically diverges to infinity. Although the previous result characterizes the behaviour of the solution of the ODE (4 . for differ-ent values of ( θ , β ) in terms of u ∗ ( θ , β ) , usually one cannot find u ∗ ( θ , β ) analytically and, given ( θ , β ) , equation (4 . must be solved numerically to know whether the solution associated to equation (4 . is bounded or not. Hence, the following corollary of Theorem 4.14 may be helpful in practice. PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 29
Corollary 4.16.
Under the hypothesis of Theorem 4.14 and for θ ∈ D gL ( δ ) fixed, a sufficient conditionfor u ∗ ( θ , β ) > is that β < κ ′′ L ( θ ) κ ′ L (1 + θ ) − κ ′ L ( θ ) . (4.25) Proof.
Assume θ ∈ D gL ( δ ) fixed. According to the discussion in the proof of Theorem 4.14, for any θ ∈ D gL ( δ ) and β ∈ (0 , there exists a unique root u ∗ = u ∗ ( θ , β ) of the vector field Λ θ ,β ( u ) defined byequation (4 . and such that Λ θ ,β ( u ) < if (0 , u ∗ ( θ , β )) and Λ θ ,β ( u ) > if ( u ∗ ( θ , β ) , Θ L − θ ) . Now, note that β ∗ (1) , κ ′′ L ( θ ) κ ′ L (1 + θ ) − κ ′ L ( θ ) , is such that u ∗ ( θ , β ∗ (1)) . If β < β ∗ (1) one has that Λ θ ,β (1) = α Y ( − β κ ′′ L ( θ ) κ ′ L (1 + θ ) − κ ′ L ( θ )) < , which yields that the unique root u ∗ = u ∗ ( θ , β ) of the vector field Λ θ ,β ( u ) must be strictly greaterthan one and, therefore, we are in the case (1) of Theorem 4.14. (cid:3) Next, we present two examples where we apply the previous results.
Example 4.17.
We start by the simplest possible case. Assume that the L´evy measure is δ { } ( dz ) , that is,the L´evy process L has only jumps of size . In this case Θ L = ∞ and, hence, D gL = R . We have that κ L ( θ ) = e θ − and κ ( n ) L ( θ ) = e θ , n ∈ N . Therefore, Λ θ ,β ( u ) = µ Y u + κ L ( u + θ ) − κ L ( θ ) = µ Y u + ( e u + θ − e θ ) , Λ θ ,β ( u ) = − α Y u + α Y β κ ′′ L ( θ ) (cid:0) κ ′ L ( u + θ ) − κ ′ L ( θ ) (cid:1) = − α Y u + α Y β ( e u − . First, we have to solve ddt Ψ θ ,β ( t ) = − α Y Ψ θ ,β ( t ) + α Y β ( e Ψ θ ,β ( t ) − , (4.26) Ψ θ ,β (0) = 1 . and then integrate Λ θ ,β (Ψ θ ,β ( s )) from to t. Although equation (4 . can be solved analytically, itssolution is given in implicit form and a numerical method is easier to use. In this example, equation (4 . reads u = β e θ (cid:16) e u + θ − e θ (cid:17) = β ( e u − , (4.27) which can only be solved numerically. Heuristically, if β is close to one the solution of the previousequation must be close to zero and, hence, the solution Ψ θ ,β ( t ) diverges to ∞ . Applying Corollary 4.16we can guarantee that Ψ θ ,β ( t ) converges to zero if β < κ ′′ L ( θ ) κ ′ L (1 + θ ) − κ ′ L ( θ ) = e θ e θ − e θ = ( e − − . Example 4.18.
