The market price of jump risk for delivery periods: pricing of electricity swaps with geometric averaging
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Kemper, Annika; Schmeck, Maren Diane Article — Published Version The market price of jump risk for delivery periods: pricing of electricity swaps with geometric averaging Mathematics and Financial Economics Provided in Cooperation with: Springer Nature Suggested Citation: Kemper, Annika; Schmeck, Maren Diane (2025) : The market price of jump risk for delivery periods: pricing of electricity swaps with geometric averaging, Mathematics and Financial Economics, ISSN 1862-9660, Springer, Berlin, Heidelberg, Vol. 19, Iss. 2, pp. 293-327, https://doi.org/10.1007/s11579-025-00383-5 This Version is available at: https://hdl.handle.net/10419/323572 Standard-Nutzungsbedingungen: Die Dokumente auf EconStor dürfen zu eigenen wissenschaftlichen Zwecken und zum Privatgebrauch gespeichert und kopiert werden. Sie dürfen die Dokumente nicht für öffentliche oder kommerzielle Zwecke vervielfältigen, öffentlich ausstellen, öffentlich zugänglich machen, vertreiben oder anderweitig nutzen. Sofern die Verfasser die Dokumente unter Open-Content-Lizenzen (insbesondere CC-Lizenzen) zur Verfügung gestellt haben sollten, gelten abweichend von diesen Nutzungsbedingungen die in der dort genannten Lizenz gewährten Nutzungsrechte. Terms of use: Documents in EconStor may be saved and copied for your personal and scholarly purposes. You are not to copy documents for public or commercial purposes, to exhibit the documents publicly, to make them publicly available on the internet, or to distribute or otherwise use the documents in public. If the documents have been made available under an Open Content Licence (especially Creative Commons Licences), you may exercise further usage rights as specified in the indicated licence. http://creativecommons.org/licenses/by/4.0/
Mathematics and Financial Economics (2025) 19:293–327 https://doi.org/10.1007/s11579-025-00383-5 The market price of jump risk for delivery periods: pricing of electricity swaps with geometric averaging Annika Kemper1·Maren Diane Schmeck1 Received: 4 January 2024 / Accepted: 19 January 2025 / Published online: 10 April 2025 © The Author(s) 2025 Abstract In this paper, we extend the market price of risk for delivery periods (MPDP) of electricity swap contracts by introducing a dimension for jump risk. As introduced by Kemper et al. [30], the MPDP arises through the use of geometric averaging while pricing electricity swaps in a geometric framework. We adjust the work by Kemper et al. [30] in two directions: First, we examine a Merton type model taking jumps into account. Second, we transfer the model to the physical measure by implementing mean-reverting behavior. We compare swap prices resulting from the arithmetic (approximated) average to the geometric weighted average. Under the physical measure, we discover a decomposition of the swap’s market price of risk into the instantaneous market price of risk and the MPDP. Keywords Electricity swaps ·Delivery period ·MPDP for diffusion and jump risk · Mean-reversion ·Jumps ·Samuelson effect ·Seasonality JEL Classification G130 ·Q400 1 Introduction With the turn of the millennium, pricing derivatives on electricity has become important through the liberalization of energy markets. Nowadays, new challenges appear due to the transition to a climate neutral energy system: Electricity generated from renewable energy sources, like wind and solar energy, clearly depends on the weather conditions of the season. Consequently, a rising share of renewable energy induces stronger intermittency and seasonality effects influencing especially delivery-dependent pricing effects. In electricity markets, such delivery-dependent futures contracts are the most important derivatives. They deliver the underlying over a period of time since electricity is not storable on a large scale. We would like to thank Christa Cuchiero for her fruitful comments and suggestions. Financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - SFB 1283/2 2021 - 3172102 26 is gratefully acknowledged. BMaren Diane Schmeck [email protected] Annika Kemper [email protected] 1Center for Mathematical Economics (IMW) at Bielefeld University, Bielefeld, Germany 123
294 Mathematics and Financial Economics (2025) 19:293–327 We therefore call them electricity swaps. The dependence on the delivery time affects the price dynamics, the pricing measure, and the swap’s market price of risk for delivery periods (MPDP) introduced by Kemper et al. [30]. In this paper, we provide an extension of the MPDP. To do so, we adjust the model to a Merton type model taking jumps into account. In addition, under the physical measure, we identify a decomposition of the swap’s market price of risk into the instantaneous market price of risk and the MPDP. The delivery period is a unique feature of electricity markets that differs from other commodities such as oil, gas, or corn. In fact, it plays a crucial role in the pricing of electricity swaps. Following the market model approach, the electricity swap price results from averaging an instantaneous stream of futures with respect to the delivery time. This approach goes back to the famous model by Heath et al. [23]. It was firstly connected to energy-related derivatives by Clewlow and Strickland [15] and to electricity derivatives by Bjerksund et al. [9] followed by a row of works (see, e.g., Koekebakker and Ollmar [34], Benth and Koekebakker [5], Bjerksund et al. [9], Benth et al. [1], and Kemper et al. [30] for geometric settings, Hinderks et al. [25] for a structural model and Cuchiero et al. [17] measure-valued processes). One stream of literature investigates spot based price dynamics and derive electricity futures based on the spot price referring to the day ahead market (see, e.g., Cartea and Figueroa, [13], Cartea and Villaplana [14], Escribano et al. [19]). In this paper, we focus on a HJM-type approach modelling