A pricing measure to explain the risk premium in power markets
In electricity markets, it is sensible to use a two-factor model with mean reversion for spot prices. One of the factors is an Ornstein-Uhlenbeck (OU) process driven by a Brownian motion and accounts for the small variations. The other factor is an OU process driven by a pure jump Lévy process and models the characteristic spikes observed in such markets. When it comes to pricing, a popular choice of pricing measure is given by the Esscher transform that preserves the probabilistic structure of the driving Lévy processes, while changing the levels of mean reversion. Using this choice one can generate stochastic risk premiums (in geometric spot models) but with (deterministically) changing sign. In this paper we introduce a pricing change of measure, which is an extension of the Esscher transform. With this new change of measure we also can slow down the speed of mean reversion and generate stochastic risk premiums with stochastic non constant sign, even in arithmetic spot models. In particular, we can generate risk profiles with positive values in the short end of the forward curve and negative values in the long end. Finally, our pricing measure allows us to have a stationary spot dynamics while still having randomly fluctuating forward prices for contracts far from maturity.
In modelling and analysis of forward and futures prices in commodity markets, the risk premium plays an important role. It is defined as the difference between the forward price and the expected commodity spot price at delivery, and the classical theory predicts a negative risk premium. The economical argument for this is that producers of the commodity is willing to pay a premium for hedging their production (see Geman  for a discussion, as well as a list of references).
Geman and Vasicek  argued that in power markets, the consumers may hedge the price risk using forward 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 (sometimes referred to as spikes). One might observe a risk premium which may be positive in the short end of the forward market, and negative in the long end where the producers are hedging their power generation. A theoretical and empirical foundation for this is provided in, for example, Bessembinder and Lemon  and Benth, Cartea and Kiesel .
When deriving the forward price, one specifies a pricing probability and computes the forward price as the conditional expected spot at delivery. In the power market, this pricing probability is not necessarily a so-called equivalent martingale measure, or a risk neutral probability (see Bingham and Kiesel ), as the spot 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 of pricing 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 the form as a two-factor mean reversion dynamics. Lucia and Schwartz  considered two-factor models for the electricity spot price dynamics in the Nordic power market NordPool. Both arithmetic and geometric models where suggested, that is, either directly modelling the spot price by a two-factor dynamics, or assuming such a model for the logarithmic spot prices. Their models were based on Brownian motion and, as such, not able to capture the extreme variations in the power spot markets. Cartea and Figueroa  used a compound Poisson process to model spikes, that is, extreme price jumps which are quickly reverted back to ”normal levels”. Benth, Šaltytė Benth and Koekebakker  give a general account on multi-factor models based on Ornstein-Uhlenbeck processes driven by both Brownian motion and Lévy processes. Empirical studies suggest a stationary power spot price dynamics after explaining deterministic seasonal variations (see e.g. Barndorff-Nielsen, Benth and Veraart  for a study of spot prices at EEX, the German power exchange). We will in this paper focus on a two-factor model for the spot, where each factor is an Ornstein-Uhlenbeck process, driven by a Brownian motion and a jump process, respectively. The first factor 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 Lévy property. This is called the Esscher transform (see Benth et al. ), and works for Lévy processes as the Girsanov transform with a constant parameter for Brownian motion. The effect of doing such a measure change is to adjust the mean reversion level, and it is known that the risk premium becomes deterministic and 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 two factors. As it turns out, in conjunction with an Esscher transform as mentioned above, we can produce a stochastically varying risk premium, where potential positive premiums in the short end of the market can be traced back to sudden jumps in the spike factor being slowed down under the pricing measure. This result holds for arithmetic spot models, whereas the geometric ones are much harder to analyse under this change of probability. The class of probabilities preserves the Ornstein-Uhlenbeck structure of the factors, and as such may be interpreted as a dynamic structure preserving measure change. For the Lévy driven component, the Lévy property is lost in general, and we obtain a rather complex jump process with state-dependent (random) compensator measure.
We can explicitly describe the density process for our measure change. The theoretical contribution of this paper, besides the new insight on risk premium, is a proof that the density process is a true martingale process, indeed verifying that we have constructed a probability measure. This verification is not straightforward because the kernels used to define the density process, through stochastic exponentiation, are stochastic and unbounded. Hence, the usual criterion by Lépingle-Mémin  is difficult to apply and, furthermore, it does not provide sharp results. We follow the same line of reasoning as in a very recent paper by Klebaner and Lipster . Although their result is more general than ours in some respects, it does not apply directly to our case because we need some additional integrability requirements. The proof is roughly as follows. First, we reduce the problem to show the uniform integrability of the sequence of random 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, combined with an easy inequality for the logarithm function, allow us to get rid of the stochastic exponential in the expression to be bounded. Finally, we can reduce the problem to get an uniform bound for the second moment of the factors under these new probability measures.