Assume that the L´evy measure is ℓ ( dz ) = ce − λz (0 , ∞ ) , that is, L is a compound Poissonprocess with intensity c/λ and exponentially distributed jumps with mean /λ. In this case Θ L = λ and,hence, D gL = ( −∞ , ( λ − ∧ ( λ/ . We have that κ L ( θ ) = cθ λ ( λ − θ ) and κ ( n ) L ( θ ) = cn !( λ − θ ) n +1 , n ∈ N . Therefore, Λ θ ,β ( u ) = µ Y u + κ L ( u + θ ) − κ L ( θ )= µ Y u + c ( u + θ ) λ ( λ − θ − u ) − cθ λ ( λ − θ ) , Λ θ ,β ( u ) = − α Y u + α Y β κ ′′ L ( θ ) (cid:0) κ ′ L ( u + θ ) − κ ′ L ( θ ) (cid:1) = − α Y u + α Y β ( λ − θ ) (cid:26) λ − θ − u ) − λ − θ ) (cid:27) . Hence, we have to solve ddt Ψ θ ,β ( t ) = − α Y Ψ θ ,β ( t ) + α Y β ( λ − θ ) ( λ − θ − Ψ θ ,β ( t )) − λ − θ ) ) , Ψ θ ,β (0) = 1 , and then integrate Λ θ ,β (Ψ θ ,β ( s )) from to t. As in the previous example, there is an analytic solutionto this equation in implicit form, but it is easier to use a numerical method. In this example, equation (4 . reads u = β ( λ − θ ) (cid:18) λ − θ − u ) − λ − θ ) (cid:19) , which has roots ( u , u − , u + ) = (0 , λ − θ (cid:18) − β − q β + 8 β (cid:19) , λ − θ (cid:18) − β + q β + 8 β (cid:19) ) . We are just interested in the root u − ∈ (0 , λ − θ ) , note that u + > λ − θ . The inequality λ − θ > u − > yields < β < λ − θ − ( λ − θ )(2( λ − θ ) − . (4.28) Hence, for any θ ∈ D gL ( δ ) and β satisfying (4 . we can ensure global existence and boundedness of Ψ θ ,β ( t ) and Ψ θ ,β ( t ) . Discussion on the risk premium.
For the study of the sign change we are going to abuse the no-tation, as in the arithmetic spot price model, and we will denote R Fg,Q ( t, τ ) , R Fg,Q ( t, t + τ ) , where τ = T − t is the time to maturity. We also fix the parameters of the model under the historical measure P, i.e., µ X , α X , σ X , µ Y , and α Y , and study the possible sign of R Fg,Q ( t, τ ) in terms of the change of measureparameters, i.e., ¯ β = ( β , β ) and ¯ θ = ( θ , θ ) and the time to maturity τ. As in the arithmetic model, thepresent time just enters into the picture through the stochastic components X and Y. We are also goingto assume µ X = µ Y = 0 . Analogously to the arithmetic case, in this way the seasonality function Λ g accounts completely for the mean price level. We also assume that α X < α Y , which means that thecomponent accounting for the jumps reverts the fastest. Finally, in the sequel, we are going to assume thatwe are in the Case 1 of Theorem 4.14, i.e., the values θ , β are such that u ∗ ( θ , β ) > , and Ψ θ ,β and Ψ θ ,β are globally defined and the exponential affine formula (4 . holds.The following lemma will help us in the discussion to follow. Lemma 4.19. If µ X = µ Y = 0 and α X < α Y , we have that the sign of the risk premium R Fg,Q ( t, τ ) willbe the same as the sign of Σ( t, τ ) , X ( t ) e − α X τ ( e α X β τ −
1) + Y ( t )(Ψ θ ,β ( τ ) − Ψ , ( τ )) (4.29) + θ α X (1 − β ) (1 − e − α X (1 − β ) τ ) + σ X α X Λ(2 α X τ, − β )+ Ψ θ ,β ( τ ) − Ψ , ( τ ) , where Λ( x, y ) is the (non-negative) function defined in Lemma 4.7. Moreover, lim τ →∞ Σ( t, τ ) = θ α X (1 − β ) + σ X α X β − β (4.30) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 31 + Z ∞ κ L (Ψ θ ,β ( t ) + θ ) − κ L ( θ ) − κ L ( e − α Y t ) dt lim τ → ∂∂τ Σ( t, τ ) = X ( t ) α X β + Y ( t ) α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) (4.31) + θ + κ L (1 + θ ) − κ L ( θ ) − κ L (1) Proof.
