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Pricing forward contracts in power markets by the certainty equivalence principle: Explaining the sign of the market risk premium q
Fred Espen Benth a,*,A´ lvaro Cartea b,1,Ru¨diger Kiesel c
a Centre of Mathematics for Applications, University of Oslo, Norway b Commodities Finance Centre Birkbeck, University of London, UK c Institute of Mathematical Finance, University of Ulm, Germany
Abstract
In this paper we provide a framework that explains how the market risk premium, defined as the difference between forward prices and spot forecasts, depends on the risk preferences of market players and the interaction between buyers and sellers. In commodities markets this premium is an important indicator of the behavior of buyers and sellers and their views on the market spanning between short-term and long-term horizons. We show that under certain assumptions it is possible to derive explicit solutions that link levels of risk aversion and market power with market prices of risk and the market risk premium. We apply our model to the German electricity market and show that the market risk premium exhibits a term structure which can be explained by the combination of two factors. Firstly, the levels of risk aversion of buyers and sellers, and secondly, how the market power of producers, relative to that of buyers, affects forward prices with different delivery periods.
JEL classification: G13; G11; D81; Q40
Keywords: Contango; Backwardation; Market price of risk; Electricity forwards; Market risk premium; Forward risk premium; Forward bias; Market power
1. Introduction and live stock. The physical nature of commodities is per- haps one of their most defining characteristics specifically Commodities are a very different asset class from the because it plays an important role in the behavior of their more traditional classes of traded assets such as equities prices in both the spot and forward markets. and bonds. Commodities normally encompass physical Let us contrast equity forwards with commodity for- goods such as oil, gas, electricity, metals, agriculturals wards. For example, if interest rates and dividends are assumed to be deterministic, the pricing of equity forwards is a straightforward exercise. Simple no-arbitrage arguments q This paper was reviewed and accepted while Prof. Giorgio Szego was are employed and the pricing is principally based on the abil- the Managing Editor of The Journal of Banking and Finance and by the ity to borrow money to purchase the underlying equity and past Editorial Board. The authors are grateful to Kevin Metka, for excellent research assistance, and to two anonymous referees for their hold it until delivery. As a result, the arbitrage-free forward helpful comments. price is the cost of borrowing net of dividends yielded by the * Corresponding author. Tel.: +47 22855892; fax: +47 22854349. equity. With commodities one can in principle apply a sim- E-mail addresses: [email protected] (F.E. Benth), a.cartea@bbk. ilar strategy to price forward contracts. However, the phys- ac.uk (A´ . Cartea), [email protected] (R. Kiesel). 1 A´ . Cartea is thankful for the hospitality and generosity shown by the ical nature of commodities makes it very difficult for two Finance Group at the Saı¨d Business School, Oxford, where part of this reasons. First, the cost-of-carry (interest plus ‘storage’ costs) research was undertaken. is not straightforward to calculate or measure. Second, it is F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2007 2 necessary to account for the convenience yield (equivalent to risk diversification. Producers have made large investments collection of dividends on equities), but this is also excep- with the aim of recouping them over a long period of time tionally difficult to quantify or model. as well as making a return on them. As with any other The shape of commodities’ forward curves for different investments, there is an incentive for producers to reduce delivery periods has always been of utmost importance to variability in their profits by trading in instruments with understand market players’ (producers, consumers and payoffs that covary with their profits. Similarly, consumers speculators) ‘attitudes’ towards bearing risk in these mar- (which might be intermediaries and/or use the commodity kets. Forwards exhibit peculiar behavior depending on in their production process) also have an incentive to hedge the time or delivery period. A situation where forward their positions in the market by contracting forwards that prices are above current spot prices is labeled contango help diversify their risks. and it is normally associated with circumstances where The relative appetite of producers and consumers for the immediate supply of the commodity is plentiful relative risk-diversification has a temporal dimension to it. Varia- to demand. Similarly, the situation where forward prices tions in this appetite for risk diversification will be evident are below spot prices is known as backwardation and it in the different levels of market exposure chosen by produc- is generally associated with circumstances of low current ers and consumers and in the different levels chosen by supply levels and/or low inventory levels. One can deter- members within each of these groups. For example a pro- mine whether contango or backwardation exists by simple ducer will generally be exposed to market uncertainty for observation of the forward markets. For example, in elec- a longer period of time, perhaps determined by the remain- tricity and gas markets one normally observes that, for ing life of its assets, whilst consumers will tend to make ‘long’ dated forward contracts, markets are in backwarda- decisions based on a shorter time scale. In other words, tion and for ‘shorter’ maturities the market is in contango the gains in terms of risk-diversification for consumers (Cartea and Figueroa, 2005; Cartea and Williams, 2007). and producers will vary across time, therefore having a first Another quantity of importance that relates forward order impact on forward clearing prices. and expected spot prices is the market risk premium or for- In this article we argue that it is precisely these differ- ward bias p(t,T). This is defined as the difference, calcu- ences in the desire to hedge positions and diversify risk that lated at time t, between the forward price F(t,T), at time explain the market risk premium and its sign. Intuitively, t with delivery at T, and the expected spot price: the further out one looks into the market, the less incentiv- ized consumers are to contract commodity forwards; how- pðt; T Þ¼F ðt; T Þ EP ½SðT ÞjF : ð1Þ t ever the producers’ desire to hedge does not diminish as Here EP is the expectation operator, under the historical quickly. We associate situations where p(t,T) > 0 (a posi- measure P, with information up until time t and S(T)is tive market risk premium) with markets where the consum- the spot price at time T.2 ers’ desire to cover their positions ‘outweighs that of the To the best of our knowledge, recent literature on com- producers. Conversely situations where p(t,T) < 0 (a nega- modities has not addressed the connection between the tive market risk premium) result when the producers’ desire market risk premium and market players’ behavior and to hedge their positions outweighs that of the consumers. risk preferences. Moreover, it has not dealt with the ques- In order to explain the market risk premium and the tion of why and how in some commodities markets we driving forces that give rise to it we organize the rest of expect the market risk premium p(t,T) to change signs in the article as follows. Section 2 discusses the notion of a rep- time T. The main contribution of this article is therefore resentative producer and a representative consumer. Based to address these questions and propose a framework that on their preferences we calculate an attainable set of for- allows us to establish explicit relationships between the ward prices where the two representative agents are willing market risk premium, the market price of risk and market to trade forward contracts. Section 3 discusses clearing mar- players’ risk preferences. By doing so, this allows us to ket forward prices and the relative ‘market power’ agents explain the interesting connections between forward price have over these prices. Section 4 examines the market price formation and its deviations from spot forecasts based on of risk and market risk premium implied by our model the consumers’ and producers’ attitudes to risk. under different assumptions. Section 5 applies our model To understand the importance of the market risk pre- to German electricity data and Section 6 concludes. mium, it is important to point out that forward curves are not forecasts of the commodity spot price in the future. 