A structural Heath-Jarrow-Morton framework for consistent intraday, spot, and futures electricity prices
aa r X i v : . [ q -f i n . M F ] J a n A structural Heath-Jarrow-Morton framework forconsistent intraday, spot, and futures electricityprices
W.J. Hinderks ∗ , A. Wagner , and R. Korn Fraunhofer ITWM, Fraunhofer-Platz 1, 67663 Kaiserslautern, Germany TU Kaiserslautern, Erwin-Schr¨odinger-Straße 1, 67663 Kaiserslautern, Germany
January 21, 2019
Abstract
In this paper we introduce a flexible HJM-type framework that allows forconsistent modelling of intraday, spot, futures, and option prices. Thisframework is based on stochastic processes with economic interpretationsand consistent with the initial term structure given in the form of a priceforward curve. Furthermore, the framework allows for existing day-aheadspot price models to be used in an HJM setting. We include several ex-plicit examples of classical spot price models but also show how structuralmodels and factor models can be formulated within the framework.
Keywords:
Heath-Jarrow-Morton framework, electricity markets, intra-day prices, day-ahead spot prices, futures prices, option prices, structuralmodel, factor model
In recent years the electricity intraday markets have gained increased popular-ity: the traded volume at the German/Austrian intraday market has grown by30.3 percent from May 2016 to May 2018 (EPEX, 2017, 2018). Since differentelectricity contracts exhibit different price behaviour such as spikes in the day-ahead spot but not in futures prices, it is a rising challenge in energy financeto define a single model that allows for a joint simulation of power prices atintraday, spot, and futures markets.In this paper we suggest a Heath-Jarrow-Morton framework for modellingelectricity prices. The framework is consistent with the current forward termstructure (i.e. the price forward curve) and we motivate each mathematicalcomponent by an economic interpretation. Furthermore, we discuss the compu-tation of intraday, spot, and futures prices within this framework and we showhow options on futures contracts can be priced. A new approach is the use of ∗ Corresponding author: [email protected] ) approach for elec-tricity prices is the fictitious forward price or forward kernel . The forwardkernel f t ( τ ), t ≤ τ , is the price at time t of a forward contract delivering elec-tricity instantly at time τ . It follows that the price at t of a futures contractdelivering from τ to τ is the averaged forward kernel during the delivery period,i.e. F t ( τ , τ ) = 1 τ − τ Z τ τ f t ( u ) du, t ≤ τ . (1.1)In the HJM framework for interest rates the forward rate is modelled insteadof the short rate (cf. Brigo and Mercurio (2006)). Therefore, modelling theforward kernel instead of the day-ahead spot price makes this an HJM approachfor power prices. Furthermore, just like in the HJM framework for interestrates, the forward kernel itself is not a traded product at the market but its(integrated) derivatives are.Several models for the forward kernel f t ( τ ) have been introduced by Benthand Koekebakker (2008); Clewlow and Strickland (1999); Hinz, von Grafen-stein, Verschuere, and Wilhelm (2005); Kiesel, Schindlmayr, and B¨orger (2009);Koekebakker and Ollmar (2005). They define the forward kernel dynamicsdriven by Brownian motions. However, since the day-ahead spot prices showspikes, these models have drawbacks. Therefore, there is a need for a forwardkernel model that allows for spikes in relatively short delivery periods (day-ahead spot contracts) but smooths these out for longer delivery periods (futurescontracts). The theoretical HJM framework of Benth, Piccirilli, and Vargiolu(2017) introduces forward kernel dynamics driven by Brownian motions andpure jump L´evy processes. However, Benth et al. (2017) assume that day-aheadspot and futures contracts are priced under two different measures. Motivatedby economic arguments, this ambiguity is avoided in our approach.In the literature the use of more than one probability measure has also beenchallenged: Caldana et al. (2017); Lyle and Elliott (2009) assume a single prob-ability measure, for example. This is supported by the fact that it is not clearwhich equivalent measure should be the pricing measure Q . Since electricity isa non-storable commodity and buy-and-hold strategy arguments are not valid,it is not clear what the relation between the price of electricity contracts andthe money market account is (Bessembinder & Lemmon, 2002). This also im-plies that the market is incomplete and that there are (possibly) infinitely manyequivalent martingale measures. Again, this leaves the choice of pricing measureunclear.We follow the idea of Caldana et al. (2017) that the prices of day-ahead spotand futures contracts both should be computed by Equation (1.1). This actuallysounds intuitively since, for example at the German markets, day-ahead spotcontracts are traded at least twelve hours before delivery. In other countries suchas the US the terminology is different: the day-ahead spot price is commonlyreferred to as the forward price (Longstaff & Wang, 2004). Even in Europe, See Heath, Jarrow, and Morton (1992) for the original paper introducing this frameworkfor interest rate modelling. Forward kernel is the name used by Caldana, Fusai, and Roncoroni (2017). Modelling of the day-ahead spot price is a common approach, for which several differentapproaches have been developed, cf. Weron (2014). − d d + 1 : E XAA : EPE X EPEX day-ahead : EPE X q u a rt e r h o u r EPEX intraday τt EEX futures