The result follows easily from Theorem 4.11 and the following computations with Ψ θ ,β ( τ ) and Ψ θ ,β ( τ ) . We have that lim τ → ddτ Ψ θ ,β ( τ ) = lim τ → Λ θ ,β (Ψ θ ,β ( τ )) = Λ θ ,β (1)= − α Y + α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) , and lim τ → ddτ Ψ θ ,β ( τ ) = lim τ → Λ θ ,β (Ψ θ ,β ( τ )) = Λ θ ,β (1)= κ L (1 + θ ) − κ L ( θ ) . In Theorem 4.14, it is proved that Ψ θ ,β ( τ ) converges to when τ tends to infinity and lim τ →∞ Ψ θ ,β ( τ ) = Z ∞ Λ θ ,β (Ψ θ ,β ( t )) dt. Hence, using the definitions of Λ θ ,β ( u ) and Λ , ( u ) , the fact that Ψ , ( t ) = e − α Y t and κ L (0) = 0 weget lim τ →∞ (Ψ θ ,β ( τ ) − Ψ , ( τ )) = Z ∞ κ L (Ψ θ ,β ( t ) + θ ) − κ L ( θ ) − κ L ( e − α Y t ) dt. (cid:3) The sign of Σ( t, τ ) is more complex to analyse than the sign of R Fa,Q ( t, τ ) , the risk premium in thearithmetic model. In the Esscher case the computations can be done quite explicitly. In the generalcase we shall make use of Lemma 4.19 to prove that one can generate the empirically observed riskpremium profile. Moreover, some additional information on Σ( t, τ ) can be deduced from classical resultson comparison of solutions of ODEs. In order to graphically illustrate the discussion we plot the riskpremium profiles obtained assuming that the subordinator L is a compound Poisson process with jumpintensity c/λ > and exponential jump sizes with mean λ. That is, L will have the L´evy measure given inExample (3 . , (1) . We shall measure the time to maturity τ in days and plot R Fg,Q ( t, τ ) for τ ∈ [0 , , roughly one year. We fix the values of the following parameters α X = 0 . , σ X = 0 . , α Y = 0 . , c = 0 . , λ = 2 . The speed of mean reversion for the base component α X yields a half-life of seven days, while the onefor the spikes α Y yields a half-life of two days. The value for σ X yields an annualised volatility of .The values for c and λ give jumps with mean . and frequency of spikes a month. • Changing the level of mean reversion (Esscher transform) , ¯ β = (0 ,
0) :
Setting ¯ β = (0 , , the probability measure Q only changes the level of mean reversion (which is assumed to be zerounder the historical measure P ). Moreover, as R Fa,Q ( t, τ ) is deterministic when ¯ β = (0 , , wehave that the randomness in R Fg,Q ( t, τ ) comes into the picture through E P [ S ( T ) |F t ] , in particularthrough the levels of the driving factors X and Y. By Proposition 4.9 we have that R Fg,Q ( t, τ ) = E P [ S ( t + τ ) |F t ] × (cid:26) exp (cid:18) R Fa,Q ( t, τ ) − κ ′ L ( θ ) − κ ′ L (0) α Y (1 − e − α Y τ ) (cid:19)
50 100 150 200 250 300 350 Τ- - - H Τ L (a) θ = − . , θ = 0 . , X ( t ) = − . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - - H Τ L (b) θ = 0 . , θ = − . , X ( t ) = 0 . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - - H Τ L (c) θ = − . , θ = 0 . , X ( t ) = − . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (d) θ = − . , θ = 0 . , X ( t ) = 0 . , Y ( t ) = 0 . F IGURE