2. Representative agents, price dynamics and forward price The clearing prices of forwards are the result of demand bounds and supply, which in turn are determined by the individual characteristics of market players. Indeed the main motiva- In this section we describe producers’ and consumers’ tion for players to engage in forward contracts is that of preferences via the utility function of two representative agents. As an example we look at the wholesale electricity 2 Note that it is incorrect to say that when p(t,T) < 0 (resp. p(t,T)>0) markets where we model the dynamics of the spot price as the forward curve is in contango (resp. backwardation). Moreover, S(t)is a stochastic process. Agents must decide how to manage not generally a martingale under P. their exposure to the spot and forward markets for every 2008 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 3 future date T. A key question for the producer is how much dX iðtÞ¼ aiX iðtÞdt þ riðtÞdBiðtÞ; ð2:2Þ of his future production, which cannot be predicted with and total certainty, will he wish to sell on the forward market or, when the time comes, sell it on the spot market. Simi- dY jðtÞ¼ bjY jðtÞdt þ dLjðtÞ: ð2:3Þ larly, the consumer must decide how much of her future Here, B (t), i =1,...,m, are standard independent Brown- needs, which cannot be predicted with full certainty either, i ian motions and L (t), j =1,...,n are independent Le´vy will be acquired via the forward markets and how much j processes.3 Let r (t) be (possibly seasonal) deterministic on the spot. Clearly, as described above, both agents have i volatility functions. The processes Y (t) are zero-mean the incentive to enter the forward market in the interest of j reverting processes responsible for the spikes or large devi- risk diversification. We approach this financial decision ations which revert at a fast rate b > 0, while X (t) are zero- and equilibrium price formation in two steps. First, we j i mean reverting processes that account for the normal determine the forward price that makes the agents indiffer- variations in the spot price evolution with mean-reversion ent between the forward and spot market and, second, we a >0. discuss how the relative willingness of producers and con- i We suppose that the Le´vy processes are exponentially sumers to hedge their exposures determines market clearing integrable in the sense that there exists a constant j >0 prices. such that We assume that the risk preferences of the representative Z agents are expressed in terms of an exponential utility func- ejz e ‘jðdzÞ < 1; ð2:4Þ tion parameterized by the risk aversion constant c >0; jzjP1 UðxÞ¼1 expð cxÞ: for all je 6 j and j =1,...,n. This implies that the spot price process S(t) has exponential moments up to order We let c:=c for the producer and c:=c for the con- p c j, and that the log-moment generating functions defined by sumer. The two agents can choose whether to act in the xLjð1Þ spot or the forward market. The forward market consists /jðxÞ¼ln E e ; j ¼ 1; ...; n; ð2:5Þ of contracts delivering the spot (physically, or in money 6 ´ terms) over a given delivery period. Typical examples can exist for jxj j, where ‘j is the Levy measure of the process be the electricity or gas markets. In the latter the forward Lj(t). In the sequel we shall assume that j is sufficiently contracts have a monthly delivery period, while in the elec- large to make the necessary exponential moments of Lj(t) tricity market, which will be the particular case discussed in finite. the remaining of this article, the contracts may have differ- Assume that the producer will deliver the spot over the ent periods of settlement, ranging from daily, through time interval [T1,T2]. He has the choice to deliver the pro- weekly and up to even yearly. duction in the spot market, where he faces uncertainty in We want to derive bounds for forward prices through the prices over the delivery period, or to sell a forward con- the principle of certainty equivalence between the two mar- tract with delivery over the same period. The producer 6 kets. In particular, we will obtain an upper bound, given by takes this decision at time t T1. the maximum price the consumer is willing to pay before We determine the forward price that makes the producer switching to the spot market, and a lower bound given indifferent between the two alternatives, denoted by Fpr(t,T1,T2), from the equation by the producer’s lowest forward price he is willing to trade Z T 2 at before switching to the spot market. These two bounds P restrict forward prices to a set of feasible forward equilib- 1 E exp cp SðuÞdu jFt T 1 rium prices and we postpone until Section 3 the discussion P of how market clearing forward prices are singled out from ¼ 1 E exp cpðT 2 T 1ÞF prðt; T 1; T 2Þ jFt ; this feasible set. or equivalently, 1 1 2.1. Producers and consumers forward price bounds F prðt; T 1; T 2Þ¼ cp T 2 T 1 Z T 2 Let ðX; F; PÞ be a probability space equipped with a fil- P ln E exp c SðuÞdu jFt ; tration . Following Lucı´a and Schwartz (2002) and p Ft T 1 Benth et al. (2007) we assume that the electricity spot price ð2:6Þ follows a mean-reverting multi-factor additive process Xm Xn where for simplicity we haveR assumed that the risk-free interest rate is zero. Note that T 2 SðuÞdu is what the pro- St ¼ KðtÞþ X iðtÞþ Y jðtÞ; ð2:1Þ T 1 i¼1 j¼1 3 In commodities markets one can expect to observe seasonal jumps. In where K(t) is the deterministic seasonal spot price level, this case we may use inhomogeneous Le´vy or Sato processes which are while Xi(t)andYj(t) are the solutions to the stochastic dif- processes with independent increments (Cont and Tankov, 2004; Sato, ferential equations 1999). Only minor technical changes in what follows are required. F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2009 4 Z ducer collects from selling the commodity on the spot T 2 E exp cpr SðuÞdu jFt market over the delivery period [T1,T2], while he receives T 1 (T T )F (t,T ,T ) from selling it on the forward market. Z 2 1 pr 1 2 T In the Proposition below we employ the spot dynamics 2 ¼ exp cpr KðuÞdu þ X ðtÞa ðt; T 1; T 2Þ (2.1) to explicitly calculate the indifference forward price. T 1 For ease of presentation we introduce the notation for the following functions. For i =1,...,m and j =1,...,n, þ Y ðtÞbðt; T 1; T 2Þ ( 1 e aiðT 1 sÞ e aiðT 2 sÞ ; s 6 T ; Z ai 1 T 2 a iðs; T 1; T 2Þ¼ ð2:7Þ 1 aiðT 2 sÞ E exp c rðsÞa ðs; T 1; T 2ÞdBs 1 e ; s P T 1: pr ai t Z and T 2 8 E exp cpr bðs; T 1; T 2ÞdLðsÞ ; < 1 bjðT 1 sÞ bjðT 2 sÞ e e ; s 6 T 1; t bj Z bjðs; T 1; T 2Þ¼ T 2 : 1 b ðT 2 sÞ 1 e j ; s P T 1: bj ¼ exp cpr KðuÞdu þ X ðtÞa ðt; T 1; T 2Þ T 1 ð2:8Þ þ Y ðtÞbðt; T 1; T 2Þ Proposition 2.1. The price for which the producer is indif- Z ferent between the forward and spot market is given by T 2 1 2 2 2 exp cpr r ðsÞa ðs; T 1; T 2Þds 2 t F prðt; T 1; T 2Þ Z Z T 2 T 2 Xm 1 a iðt; T 1; T 2Þ exp /ð c b ðs; T ; T ÞÞds : ¼ KðuÞdu þ X ðtÞ pr 1 2 T T T T i t 2 1 T 1 i¼1 2 1 Xn bjðt; T 1; T 2Þ cp þ Y jðtÞ Thus, the Proposition is proved after taking logarithms and T 2 T 1 2ðT 2 T 1Þ Zj¼1 dividing by the risk aversion and length of the delivery per- T 2 Xm 2 2 1 1 iod. h ri ðsÞa i ðs; T 1; T 2Þds c T 2 T 1 Zt i¼1 p Before proceeding we can interpret how jumps in the T Xn 2 model affect the indifference price calculated in Proposition / c b ðs; T ; T Þ ds; j p j 1 2 2.1 for the producer. For simplicity, if we assume that