Observation structure of f t ( τ )Figure 1: Observation structure of f t ( τ ) for the German electricity market anda fixed delivery time τ . The red marked lines and time points are the (indirect)observation moments. The lines with d − d stand for the start of day d − d .with the increasing popularity of the intraday markets, we observe a shift interminology: Weron (2014) remarks that the term spot is used more and morefrequently for the real-time or intraday market. We will always explicitly stateto which spot market we refer.In this paper we even propose to extend Equation (1.1) to the intradaymarket. Figure 1 gives an example of the development of the forward kernel f t ( τ )and how it becomes observable at the German/Austrian market. First theforward kernel f t ( τ ) is only (partly) observable through EEX futures contracts.Then the Austrian EXAA and two German EPEX day-ahead spot auctions are held, after which the EPEX intraday spot market opens.Furthermore, we show how the classical models described by Lucia andSchwartz (2002); Schwartz and Smith (2000) fit into our framework. We alsoshow how other more general day-ahead spot price models can be used to fitinto our model. A particular new example we introduce in this paper, is touse structural models in the context of an HJM framework. We also apply ourframework to the setting of multi-factor models.This paper is structured as follows: Section 2 introduces a model for theforward kernel based on the economic intuition that there are two driving com-ponents behind the forward kernel. The first component is the equilibrium ofsupply and demand at delivery time and the second is a general noise frompartially informed traders or illiquidity at trading time t . Successively, in Sec-tion 2.2 and Section 2.3 the futures and option prices are computed, respectively.Section 3 contains the above explicitly mentioned examples for the market equi-librium process, while Section 4 concludes. In Section 2.1 we will define a model for the forward kernel motivated by eco-nomic interpretations. Using this model in Sections 2.2 and 2.3 we derive theprices of futures contracts and options on futures contracts, respectively. Sec-tion 2.4 gives an overview of the prices for different electricity contracts for theexample of the German market. Of course, it is not clear what the roles the EXAA and EPEX willplay for each other after the announced market division. Press release: (visited on March 26 2018). .1 Forward kernel The forward kernel f t ( τ ) is the price at time t of a forward contract delivering1 MW instantly at time τ . Throughout the rest of this paper we interpret t asthe trading time and τ as the delivery time .For τ ≥ X τ = { X τt ; t ≥ } and Y = { Y t ; t ≥ } be two independent, a.s.c`adl`ag stochastic processes on the complete probability space (Ω , F , P ) takingvalues in R and R n , respectively. Furthermore, assume that the processes X τ for each τ ≥ Y are adapted to the filtration {F t ; t ≥ } , which satisfiesthe usual conditions, i.e. {F t ; t ≥ } is right-continuous and F contains all P -null sets. The filtration generated by Y and X τ augmented by all P -null setsautomatically fulfills these conditions. Finally, let g : R n → R be a functionsuch that g ( Y t ) is real-valued stochastic process.We have two strong economic interpretations for these two stochastic pro-cesses: we interpret the n -dimensional process Y t as the randomness or the stateof the market, where each component of Y t stands for a (random) facet of themarket, e.g. demand, load, or weather predictions. The function g maps thestate of the market state Y t to its corresponding price. Combining the fact thatour inspiration came from the class of structural models for day-ahead spot pricemodelling and the fact that it gives the basic structure to the forward kernel,we call the pair ( g, Y t ) the structural component . Often we will also only call Y t the structural component.The process X τt is called the market noise because it accounts for the incom-plete market information of all market participants and illiquidity of the market.An example of incomplete market information is the uncertainty of weather pre-dictions: nobody knows with complete certainty about the future weather ortemperature. With these interpretations we define the forward kernel: Definition 2.1 (Forward kernel) . We define the forward kernel at trading time t and delivery time τ as f t ( τ ) := X τt E [ g ( Y τ ) | F t ] , where X τt is the market noise at trading time t for the delivery time τ and( g, Y τ ) the structural component at delivery time τ .We use the notation X τt to emphasize that the market noise is a stochasticprocess in the trading time t but can (deterministically) depend on the deliverytime τ , whereas the structural component Y τ only depends on delivery time.Economically, this makes sense since the imbalance of supply and demand atdelivery time τ determines the price independent of the trading time t at whichwe predict this imbalance. However, the market noise is the disturbance of thisprediction originating from market participants with incomplete market infor-mation, which intuitively depends on both the trading time t and the deliverytime τ they are trying to predict. Although we call X τt the market noise, itcan also be interpreted as a measure transformation (or Radon-Nikodym deriva-tive, see Remark 2.7) or as a general additional component that introduces anadditional degree of freedom in the modelling process. Assumption 2.2 (Market noise) . The process X τ = { X τt ; t ≥ } with itsinterpretation as market noise for delivery time τ is defined as multiplicative This also allows for seasonal volatility in the market noise. E X τt = 1 for all τ ≥ t ≥
0. In particular, we assume thatthe initial value X τ = 1 a.s. for all τ ≥ Assumption 2.3 (Structural component) . We assume that Y = { Y t ; t ≥ } isa R n -valued c`adl`ag stochastic process. In particular, we assume that the initialvalue equals Y = y ∈ R n a.s. such that g ( y ) = f (0), where f ( τ ) is the priceforward curve (PFC) for delivery time τ , which we assume to be known (cf.Remark 2.4). Furthermore, as a technical assumption we need that E | g ( Y t ) | < ∞ for all t ≥
0. Finally, although we assume that g ( Y t ) can take all values in R ,including negative values, we assume that its expectation E g ( Y t ) > expect negative forward prices to occur.With these assumptions the sign of the forward kernel is uniquely deter-mined by the structural component Y and the process X τ cannot influenceit. Furthermore, the expectation E f t ( τ ) is fully determined by the structuralcomponent Y τ and independent of trading time t (cf. Lemma 2.6). Remark f ( τ )) . In the framework the price forwardcurve (PFC), denoted by f ( τ ), plays an important role: it determines theexpectation of the forward kernel f t ( τ ). There are many studies that describehow one can construct a PFC from market prices such as Caldana et al. (2017);Kiesel, Paraschiv, and Sætherø (2018), for example. In practice every energyutility has an in-house PFC. In the following we will therefore assume that thePFC is known. Theorem 2.5.