Interestingly, as our pricing probability is reducing the speed of mean reversion, we might in the extreme situation ”turn off” the mean reversion completely (by reducing it to zero). For example, if we take the Brownian factor as the case, we can have a stationary dynamics of the ”normal variations” in the market, but when looking at the process under the pricing probability the factor can be non-stationary, that is, a drifted Brownian motion. A purely stationary dynamics for the spot will produce constant forward prices in the long end of the market, something which is not observed empirically. Hence, the inclusion of non-stationary factors are popular in modelling the spot-forward markets. In many studies of commodity spot and forward markets, one is considering a two-factor model with one non-stationary and one stationary component. The stationary part explains the short term variations, while the non-stationary is supposed to account for long-term price fluctuations in the spot (see Gibson and Schwartz  and Schwartz and Smith  for such models applied to oil markets). Indeed, the power spot models in Lucia and Schwartz  are of this type. It is hard to detect the long term factor in spot price data, and one is usually filtering it out from the forward prices using contracts far from delivery. Theoretically, such contracts 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 measure which turns one of the factors into a non-stationary dynamics. This would imply that one could directly fit a two-factor stationary spot model to power data, and next calibrate a measure change to account for the long term variations in the forward prices by turning off (or significantly slow down) the speed of mean reversion.
Our results are presented as follows: in the next section we introduce the basic assumptions and properties satisfied by the factors in our model. Then, in Section 3, we define the new change of measure and prove the main results regarding the uniform integrability of its density process. We deal with the Brownian and pure jump case separately. Finally, in Section 4, we recall the arithmetic and geometric spot price models. We compute the forward price processes induced by this change of measure and we discuss the risk premium profiles that can be obtained.
2. The mathematical set up
Suppose that is a complete filtered probability space, where is a fixed finite time horizon. On this probability space there are defined , a standard Wiener process, and a pure jump Lévy subordinator with finite expectation, that is a Lévy process with the following Lévy-Itô representation where is a Poisson random measure with Lévy measure satisfying We shall suppose that and are independent of each other. The following assumption is minimal, having in mind, on the one hand, that our change of measure extends the Esscher transform and, on the other hand, that we are going to consider a geometric spot price model.
We assume that
is strictly positive constant, which may be
Actually, to have the geometric model well defined we will need to assume later that Some remarks are in order.
In the cumulant (or log moment generating) function is well defined and analytic. As , has moments of all orders. Also, is convex, which yields that and, hence, that is non decreasing. Finally, as a consequence of a.s., we have that is non negative.
Thanks to the Lévy-Kintchine representation of we can express and its derivatives in terms of the Lévy measure We have that for
showing, in fact, that
Consider the OU processes
with Note that, in equation is written as a sum of a finite variation process and a martingale. We may also rewrite equation as a sum of a finite variation part and pure jump martingale
where is the compensated version of . In the notation of Shiryaev , page 669, the predictable characteristic triplets (with respect to the pseudo truncation function ) of and are given by
respectively. In addition, applying Itô formula to and one can find the following explicit expressions for and
Using that the stochastic integral of a deterministic function is Gaussian, one easily gets that is a Gaussian process and with
3. The change of measure
We will consider a parametrized family of measure changes which will allow us to simultaneously modify the speed and the level of mean reversion in equations and . The density processes of these measure changes will be determined by the stochastic exponential of certain martingales. To this end consider the following families of kernels
Under the assumption which is stronger than one can consider the set cl( and our results still hold by changing and by its left derivatives at the rigth end of
Typical examples of and are the following:
Bounded support: has a jump of size i.e. In this case and
Finite activity: is a compound Poisson process with exponential jumps, i.e., for some and In this case and
Infinite activity: is a tempered stable subordinator, i.e., for some and In this case also and
Next, for define the following family of Wiener and Poisson integrals
associated to the kernels and respectively.
Let be a semimartingale on and denote by the stochastic exponential of that is, the unique strong solution of
When is a local martingale, is also a local martingale. If is positive, then is also a supermartingale and In that case, one has that is a true martingale if and only If is a positive true martingale, it can be used as a density process to define a new probability measure equivalent to that is,
The desired family of measure changes is given by with
where we are implicitly assuming that is a strictly positive true martingale. Then, by Girsanov’s theorem for semimartingales (Thm. 1 and 3, p. 702 and 703 in Shiryaev ), the process and become
where is a -standard Wiener process and the -compensator measure of (and ) is
In conclusion, the semimartingale triplet for and under are given by and respectively.