for t j¼1 the jump processes L ðtÞ; j ¼ 1; ; n, each process can only j where a i and bj are given by (2.7) and (2.8), respectively. jump either up or down it is straightforward to see how F prðt; T 1; T 2Þ is affected by each jump process. Suppose Proof. Suppose for simplicity that m = n = 1. We calculate LjðtÞ is a process of only positive jumps. Then, the log- the conditional expectation in (2.6). First observe that moment generating function / ðxÞ of L ðtÞ is an increasing Z Z Z Z j j T T T T 2 2 2 2 function with /jð0Þ¼0. Thus, when x < 0, /jðxÞ < 0, and S u du K u du X u du Y u du: ð Þ ¼ ð Þ þ ð Þ þ ð Þ since bj is positive, we have that the argument of /jð Þ in T 1 T 1 T 1 T 1 the indifference price of the producer is negative, and thus Inserting the explicit dynamics of X(u) and appealing to the the jump process LjðtÞ causes an increase in the indifference stochastic Fubini Theorem (see e.g. Protter, 1992), we find forward price. On the other hand, if LjðtÞ only exhibits neg- Z Z Z T 2 T 2 u ative jumps, we see that the indifference price is pushed aðu tÞ aðu sÞ X ðuÞdu ¼ X ðtÞe þ rðsÞe dBs du downwards. This is intuitively clear, because the producer T 1 T 1 Z t Z is willing to accept lower forward prices when there is a risk T 2 u of price drops in the spot market, whereas positive price ¼ X ðtÞa ðt; T ; T Þþ rðsÞe aðu sÞdB du 1 2 s spikes work to the advantage of the producer, and he will ZT 1 t T 2 be more reluctant to enter forward contracts that miss ¼ X ðtÞa ðt; T 1; T 2Þþ rðsÞa ðs; T 1; T 2ÞdBs: opportunities where he might be better-off selling in the t spot market. R T The consumer will derive the indifference price from the A similar calculation for 2 Y ðuÞdu yields, Z T 1 Z incurred expenses in the spot or forward market, which T 2 T 2 entails Y ðuÞdu ¼ Y ðtÞbðt; T 1; T 2Þþ bðs; T 1; T 2ÞdLðsÞ: T t Z 1 T 2 1 EP exp c SðuÞdu j Thus, since X(t) and Y(t) are measurable with respect to Ft c Ft T and using the independent increment properties of the 1 P Brownian motion and the Le´vy process, we get, ¼ 1 E ½ exp ð ccð ðT 2 T 1ÞF cðt; T 1; T 2Þ ÞÞjFt ; ð2:9Þ 2010 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 5 or, 3. Forward price and the market power 1 1 F cðt; T 1; T 2Þ¼ Inequality (2.11) clearly indicates a range of prices c T T c 2 1 Z where the producer and consumer are willing to attain a T 2 P deal. Aggregate demand and supply will ultimately deter- ln E exp cc SðuÞdu jFt : ð2:10Þ T 1 mine the clearing forward prices within this range. Previously we mentioned that the ‘appetite’ for risk diver- We calculate the following price for the consumer. sification varies across consumers and producers, depend- Proposition 2.2. The price that makes the consumer indif- ing on their degree of risk-aversion. Moreover, it seems ferent between the forward and the spot market is given by reasonable to assume that the desire to hedge exposure to Z market uncertainties will also vary with the horizon agents T 2 Xm 1 a iðt; T 1; T 2Þ are looking at. In circumstances where there are not a large F ðt; T ; T Þ¼ KðuÞdu þ X ðtÞ c 1 2 T T T T i number of consumers hedging long-term positions, whilst 2 1 T 1 i¼1 2 1 Xn at the same time producers are eager to hedge their expo- b ðt; T ; T Þ c þ j 1 2 Y ðtÞþ c sure, we say that consumers have market power. Similarly, T T j 2ðT T Þ j¼1 2 1 2 1 in situations (usually short-term horizons) where a large Z T Xm 2 1 1 amount of consumers come to market to cover their posi- r2ðsÞa 2ðs; T ; T Þds þ i i 1 2 c T T tions, the balance of power tilts over to the producers. t i¼1 c 2 1 Z We introduce the deterministic function pðt; T 1; T 2Þ2 T 2 Xn ½0; 1 describing the market power of the representative pro- /j ccbjðs; T 1; T 2Þ ds: t j¼1 ducer which therefore depends on time t and delivery per- iod. If the producer has full market power, corresponding to pðt; T 1; T 2Þ¼1, he can charge the maximum price possi- Proof. The proof is similar as in the producer’s case. h ble in the forward market. This will be equal to the maxi- Note that the producer prefers to sell his production in mum price that the consumer can accept, namely the forward market as long as the market forward price F cðt; T 1; T 2Þ, since the consumer will leave the forward mar- ket for any higher price. On the other hand, if the consumer F ðt; T 1; T 2Þ is higher than F prðt; T 1; T 2Þ. On the other hand, the consumer prefers the spot market if the market forward has full power, ie pðt; T 1; T 2Þ¼0, she will drive the forward price is more expensive than his indifference price price as far down as possible which corresponds to F t; T ; T . Thus, we have the bounds F prðt; T 1; T 2Þ. For any market power 0 < pðt; T 1; T 2Þ < 1, cð 1 2Þ p the forward price F ðt; T 1; T 2Þ is defined to be F prðt; T 1; T 2Þ 6 F ðt; T 1; T 2Þ 6 F cðt; T 1; T 2Þ: ð2:11Þ p F ðt; T 1; T 2Þ¼pðt; T 1; T 2ÞF cðt; T 1; T 2Þ Letting the risk aversion of the producer go to zero, we þð1 pðt; T ; T ÞÞF ðt; T ; T Þ: ð3:1Þ end up with the expected earnings from selling in the spot 1 2 pr 1 2 market, also known as the forecasted forward price. We In the most general setting, it would be possible to allow observe the same with the indifference price of the con- for a stochastic market power, being for instance depen- sumer when her risk aversion tends to zero. dent on the spot price dynamics. However, in this paper we shall constrain ourselves to the much simpler case of Proposition 2.3. It holds a deterministic market power. In some examples we con- lim F pr;cðt; T 1; T 2Þ c #0 sider it as a constant for simplicity, but in the empirical p;c Z Z T 2 T 2 study of Section 5 we find evidence of a term structure P 1 1 ¼ E SðuÞdujFt ¼ KðuÞdu for the market power. T T T T 2 1 T 1 2 1 T 1 The explicit dynamics for the forward price are easily Xm Xn a ðt; T ; T Þ b ðt; T ; T Þ stated as: þ i 1 2 X ðtÞþ j 1 2 Y ðtÞ T T i T T j i¼1 2 1 j¼1 2 1 Proposition 3.1. The forward price dynamics are given by Z Xm 0 T 2 /jð0Þ Z m þ bjðs; T 1; T 2Þds: T 2 X p 1 a iðt; T 1; T 2Þ j 1 T 2 T 1 t F ðt; T ; T Þ¼ KðuÞdu þ X ðtÞ ¼ 1 2 T T T T i 2 1 T 1 i¼1 2 1 Moreover, we find that Xn Z bjðt; T 1; T 2Þ T 2 þ Y jðtÞ P 1 T 2 T 1 F prðt; T 1; T 2Þ 6 E SðuÞdujSðtÞ j¼1 T 2 T 1 T 1 pðt; T 1; T 2Þðcpr þ ccÞ cpr 6 þ F cðt; T 1; T 2Þ: 2ðT T Þ Z 2 1 T 2 Xm 2 2 Proof. The first part is straightforward. The second part ri ðsÞa i ðs; T 1; T 2Þds results from applying Jensen’s inequality. h t i¼1 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2011 6 Z T Xn pðt; T ; T Þ 2 price of risk may be either positive or negative depending þ 1 2 / ðc b ðs; T ; T ÞÞds c T T j c j 1 2 on the time horizon considered. In Schwartz (1997) the cal- cð 2 1Þ t j¼1 ibration of one-factor models to futures prices of oil and 1 pðt; T ; T Þ 1 2 copper delivered negative market prices of risk in both c ðT T Þ Z pr 2 1 cases. Cartea and Figueroa (2005) model England and T 2 Xn Wales wholesale electricity prices and estimate a negative /jð ccbjðs; T 1; T 2ÞÞds market price of risk. Cartea and Williams (2007) model t j¼1 gas prices and forward contracts where a positive market for 0 6 t 6 T 1 < T 2. price of risk for long-term contracts is observed and for short-term contracts the market price of risk, although p Given the market power pðt; T 1; T 2Þ, F ðt; T 1; T 2Þ is the positive on average, changes signs across time. In this sec- price that the consumer and producer agree upon in the tion we want to relate the market power pðt; T 1; T 2Þ to the market. Since pðt; T 1; T 2Þ2½0; 1 , this forward price will market price of risk. By working with a parametrization of be in between the producer’s and the consumer’s indiffer- the market price of risk via a class of risk-neutral probabil- ence price, so both are willing to accept such a price. ities introduced by an Esscher transform, we shall see that Let us discuss the correlation structure between forward there are explicit connections between the market power contracts with different delivery periods. For simplicity we and both the market price of risk and the market risk pre- start with the case of no jumps (m = 0), and consider for- mium. Further, to ensure an arbitrage-free forward mar- ward