For fixed τ ≥ the forward kernel process f ( τ ) := { f t ( τ ); t ≥ } is an adapted stochastic process. Furthermore, f ( τ ) is a.s. c`adl`ag.Proof. By definition f ( τ ) is a stochastic process. Moreover, since we assumed X τt to be F t -measurable and since the conditional expectation Z τt := E [ g ( Y τ ) | F t ]is always F t -measurable, the F t -measurability of f t ( τ ) follows immediately. Be-cause the filtration satisfies the usual conditions, Z τt has a c`adl`ag modifica-tion (Karatzas & Shreve, 1998, Chapter 1, Theorem 3.13). Since the condi-tional expectation Z τt is uniquely defined up to null sets, we can choose thismodification and the result follows by the assumption that X τt is c`adl`ag.Since we assume that X τ and Y both a.s. start at a deterministic value, weassume without loss of generality that F is generated by Ω and all P -null sets.This in particular implies that E g ( Y τ ) = E [ g ( Y τ ) | F ], a fact we will exploit inthe next lemma. Lemma 2.6.
For fixed τ ≥ the forward kernel process f ( τ ) := { f t ( τ ); t ≥ } is a martingale. Furthermore, its expectation is given by E f t ( τ ) = E g ( Y τ ) = f ( τ ) for all ≤ t ≤ τ .Proof. The product of two independent martingales clearly is a martingale.Furthermore, it follows immediately from Assumption 2.2 and 2.3 that E f t ( τ ) = E [ X τt ] E [ E [ g ( Y τ ) | F t ]] = E g ( Y τ ) = X τ E [ g ( Y τ ) | F ] = f ( τ )by the independence of X τt and Y t . 5emma 2.6 also imposes a condition for the expectation E g ( Y τ ) of the struc-tural component, which can be used to calibrate the structural component Y and function g after the PFC f ( τ ) has been determined. If one wants to obtaina model that is consistent with an existing PFC f ( τ ), one needs to choose andcalibrate g and Y such that E g ( Y τ ) = f ( τ ). Remark . In the previous discussion we considered themeasure space (Ω , F , P ) equipped with the real-world measure P . However, inarbitrage-free markets there is a pricing measure under which derivatives arevalued. The τ -forward measure Q τ defined by its Radon-Nikodym derivative dPdQ τ (cid:12)(cid:12)(cid:12) F t = X τt (2.1)could be used for this purpose. Using the τ -forward measure and Bayes’ theoremfor conditional expectations we can rewrite Definition 2.1 f t ( τ ) = X τt E P [ g ( Y τ ) | F t ] = E Q τ [ X ττ g ( Y τ ) | F t ] . The latter term can be defined S t := X tt g ( Y t ) , which yields a general spot price model. The choice of the stochastic process X τt can be viewed as the choice of a pricing measure Q τ in light of Equation (2.1).If the noise X t := X τt is chosen to be independent of the delivery time τ , so isthe forward measure Q := Q τ . As discussed in Section 1 the forward kernel can be used to compute the priceof futures contracts. In the following we assume the interest rate to equal r = 0for notational convenience. Of course, when one assumes r = 0, discountinghas to be taken into account. In Remark 2.13 we have some notes on how tochange our framework to include discounting. Furthermore, we assume that allprices are normalized, meaning that we assume all prices to be in Euro/MWhas usual. Definition 2.8 (Futures contract price) . For 0 ≤ t ≤ τ < τ we call F t ( τ , τ ) := 1 τ − τ Z τ τ f t ( u ) du the price of a futures contract at time t delivering 1 MW continuously from τ to τ .Since we denote all prices in Euro/MWh, the price that one pays at time t when one buys a futures contract delivering 1 MW from τ to τ is given by( τ − τ ) F t ( τ , τ ), where we assume that τ − τ is measured in hours. Example 2.9 (Day-ahead spot price) . We compute the day-ahead spot priceas a futures contract. It is auctioned at day d − a and delivered atday d from h :00 until ( h + 1):00 o’clock, i.e. S ( d, h ) := F t ad − (cid:0) t hd , t h +1 d (cid:1) . Here t hd denotes the time at day d and hour h .6he next theorem shows that the framework is consistent with cascading. It also shows that there are no arbitrage opportunities in the sense that the costof a futures contract delivering for one year is the same as the cost of its fourquarters, for example.
Proposition 2.10 (Consistency of cascading) . Let ≤ τ < τ < τ < · · · < τ n be delivery times, then we have ( τ n − τ ) F t ( τ , τ n ) = n X i =1 ( τ i − τ i − ) F t ( τ i − , τ i ) for all t ≥ .Proof. This follows directly from Definition 2.8 and the countable additivity ofthe Lebesgue integral.
Lemma 2.11.