the futures market directly. That is we consider so-called atomic swap contracts inducing a delivery period of a month, that are used to price overlapping swap contracts delivering for example over a quarter or a year. We refer to Benth et al. [1]and Benth and Koekebakker [5] and Kemper et al. [30] for a construction of overlapping swap contracts based on atomic swaps. The delivery period can be incorporated in different ways of averaging. We distinguish between three types of averaging: Arithmetic, approximated, and geometric averaging. Arithmetic averaging is theclassicalwaytoimplementtheswap’sdeliveryperiodandisconvenient for arithmetic price dynamics. In particular, continuous arithmetic averaging is applied by Benth et al. [3], Benth and Koekebakker [5] Benth et al. [1], Benth et al. [4], Benth et al. [7], Kleisinger-Yu et al. [32], Koekebakker and Ollmar [34], and Latini et al. [35], among others. For discrete arithmetic averaging, we refer to Lucia and Schwartz [37] and Burger et al. [12]. Instead, arithmetic averaging of geometric price dynamics is poorly suited since the resulting swap price dynamics are neither geometric nor Markovian. It requires, e.g., an approximation of the swap price volatility introduced by Bjerksund et al. [9] whenever we want to consider tractable swap price dynamics (see also Benth et al. [1], Benth and Koekebakker [5]). We call this procedure approximated averaging.Geometric averaging, instead, does not require any approximations whenever the price dynamics are of geometric type and lead to suitable geometric dynamics (see Kemper et al. [30]). Hence, the geometric average is tailor-made for relative growth rate models. Nevertheless, the geometric average does not preserve the martingale property. This issue is tackled by Kemper et al. [30] using a measure change with their MPDP. Usually, negative prices are not observable in the data of the futures prices, such that we stick to a geometric setting and compare the latter averaging procedures while adjusting the MPDP to a Merton type model. Both papers, Kemper et al. [30] and Bjerksund et al. [9], investigate the modeling of the delivery period explicitly through a continuous weighted averaging approach for geometric futures prices. Both approaches lead to Markovian and geometric swap price dynamics. We discuss similarities and differences between these approaches and introduce a numéraire caused by the different averaging techniques in Sect.2. In line with the market model approach, we base the averaging procedure on a continuous stream of futures contracts that is a martingale under the futures risk-neutral measure Q. As the futures have instantaneous 123
Mathematics and Financial Economics (2025) 19:293–327 295 Fig. 1 Measure changes between the physical measure P, the instantaneous risk-neutral measure Q,and the swap’s pricing measure Qas well as their connections with the swap’s market prices of risk PQ,the instantaneous market price of risk P Q, and the MPDP denoted by Q Q delivery, we refer to Qas the instantaneous risk neutral measure. The resulting swap price dynamics based on geometric averaging are not a martingale under Q.Wethendefinethe MPDP of diffusion and jump risk and a new pricing measure Q, which can thus be used to price derivatives on the swap. We may refer to Qas the “swap’s” risk-neutral measure since the swap price is a Q-martingale without any approximations. Therefore, we call Qalso the “instantaneous” risk-neutral measure. It is a clear advantage that the approximated average preserves the martingale property of the swap under the measure Q. A decomposition of the market price of risk for electricity swaps arises when turning to the physical measure P. Figure1gives an overview over the connections between the different measures P,Q,and Qand the swap’s market price of risk PQ, the instantaneous market price of risk P Q, and the MPDP denoted by Q Q. Indeed, the MPDP is triggered by typical features of the electricity market entering the swap’s volatility. In particular, delivery-dependent effects like seasonalities and term-structure effects play a crucial role. Fanelli and Schmeck [20] empirically identify seasonalities in the swap’s delivery period by considering implied volatilities of electricity options. Renewable energy, like wind and solar energy, intensify especially the seasonal effects mentioned before. An additional property of electricity and commodity markets is the Samuelson effect (see Samuelson [43]): The closer we reach the end of the maturity, the more effect the volatility has. Benth and Paraschiv [6] and Jaeck and Lautier [27] provide empirical evidence for the Samuelson effect in the volatility term-structure of electricity swaps. It can also be observed in the implied volatilities of electricity options, especially far out and in the money (see Kiesel et al. [31]). Kemper et al. [30] characterize the MPDP for such seasonalities and term-structure effects within a stochastic volatility model through the variance per unit of expectation of the delivery-dependent effects. We contribute to the literature by investigating the MPDP analytically, affected by seasonalities and the Samuelson effect. Moreover, we lay the foundation for the empirical analysis of the MPDP by specifying the model under the real world measure P. Further characteristics of the observerd electricity swap prices are mean-reversion and jump behavior. As mentioned by Latini et al. [35] and Kleisinger-Yu et al. [32] among others, mean-reversion is an important property of the electricity swap prices. Koekebakker and Ollmar [34] empirically validate that the short-term price varies around the long-term price, which confirms mean-reverting behavior. As Benth et al. [7], we face the problem of changing a mean-reverting process to the risk-neutral measure. We extend their measure change to the geometric setting. We even provide a proof for stochastic volatility settings in order to address models such as Kemper et al. [30] and Schneider and Tavin [44]. Besides mean-reversion, Benth et al. [7] include jumps as an outstanding characteristic of electricity prices. They consider compound Poisson processes under the physical measure in a meanreverting, arithmetic setting. While adjusting the paper by Kemper et al. [30] to jumps, we 123