prices for two contracts with non-overlapping deliv- ket, we need to have certain conditions on the number of ery periods ½T 1; T 2 and ½T 3; T 4 . By the explicit form of factors and contracts traded in the market. In commodities p F ðt; T 1; T 2Þ and the independence of the Brownian markets, especially electricity models, the connection motions, we easily calculate between the physical and risk-neutral measure is usually p p performed by introducing a correction in the drift of the CovðÞF ðt; T 1; T 2Þ; F ðt; T 3; T 4Þ Z Xm t physical process to reflect how market participants are a iðt; T 1; T 2Þa iðt; T 3; T 4Þ ¼ r2ðsÞe 2aiðt sÞds: compensated for bearing risk, see for example (Schwartz, T T T T i i¼1 ð 2 1Þð 4 3Þ 0 1997; Schwartz and Smith, 2000; Lucı´a and Schwartz, 2002; Cartea and Figueroa, 2005; Cartea and Williams, The correlation between two contracts can be estimated 2007). We point out that this widely employed change of from market price data, and thus we can estimate the measure in the literature is the result of applying the speeds of mean reversion ai by calibrating the theoretical Esscher transform to the physical measure which is what correlation to the empirical. Taking jumps into account we propose to use in this article. will give rise to a structure where we can include the possi- Suppose that we want to price a forward contract with bility of jump correlation between forward contracts. We delivery over the period [T1,T2]. The forward price is see that it is the multi-factor spot model which implies defined as the correlation structure for the forward contracts. This Z resembles the market models in fixed-income theory, the 1 T 2 F Qðt; T ; T Þ¼EQ SðuÞduj ; so-called LIBOR models, where one models each LIBOR 1 2 Ft T 2 T 1 T 1 rate separately, and include a correlation structure among the different rates. Further, it is also a known fact that for- where we use FQ to indicate the dependency on the chosen ward contracts in energy markets are not perfectly corre- risk-neutral probability Q. lated, but each contract has its intrinsic risk. Note that We parameterize the market price of risk by introducing h the market power function is not contributing to the corre- a probability measure Q :¼¼ QB QL, where QB is a lation between two contracts. Girsanov transform of the Brownian motions BiðtÞ, QL is an Esscher transform of the jump processes LjðtÞ,andh nþm 4. The market price of risk and market risk premium is an R -valued function describing the market price of risk. We define the measure change as follows. For t 6 T , The standard way to price a forward contract is to find with T P T 2 being a finite time horizon encapsulating all the conditional risk-neutral expected value of the future the delivery periods in the market, let the probability QB delivery from the contract. The risk-neutral probability is have the density process Z Z ! usually chosen to be related to what is called the market t Xm t Xm 2 h ðtÞ 1 h ðsÞ price of risk, which can be seen as a drift adjustment in Z ðtÞ¼exp B;i dB ðsÞ B;i ds ; B r ðsÞ i 2 r2ðsÞ the dynamics of an asset to reflect how investors are com- 0 i¼1 i 0 i¼1 i pensated for bearing risk when holding the asset.4 One of the peculiarities of commodities markets is that the market where we have supposed that the functions hB;i=ri, i ¼ 1; ...; m, are square integrable over ½0; T . This measure 4 Note that the market price of risk is not what we have defined as the change in the Wiener coordinates is given by the Girsanov market risk premium. transform, 2012 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 7
hB;iðtÞ prices, and from the explicit representation of X ðtÞ under dW ðtÞ¼ dt þ dB ðtÞ; i r ðtÞ i Qh as i Z u where W iðtÞ become Brownian motions on ½0; T ; i ¼ aðu tÞ aðu sÞ X ðuÞ¼X ðtÞe þ hBðsÞe ds 1; ...; m. The functions hB;i represent the compensation Z t market players obtain for bearing the risk introduced by u aðu sÞ the non-extreme variations in the market, i.e. the diffusion þ rðuÞe dW ðsÞ; t component. We let it be time dependent to allow for vari- ations across different seasons throughout the year. Later, for u P t, we find Z we shall see that it could even be dependent on the delivery T 2 Qh 1 period, indicating that the market price of diffusion risk de- E SðuÞdujFt T 2 T 1 T pends on the forward under consideration. This Girsanov Z1 1 T 2 a ðt; T ; T Þ change gives the dynamics (for 1 6 i 6 m) ¼ KðuÞdu þ X ðtÞ 1 2 T 2 T 1 T 1 T 2 T 1 dX iðtÞ¼ðÞhB;iðtÞ aiX iðtÞ dt þ riðtÞdW iðtÞ; Z Z T 2 u bðt; T 1; T 2Þ 1 aðu sÞ and thus we have added a time-dependent level of mean- þ Y ðtÞ þ hBðsÞe dsdu T 2 T 1 T 2 T 1 T 1 t reversion to the processes X iðtÞ. Z Z 1 T 2 u Further, define for bounded functions h ; j ¼ 1; ...; n, QL bðu sÞ L;j !þ E e dLsjFt du: Z Z T 2 T 1 t Xn t Xn T 1 t ZLðtÞ¼exp hL;jðsÞdLjðsÞ /jðhL;jðsÞÞds ; By Bayes’ Theorem and the independent increment prop- 0 j¼1 0 j¼1 erty of the Le´vy process, we see that the expectation in the last integral is for t 6 T 2, and let the density process for the Radon–Niko- Z u dym derivative of the measure change in the jump compo- QL bðu sÞ E e dLðsÞjFt nent be t Z u dQ P bðu sÞ ZLðuÞ L ¼ E e dLðsÞ jFt ¼ ZLðtÞ: dP t ZLðtÞ Ft Z R R u u u P bðu sÞ hLðsÞdLðsÞ /ðhLðsÞÞds This is the so-called Esscher transform, and the time depen- ¼ E e dLðsÞe t t dent functions h ðtÞ are the market prices of jump risk. We t R R L;j u u m n bðu sÞ d P ðÞxe þhLðsÞ dLðsÞ /ðhLðsÞÞds let h :¼ðhB; hLÞ, where hB :¼ðhB;iÞi¼1 and hL :¼ðhL;jÞj¼1. ¼ E e t j e t h dx x¼0 The density process of the probability Q becomes R R h u u Q bðu sÞ ZðtÞ :¼ ZBðtÞZLðtÞ. Further, we denote by E the expecta- d / xe þhLðsÞ ds /ðhLðsÞÞds t ðÞ t h ¼ e jx¼0 e tion with respect to the probability measure Q . dZx h u The forward price F resulting from the market price of 0 bðu sÞ h ¼ / ðhLÞe ds: risk specification given by Q is derived in the next t Proposition. After reorganizing the integrals the result follows. h Proposition 4.1. The forward price F hðt; T ; T Þ is given by 1 2 To gain insight into how the market risk premium Z depends on diffusion and jump risk we look at the follow- 1 T 2 ing two corollaries. The first one assumes that the market F hðt; T ; T Þ¼ KðuÞdu 1 2 price of jump risk be zero. The second one assumes that T 2 T 1 T 1 Xm the market price of diffusion risk is zero. a ðt; T ; T Þ þ i 1 2 X ðtÞ T T i Corollary 4.2. Suppose that the market price of jump risk is i¼1 2 1 Xn zero, i.e. hL;jðsÞ¼0 for j ¼ 1; ...; n. Then b ðt; T ; T Þ þ j 1 2 Y ðtÞ T T j j¼1 2 1 Z Z 1 T 2 T 2 Xm h P a iðs; T 1; T 2Þ F ðt; T 1; T 2Þ¼E SðuÞdujFt þ hB;iðsÞ ds T 2 T 1 T 1 T 2 T 1 Z t i¼1 T 2 Xm Z a iðs; T 1; T 2Þ T 2 Xn þ hB;iðsÞ ds: 0 bjðs; T 1; T 2Þ T T þ / ðh ðsÞÞ ds: ð4:1Þ t i¼1 2 1 j L;j T T t j¼1 2 1 Proof. This is straightforward from the results above. h for 0 6 t 6 T 1 < T 2. Thus, from this we see that when market players are not Proof. For simplicity suppose m ¼ n ¼ 1. In line with the compensated for bearing jump risk, the market risk pre- calculations for the producer’s and consumer’s indifference mium is positive as long as F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2013 8 Z Z T 2 Xm T 2 Xm a iðs; T 1; T 2Þ pðt; T 1; T 2Þ¼ hB;iðsÞ ds hB;iðsÞa iðs; T 1; T 2Þds t T 2 T 1 t i¼1 i¼1Z T 2 Xn /0 h s b s; T ; T s is positive. As a particular case we can assume all h ðtÞ’s to þ jð L;jð ÞÞ jð 1 2Þd B;i t be positive constants which yields a positive market price j¼1 1 of risk since a i are positive functions for all s 6 T 2. How- ¼ pðt; T 1; T 2Þðcpr þ ccÞ cpr ever, in the more general setting, one can obtain a change Z2 T 2 Xm in the sign of the market risk premium over time t by 2 2 pðt; T 1; T 2Þ ri ðsÞa i ðs; T 1; T 2Þds þ appropriate specification of the functions hB;iðtÞ. Further- t cc Z i¼1 more, although changes in the sign of the market price of T 2 Xn 1 pðt; T 1; T 2Þ risk are also of particular interest, it is clear that a change /jðccbjðs; T 1; T 2ÞÞds t j cpr in the sign of the market prices of risk hB;iðtÞ does not al- Z ¼1 ways imply a change in the sign of the market risk T 2 Xn premium. /jð cprbjðs; T 1; T 2ÞÞds: ð4:3Þ t Now we turn our attention to the case with no