Fix ≤ t < τ . If u f t ( u ) is almost surely continuous on ( τ − ǫ, τ ] for some ǫ > , then we have lim s → τ − F t ( s, τ ) = f t ( τ ) almost surely.Proof. We computelim s → τ − F t ( s, τ ) = lim s → τ − R τs f t ( u ) du lim s → τ − τ − s = lim s → τ − − f t ( s ) − f t ( τ )where we used L’Hˆopital’s rule for the second equality.The previous lemma shows that the price of a futures contract deliveringfor just an instant equals the forward kernel. This supports the naming of thequantity f t ( τ ) as forward kernel. Lemma 2.12.
Assume that the price forward curve τ f ( τ ) is continuous.The futures price process F ( τ , τ ) := { F t ( τ , τ ); t ≥ } is a martingale. Itsexpectation is given by E F t ( τ , τ ) = F ( τ , τ ) = 1 τ − τ Z τ τ f ( u ) du for all ≤ t ≤ τ < τ .Proof. Since the price forward curve is continuous, it is bounded on any com-pact set, in particular intervals of the form [ τ , τ ], and therefore integrable oncompacts. Direct computation with Fubini’s Theorem shows that for 0 ≤ t < s E [ F s ( τ , τ ) | F t ] = E (cid:20) τ − τ Z τ τ f s ( u ) du | F t (cid:21) = 1 τ − τ Z τ τ E [ f s ( u ) | F t ] du, By cascading we mean the way how futures with a longer delivery period are settled.For example, a calendar year futures contract cascades (or splits up) into three monthlyfutures (January, February, and March) and three quarterly futures (Q2, Q3, and Q4) uponstart of delivery. This way, these can be traded independently again. In the German marketmonthly futures do not cascade. However, the settlement price at the end of the delivery isexactly the average of the day-ahead spot prices during delivery. This could be interpretedthat also monthly futures are cascading to the hourly (day-ahead) spot contracts, since theirprice converges to this average.
Remark r = 0) . If we assume that r = 0, the futures price depends onthe settlement date. There are two possibilities: settlement takes place eitherthrough continuous payments during the delivery period or at once at the endof the delivery period. If d t ( τ ) denotes the discount factor of a future paymentat time τ to an earlier time t , the price of a futures contract is given by F t ( τ , τ ) = 1 R τ τ d t ( u ) du Z τ τ d t ( u ) f t ( u ) du for continuous settlement and by F t ( τ , τ ) = 1( τ − τ ) d t ( τ ) Z τ τ d t ( u ) f t ( u ) du for settlement at the end of delivery. In this section we assume that the market noise is given by a geometric Brownianmotion (GBM) without drift, i.e. dX τt = X τt Σ( t, τ ) T dW t where Σ( t, τ ) is a deterministic m -dimensional volatility vector and W t is an m -dimensional Brownian motion. The strong solution of X τt is given by X τt = exp (cid:18)Z t Σ( u, τ ) T dW u − Z t Σ( u, τ ) T Σ( u, τ ) du (cid:19) . In this case, X τt satisfies Assumption 2.2 if Σ( u, τ ) is square integrable in u .But this is already a requirement for the stochastic integral to be defined. Example 2.14 (Hull-White market noise dynamics) . A possible choice for Σis a two-factor forward dynamic similar to Kiesel et al. (2009), which is alsodiscussed in a geometric setting by Fanelli and Schmeck (2018) for pricing op-tions on futures. This volatility structure is extended by Latini, Piccirilli, andVargiolu (2018) in an additive setting. They discussed a 2-factor volatilitystructure comparable to the two-factor Hull-White model for interest rate mod-elling (Brigo & Mercurio, 2006, Section 4.2.5). It is given byΣ( t, τ ) T := ( e − κ ( τ − t ) σ , σ ( τ )) , where σ > κ > σ ( τ ) > τ . A convenient choice for σ is a piecewise constant function, beingconstant on delivery periods of tradable futures contracts. An advantage of thischoice is that we can use the calibration methods for X τt as discussed by Fanelliand Schmeck (2018); Kiesel et al. (2009); Latini et al. (2018). Continuous settlement of the futures contract makes it more like a swap contract on theforward kernel.
Definition 2.15 (Affine structural component decomposition) . We say thestructural component ( g, Y t ) allows for the affine structural component decom-position , if there exist deterministic functions ( t, τ ) A τt ∈ R n × n and ( t, τ ) B τt ∈ R n such that the following decomposition holds E [ g ( Y τ ) | F t ] = g ( A τt Y t + B τt ) (2.2)a.s. for all τ ≥ t ≥ t for the state of market Y τ at time τ is an affine transformation of the currentstate of the market Y t . This is also the main idea behind Kalman filtering, forexample. If the decomposition holds, this merely states that this best guessshould hold under the transformation g , which transforms the market state intoa price.It follows immediately that the forward kernel is given by f t ( τ ) = X τt g ( A τt Y t + B τt ) , (2.3)when the affine structural component decomposition assumption is satisfied.Furthermore, the futures price of Definition 2.8 can be rewritten as F t ( τ , τ ) = 1 τ − τ Z τ τ X ut g ( A ut Y t + B ut ) du for all 0 ≤ t ≤ τ < τ . As immediate consequences we obtain: Lemma 2.16. If ( g, Y t ) allows for the affine structural component decomposi-tion, then E [ g ( Y τ ) | F t ] = E [ g ( Y τ ) | Y t ] . Lemma 2.17.