296 Mathematics and Financial Economics (2025) 19:293–327 establish the MPDP of jump risk whenever the jump coefficient relies on delivery-dependent effects. In this paper, we follow a so-called Heath-Jarrow-Morton approach to model forward markets,thatiswedefinetheswapwithdeliveryperiodasaverageoveraninfinitedimensional stream of futures with instantaneous delivery. Note that these futures are not traded at the market, only the swaps with delivery period are traded. In particular, it is not possible to observe traded quotes of the futures and it is not possible to see if the spot price converges to the futures if time approaches maturity of a futures. In fact, this relationship typically does not hold true for the traded swaps either due to the delivery period (see e.g. Benth et al. [1]). Another approach to model the forward market is to start with a model for the spot and define the swap price as conditional expectation of the average spot during the delivery period, where the expectation is taken under some probability measure that is equivalent to the real world measure P. Note that it is not necessary to have a martingale measure for the spot: as the underlying electricity is not storable on a large scale, it has to be consumed once purchased. In particular, it is not possible to set up buyand hold-strategies, that are required in no-arbitrage portfolios. In this sense, the electricity spot is said to be “not tradable” (see e.g. Benth et al. [1]). Our contribution to the literature is twofold: First, we adjust the paper by Kemper et al. [30] to the jump case under the instantaneous risk-neutral measure leading to an extended characterization of the MPDP regarding diffusion and jump risk. Second, we transfer the model to the physical measure P. Under Pwe compare the swap prices resulting from geometric and approximated averaging as well as their risk-neutral measures revealing the decomposition of the swap’s market price of risk into the instantaneous market price of risk and the MPDP. Consequently, the model lays the foundation for empirical investigations in the future. Thepaperis organizedas follows: Sect.2presentsthe geometricaveragingapproachunder the instantaneous risk-neutral measure applied to the jump-type futures curve. In addition, it presents the MPDP of diffusion and jump risk. Section4introduces the model under the physical measure and identifies the decomposition of the swap’s market price of risk. An example under the physical measure closes the section. Finally, Sect.5concludes our main findings. 2 On the MPDP of diffusion and jump risk We particularly focus on an electricity swap contract delivering 1 MWh of electricity during the agreed delivery period (τ1,τ 2]. At a trading day t≤τ1before the contract expires, we denote the swap price by F(t,τ 1,τ 2)settled such that the contract is entered at no cost. It can be interpreted as an average price of instantaneous delivery. Motivated by this interpretation, we consider a futures contract with price f(t,τ) that stands for instantaneous delivery at time τ∈(τ1,τ 2]. Note that such a contract does not exist on the market but it turns out to be useful for modeling purposes when considering delivery periods (see, e.g., Benth et al. [7] and Kemper et al. [30]). Following the approach by Heath et al. [23], we derive the price of an electricity swap contract based on an instantaneous futures price model. More precisely, we compare two types of swap prices resulting from geometric and approximated averaging. The goal of this section is to investigate the pricing spread between both approaches in order to quantify the 123
Mathematics and Financial Economics (2025) 19:293–327 297 consequencesofthe approximationandthus the effect oftheprecise geometricaveragingprocedure. As the pricing spread goes along with different risk-neutral measures, we additionally investigate the distance of both risk-neutral measures quantified by the MPDP. Moreover, we characterize the MPDP for specific volatility functions and different jump size distributions. Before doing so, we would like to repeat the main ideas of the MPDP introduced in Kemper et al. [30]. 2.1 The idea of the MPDP The procedure for the derivation of the swap’s martingale measure used in this paper goes back to Kemper et al. [30] starting with a futures contract with instantaneous delivery. The dynamics for the futures price f(t,τ)is given by df(t,τ)=σ(ω,t,τ)f(t,τ)dWQ t.(2.1) Kemper et al. [30] investigate the gap between the swap’s martingale measure associated with two methodologies for implementing the delivery period: arithmetic weighted average with approximation and the geometric weighted average without approximation. The geometric weighted average creates a drift term. More precisely, the swap price dynamics under Q resulting from geometric averaging without approximations evolve as dF(t,τ 1,τ 2) F(t,τ 1,τ 2)=−1 2EUσ(ω,t,U)2−EU[σ(ω,t,U)]2dt +EU[σ(ω,t,U)]dWQ t, (2.2) where the swap’s volatility is given by EU[σ(ω,t,U)]=τ2 τ1 w(u,τ 1,τ 2)σ(ω, t,u)du .(2.3) We refer to a more detailed discussion and motivation of the notation to Sect.2.3 below. This additional drift term gives reason for the existence of the MPDP whenever the futures volatility depends on the delivery period. In particular, the MPDP is defined by Q Q 1:= −1 2 VU[σ(ω,t,U)] EU[σ(ω,t,U)],(2.4) affecting the Brownian motion WQ. The MPDP introduced by Kemper et al. [30] is essential to change the measure to the swap’s risk-neutral measure Q. While Kemper et al. [30]base the derivation of the electricity swap and the MPDP on a geometric dynamics with stochastic volatility in the spirit of Heston [24], this paper focuses on a geometric jump diffusion with deterministic volatility. Moreover, this paper investigates not only the connection between the futures’ and swap’s risk neutral measure as in Kemper et al. [30] but creates also a bridge to the physical measure. 2.2 The model Consider a filtered probability space (, F,(Ft)t∈[0,τ],Q), where the filtration satisfies the usual conditions. We first model the solution of a futures contract and then derive the corresponding dynamics to avoid lacks of existence in the presence of jumps (see Papapantoleon 123
298 Mathematics and Financial Economics (2025) 19:293–327 [40]). At time t≤τ, let the logarithmic price process of the futures contract be defined as ln f(t,τ)=ln f(0,τ)+t 0 σ(s,τ)dWQ s+t 0 η(s,τ)d JQ s−t 0 cQ(s,τ)ds ,(2.5) with initial non-random conditions f(0,τ) > 0. Moreover, WQis a one-dimensional standard Brownian motion under Qindependent of the jump process JQ. In particular, JQis a compound compensated jump process defined through the compensated Poisson random measure NQ(dt,dz)=N(dt,dz)−Q(dz)dt: JQ t=t 0R z NQ(ds,dz), (2.6) with Lévy measure Q(dz)=λQG(dz), which is independent of the delivery time, and where λQ>0 indicates the jump intensity and G(dz)the jump size distribution. The last term in Equation (2.5) defines the compensator of the logarithmic return under the current measure Q: cQ(t,τ)=1 2σ2(t,τ)+ψQ(iη(t,τ)), (2.7) where ψQ(ir)is the integrand of the Lévy-Khintchine exponential defined through the moment generating function ψQ(r):= Rerz −1−rzQ(dz). (2.8) We assume that the futures price volatility and jump coefficients, σ(t,τ) and η(t,τ),are deterministic and that the futures price f(t,τ) is Ft-adapted for t∈[0,τ].Wefurther assumethattheysatisfysuitable integrabilityand measurabilityconditions(see Assumption 1 in Appendix A for details) to ensure that the process in Equation (2.5)isaQ-martingale, and that Equation (2.5) gives the unique solution to the process evolving as df(t,τ) f(t−,τ) =σ(t,τ)dWQ t+Reη(t,τ )z−1 NQ(dt,dz). (2.9) As σ(t,τ) depends on both, trading time tand delivery time τ, we allow for volatility structures as the Samuelson effect or seasonalities in the delivery time, which are addressed in Examples 3.1 and 3.2. 2.3 Implementing the delivery period Following the Heath-Jarrow-Morton approach to price futures and swaps in electricity markets, the swap price is usually defined as the arithmetic weighted average of futures prices (see, e.g., Benth et al. [1], Bjerksund et al. [9], and Benth et al. [7]): FA(t,τ 1,τ 2):= τ2 τ1 w(u,τ 1,τ 2)f(t,u)du,(2.10) for a general weight function w(u,τ 1,τ 2):= ˆw(u) τ2 τ1ˆw(v)dv,for u∈(τ1,τ 2],(2.11) 123
Mathematics and Financial Economics (2025) 19:293–327 299 where ˆw:(τ1,τ 2]→R+ 0is the corresponding settlement function which is deterministic, integrable and non-negative. Note that wdefines a probability density function with support on (τ1,τ 2]since it is positive and integrates to one, that is τ2 τ1w(u,τ 1,τ 2)du =1. Hence, we denote Uas a random delivery variable with density w(u,τ 1,τ 2)(see also Kemper et al. [30]). The most popular example is given by a constant settlement type ˆw(u)=1, such that the density becomes w(u,τ 1,τ 2)=1 τ2−τ1and U∼U((τ1,τ 2])is uniformly distributed over the delivery period. This corresponds to a one-time settlement. A continuous settlement over the time interval (τ1,τ 2]is covered by a continuous discount function ˆw(u)=e−ru,where ris the constant interest rate (see, e.g., Benth et al. [1]). ThearithmeticaverageofthefuturespriceasinEquation (2.10)leadstotractabledynamics for the swap as long as one assumes an arithmetic structure of the futures prices as well. This is based on the fact that arithmetic averaging is tailor-made for absolute growth rate models. Nevertheless, if one defines the futures price as a geometric process as in Equation (2.9), one can show that the dynamics of the swap price FAdefined through Equation (2.10)aregiven by dFA(t,τ 1,τ 2) FA(t−,τ 1,τ 2)=σ(t,τ 2)−τ2 τ1 ∂σ ∂u(t,u)w(τ, τ1,τ 2) w(τ, τ1,u) FA(t,τ 1,u) FA(t,τ 1,τ 2)dudWQ t +Reη(t,τ2)z−1−τ2 τ1 ∂eη(s,u)z ∂u w(τ, τ1,τ 2) w(τ, τ1,u) FA(t,τ 1,u) FA(t,τ 1,τ 2)du NQ(dz,dt), (2.12) for any τ∈(τ1,τ 2](see Benth et al. [1], cf. Chapter 6.3.1). Thus, the dynamics of the swap price is neither a geometric process nor Markovian, which makes it unhandy for further analysis. To overcome this issue, Bjerksund et al. [9] suggest an approximation in the setup without jumps, which we call approximated averaging since it is the arithmetic average of approximated logarithmic returns. Approximated averaging maintains the martingale property meaning that the swap is a martingale whenever fis a martingale. If we transfer the approximated averaging procedure to our jump setting, we can define the swap price process based on approximated averaging by dFa(t,τ 1,τ 2) Fa(t−,τ 1,τ 2):= τ2 τ1 w(u,τ 1,τ 2)df(t,u) f(t−,u)du .(2.13) Incontrast,geometric averaging originatesfromthe arithmeticaverageoflogarithmic returns without any need for approximations. Hence, in line with Kemper et al. [30], we define the swap price originating from geometric averaging by F(t,τ 1,τ 2):= eτ2 τ1w(u,τ1,τ2)ln f(t,u)du ,(2.14) (see also Kemna and Vorst [29]). Assume that the volatility and jump coefficients satisfy further integrability conditions (see Assumption 2in Appendix A). It turns out, that the resulting swap price dynamics is a geometric process with a non-zero drift term: Lemma 2.1 (The Swap Price under Q) Let Assumption 2in Appendix A be satisfied. Under the instantaneous pricing measure Q, the dynamics of the swap price process F(·,τ 1,τ 2), defined in Equation (2.14), are given by dF(t,τ 1,τ 2) F(t−,τ 1,τ 2)=E[σ(t,U)]dWQ t+ReE[η(t,U)]z−1 NQ(dt,dz) −1 2V[σ(t,U)]+E[ψQ(η(t,U))]−ψQ(E[η(t,U)])dt, (2.15) 123