Under and still satisfy Langevin equations with different parameters, that is, the measure change preserves the structure of the equations. The process is not a Lévy process under , but it remains a semimartingale. Therefore, one can use Itô formula again to obtain the following explicit expressions for and
Looking at equations and , one can see how the values of the parameters and change the drift. Setting we keep fixed the level to which the process reverts and change the speed of mean reversion by changing . If we fix the speed of mean reversion and change the level by changing By choosing , say, we observe that in (3.9) becomes (using a limit consideration in the second term)
Hence, is a drifted Brownian motion and we have a non-stationary dynamics under the pricing measure with this choice of . Obviously, we can choose and obtain similarly a non-stationary dynamics for the jump component as well, however, this will not be driven by a Lévy process under .
The previous reasonings rely crucially on the assumption that is a probability measure. Hence, we have to find sufficient conditions on the Lévy process and the possible values of the parameters and that ensure to be a true martingale with strictly positive values. As the quadratic co-variation between and is identically zero, by Yor’s formula (equation II.8.19 in ) we can write
and, as the stochastic exponential of a continuous process is always positive, we just need to ensure the positivity of Assume that is positive, then remark 3.3 yields that is a true martingale if and only if Using the independence of and and the identity we get
showing that is a martingale if and only if and are also martingales. Hence, we can write
The previous reasonings allow us to reduce the proof that is a probability measure equivalent to , to prove that is martingale (or ) and is a martingale with strictly positive values (or ). The literature on this topic is huge, see for instance Kazamaki , Novikov , Lépingle and Mémin  and Kallsen and Shiryaev . The main difficulty when trying to use the classical criteria is that our kernels depend on the processes and which are unbounded. To prove that is a martingale one could use a localized version of Novikov’s criterion. However, this approach would entail to show that the expectation of the exponential of the integral of a stochastic iterated integral of order two is finite. Although these computations seem feasible, they are definitely very stodgy. On the other hand, the most widely used sufficient criterion for martingales with jumps is the Lépingle-Mémin criterion. This criterion is very general but the conditions obtained are far from optimal. Using this criterion we are only able to prove the result by requiring the Lévy process to have bounded jumps.
In a very recent paper, assuming some structure on the processes, Klebaner and Lipster  give a fairly general criterion which seems easier to apply than those of Novikov and Lépingle-Mémin. Although we can not apply directly their criteria, at least not in the pure jump case, we can reason similarly to prove the desired result for and
Finally, note that these results can be extended, in a straightforward manner, to any finite number of Langevin equations driven by Brownian motions and Lévy processes, independent of each other. In the following two subsections, we will drop the subindices in the parameters and
3.1. Brownian driven OU-process
We first show that the process is a martingale under .
Let and . Then, , defined by is a square integrable martingale under .
We have to show that We get
By remark 2.3 and the properties of the Gaussian distribution, one has
because and are continuous functions on ∎
Let and . Then is a martingale under
As is a martingale with continuous paths, we have that is a positive local martingale. By remark 3.3, it suffices to prove that Note that the sequence of stopping times is a reducing sequence for That is, converges a.s. to and, for every fixed, the stopped process is a (bounded) martingale on . Therefore, and if we show that
we will have finished. To show is equivalent to show the uniform integrability of the sequence of random variable that is, to show
It is not difficult to prove that if is a non-negative function such that and
then is uniformly integrable. We consider the test function Hence, it suffices to prove that
Note that we can use the sequence of martingales on given by to define a sequence of probability measures with Radon-Nykodim densities given by In addition, one has that
where On the other hand, from we have the trivial bound Combining the last bound with the change of measure given by we get that
implies that holds. Applying Girsanov’s Theorem, we can write
where is a -Brownian motion. Therefore, it suffices to prove that
because this imply that is a -martingale with zero expectation and, in passing, that holds. Now we proceed as in the proof of Proposition 3.6. We have that
but now the term with is more delicate to treat. Using Remark 2.3, we know that conditioned to is Gaussian, but we do not know the distribution of and, hence, a direct computation of is not possible. However, we have that
where we have used that the function for and that
Hence, we have shown and the result follows. ∎
3.2. Lévy driven OU-processes
First we will prove that is a square integrable martingale.
Let . Then defined by , is a square integrable martingale under
According to Ikeda-Watanabe , p. 59-63, we have to check that We can write