Under assumption of the decomposition of Definition 2.15 theforward kernel conditioned on Y t is lognormally distributed, i.e. ( f t ( τ ) | Y t = y ) ∼ LN (cid:18) ln[ g ( A τt y + B τt )] , Z t Σ( u, τ ) T Σ( u, τ ) du (cid:19) . Proof.
Using Equation (2.3) we compute P ( f t ( τ ) ≤ x | Y t = y ) = P ( X τt g ( A τt y + B τt ) ≤ x ) , which shows the result since X τt ∼ LN (cid:16) , R t Σ( u, τ ) T Σ( u, τ ) du (cid:17) . Theorem 2.18. If ( g, Y t ) allows for the affine structural component decompo-sition, then the first two moments of the futures price F t ( τ , τ ) exist and aregiven by E [ F t ( τ , τ ) | Y t = y ] = 1 τ − τ Z τ τ g ( A ut y + B ut ) du and E [ F t ( τ , τ ) | Y t = y ] = 1( τ − τ ) Z τ τ Z τ τ w Xt ( u, s ) w Yt ( u, s, y ) du ds, here w Xt ( u, s ) := exp (cid:18) Z t (cid:0) Σ( v, u ) T Σ( v, s ) + Σ( v, s ) T Σ( v, u ) (cid:1) dv (cid:19) (2.4) and w Yt ( u, s, y ) := g ( A ut y + B ut ) g ( A st y + B st ) . (2.5) Proof.
We see that the expectation follows immediately by an Fubini argumentcombined with the fact that E X τt = 1 for all τ ≥
0. Applying Fubini twice wefind E [ F t ( τ , τ ) | Y t = y ] = R τ τ R τ τ E [ X ut X st ] E [ Y u Y s | Y t = y ] du ds ( τ − τ ) , where it is easy to verify that the expectations equal E [ X ut X st ] = w Xt ( u, s ) and E [ Y u Y s | Y t = y ] = w Yt ( u, s, y ) using Equation (2.2). Corollary 2.19. If ( g, Y t ) allows for the affine structural component decom-position, then the conditional variance of the futures price F t ( τ , τ ) is givenby Var[ F t ( τ , τ ) | Y t = y ] = 1( τ − τ ) Z τ τ Z τ τ (cid:0) w Xt ( u, s ) − (cid:1) w Yt ( u, s, y ) du ds, where w X and w Y are given by Equation (2.4) and Equation (2.5) , respectively.Proof. We directly computeVar[ F t ( τ , τ ) | Y t = y ] = E [ F t ( τ , τ ) | Y t = y ] − E [ F t ( τ , τ ) | Y t = y ] . Using Theorem 2.18 the first term is immediately given and the second termcan be computed using Fubini’s Theorem E [ F t ( τ , τ ) | Y t = y ] = (cid:18) τ − τ Z τ τ g ( A ut y + B ut ) du (cid:19) = 1( τ − τ ) Z τ τ Z τ τ g ( A ut y + B ut ) g ( A st y + B st ) du ds, from which the result follows. Remark . Similar to the discrete approach usedby Kiesel et al. (2009) we have that the futures price is an integral of lognor-mally distributed variables, which can be approximated by a lognormal randomvariable with the same mean and standard deviation. Since there is no simpleexpression for the convolution of lognormal distributions, this approximation ofthe integral (or sum) of lognormal random variables is widely used in finance,e.g. in the context of LIBOR market models by Brigo and Mercurio (2006). Ananalysis of this approximation, also with regard to Asian options (which may becompared to an option on a futures with delivery period), is found in Dufresne(2004), for example. 10 ssumption 2.21 (Lognormal approximation) . Assume that the first two mo-ments of the futures price F t ( τ , τ ) exist. Justified by Remark 2.20, we thenassume that( F t ( τ , τ ) | Y t = y ) ≈ ( ˜ F t ( τ , τ ) | Y t = y ) ∼ LN (cid:0) µ F ( y ) , σ F ( y ) (cid:1) , i.e. the futures price is approximately lognormally distributed.As stated in Remark 2.20 we need that the first two moments of F and ˜ F match, which is resolved by the following lemma: Lemma 2.22. If ( g, Y t ) allows for the affine structural component decomposi-tion and Assumption 2.21 holds, then the mean and standard deviation of thelognormal distribution are given by µ F ( y ) := ln Z τ τ g ( A ut y + B ut ) du − ln( τ − τ ) − σ F ( y ) and σ F ( y ) := ln R τ τ R τ τ (cid:0) w Xt ( u, s ) − (cid:1) w Yt ( u, s, y ) du ds R τ τ R τ τ w Yt ( u, s, y ) du ds ! , where w X and w Y are given by Equation (2.4) and Equation (2.5) , respectively.Proof. For a lognormal random variable Z ∼ LN ( m, s ), the expectation andvariance are given by E Z = exp( m + s /
2) and Var Z = ( E Z ) (exp( s ) − Y t ) of call (andput) options on futures contracts by the Black-Scholes formula. A call optionwith strike price K and maturity T < τ has a pay-off equal to( τ − τ ) ( F T ( τ , τ ) − K ) + . (2.6)Recall that, as stated in Section 2.2, the price one has to pay for a futurescontract at time T equals ( τ − τ ) F T ( τ , τ ), since we consider normalizedprices. Proposition 2.23 (Conditional call option price) . Assume that ( g, Y t ) allowsfor the affine structural component decomposition and let Assumption 2.21 hold.Denote the futures price at maturity by F := F T ( τ , τ ) . Let µ F and σ F be givenby Lemma 2.22. The price of a call option at t = 0 with pay-off given by (2.6) conditioned on Y T = y equals C ( T, K, τ , τ ; y ) = Φ( δ ( y )) Z τ τ g ( A uT y + B uT ) du − ( τ − τ ) K Φ( δ ( y )) , where Φ is the cumulative distribution function of the standard normal distri-bution, δ ( y ) := µ F ( y ) − ln Kσ F ( y ) , and δ ( y ) := δ ( y ) + σ F ( y ) . roof. Using the discounted conditional expectation of the pay-off given in (2.6)yields C ( T, K, τ , τ ; y ) τ − τ = E (cid:2) ( F − K ) + | Y T = y (cid:3) = E [ F { F ≥ K } | Y T = y ] − K P ( F ≥ K | Y T = y ) , where noting that we have ( F | Y T = y ) ∼ LN ( µ F ( y ) , σ F ( y )), yields the resultby direct computation.As an immediate consequence we have: Corollary 2.24 (Call option price) . Assume that ( g, Y t ) allows for the affinestructural component decomposition and let Assumption 2.21 hold. Let µ F and σ F be given by Lemma 2.22. The price of a call option at t = 0 with pay-offgiven by (2.6) equals C ( T, K, τ , τ ) = E C ( T, K, τ , τ ; Y T ) , (2.7) where the conditional call option price C ( T, K, τ , τ ; y ) is given in Proposi-tion 2.23. When the distribution of Y T is specified the price of a call option given byEquation (2.7) might be evaluated analytically, numerically, or through sim-ulative methods such as Monte Carlo estimation. Alternatively, with furtherassumptions on the distribution of Y T this expectation could also be approxi-mated differently. In this section we give an overview of the prices of several different electricitycontracts in this HJM framework. Although there is not a single unique quotedcontinuous electricity price we regard F t ( τ , τ ) as the true fair price for thedelivery period from τ to τ at any trading time t . Futures price
The price of a futures contract at time t delivering 1 MWcontinuously from τ to τ is given by Definition 2.8 and denoted by F t ( τ , τ ). Options on futures
In the setting of Section 2.3 the price of call and putoptions on futures contracts can be computed by the Black-Scholes formula asgiven by Proposition 2.23 or Corollary 2.24.
Day-ahead spot prices
The day-ahead spot price equals the futures pricewithin this framework as discussed in Example 2.9. ID and ID price The ID and ID price indices on the German intradaymarket are given as the one and three hour volume-weighted average of allintraday trades before delivery. Therefore, we suggest the ID n price for thedelivery period from τ to τ to equalID n ( τ , τ ) := 22 n − Z τ − . τ − n F u ( τ , τ ) du, where n = 1 or n = 3 and the subtraction of τ is meant in hours.12 Examples of the structural component
First we show how two classical day-ahead spot price models can be used in thisHJM framework. Then we also introduce a structural model approach as wellas a multi-factor model approach for Y .To make defining a model easier in this framework we introduce the rela-tive structural component, which can be used to set the initial price forwardcurve (PFC) to an existing one: Definition 3.1 (Relative structural component) . The additive mean-normalizedversion of g ( Y τ ) I aτ := g ( Y τ ) − E g ( Y τ )is called the additive relative structural component and its multiplicative mean-normalized version I mτ := g ( Y τ ) E g ( Y τ )is called the multiplicative relative structural component .We directly obtain from these definitions: Corollary 3.2.
The relative structural components I a and I m are stochasticprocesses with constant expectation E I aτ = 0 and E I mτ = 1 for all τ ≥ . Corollary 3.3 (Arithmetic PFC decomposition) . For a given initial price for-ward curve f ( τ ) the forward kernel equals f t ( τ ) = X τt ( f ( τ ) + E [ I aτ | F t ]) , where I aτ is the arithmetic relative structural component given in Definition 3.1.Proof. Define an extended structural component ˜ Y τ = ( Y τ , f ( τ )) ∈ R n +1 ,where f ( τ ) is the constructed PFC, and another function ˜ g ( y, x ) = x + g ( y ) − E g ( y ). It is clear that ˜ Y and ˜ g satisfy Assumption 2.3. It follows immediatelythat ˜ g ( ˜ Y ( τ )) = f ( τ ) + I aτ , which proves the result. Corollary 3.4 (Geometric PFC decomposition) . For a given initial price for-ward curve f ( τ ) the forward kernel equals f t ( τ ) = f ( τ ) X τt E [ I mτ | F t ] , where I mτ is the geometric relative structural component given in Definition 3.1.Proof. The result can be shown analogously to the proof of Corollary 3.3.The interpretation of these decompositions is that today’s price forwardcurve is the expectation of the forward kernel that is being disturbed by themarket noise X τt in trading time t and by the structural component in deliv-ery time τ . Depending on the choice of the structural component ( g, Y τ ) thisdisturbance can be chosen to be multiplicatively in case of the geometric PFCdecomposition or additively in case of the arithmetic PFC decomposition.13 .1 Classical spot models We can use classical day-ahead spot price models in our framework by choosing g ( Y t ) = S t , where S t denotes the spot price at time t . Two examples of spotprice models that we explicitly compute in this section are the spot price modelsby Schwartz and Smith (2000) and Lucia and Schwartz (2002).For both examples we need the same structural component and thereforewe assume in this subsection that it is given by Y t = ( y t , y t ) ∈ R . The firstprocess is an Ornstein-Uhlenbeck process, i.e. dy t = − κ y t dt + σ dW t , y = 0 , (3.1)and the second y t = µ t + σ ρW t + σ p − ρ W t (3.2)is a (correlated) Brownian motion with drift. The standard one-dimensionalBrownian motions W and W are assumed to be independent. The parameters κ > σ , σ > − ≤ ρ ≤