300 Mathematics and Financial Economics (2025) 19:293–327 where U denotes the random delivery variable with density w(u,τ 1,τ 2). Proof Plugging the integral representation of the futures rate process from Equation (2.5) into Equation (2.14)givesusF(t,τ 1,τ 2)=F(0,τ 1,τ 2)e¯ X(t,τ1,τ2), where an application of the stochastic Fubini Theorem (see Protter [42], cf. Theorem 65, Chapter IV.6) leads to ¯ X(t,τ 1,τ 2)=t 0 Eσ(s,U)dWQ s +t 0 E[η(s,U)]d JQ s−1 2t 0 Eσ2(s,U)ds −t 0 E[ψQ(η(s,U))]ds .(2.16) Then, Equation (2.15) follows using Itô’s formula (see, e.g., Øksendal and Sulem [39]). Having presented the three procedures of continuous time averaging that are used to derive the swap from an underlying futures curve, we would like to compare them: Arithmetic averaging, defined by Equation (2.10), is tractable for arithmetic futures curves, whereas approximated averaging, defined by Equation (2.13), and geometric averaging, defined by Equation (2.14), are well suited for geometric futures curves. In line with a series of literature (see Koekebakker and Ollmar [34], Benth and Koekebakker [5], Bjerksund et al. [9], Benth et al. [1], and Kemper et al. [30]), we follow the geometric approach. Our goal throughout this paper is to investigate the pricing spread between geometric and approximated averaging analytically. 3 The MPDP Although the futures price fand the approximated Faare martingales under the pricing measure Q, the swap price Fis not a Q-martingale: Indeed, the swap price process under Q hasanegativedrifttermconsistingoftwopartsgivenbytheswap’svarianceandthedifference between the averaged Lévy-Khintchine integrand and the Lévy-Khintchine integrand of the averaged jump coefficient. Hence, using geometric averaging leads to a new interpretation of risk related to the delivery period as we will analyze in the following. Analogous to Kemper et al. [30], we derive the corresponding risk-neutral measure Q under which the electricity swap price Fis a martingale. For deriving the swap’s risk-neutral measure, we thus define the MPDP extended to jumps in the following. Definition 3.1 [(The MPDP)] At time t∈[0,τ 1],themarket price of diffusion and jump risk for delivery periods associated to the delivery period (τ1,τ 2]is defined by Q Q:= (Q Q 1, Q Q 2),where Q Q 1(t,τ 1,τ 2):= −1 2 V[σ(t,U)] E[σ(t,U)],(3.1) Q Q 2(t,τ 1,τ 2):= −RE[eη(t,U)z]−eE[η(t,U)]zQ(dz) ReE[η(t,U)]z−1Q(dz).(3.2) In general, the MPDP does not coincide with the market price of risk. In fact, it is an additional risk that has to be taken into account whenever approximated averaging is conducted. Technically speaking, the MPDP characterizes the distance between the martingale measure of the swaps, Fand Fa, resulting from geometric and approximated averaging. In particular, 1refers to the additional diffusion risk, which is measurable and Ft-adapted as σ(t,u)is. 123
Mathematics and Financial Economics (2025) 19:293–327 307 −1 2E[σ(t,U)]2+R E[eη(t,U)z]−1−ln E[eη(t,U)z]dt . Taking Eq.2.16 into account, the pricing spread is given by Fa(t,τ 1,τ 2)−F(t,τ 1,τ 2)=Fa(t,τ 1,τ 2)(1−D(t,τ 1,τ 2)), such that F(t,τ 1,τ 2)=Fa(t,τ 1,τ 2)D(t,τ 1,τ 2),where D(t,τ 1,τ 2)=e¯ X(t,τ1,τ2)−¯ Xa(t,τ1,τ2)=e−1 2t 0V[σ(s,U)]ds−t 0Rln E[eη(s,U)z]−E[η(s,U)]zN(ds,dz). (iii)If η(t,u)⊥⊥ u,then D(t,τ 1,τ 2)=e−1 2t 0V[σ(s,U)]ds . Since V[σ(·,U)]≥0 by Jensen, it follows that D(t,τ 1,τ 2)∈(0,1]and thus F≤Fa. We conclude that arithmetic and in specific cases approximated averaging lead to higher swap prices than the geometric average. We would like to stress that Din Equation (3.26) is not affected by measure changes since it is characterized by a drift component and a pure jump component exclusively (see also Equation (3.28)). Moreover, note that Dcan be seen as stochastic discount factor, which can be used to derive the swap price Fgiven Fa.Vice versa, consider Fa(t,τ 1,τ 2)=F(t,τ 1,τ 2)D−1(t,τ 1,τ 2). (3.27) The exponential part of D−1can be interpreted as a price (premium) per share, which we pay for an imprecise averaged swap. Moreover, we can see Das the price process of a non-dividend paying asset evolving as dD(t,τ 1,τ 2) D(t−,τ 1,τ 2)=−1 2V[σ(t,U)]dt +ReE[η(t,U)]z E[eη(t,U)z]−1N(dt,dz), (3.28) such that we can interpret Das a numéraire. If Fais a martingale, then F Dis also a martingale. If Fis a martingale, then FaDis a martingale (see, e.g., Shreve [46], cf. Theorem 9.2.2). We can thus use it to price options and other derivatives on the swap. In the subsequent section, we introduce the model under its physical measure P. Remark 3.2 (Delivery-Dependent Intensity) Let us consider an adjusted version of the futures price under the instantaneous pricing measure Qsimilar to Equation (2.9)givenby df(t,τ) f(t−,τ) =σ(t,τ)dWQ t+Reηz−1 Nτ(dt,dz), (3.29) where η∈Rand the compensated Poisson random measure is defined by Nτ(dt,dz):= N(dt,dz)−λQ(τ)G(dz)dt, with a jump intensity adjusted to a deterministic, positive, and bounded function of the delivery time. (i)Analogous to Lemma 2.1, the dynamics of the swap price, defined by geometric averaging, are given by dF(t,τ 1,τ 2) F(t−,τ 1,τ 2)=E[σ(t,U)]dWQ t+Reηz−1N(dt,dz) −1 2V[σ(t,U)]+E[λQ(U)]Reηz−1G(dz)dt, (3.30) 123
308 Mathematics and Financial Economics (2025) 19:293–327 where Uis the random delivery variable with density w(u,τ 1,τ 2). (ii)If the volatility is independent of delivery time, then the approximated and geometric average coincide and so their risk-neutral pricing measure. However, the resulting swap price process in Equation (3.30) is not a martingale under Qsince the intensity is affected by the averaging procedure. (iii)In the case of delivery-dependent volatility, the MPDP from Definition 3.1 adjusts to (Q Q 1,0). Hence, the MPDP associated to the Brownian motion stays the same and its second dimension becomes zero since the jump coefficient ηis independent of delivery time. However, under the assumption that the intensity is delivery-dependent, the swap price is in general not a martingale under Q. (iv) The swap price is a Q−martingale, only if the swap’s jump intensity under Qis given by E[λQ(U)], i.e., if Nτ1,τ2(dt,dz):= N(dt,dz)−E[λQ(U)]G(dz)dt,isa compensated Poisson random measure under Q. 4 The real-world model A typical feature of electricity prices beyond seasonalities and the Samuelson effect is the mean-reverting behavior (see, e.g., Benth et al. [1] and Benth et al. [7]). In order to implement the drift feature, we derive the futures under the physical measure P. Note that we will include mean-reversion at the futures and thus the swap’s rate level. We then consider the resulting market prices of risk transferring to the instantaneous and the swap’s risk-neutral measure. 