5. Risk premium profiles when L is a compound Poisson process with exponen-tially distributed jumps. Esscher transform: case ¯ β = (0 , . Geometric spot model × exp (cid:18)Z τ Z ∞ ( e θ z − e ze − αY s − ℓ ( dz ) ds (cid:19) − (cid:27) , and the sign of R Fg,Q ( t, τ ) is the same as the sign of R Fa,Q ( t, τ ) − κ ′ L ( θ ) − κ ′ L (0) α Y (1 − e − α Y τ ) + Z τ Z ∞ ( e θ z − e ze − αY s − ℓ ( dz ) ds = θ α X (1 − e − α X τ ) + Z τ Z ∞ ( e θ z − e ze − αY s − ℓ ( dz ) ds, which is equal to Σ( t, τ ) in Lemma 4.19.If θ = 0 , then the sign of R Fg,Q ( t, τ ) is the same as the sign of θ and it is constant over alltimes to maturity τ. Similarly, if θ = 0 , the sign R Fg,Q ( t, τ ) is the same as the sign of θ and it isalso constant. If both θ and θ are different from zero we can get risk premium profiles with nonconstant sign. By Lemma 4.19, we have that lim τ → ∂∂τ Σ( t, τ ) = θ + κ L (1 + θ ) − κ L ( θ ) − κ L (1)= θ + Z ∞ ( e θ z − e z − ℓ ( dz ) . Hence, if we want the sign of R Fg,Q ( t, τ ) to be positive when τ is close to zero we have to impose θ + Z ∞ ( e θ z − e z − ℓ ( dz ) > . (4.32) PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 33
For large times to maturity, Lemma 4.19 yields lim τ →∞ Σ( t, τ ) = θ α X + Z ∞ κ L ( e − α Y t + θ ) − κ L ( θ ) − κ L ( e − α Y t ) dt = θ α X + Z ∞ Z ∞ ( e θ z − e ze − αY t − ℓ ( dz ) dt. Using Fubini’s theorem we get that Z ∞ Z ∞ ( e θ z − e ze − αY t − ℓ ( dz ) dt = Z ∞ ( e θ z − Z ∞ ( e ze − αY t − dtℓ ( dz )= Z ∞ ( e θ z − α Y (Ei( z ) − log( z ) − γ ) ℓ ( dz ) , where Ei( z ) = R z −∞ e t t dt is the exponential integral function and γ is the Euler-Mascheroni con-stant. Hence, if we want R Fg,Q ( t, τ ) to be negative when τ is large we have to impose θ + α X α Y Z ∞ ( e θ z −
1) (Ei( z ) − log( z ) − γ ) ℓ ( dz ) < . (4.33)Note that Ei( z ) − log( z ) − γ ≥ , ∀ z ≥ and e z − − α X α Y (Ei( z ) − log( z ) − γ ) > , for all z > and α X < α Y . Therefore, for all θ > one has that < Z ∞ ( e θ z −
1) (Ei( z ) − log( z ) − γ ) ℓ ( dz ) < α Y α X Z ∞ ( e θ z −
1) ( e z − ℓ ( dz ) . (4.34)Combining equations (4 . , (4 . and (4 . we can conclude that it is possible to choose θ < and θ > such that R Fg,Q ( t, τ ) > when the time to maturity is close to zero and R Fg,Q ( t, τ ) < when the time to maturity is large. • Changing the speed of mean reversion, ¯ θ = (0 ,
0) :
Setting ¯ θ = (0 , , the probability measure Q only changes the speed of mean reversion. By Lemma 4.19 we have that the sign of R Fg,Q ( t, τ ) will coincide with the sign of Σ( t, τ ) = X ( t ) e − α X τ ( e α X β τ −