1, and µ ∈ R are assumed to be real-valued. Example 3.5 (Schwartz and Smith) . Schwartz and Smith (2000) define theday-ahead spot price using the function g ( y , y ) = e y + y , i.e. they chose theprice to equal S t := g ( Y t ) = exp( y τ + y τ ). In the HJM framework this transfersto the following forward kernel f t ( τ ) = X τt E [ e y τ + y τ | F t ] , where we do not assume any extra conditions on X τ apart from Assumption 2.2.In this setting we can explicitly compute the conditional expectation on g ( Y τ ) and we findln E [ e y τ + y τ | F t ] = e − κ ( τ − t ) y t + y t + (cid:16) µ + σ (cid:17) ( τ − t )+ σ κ (cid:16) − e − κ ( τ − t ) (cid:17) + ρ σ σ κ (cid:16) − e − κ ( τ − t ) (cid:17) . This implies that this model for g and Y τ satisfies the affine structural compo-nent decomposition of Definition 2.15. The coefficient A τt of the decompositionis given by A τt = (cid:18) e − κ ( τ − t )
00 1 (cid:19) (3.3)and B τt can be chosen to be any vector in R such thatln g ( B τt ) = (cid:18) µ + σ (cid:19) ( τ − t ) + σ κ (cid:16) − e − κ ( τ − t ) (cid:17) + ρ σ σ κ (cid:16) − e − κ ( τ − t ) (cid:17) holds.Since the function g is multiplicative in nature, the geometric PFC decom-position, Corollary 3.4, is especially suited for this model. The conditionalexpectation of the multiplicative relative structural component is given byln E [ I mτ | F t ] = ln g ( A τt Y t + B τt ) E g ( Y τ )= e − κ ( τ − t ) y t + y t − (cid:16) µ + σ (cid:17) t + σ e − κτ κ (cid:0) − e κt (cid:1) + ρ σ σ e − κτ κ (cid:0) − e κt (cid:1) , f t ( τ ) = f ( τ ) X τt e e − κ ( τ − t ) y t + y t − µ + σ ! t + σ e − κτ κ ( − e κt ) + ρ σ σ e − κτκ ( − e κt ) , where any initial price forward curve f ( τ ) can be used. Example 3.6 (Lucia and Schwartz) . Lucia and Schwartz (2002) discuss fourdifferent models. Here, we highlight the arithmetic two factor model for thespot price. This model is defined by the function g ( y , y ) = y + y and theforward kernel equals f t ( τ ) = X τt E [ y τ + y τ | F t ] . Again, apart from Assumption 2.2 the process X τ can be chosen freely.The conditional expectation can easily be computed as E [ y τ + y τ | F t ] = e − κ ( τ t ) y t + y t + µ ( τ − t )and the affine structural component decomposition of Definition 2.15 followsimmediately with the coefficient A τt given by Equation (3.3) and B τt can be anyvector in R such that g ( B τt ) = µ ( τ − t ).The additive nature of g makes the arithmetic PFC decomposition, Corol-lary 3.3, the best suited candidate for this model. It follows that f t ( τ ) = X τt (cid:16) f ( τ ) + e − κ ( τ t ) y t + y t − µ t (cid:17) for any initial price forward curve f ( τ ). We continue the study of this type offorward kernel in Section 3.3 with a factor model approach.In the rest of this section we will give two further examples of the structuralcomponent Y . The first is based on the structural model approach for day-ahead spot prices and the other uses multi-factor models, which are the sum ofOrnstein-Uhlenbeck type processes, cf. Benth, Benth, and Koekebakker (2008). We will use the HJM framework to model the structural component by a struc-tural model approach: a spot price modelling technique started by Barlow (2002)which uses the idea of equilibrium of supply and demand to derive a spot price.In contrast to reduced-form models which need to implement a jump compo-nent to model spikes, structural models use a non-linear transformation of a(Gaussian) diffusion process to reach this goal. This method has been devel-oped further by many authors, e.g. A¨ıd, Campi, Huu, and Touzi (2009); Wagner(2014).For the real-valued demand process D we use a Gaussian Ornstein-Uhlenbeckprocess, i.e. dD t = − λ D t dt + σ dW t , D = 0 . We choose the structural component to equal Y t := (cid:18) β ( t ) D t (cid:19) , β ( t ) is a real-valued deterministic function. Furthermore, we define thefunction g as follows g ( y , y ) = γ + y sinh( α y ) = γ + y e α y − e − α y α > γ >
0. Through the first coordinate of Y t , i.e. β ( t ), we associate y with the evolution of time and y through the second coordinate of Y t , namely D t , with the demand. Therefore, g ( Y t ) represents the price at time t for a loadof D t through the merit order curve . Remark . It might be convenient to use more re-alistic models, such as described by Wagner (2014). This is an extension of theOU model, where stochastic processes for wind and solar infeed are subtractedfrom the demand process D . This difference is seen to model power prices evenmore accurately. It can easily be seen that the structural component Y t andfunction g can be extended for these processes.Using the auxiliary function ν ( s ) := σ λ (1 − e − λs ) the affine structuralcomponent decomposition of Definition 2.15 can be derived from the followingtheorem: Theorem 3.8.