4.1 The swap price under the physical measure We now derive the price of a swap contract that delivers one unit of electricity during the fixed delivery period (τ1,τ 2], similar to Sect.2but now under the physical measure P. Hence, starting from the physical measure P, the logarithmic futures price process from Equation (2.5), given by ln f(t,τ)=e−t 0κ(s)ds ln f(0,τ)+t 0 e−t vκ(q)dqμ(v, τ)dv +t 0 e−t vκ(q)dqσ(v,τ)dWP v+t 0 e−t vκ(q)dqη(v, τ)d JP v, (4.1) where WPis a Brownian motion under the physical measure Pindependent of the compound compensated jump process JP. In particular, JPis defined through the P-compensated Poisson random measure NP(dt,dz)=N(dt,dz)−P(dz)dt with Lévy measure P(dz)= λPG(dz)that is independent of delivery time. Note that λP>0 indicates the jump intensity under the physical measure and G(dz)is the jump size distribution. In order to characterize the futures price in more detail, we introduce the following lemma. Lemma 4.1 We assume that the coefficients satisfy suitable integrability and measurability conditions (see Assumption 4in Appendix A) such that Equation 4.1 is the unique strong solution to the dynamics df(t,τ) f(t−,τ) =σ(t,τ)dWP t+Reη(t,τ )z−1 NP(dt,dz)+cP(t,τ,ln f(t,τ))dt,(4.2) 123
Mathematics and Financial Economics (2025) 19:293–327 309 where the drift-term is characterized by cP(t,τ,Y)=μ(t,τ)−κ(t)Y+1 2σ(t,τ) 2+ψP(η(t,τ)). (4.3) Hence, the logarithmic futures evolves as dln f(t,τ)=(μ(t,τ)−κ(t)ln f(t,τ) )dt +σ(t,τ)dWP t+η(t,τ)d JP t.(4.4) Proof The unique strong solution follows from Benth et al. [1] (cf. Proposition 3.1). Applying Ito’s formula leads to the desired dynamics (see Øksendal and Sulem [39], cf. Theorem 1.16). Note that the assumption behind the model induces a finite second moment as well as a finite moment generating function of the jump size distribution. In Examples 3.3 to 3.6,we consider suitable distributions for these jump sizes. Remark 4.1 In the literature, we sometimes find the application of lognormal distributed jump sizes (see, e.g., Borovkova and Permana [10] and Borovkova and Permana [11]). This distribution, however, is not suitable for our setting since its moment generating function E[eηZ]is not finite at any positive value η(see, e.g., Gray and Pitts [22], cf. Chapter 2.2.6). Hence,thelognormaldistributioncontradictstheintegrabilityassumptioninAssumption4(i) under the physical measure in Appendix A. As in the previous section, we now derive the swap prices resulting from geometric and approximated averaging. Lemma 4.2 [The Swap Price under P] Let Assumption 4and 5in Appendix A be satisfied. Then, the swap price based on geometric averaging evolves as dF(t,τ1,τ2) F(t−,τ1,τ2)=E[σ(t,U)]dWP t +ReE[η(t,U)]z−1 NP(dt,dz)+cP(t,τ 1,τ 2,ln F(t,τ 1,τ 2))dt,(4.5) where the drift term is given by cP(t,τ 1,τ 2,¯ Y)=E[μ(t,U)]−κ(t)¯ Y+1 2E[σ(t,U)]2+ψP(E[η(t,U)]). (4.6) Proof Following the considerations in the previous section, the swap price is defined by the geometric average in Equation (2.14). Using the integral representation of Eq.4.4 and the stochastic Fubini theorem (see Protter [42], cf. Theorem 65), we can introduce the dynamics of the swap’s logarithmic return by dln F(t,τ 1,τ 2) =(E[μ(t,U)]−κ(t)ln F(t,τ 1,τ 2))dt +E[σ(t,U)]dWP t+E[η(t,U)]d JP t. (4.7) An application of Ito’s formula (see Øksendal and Sulem [39], cf. Theorem 1.16) yields the desired swap dynamics. Note that the speed of mean-reversion κ(t)has to be independent of the delivery time. This assumption ensures that ln F, in Equation (4.7), is again an Ornstein-Uhlenbeck process and that the swap’s price dynamics in Equation (4.5) stay tractable. This is also in line with the 123
310 Mathematics and Financial Economics (2025) 19:293–327 findings in Benth et al. [7] (cf. Proposition 2.2) and Latini et al. [35]. In particular, the meanreverting effect comprises the jump component as well, even if we implement it through a measure change of the Brownian part. More precisely, mean-reversion connected to jumps covers indeed a unique feature of electricity markets known as spikes: Spikes are large jumps quickly returning to the “normal” level (see, e.g., Klüppelberg et al. [33]). They arise as electricity is not storable on a large scale and since the electricity demand is not elastic (see Borovkova and Schmeck [11]). Let us now investigate the swap price under the physical measure resulting from approximated averaging (see Eq.2.13 in order to compare the pricing spread between both approaches. Lemma 4.3 Let Assumption 4and 5in Appendix A be satisfied. Then, the swap price dynamics based on approximated averaging evolve as dFa(t,τ 1,τ 2) Fa(t−,τ 1,τ 2) =E[σ(t,U)]dWP t+RE[eη(t,U)z]−1 NP(dt,dz)+EU[cP(t,U,ln f(t,U))]dt, (4.8) with EUdenoting the expectation with respect to the random delivery variable U having density w(u,τ 1,τ 2). Proof We use the approximated averaging methodology (see Equation (2.13)) in order to derive the swap price evolution and apply the stochastic Fubini theorem (see Protter [42], cf. Theorem 65) leading to Equation (4.8). 4.2 The risk premium We would like to close this section with a discussion of the risk premium in our setting. The risk premium usually represents the difference between the forward price and the spot price prediction at delivery time (cf. for instance Benth et al. [2]). In the case of swaps, Benth et al. [7] extend the definition of the risk premium as the difference between the swap price and the expected value of the spot price weighted over the delivery period. Inspired by the extended definition of the risk premium to swaps by Benth et al. [7], we consider the risk premium as the differences between the swap price and the expected value of the geometrically averaged futurespriceweighted over the deliveryperiod.Additionally,we consider theriskpremium as a pricing spread (similar to in Corollary 3.1) resulting from different pricing methodologies, i.e. the difference between two prices under the same measure. Corollary 4.1 (Risk Premium) (i)The risk premium between the swap price F and F Ais always non-positive under the physical measure, i.e., RPF,FA(t,τ 1,τ 2):= F(t,τ 1,τ 2)−FA(t,τ 1,τ 2)≤0.(4.9) 123