1) + Y ( t ) (cid:0) Ψ ,β ( τ ) − Ψ , ( τ ) (cid:1) + σ X α X Λ(2 α X τ, − β ) + (cid:0) Ψ ,β ( τ ) − Ψ , ( τ ) (cid:1) , Σ ( t, τ ) + Σ ( t, τ ) + Σ ( t, τ ) + Σ ( t, τ ) , and lim τ →∞ Σ( t, τ ) = σ X α X β − β ≥ τ → ∂∂τ Σ( t, τ ) = X ( t ) α X β + Y ( t ) α Y β κ ′ L (1) − κ ′ L (0) κ ′′ L (0) , where κ ′ L (1) − κ ′ L (0) and κ ′′ L (0) are strictly positive. Hence the risk premium will approachto a non negative value in the long end of the market. In the short end, it can be both positiveor negative and stochastically varying with X ( t ) and Y ( t ) , but Y ( t ) will always contribute to apositive sign. For any τ, the sign of Σ ( t, τ ) will be the sign of X ( t ) , that can be positive ornegative. As the function Λ( x, y ) is positive, the term Σ ( t, τ ) is always positive. To analyse thesign of Σ ( t, τ ) , note that Λ ,β ( u ) − Λ , ( u ) = α Y β κ ′′ L (0) Z ∞ ( e uz − zℓ ( dz ) ≥ , u ≥ , and Ψ ,β (1) = Ψ , ( τ ) . Hence, applying a comparison theorem for ODEs, see Theorem 6.1,pag.31, in Hale [12], we have that Ψ ,β ( τ ) − Ψ , ( τ ) ≥ , for all τ, and, as Y ( t ) is alwayspositive, the term Σ ( t, τ ) is also always positive. Finally, as Λ ( u ) , Λ ,β ( u ) = Λ , ( u ) = Z ∞ ( e uz − ℓ ( dz ) , is an strictly increasing function and Ψ ,β ( t ) ≥ Ψ , ( t ) we get that Σ ( t, τ ) = Ψ ,β ( τ ) − Ψ , ( τ ) = Z τ { Λ (Ψ ,β ( t )) − Λ (Ψ , ( t )) } dt ≥ . Hence, if β = 0 or X ( t ) ≥ , then R Fg,Q ( t, τ ) will be positive for all times to maturity. Some ofthe possible risk profiles that can be obtained are plotted in Figure 6. • Changing the level and speed of mean reversion simultaneously : We proceed as in the arith-metic case. As we are more interested in how the change of measure Q influence the component Y ( t ) , responsible for the spikes in the prices, we are going to assume that β = 0 . This meansthat Q may change the level of mean reversion of the regular component X ( t ) , but not the speedat which this component reverts to that level. According to Lemma 4.19 we have that the sign of R Fg,Q ( t, τ ) will coincide with the sign of Σ( t, τ ) = Y ( t )(Ψ θ ,β ( τ ) − e − α Y τ ) + θ α X (1 − e − α X τ ) + Ψ θ ,β ( τ )
50 100 150 200 250 300 350 Τ- - H Τ L (a) β = 0 . , β = 0 . , X ( t ) = 1 . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (b) β = 0 . , β = 0 . , X ( t ) = − . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (c) β = 0 . , β = 0 . , X ( t ) = − . , Y ( t ) = 0 .
50 100 150 200 250 300 350 Τ- - H Τ L (d) β = 0 . , β = 0 . , X ( t ) = − . , Y ( t ) = 2 . F IGURE
6. Risk premium profiles when L is a compound Poisson process with exponen-tially distributed jumps. Case ¯ θ = (0 , . Geometric spot price model
PRICING MEASURE TO EXPLAIN THE RISK PREMIUM IN POWER MARKETS 35
50 100 150 200 250 300 350 Τ- - H Τ L (a) β = 0 , β = 0 . , θ = − . , θ = 0 . , X ( t ) =1 . , Y ( t ) = 1 . F IGURE