The conditional expectation of the structural component is givenby E [ g ( Y τ ) | F t ] = γ + β ( τ ) e α ν ( τ − t ) sinh( α e − λ ( τ − t ) D t ) for all τ ≥ t ≥ .Proof. For Gaussian OU processes we have the following decomposition D τ d = e − λ ( τ − t ) D t + ν ( τ − t ) ε, ε ∼ N (0 , . Now, exploiting the decomposition and plugging it into the definition we get E [ g ( Y τ ) | F t ] = γ + β ( τ ) E [sinh( α D τ ) | F t ]= γ + β ( τ ) sinh( α e − λ ( τ − t ) D t ) E h e αν ( τ − t ) ε i = γ + β ( τ ) e α ν ( τ − t ) sinh( α e − λ ( τ − t ) D t )by symmetry of the normal distribution. Corollary 3.9 (Affine structural component decomposition) . With coefficientsgiven by A τt = β ( τ ) β ( t ) e α ν ( τ − t ) αe − λ ( τ − t ) ! and B τt = 0 ∈ R the affine structural component decomposition of Defini-tion 2.15 holds. By Theorem 3.8 it follows immediately by taking t = 0 that the expecta-tion E g ( Y τ ) = γ > τ ≥
0. Therefore we can use both the additive and16eometric PFC decomposition, i.e. Corollary 3.3 and Corollary 3.4, respectively.In the additive case the forward kernel equals f t ( τ ) = X τt ( f ( τ ) + g ( A τt Y t ) − γ )= X τt (cid:16) f ( τ ) + β ( τ ) e α ν ( τ − t ) sinh( α e − λ ( τ − t ) D t ) (cid:17) , whereas in the multiplicative case it equals f t ( τ ) = f ( τ ) X τt g ( A τt Y t ) γ = f ( τ ) X τt (cid:18) β ( τ ) γ e α ν ( τ − t ) sinh( α e − λ ( τ − t ) D t ) (cid:19) . For both decompositions any initial price forward kernel can be used.
In this section we use an arithmetic factor model approach for the structuralcomponent in the HJM framework. More precisely, the structural component isgiven by an n -dimensional L´evy driven Ornstein-Uhlenbeck process dY t = − Λ Y t dt + dL t , Y = y , where Λ = diag( λ , λ , . . . , λ n ) ∈ R n × n with λ , λ , . . . , λ n > L is an n -dimensional L´evy process. For more information on this type of moving averageprocess we refer the interested reader to Applebaum (2009); Barndorff-Nielsenand Shephard (2001); Jurek and Vervaat (1983); Sato (2013); Wolfe (1982). Foran application of OU processes in the form of multi-factor models for energyprices we refer to Benth et al. (2008).The function g is given by the summation of all the coefficients, i.e. weassume that g ( y ) = P ni =1 y i . If Y t satisfies Assumption 2.3 we can explicitlycompute the conditional expectation: Theorem 3.10.
The conditional expectation of the structural component isgiven by E [ g ( Y τ ) | F t ] = g (cid:18) e − Λ( τ − t ) Y t + E Z τt e − Λ( τ − u ) dL u (cid:19) for all τ ≥ t ≥ .Proof. For general OU processes the same decomposition holds as was used inthe proof of Theorem 3.8, i.e. Y τ = e − Λ( τ − t ) Y t + Z τt e − Λ( τ − u ) dL u . Noting that the first term is F t -measurable and the second term is independentof F t yields the result, as the sum g and E commute.As a direct consequence we obtain: Corollary 3.11 (Affine structural component decomposition) . With coeffi-cients given by A τt = e − Λ( τ − t ) and B τt = E R τt e − Λ( τ − u ) dL u the affine structuralcomponent decomposition of Definition 2.15 holds. g the logical PFC decomposition to choosein this setting is the arithmetic one, i.e. Corollary 3.3. From Theorem 3.10 wefind that the expectation is given by E g ( Y τ ) = g (cid:18) e − Λ τ y + E Z τ e − Λ( τ − u ) dL u (cid:19) . It follows that the forward kernel is given by f t ( τ ) = X τt (cid:18) f ( τ ) + g (cid:18) e − Λ( τ − t ) Y t − e − Λ τ y − E Z t e − Λ( τ − u ) dL u (cid:19)(cid:19) , where f ( τ ) can be any initial price forward curve. In this paper we have developed a unifying Heath-Jarrow-Morton (HJM) frame-work that • models intraday, spot, and futures prices, • is based on two stochastic processes motivated by economic interpreta-tions, • separates the stochastic dynamics in trading and delivery time, • is consistent with the initial term structure (i.e. the price forward curve), • is able to price options on futures by means of the Black-Scholes formula, • allows for the use of classical day-ahead spot price models such as Luciaand Schwartz (2002); Schwartz and Smith (2000), • includes many model classes such as structural models and factor models.To further the development of this framework empirical studies are needed:statistical evaluations but also calibration methods need to be discussed. Thetheoretical applications of Section 3 need to be specified and calibrated to realdata from intraday, spot, futures, and option prices. This is subject of futureresearch. Acknowledgments
WJH is grateful for the financial support from Fraunhofer ITWM (
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