Mathematics and Financial Economics (2025) 19:293–327 311 (ii)The risk premium between the swap price F and Faunder the physical measure is determined by RPF,Fa(t,τ 1,τ 2):= F(t,τ 1,τ 2)−Fa(t,τ 1,τ 2)=Fa(t,τ 1,τ 2)(D(t,τ 1,τ 2)−1), (4.10) where D(t,τ 1,τ 2)is defined in Equation (3.26). Hence, the distance between F and Fais not affected by the measure change. (iii)The risk premium of the geometric swap price between the physical and the swap’s risk neutral measure is defined by RPF(t,τ 1,τ 2):= F(t,τ 1,τ 2)−EP[F(τ1,τ 1,τ 2)|Ft](4.11) and is explicitly determined by RPF(t,τ 1,τ 2)=F(t,τ 1,τ 2)(1−DRP(t,τ 1,τ 2)),(4.12) where DRP(t,τ 1,τ 2) := exp τ1 0e−τ1 sκ(v)dvE[μ(s,U)]ds +1 2τ1 t E[σ(s,U)]2e−2τ1 sκ(v)dv+1ds +τ1 t ψPE[η(s,U)]e−τ1 sκ(v)dv+ψ Q(E[η(s,U)])ds +t 0e−t sκ(v)dvE[σ(s,U)]dWP s−t 0E[σ(s,U)]dW Q s +t 0e−t sκ(v)dvE[η(s,U)]d JP s−t 0E[η(s,U)]d J Q s. (4.13) Proof (i)Fisalwayssmaller orequal than FAanalogousto Corollary3.1(i)sinceJensen’s inequality also holds under the physical measure. (ii)Analogous to Corollary 3.1 (ii). (iii)A straightforward evaluation of EP[F(τ1,τ 1,τ 2)|Ft]leads to EP[F(τ1,τ 1,τ 2)|Ft]=F(t,τ 1,τ 2)DRP(t,τ 1,τ 2). (4.14) So that Equation (4.12) directly follows. Hence, the spread between the swap prices Fand Faunder the physical measure P coincides with the pricing spread from Corollary 3.1 (ii) under the instantaneous risk measure Q, as the numéraire in Equation (3.28) is not affected by a change of measure. 4.3 The swap price Funder its risk-neutral measure Q In order to derive the swap’s martingale measure Q, we introduce the swap’s market price of risk for the swap price resulting from geometric averaging in the next definition: Definition 4.1 We define the swap’s market price of risk by P Q:= (P Q 1, P Q 2),where P Q 1(t,τ 1,τ 2):= E[μ(t,U)]−κ(t)ln F(t,τ 1,τ 2)+1 2E[σ(t,U)]2 E[σ(t,U)],(4.15) 123
312 Mathematics and Financial Economics (2025) 19:293–327 P Q 2(t,τ 1,τ 2):= 1−R zP(dz)E[η(t,U)] ReE[η(t,U)]z−1P(dz).(4.16) Note that the market price of risk does not enter the jump size distribution since we restrict P Q 2to depend on trading time and delivery period. Hence, the market price of jump risk affects the jump intensity only. We follow the methodology of Benth et al. [7] to change the measure from the physical measure Pto the swap’s risk-neutral measure Q. Therefore, let π=(π1,π 2)be a predictable process satisfying Eτ1 0π(s,τ 1,τ 2)2ds<∞.(4.17) We define a new process ZP Qbeing the unique strong solution of dZP Q(t,τ 1,τ 2)=ZP Q(t−,τ 1,τ 2)dH(t,τ 1,τ 2), (4.18) such that ZP Q(0,τ 1,τ 2)=1, where dH(t,τ 1,τ 2)=π1(t,τ 1,τ 2)dWP t+π2(t,τ 1,τ 2)d JP t.(4.19) If πjsatisfies Equation (4.17), then His a well-defined square integrable martingale. Note that the process ZP Qis known as the Doléans-Dade exponential of Hthat is explicitly given by ZP Q(t,τ 1,τ 2)=eH(t,τ1,τ2)−1 2t 0π1(s,τ1,τ2)2ds 0<s≤t (1+H(s,τ 1,τ 2))e−H(s,τ1,τ2). (4.20) If ZP Qis a strictly positive martingale, then we can define the equivalent probability measure Qby d Q dP=ZP Q(τ1,τ 1,τ 2), (4.21) where ZP Qfunctions as the Radon-Nikodym derivative. If we further assume that EP[ZP Q(τ1,τ 1,τ 2)]=1, then Girsanov’s theorem (see Øksendal and Sulem [39], cf. Theorem 1.35) states for π:= −P Qthat W Q t=WP t+t 0 P Q 1(s,τ 1,τ 2)ds ,(4.22) is a Brownian motion with respect to Qand N Q(dt,dz)= NP(dt,dz)+P Q 2(t,τ 1,τ 2)P(dz)dt ,(4.23) is a Q-compensated Poisson random measure of N(·,·). Under the above assumptions specified later a straightforward valuation leads to the following result: Proposition 4.1 The swap price process F defined in Equation (2.14)is a martingale under Qgiven by dF(t,τ 1,τ 2) F(t−,τ 1,τ 2)=E[σ(t,U)]dW Q t+ReE[η(t,U)]z−1 N Q(dt,dz). (4.24) 123
Mathematics and Financial Economics (2025) 19:293–327 313 We would like to investigate the consequences of our previous assumptions. Remark 4.2 (i)The Doléans-Dade exponential in Equation (4.20) is positive if π2(s−)J>−1, i.e., if P Q 2J<1. Hence, similar to Benth et al. [7], we need to assume that the market price of jump risk is bounded and deterministic over the entire time period such that P Q 2(t,τ 1,τ 2)z<1forP-a.e. z∈Rand for each t∈[0,τ 1]. (ii)If ln f,andsolnF, is driven by a compensated Poisson process only, then the swap’s market price of risk is attained by P Q:= (0, P Q 2),where P Q 2(t,τ 1,τ 2):= 1−E[η(t,U)]RzP(dz) ReE[η(t,U)]z−1P(dz)+E[μ(t,U)]−κ(t)ln F(t,τ 1,τ 2) ReE[η(t,U)]z−1P(dz). (4.25) In this setting, we need to require that κ(t)≡0. Note that a positive local martingale is a supermartingale. Hence, in order to prove that the Radon-Nikodym density ZP Qis a true martingale, it is sufficient to verify that EP[ZP Q(τ1,τ 1,τ 2)]=1 is satisfied, which is proven in the next proposition. Proposition 4.2 Under Assumption 6in Appendix A, the process ZP Qdefined by Equation (4.18)is a strictly positive true martingale. Proof In Appendix B, we prove this proposition even in a stochastic volatility framework. 4.4 The approximated swap price Faunder the instantaneous risk-neutral measure We introduce the instantaneous market price of risk for the approximated swap price in the next definition. Definition 4.2 We define the instantaneous market price of risk for the approximated swap by PQ := (PQ 1, PQ 2),where PQ 1(t,τ 1,τ 2):= E[μ(t,U)]−κ(t)ln F(t,τ 1,τ 2)+1 2E[σ2(t,U)] E[σ(t,U)],(4.26) PQ 2(t,τ 1,τ 2):= 1−R zP(dz)E[η(t,U)] RE[eη(t,U)z]−1P(dz).(4.27) Note that we assume that the market price of jump risk affects the jump intensity only. The market price of risk does not enter the jump size distribution since we restrict PQ 2to depend on trading and delivery period. Similar to the last subsection, we can define the equivalent (instantaneous) probability measure Qby dQ dP=ZPQ(τ1,τ 1,τ 2), (4.28) where ZPQ functions as the Radon-Nikodym derivative characterized by π:= −PQ.Ifwe further assume that EP[ZPQ(τ1,τ 1,τ 2)]=1, then Girsanov’s theorem (see Øksendal and Sulem [39], cf. Theorem 1.35) states that WQ t=WP t+t 0 PQ 1(s,τ 1,τ 2)ds ,(4.29) 123