diffusion j¼1 market price of risk, i.e. hB;iðtÞ¼0 for all i ¼ 1; ...; m. We have n þ m unknown functions hB;i and hL;j, i ¼ 1; ...; m; j ¼ 1; ...; n. If the market consists of k con- Corollary 4.3. Suppose that hB;iðtÞ¼0 for all i ¼ 1; ...; m. Then tracts with non-overlapping delivery periods, we may find a solution as long as k 6 m þ n. We need at least one solu- Z T 2 tion h ¼ðhB; hLÞ in order to have an arbitrage-free market h P 1 F ðt;T 1;T 2Þ¼E SðuÞdujFt of forward contracts. Suppose for instance that n ¼ 0, that T 2 T 1 T Z 1 is, there are no jumps in the market. In this case we have k T Xn no 2 b ðs;T ;T Þ linear equations for the m unknown functions where we þ /0 ðh ðsÞÞ /0 ð0Þ j 1 2 ds: j L;j j T T find at least one solution as long as k 6 m and the mean t j¼1 2 1 reversion coefficients ai are different. If m > 0 and n > 0 we have both diffusion and jumps in the model, and we Proof. Follows from the results above. h may simply choose hL;j freely, and then solve for the Note that the market risk premium in this case is given remaining unknowns hB;j given by Eq. (4.3) for each k by delivery periods. There is at least one solution when k 6 m Z no for this situation whenever the ai’s are different. T 2 Xn bjðs; T 1; T 2Þ One way is to choose h ðtÞ¼0, which means that there pðt; T ; T Þ¼ /0 ðh ðsÞÞ /0 ð0Þ ds; L;j 1 2 j L;j j T T is no price for jump risk incurred by the market, see for t j¼1 2 1 example (Merton, 1990). We remark in passing that if we chose a spot model with just two factors (for example and even if we assume constant and positive hL;j > 0, the m ¼ n ¼ 1), and the market trades in more than two for- 0 sign of pðt; T 1; T 2Þ will depend on the monotonicity of /j. ward contracts, there may be no solution to the Eq. (4.3) In general, the sign of the market risk premium will result for all the different contracts at once. Solving the equations from a combination of hL;jðtÞ and the monotonicity proper- in (4.3) for each contract separately leads to solutions 0 ties of /j. which are dependent on the delivery period, and will not Finally, when both the market price of jump and diffu- give one risk neutral measure for the market as a whole, sion risk are taken into account, the market risk premium but one measure for each contract separately; allowing is given by for arbitrage opportunities. Z In the rest of this section we look at two illustrative T 2 Xm a iðs; T 1; T 2Þ examples to gain further insights into the model before pðt; T 1; T 2Þ¼ hB;iðsÞ ds applying it to German data in Section 5. First we shall t T 2 T 1 i¼1 demonstrate that for a simple Poisson jump model and Z T Xn no constant market power, the market risk premium changes 2 b ðs; T ; T Þ þ /0 ðh ðsÞÞ /0 ð0Þ j 1 2 ds: sign across time. Second, we explore the case of fixed time j L;j j T T t j¼1 2 1 delivery without the presence of jumps and look explicitly
at how pðt; T Þ depends on the parameters pðt; T Þ, cpr and cc. Now we proceed to relate the market risk premium (4.2) to the market power pðt; T 1; T 2Þ. Comparing the expressions 4.1. An example with constant market power and poisson of F h given by (4.1) and F p calculated in Proposition 3.1, jumps we have that the sum of the last two terms of F p must match the sum of the two last terms of F h. Hence, we must We consider a forward market consisting of 52 contracts find a solution h ¼ðhB; hLÞ to the equation with weekly delivery. The market power is supposed to be 2014 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 9 constant pðt; T 1; T 2Þ¼p 2½0; 1 . Assume that the spot curves have very different shapes in the short end. As model has m ¼ 52 diffusion components X iðtÞ, and one expected, market clearing forward prices are increasing (n ¼ 1) jump component Y ðtÞ. Suppose that the seasonal with increasing market power, since the producer will com- function is mand higher prices with more power. Note also that for a low market power of 0.25, we still observe that the fore- KðtÞ¼150 þ 20 cosð2pt=365Þ; casted price curve is below the forward curve in the shorter and the mean-reversion parameters for the diffusionpffiffi com- end, while in the medium to long end we see the opposite. ponents are ai ¼ 0:067=i, with volatility ri ¼ 0:3= i, for This corresponds to a positive market risk premium in the i ¼ 1; ...; 52. We mimic here a sequence of mean-reverting shorter end, whereas it becomes negative in the medium processes with decreasing speeds of mean reversion and and longer end. Both players have the same risk aversion, with decreasing volatility. Note that a speed of mean rever- and the consumer wishes to avoid upward jumps in the sion equal to 0.067 means that a shock will be halved over price. Hence, even for a weak producer, the consumer is 10 days. The jump process is driven by LðtÞ¼gNðtÞ, where willing to accept a positive market risk premium in the NðtÞ is a Poisson process with intensity k and the jump size short end. In the long end, the effect of jumps vanish as a is constant, equal to g. The mean-reversion for the jump consequence of mean reversion and the consumer will have component is b ¼ 0:5, meaning that a jump will, on aver- more power driving the market risk premium below zero. age, revert back in two days. Thus, we have a combination To illustrate this particular example with p = 0.5 we have of slow mean reverting normal variations and fast mean plotted the difference of the forward curve with market reverting spikes in the spot market. The frequency of spikes power 0.25 and the forecasted curve in Fig. 2. For the con- is set to k ¼ 2=365, i.e. two spikes, on average, per year. tracts with delivery up to approximately week 20, the mar- Time t ¼ 0 corresponds to January 1, and we assume that ket premium is positive. The premium decreases with time the initial spot price is Sð0Þ¼172. In our empirical inves- to delivery, and becomes negative in the medium and long tigations, we let X 1ð0Þ¼2, and X ið0Þ¼Y ð0Þ¼0 for end. i ¼ 2; ...; 52 to achieve this. The risk aversion coefficients Turning our attention to the case of negative jumps, we of the producer and consumer are set equal to observe the reverse picture. Suppose that jumps sizes are cc ¼ cpr ¼ 0:5. In the examples below we derive forward fixed at g = 10. Fig. 3 shows the corresponding forward curves for weakly settled forward contracts over a year. and indifference curves together with the forecasted price. We remark that this model is chosen for its simplicity We observe first of all that all curves are shifted down- and to illustrate the approach in this paper. wards, indicating that the producer is willing to accept Consider first a positive jump of size g ¼ 10. In Fig. 1 we lower forward prices to hedge the possibility of sudden have plotted the indifference forward curves for the pro- drops in prices. In the short-term we observe, for all cases ducer and consumer (‘ ’), together with the forward curves of market power, that the forecasted spot price is above with constant market power equal to p = 0.25, 0.5 and 0.75 forward prices, i.e. negative market risk premium. In the (marked as dashed lines, in increasing order). Finally, we long-term, only when producer’s market power is high, have included the forecasted spot price curve as ‘+’. We that is 0.75, we have the situation where the forecasted clearly see that the forecasted price curve follows a shape curve is below the forward curve signaling that the con- similar to the seasonal function, while the indifference price sumer bears a positive risk premium. Moreover, Fig. 4
Positive jumps 300
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0 10 20 30 40 50 week of delivery
Fig. 1. The indifference price curves together with the forward curves for market powers equal to p ¼ 0:25; 0:5 and p ¼ 0:75, in increasing order. The forecasted curve is depicted ‘+’. The jumps are positive of size 10. F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2015 10
Market risk premium 20
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−5
−10 0 10 20 30 40 50 Week of delivery
Fig. 2. The market risk premium given by the difference of the forward curve with market power 0.25 and the forecasted curve.