7. Risk premium profiles when L is a compound Poisson process with exponen-tially distributed jumps. Geometric spot model − Z τ Z ∞ ( e ze − αY s − ℓ ( dz ) ds = Y ( t ) (cid:0) Ψ θ ,β ( τ ) − Ψ , ( τ ) (cid:1) + θ α X (1 − e − α X τ ) + (cid:0) Ψ θ ,β ( τ ) − Ψ , ( τ ) (cid:1) , Σ ( t, τ ) + Σ ( t, τ ) + Σ ( t, τ ) , and lim τ →∞ Σ( t, τ ) = θ α X + Z ∞ κ L (Ψ θ ,β ( t ) + θ ) − κ L ( θ ) − κ L ( e − α Y t ) dt (4.35) = θ α X + Z ∞ Z κ ′ L ( θ + λ Ψ θ ,β ( t )) dλ Ψ θ ,β ( t ) dt − Z ∞ Z κ ′ L ( θ + λe − α Y t ) dλe − α Y t dt lim τ → ∂∂τ Σ( t, τ ) = Y ( t ) α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) (4.36) + θ + κ L (1 + θ ) − κ L ( θ ) − κ L (1)= Y ( t ) α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) + θ + Z κ ′ L ( θ + λ ) dλ − κ L (1) Note that we can make equation (4 . negative by simply choosing θ θ < − α X Z ∞ Z κ ′ L ( θ + λ Ψ θ ,β ( t )) dλ Ψ θ ,β ( t ) dt (4.37) + α X Z ∞ Z κ ′ L ( θ + λe − α Y t ) dλe − α Y t dt. On the other hand, to make equation (4 . positive, we have to choose θ satisfying θ > − Z κ ′ L ( θ + λ ) dλ + κ L (1) − Y ( t ) α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) . (4.38) Equations (4 . and (4 . are compatible if the following equation is satisfied U + ( θ , β ) , Z κ ′ L ( θ + λ ) dλ + α X Z ∞ Z κ ′ L ( θ + λe − α Y t ) dλe − α Y t dt + Y ( t ) α Y β κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) > α X Z ∞ Z κ ′ L ( θ + λ Ψ θ ,β ( t )) dλ Ψ θ ,β ( t ) dt + κ L (1) , U − ( θ , β ) . (4.39)As e − α Y t ≤ , Ψ θ ,β ( t ) ≤ , κ ′ L ( θ ) > and κ ′′ L ( θ ) > we have that κ ′ L (1 + θ ) − κ ′ L ( θ ) κ ′′ L ( θ ) = R κ ′′ L ( θ + λ ) dλκ ′′ L ( θ ) > , Z ∞ Z κ ′ L ( θ + λe − α Y t ) dλe − α Y t dt ≥ κ ′ L ( θ ) α Y , and Z ∞ Z κ ′ L ( θ + λ Ψ θ ,β ( t )) dλ Ψ θ ,β ( t ) dt ≤ Z κ ′ L ( θ + λ ) dλ Z ∞ Ψ θ ,β ( t ) dt, As Ψ θ ,β ( t ) converges to zero exponentially fast, see equation (4 . , we have that Z ∞ Ψ θ ,β ( t ) dt < ∞ . Actually, as Λ θ ,β ( u ) < Λ θ ,β (1) < , < u < , we can use a comparison theorem for ODEsto obtain that Ψ θ ,β ( t ) ≤ e Λ θ ,β (1) t = exp (cid:18) − α Y (1 − β κ ′′ L ( θ ) ( κ ′ L (1 + θ ) − κ ′ L ( θ ))) t (cid:19) , which yields Z ∞ Ψ θ ,β ( t ) dt ≤ α Y (1 − β R κ ′′ L ( θ + λ ) dλκ ′′ L ( θ ) ) − . Hence, U + ( θ , β ) ≥ Z κ ′ L ( θ + λ ) dλ + α X α Y κ ′ L ( θ ) + Y ( t ) α Y β , V + ( θ , β ) ,U − ( θ , β ) ≤ α X α Y Z κ ′ L ( θ + λ ) dλ (1 − β R κ ′′ L ( θ + λ ) dλκ ′′ L ( θ ) ) − + κ L (1) , V − ( θ , β ) , and if we can find θ ∈ D gL ( δ ) for some δ > and β ∈ (0 , such that V + ( θ , β ) > V − ( θ , β ) then equation (4 . will be satisfied. Note that the larger the value of Y ( t ) the easier to find such θ and β . Even in the case that Y ( t ) = 0 , by choosing β close to zero and θ large enough wecan get V + ( θ , β ) > V − ( θ , β ) . This shows that we can create a change of measure Q generatingthe empirically observed risk premium profile, see Figure 7.R EFERENCES [1] Barndorff-Nielsen, O.E., Benth, F.E., and Veraart, A. (2010). Modelling energy spot prices by volatility modulated Levy-driven Volterra processes.
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