Negative jumps 300
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0 10 20 30 40 50 Week of delivery
Fig. 3. The indifference price curves together with the forward curves for market powers equal to p ¼ 0:25; 0:5 and p ¼ 0:75, in increasing order. The forecasted curve is depicted with ‘*’. The jumps are negative of size 10. shows the difference between the forward curve and the Consider Eq. (4.3). One way to solve this is to separate forecasted curve when the producer’s market power is the Wiener and jump part, and solve the two resulting p = 0.75. equations. We find the solution We now proceed to analyze more closely the implica- 1 tions of jumps and normal variations of the model. We h ðt; T ; T Þ¼ pðc þ c Þ c r2ðtÞa ðt; T ; T Þ; ð4:4Þ B 1 2 2 pr c pr 1 2 consider the case with m = n = 1 and constant market 6 power pðt; T 1; T 2Þ¼p for p 2½0; 1 . Further, let for t T 2. Note that the sign of hB depends on the sign of 2 LðtÞ¼NðtÞ, a Poisson process with constant jump intensity pðcpr þ ccÞ cpr, since r ðtÞ and a ðt; T 1; T 2Þ are positive. We k > 0. Note that this model has only two factors, and in have a negative market price of risk hB whenever general it will not give an arbitrage-free forward curve c p < pr : ð4:5Þ dynamics for a market which trades in contracts with many c þ c different delivery periods. However, this simplification pro- pr c vides us with some insight into how the sign of the market If for instance cpr ¼ cc, the market price of risk hB becomes risk premium may change, and we include it with the negative whenever p < 0.5, which corresponds to the con- assumption that we have one forward contract with deliv- sumer being the strongest. If the producer is stronger, i.e. ery period ½T 1; T 2 traded in the market. p > 0.5, he is the superior power in forming prices and the 2016 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 11
Marke trisk premium 10
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−5
−10
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Fig. 4. The market risk premium given by the difference of the forward curve with market power 0.75 and the forecasted curve.
market price of risk becomes positive. If cpr 6¼ cc, the market Lemma 4.4. The non-negative function f :¼ Rþ7!Rþ power needs to be less than the relative risk aversion of the defined by producer against the total risk aversion for hB to be negative. Let us consider the market price of jump risk. Since L t p c z 1 p c z ð Þ f ðzÞ¼ ðe c 1Þþ ð1 e pr Þ; is assumed to be a Poisson process, the log-moment gener- cc cpr ating function is given by satisfies f ðzÞ P z for all z P 0 when /ðxÞ¼kðex 1Þ and /0ðxÞ¼kex: c p > pr : Note that cpr þ cc
0 0 hLðtÞ Moreover, if / ðhLðtÞÞ / ð0Þ¼kðe 1Þ cpr which is positive whenever hLðtÞ > 0, and negative if p < ; cpr þ cc hLðtÞ < 0, as expected following the interpretation of Cor- ollary 4.3. The equation for the jump risk derived from then f ðzÞ < z for z 6 z0, and f ðzÞ P z otherwise, where z0 is splitting (4.3) into two equations becomes (after differenti- defined by f ðzÞ¼z. ating with respect to t) p Proof. Observe that f ð0Þ¼0, and f ðzÞ!1 whenever hLðtÞ ccbðt;T 1;T 2Þ ke bðt; T 1; T 2Þ¼ k e 1 z !1. Moreover, f is monotonically increasing since cc 0 ccz ccz 1 p f ðzÞ¼pe þð1 pÞe P 0: k e cprbðt;T 1;T 2Þ 1 : cpr Consider f 00ðzÞ:
00 ccz cprz Or, equivalently, f ðzÞ¼pcce ð1 pÞcpre ; p hLðt;T 1;T 2Þ ccbðt;T 1;T 2Þ which is positive whenever p > c =ðc þ c Þ. In that case, bðt; T 1; T 2Þe ¼ e 1 pr c pr 0 0 cc f ðzÞ is an increasing function, and since f ð0Þ¼1, we find 0 1 p that f ðzÞ P 1, and therefore f ðzÞ P z for all z P 0. This 1 e cprbðt;T 1;T 2Þ : 4:6 þ ð Þ proves the first claim. When p < c =ðc þ c Þ, we will have cpr pr c pr that f 00ðzÞ < 0 for z 6 bz, where bz is some positive constant, Note that the right-hand side of (4.6) is positive since while f 00ðzÞ > 0 elsewhere. Thus, f 0ðzÞ is decreasing, and bðt; T 1; T 2Þ > 0. Thus, the market price of jump risk is neg- next increasing. Since it goes to infinity as an exponential, ative whenever we need to have that there exists z0 > 0 for which f z z . The second claim follows. h p 1 p ð 0Þ¼ 0 eccbðt;T 1;T 2Þ 1 1 e cprbðt;T 1;T 2Þ < b t; T ; T ; þ ð 1 2Þ cc cpr Let z ¼ bðt; T 1; T 2Þ in the Lemma above, and recall that by the definition of bðt; T 1; T 2Þ it is increasing in t 6 T 1 and and positive otherwise. The following Lemma is helpful in decreasing in T 1 < t 6 T 2. Its maximum is in t ¼ T 1, where bðT 2 T 1Þ understanding when the market price of jump risk is it takes the value bðT 1; T 1; T 2Þ¼ð1 e Þ=b. If this negative. maximum is less than z0, the jump risk hLðt; T 1; T 2Þ will F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2017 12 be negative for all t 6 T 2. Consider the situation where the A straightforward calculation gives maximum is greater than z0. Observe that bð0; T 1; T 2Þ¼ cp aðT tÞ l aðT tÞ bT 1 bT 2 F prðt; T Þ¼SðtÞe þ 1 e ðe e Þ=b and bðT 2; T 1; T 2Þ¼0. If bð0; T 1; T 2Þ P a z0, there exists one t0 such that bðt0; T 1; T 2Þ¼z0. In this 2 r 2aðT tÞ case we find that hLðt; T 1; T 2Þ > 0 for t < t0, and cp 1 e ð4:7Þ 4a hLðt; T 1; T 2Þ < 0 for t > t0.Ifbð0; T 1; T 2Þ < z0, we have and the existence of t0 < t1 being such that bðt; T 1; T 2Þ¼z0, 6 l t ¼ t0; t1. Then hLðt; T 1; T 2Þ is negative for t t0, positive cc aðT tÞ aðT tÞ F c ðt; T Þ¼SðtÞe þ 1 e on the interval t 2ðt0; t1Þ and negative again on t 2ðt1; T 2Þ. a Therefore, we may have a situation where the market r2 þ c 1 e 2aðT tÞ : ð4:8Þ price of jump risk becomes positive giving a positive con- c 4a tribution to the forward price which makes it larger than For fixed delivery contracts the analogous expression to the forecasted spot, even in the presence of a negative (3.1) becomes contribution to the forward price coming from hB.Itis c p cc p also interesting to see that on the forward curve we can F ðt; T Þ¼pðt; T ÞF c ðt; T Þþð1 pðt; T ÞÞF prðt; T Þð4:9Þ have different signs of the market risk premium depending and a simple calculation implies that on how far away from maturity we are on the curve. This l change of sign in the market risk premium may only take F pðt; T Þ¼SðtÞe aðT tÞ þ 1 e aðT tÞ place when jumps are present in the model, and when the a r2 market power of the producer is weaker than his relative pðt; T Þc 1 e 2aðT tÞ risk aversion to the sum of both risk aversion coefficients, p 4a see (4.5). Note that when jumps are not present in the r2 þð1 pðt; T ÞÞc 1 e 2aðT tÞ ; spot price, and we have assumed that the market price c 4a of risk is constant, the market risk premium will only and the market risk premium becomes be either positive or negative, depending on the size of r2 the market power. In general, if we assume that there is pðt; T Þ¼ c pðt; T Þðc þ c Þ 1 e 2aðT tÞ : only one factor driving the dynamics of the spot price, c p c 4a then the market price of risk ought to change signs in It is straightforward to see that the sign of pðt; T Þ is given order to get a change in the sign of the market risk pre- by the sign of ðcc pðt; T Þðcp þ ccÞÞ. Therefore, when mium. Furthermore, if we assume that the market price pðt; T Þ < cc=ðcp þ ccÞ we have that pðt; T Þ > 0 and vice of risk (per factor) is constant, then we need at least versa. two factors to observe a change in the sign of the market Note that we may also write the expression of the for- risk premium. ward as:
2 p l r 4.2. Fixed-delivery forwards without spike risk F ðt; T Þ¼ þ ðpðt; T Þðcp þ ccÞ cpÞ a 4a l þ SðtÞ e aðT tÞ pðt; T Þðc þ c Þ c e 2aðT tÞ; To gain further insight into the forward curves implied a p c p by the certainty equivalence principle and market power, and observe that the forward price consists of three terms, we consider a spot market without spikes and with a con- a ‘‘constant” level, a ‘‘slow” mean-reversion level stant level to which the prices mean-revert. Thus, we con- expð aðT tÞÞ and a ‘‘fast” mean-reversion level sider the spot price model expð 2aðT tÞÞ. Moreover, if we assume a constant mar- dSðtÞ¼ðl aSðtÞÞdt þ rdBðtÞ; ket power p with cpr l a > r P p < ; where is a constant, 0, 0, which yields the expli- c þ c cit solution pr c Z then the last term in (4.10) will be exponentially increasing l T SðT Þ¼SðtÞe aðT tÞ þ 1 e aðT tÞ þ r e aðT sÞdBðsÞ: towards zero. Further, if SðtÞ > l=a, then the a t expð aðT tÞÞ-term will be ‘‘slowly” decreasing to zero. In effect, we produce a hump in the forward curve. This The indifference price of the producer and consumer are hump will be in the short end of the curve. defined as 5. Empirical evidence: The German market cp 1 P F prðt; T Þ¼ E exp c SðT Þ jFt and c p p In this section we apply our model to the German elec- 1 cc P tricity market. We do this in two steps. First we estimate F c ðt; T Þ¼ E ½ exp ðÞjccSðT Þ Ft : cc the physical parameters of a two-factor model. Second, 2018 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 13 using forward market data, denoted by F ðt; T 1; T 2Þ, we esti- Table 1 mate the risk-aversion coefficients for both producers and Estimated coefficients of KðtÞ consumers and estimate the producer’s market power. a0 a1 a2 a3 a4 a5 a6 a7 Squared We employ daily spot prices, the so-called Phelix base error load traded at the EEX, and prices for forward contracts 19.43 11.43 0.78 2.13 6.13 1.2 53.64 0.016 39420.74 with different delivery periods: monthly, quarterly and yearly. Our data covers the period January 2 2002 to Jan- Table 2 uary 1 2006 where we have 1461 spot price observations. Parameter estimates for OU and jump components The forward data consist of 108 contracts with monthly arp kkk b delivery, 35 contracts with quarterly delivery and 12 con- 1 2 tracts with yearly delivery. 0.44 5.2 0.72 0.054 0.031 0.053 0.2 We apply the model to SðtÞ¼KðtÞþX ðtÞþY ðtÞ Table 3 Empirical ad simulated moments (1000 paths) where, as described above, KðtÞ is the seasonal component, Mean Standard deviation Skewness Kurtosis dX ðtÞ¼ aX ðtÞdt þ rdBðtÞ Empirical SðtÞ 31.6 15.2 2.7 14.5 Simulated SðtÞ 32.1 15.6 2.2 13.8 dY ðtÞ¼ bY ðtÞdt þ dLðtÞ where a P 0, b P 0 r P 0, BðtÞ is a standard Brownian motion and where the indicator function is acting on the different days of the week. The parameter estimates for the seasonal com- XNðtÞ ponent are shown in Table 1 and the estimates for the OU LðtÞ¼ J i ð5:1Þ and Jump components are shown in Table 2.5 Moreover, i Table 3 shows the mean, standard deviation, skewness is a compound Poisson process. NðtÞ is a homogeneous and kurtosis of the spot prices and those resulting from Poisson process with intensity k and J i’s are i.i.d. with 10,000 paths using our model and Fig. 5 shows the realized exponential density function and a simulated path. It is clear that our model captures both statistical and trajectile properties. k1j k2jjj f ðjÞ¼pk1e 1j>0 þð1 pÞk2e 1j<0; In order to calculate the market power pðt; T 1; T 2Þ and the forward premium p t; T ; T we need to choose the risk where k1 > 0 and k2 > 0 are responsible for the decay of ð 1 2Þ the tails for the distribution of positive and negative jump aversion coefficients for the consumer and the producer, cc sizes and 1 is the indicator function. Finally we assume that and cpr, respectively. Since F cðt; T 1; T 2Þ (upper bound) and NðtÞ, J and BðtÞ are independent. F prðt; T 1; T 2Þ (lower bound) depend on the choice of cc and We remark in passing that a two-factor model as we cpr, we estimate cpr and cc by minimizing the distance consider here will in general violate the no-arbitrage condi- between F cðt; T 1; T 2Þ, F prðt; T 1; T 2Þ and the market prices tion for the forward market if this consists of more than of forwards F ðt; T 1; T 2Þ, respectively, in the following way. two contracts (recall the discussion in Section 4). However, We examine the time range t ¼ 1 ¼ 02=Jan=2002 until the purpose of the empirical study is to provide an insight t ¼ 1461 ¼ 31=Dec=2005. For most days (excluding week- into the market power and the market risk premium, and ends and holidays) we have prices for forward contracts thus we choose a simple and tractable model to analyze. with delivery one month, three months (quarter) and We may, on the other hand, to be consistent with the no- twelve months (year). As long as there is a price on day arbitrage conditions mentioned above, split the Brownian t, we determine all values of cpr and cc such that motion part of X ðtÞ into several factors to ensure an arbi- F prðt; T 1; T 2Þ 6 F ðt; T 1; T 2Þ 6 F prðt; T 1; T 2Þð5:2Þ trage-free market as required by our conditions above. Therefore, this means that we must choose the same num- and introduce intervals which contain the risk aversion ber of Brownian motions as number of contracts and our parameters. For all trading days t 2½1; 1461 , we define the intervals I t and It containing values for c and c by results to follow will not change with this modification. pr c pr c To be able to estimate the seasonal component and the guaranteeing that (5.2) holds. Thus, to find the ranges for parameters of the OU and jump processes we follow a pro- the parameters of risk aversion we implement the following cedure similar to that in Cartea and Figueroa (2005) and algorithm: Lucı´a and Schwartz (2002). Therefore, for the seasonal t I1 I1 component we assume ¼ 1: Determine valid intervals pr and c such that F prð1; T 1; T 2Þ 6 F ð1; T 1; T 2Þ 6 F cð1; T 1; T 2Þ, for all deliv- KðtÞ¼a þ a 1 þ a 1 þ a 1 þ a 1 ery periods ½T 1; T 2 , traded on day 1. 0 1 fg t¼Su 2 fgt¼Mo 3 fgt¼Tu;We;Th 4 fgt¼Sa 6p þ a5 cos ðÞt þ a6 þ a7t; 365 5 For more details on the estimation procedure see (Metka, 2007). F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2019 14
Spot Prices
150
100
Euros/MWh 50
Jan 02 Jul 02 Jan 03 Jul 03 Jan 04 Jul 04 Jan 05 Jul 05
Simulated Spot Prices 250
200
150
100 Euros/MWh 50
Jan 02 Jul 02 Jan 03 Jul 03 Jan 04 Jul 04 Jan 05 Jul 05
Fig. 5. Spot and simulated spot prices.
2 2 t ¼ 2: Determine valid intervals Ipr and Ic such that To calculate pðt; T 1; T 2Þ and pðt; T 1; T 2Þ we split the data F prð2; T 1; T 2Þ 6 F ð2; T 1; T 2Þ 6 F cð2; T 1; T 2Þ, for all deliv- in three non-overlapping periods. Table 4 lists the contracts ery periods ½T 1; T 2 , traded on day 2. we employ in every period. The key criterion is to include ... in the first period all contracts that were being traded on 1461 1461 t ¼ 1461: Determine valid intervals Ipr and I c such January 2 2002. Then the second period starts when all that F prð1461;T 1;T 2Þ6F ð1461;T 1;T 2Þ6F cð1461;T 1;T 2Þ, contracts that were being traded on January 2 2002 are for all delivery periods ½T 1;T 2 , traded on day 1461. no longer traded. Finally, the third period is defined in a similar way. This algorithm guarantees that no forward prices Figs. 6–8 show our results when all forward contracts F ðt; T 1; T 2Þ will lay outside the bounds F prðt; T 1; T 2Þ and that were being traded on January 2 2002 are taken into F cðt; T 1; T 2Þ. Consequently the results show that account. In other words we show the results from 18 cpr 2½0:421; 1Þ and cc 2½0:701; 1Þ. In our calculations monthly contracts, 7 quarterly contracts and 3 yearly con- we choose cpr ¼ 0:421 and cc ¼ 0:701 which seems a rea- tracts (see Table 4). sonable working assumption where producers are less In particular, Fig. 6 shows the results for the 18 monthly risk-averse than consumers. contracts that were trading on January 2 2002. As expected We continue by recalling that the market power and we observe a decline in the producer’s market power as market risk premium are given by time to delivery increases. For example, in the contract with delivery period closest to January 2 2002 the pro- F ðt; T 1; T 2Þ F prðt; T 1; T 2Þ pðt; T 1; T 2Þ¼ ducer’s market power is slightly over 0.80 and decreases F cðt; T 1; T 2Þ F prðt; T 1; T 2Þ to values below 0.30 corresponding to contracts that start and delivery on or after March 2003. Moreover, Figs. 7 and 8 Z show the same behavior: a decaying market power for pro- T 2 P 1 ducers, as time to delivery increases, for both quarterly and pðt; T 1; T 2Þ¼F ðt; T 1; T 2Þ E SðuÞdujFt : T 2 T 1 T 1 yearly forward contracts.
Table 4 Forward contracts t Type # Contracts Delivery periods F ðt; T 1; T 2Þ
01/Jan/2002 Monthly 18 Jan 2002–May 2003 F ð2; T 1; T 2Þ 01/Jan/2002 Quarterly 7 2nd qtr 2002–4th qtr 2003 F ð2; T 1; T 2Þ 01/Jan/2002 Yearly 3 2003–2005 F ð2; T 1; T 2Þ 03/Mar/2003 Monthly 7 Feb 2003–Aug 2003 F ð400; T 1; T 2Þ 03/Mar/2003 Quarterly 7 2nd qtr 2003–4th qtr 2004 F ð400; T 1; T 2Þ 03/Mar/2003 Yearly 3 2004–2006 F ð400; T 1; T 2Þ 04/Oct/2005 Monthly 7 Oct 2005–Apr 2006 F ð1373; T 1; T 2Þ 04/Oct/2005 Quarterly 7 1st qtr 2006–3rd qtr 2007 F ð1373; T 1; T 2Þ 04/Oct/2005 Yearly 6 2006–2011 F ð1373; T 1; T 2Þ 2020 F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 15
0.8
20 0.7
15 0.6
10
0.5 5
0.4
Producer’s Market Power 0 Market Risk Premium (Euros)
0.3 −5
0.2 −10 Jan 02 Jul 02 Jan 03 Jan 02 Jul 02 Jan 03
Fig. 6. Producer’s market power and market risk premium, 18 monthly contracts with t = January 2 2002.
1 0.4
0 0.38
−1
0.36 −2
0.34 −3
0.32 −4 Producer’s Market Power
Market Risk Premium (Euros) − 0.3 5
−6 0.28
−7 2 4 6 2 4 6
Fig. 7. Producer’s market power and market risk premium, 7 quarterly contracts with t = second quarter 2002.
1 0.4
0 0.38 −1
0.36 −2
0.34 −3
0.32 −4 Producer’s Market Power
Market Risk Premium (Euros) −5 0.3
−6 0.28
−7 2 4 6 2 4 6
Fig. 8. Producer’s market power and market risk premium, 3 yearly contracts with t = 2002. F.E. Benth et al. / Journal of Banking & Finance 32 (2008) 2006–2021 2021 16 Moreover, as predicted by our theoretical examples cit analytical connection between the market prices of risk above, the producer’s market power is much higher in and market power with the degrees of risk aversion of the the short end for monthly contracts, Fig. 6, than for quar- representative agents. To exemplify our methodology fur- terly and yearly, Figs. 7 and 8. In the short-term, the pres- ther, we looked at particular examples where it was ence of jumps incentivies consumer’s to hedge against risk straightforward to see how different sources of risk, for produced by upward spikes, i.e. exert upward pressure on instance jump or diffusion risk, contribute to the market demand for forwards, whilst at the same time, producer’s risk premium and we were able to obtain and explain the have less incentives to sell forwards when positive spikes very distinctive characteristics observed in electricity mar- are present. Hence, we see that monthly forward contracts, kets as well as in other markets such as gas, oil, etc. that start delivery relatively soon, trade at a high market We apply our model to the German electricity market. risk premium which in our framework is also reflected in Our empirical results endorse our theoretical predictions. a high market power for producers. Furthermore, the fur- For instance, we find that over short-term horizons, and ther away the start of the delivery period is, the presence of in the presence of spike risk, producer’s market power is price spikes, due to the fast mean reverting nature of the at its highest. For example, in the contracts with delivery spot price, poses negligible risks. This situation is evident period closest to January 2 2002 the producer’s market in the term structure of the producer’s market power and power is around 0.80 and decreases to values below 0.30 market risk premium obtained from the quarterly and corresponding to contracts that start delivery on or after yearly contracts which are roughly the same in both cases March 2003. This situation is also reflected in the market (see Figs. 7 and 8). risk premium. Monthly contracts that mature in the near We also depict results for the market risk premium. We future, trade at a high premium, figures in excess of 15 see that the risk premium pðt; T 1; T 2Þ shows a clear term Euros. And monthly contracts that start delivery in a rela- structure. As expected, pðt; T 1; T 2Þ is decreasing in tively long period of time, for instance in 6 months or ½T 1; T 2 , i.e. the further the delivery of the contract is, the longer, trade most of the time, at a high discount. smaller the risk premium. In our model, this result can be Finally, we generally observe that for each class of con- explained by the decreasing market power that producers tract (monthly, quarterly or yearly) the producer’s market have; the larger the difference T 1 t, the keener producers power and the market risk premium show a term structure will be, relative to consumers, to trade forwards. Hence, that is decreasing as time to maturity of the forward con- forward prices will move away from the upper bound tract increases. F cðt; T 1; T 2Þ. As a result, the risk premium pðt; T 1; T 2Þ is decreasing, and at some point in time, it becomes negative. References Finally, it is interesting to compare the point in time where the risk premium changes its sign. When looking at the Benth, Fred Espen, Kallsen, Jan, Meyer-Brandis, Thilo, 2007. A non- term structure of pðt; T 1; T 2Þ obtained from monthly and Gaussian Ornstein–Uhlenbeck process for electricity spot price mod- quarterly contracts we see that the change of sign occurs elling and derivatives pricing. Applied Mathematical Finance 14 (2), around the months of April and May. 153–169. Cartea, A´ lvaro, Figueroa, Marcelo G., 2005. Pricing in electricity markets: A mean reverting jump diffusion model with seasonality. Applied 6. Conclusions Mathematical Finance 12 (4), 313–335. Cartea A´ lvaro, Williams Thomas, 2007. UK gas markets: The market price of risk and applications to multiple interruptible supply In this article we address the important question of what contracts. Energy Economics, in press. gives rise to the market risk premium. We provide a frame- Cont, R., Tankov, P., 2004. Financial modelling with jump processes, first work that allows us to explain how risk preferences of mar- ed.. In: Financial Mathematics Series Chapman and Hall, London. ket players explain the sign and magnitude of the market Lucı´a, J., Schwartz, E., 2002. Electricity prices and power derivatives: Evidence from the nordic power exchange. Review of Derivatives risk premium across different forward contract maturities. Research 5, 5–50. A crucial step in our framework is to be able to incorporate Merton, R., 1990. Continuous – Time Finance, first ed. Blackwell. the relative eagerness consumers and producers show, via Metka, K., 2007. Pricing Forward Contracts in Power Markets by the their respective representative agents, to enter forward con- Certainty Equivalent Principle: Explaining the Sign of the Market Risk tracts. We show that there is an attainable set where con- Premium. Diplomarbeit, University of ULM. Protter, Philip, 1992. Stochastic integration and differential equations, a sumers and producers are willing to strike a deal, but it is new approach, second ed.. In: Applications of Mathematics, vol. 21 this eagerness to hedge risk across different points in time, Springer-Verlag. which we label market power, what singles out a unique Sato, K.I., 1999. Le´vy processes and infinitely divisible distributions, first equilibrium price for each forward contract. ed.. In: Cambridge Studies in advanced mathematics, vol. 68 Furthermore, these equilibrium prices that belong to the Cambridge University Press, Cambridge. Schwartz, E., Smith, J., 2000. Short-term variations and long-term attainable set must clearly correspond to those obtained dynamics in commodity prices. Management Science 46 (7), 893–911. from pricing forwards under a risk-neutral measure. There- Schwartz, Eduardo, 1997. The stochastic behavior of commodity prices: fore, as an illustration of our approach, we looked at Implications for valuation and hedging. Journal of Finance 52 (3), wholesale electricity prices and were able to make an expli- 923–973.