314 Mathematics and Financial Economics (2025) 19:293–327 is a Brownian motion with respect to Qand NQ(dt,dz)= NP(dt,dz)+PQ 2(t,τ 1,τ 2)P(dz)dt ,(4.30) is a Q-compensated Poisson random measure of N(·,·). Under the above assumptions a straightforward valuation leads to the following result: Proposition 4.3 The approximated swap price process Fadefined in Equation (2.13)is a martingale under Qgiven by dFa(t,τ 1,τ 2) Fa(t−,τ 1,τ 2)=E[σ(t,U)]dWQ t+RE[eη(t,U)z]−1 NQ(dt,dz). (4.31) We refer to Sect.4.3 for the consequences of the assumptions made above. 4.5 The decomposition of the market price of risk From the previous subsections, we know the corresponding market prices of risk for the swap price resulting from geometric averaging (see Definition 4.1) and from approximated averaging (see Definition 4.2). In this subsection, we identify a clear distinction between both market prices of risk leading to a specific decomposition that is strongly connected to the MPDP. We now introduce the decomposition of the swap’s market price of risk, from Definition 4.1, which finally connects the instantaneous market price of risk, specified in Definition 4.2, and the MPDP, defined in Definition 3.1. The decomposition result is stated in the next proposition. Proposition 4.4 The swap’s market price of risk, P Q, resulting from geometric averaging (see Definition 4.1), decomposes into P Q j(t,τ 1,τ 2)=PQ j(t,τ 1,τ 2)+¯ Q Q j(t,τ 1,τ 2), for j =1,2,(4.32) where PQ jis specified in Definition 4.2 and ¯ Q Q jdefines the spread of diffusion and jump risk. More precisely, ¯ Q Q 1(t,τ 1,τ 2)=−1 2 V[σ(t,U)] E[σ(t,U)],(4.33) ¯ Q Q 2(t,τ 1,τ 2)=−E[η(t,U)]R zG(dz)RE[eη(t,U)z]−eE[η(t,U)]zG(dz) RE[eη(t,U)z]−1G(dz)ReE[η(t,U)]z−1G(dz), (4.34) where ¯ Q Q 2is independent of the jump intensity. Proof The result is attained by subtracting the swap’s market price of risk P Qdefined in Definition 4.1 from the instantaneous market price of risk PQ defined in Definition 4.2. Hence, we found a representation of the swap’s market price of risk of the swap price F, characterized by the instantaneous market price of risk of the approximated swap Faand the spread ¯ Q Q=¯ Q Q 1,¯ Q Q 2. We further investigate the spread in the next lemma. Lemma 4.4 (i)The spread of diffusion risk, ¯ Q Q 1(t,τ 1,τ 2), is negative for all trading times t∈[0,τ 1]. 123
Mathematics and Financial Economics (2025) 19:293–327 315 (ii)If the average jump size is positive, i.e., if RzG(dz)>0, then the spread of jump risk, ¯ Q Q 2, is negative. (iii)If the average jump size is zero, i.e., if RzG(dz)=0, then the spread of jump risk, ¯ Q Q 2,iszero. (iv) If the volatility is independent of the delivery, i.e., if σ(t,u)⊥⊥ u, then the spread of diffusion risk is zero, i.e., ¯ Q Q 1(t,τ 1,τ 2)≡0. (v) If the jump coefficient is independent of the delivery, i.e., if η(t,u)⊥⊥ u, then the spread of jump risk is zero, i.e., ¯ Q Q 2(t,τ 1,τ 2)≡0. Proof The results in (i)and (ii)follow directly from Jensen’s inequality. The results in (iii) and (iv) follow from the fact that the numerator becomes zero whenever the delivery period disappears. As a result, whenever the spread ¯ Q Q jis negative for j=1,2, then the approximated swap induces more risk than the swap price based on geometric averaging. In particular, the considered spread has the same properties as the MPDP (see Kemper et al. [30]). Indeed, a comparison with our previous considerations in Sect.2gives the following insights: Remark 4.3 (i)The spread of diffusion risk, ¯ Q Q 1, coincides with the MPDP of diffusion risk, Q Q 1, in Equation (3.1) from Sect.2. (ii)The spread of jump risk, ¯ Q Q 2, does not coincide with the MPDP of jump risk, Q Q 2, from Equation (3.2) but with Q Q 2(1−PQ 2). This connection occurs naturally by the change of measure. (iii)The condition in Lemma 4.4(iii)holds true, for example, when jump sizes follow are standard normal distribution. Hence, starting from the physical measure, we can find the swaps true martingale measure based on the swap’s market price of risk defined in Definition 4.1. If we would like to adjust already existing models using the instantaneous market price of risk, we can easily adjust the model through the spread defined in Proposition 4.4 that is strongly connected to the MPDP defined in Definition 3.1. 4.6 An example based on short-term long-term evolution In this section, we give an example in the spirit of the popular short-term long-term models based on Gibson and Schwartz [21] and Schwartz and Smith [45]. On the spot level, these modelstypicallydividethepriceevolutionintoanon-stationaryGaussiancomponent,ginving the long term mean of spot prices. It is influenced e.g. by political or regulatory decisions. Furthermore, a stationary mean reverting component describes short term price fluctuations due to imbalances in supply and demand. This mean reverting short term component leads to a Samuelson effect in the futures price dynamics, see e.g. Benth and Schmeck [8]. Then, the futures prices evolve according to df(t,τ) f(t−,τ) =σdWP t+Reη(t,τ )z−1 NP(dt,dz) +μ−κln f(t,τ)+1 2σ2+ψP(η(t,τ)) dt,(4.35) 123
316 Mathematics and Financial Economics (2025) 19:293–327 where η(t,τ) =e−(τ−t)with >0 captures the term structure effect in the spirit of Samuelson (cf. the volatility in Example 3.2), and NPis a compund Poisson process. As we want to give an example under the physical measure, we have added a drift term similar to Equation (4.3) and assume constant drift and mean-reversion parameters μand κ>0 for simplicity. Assuming a gradual inflow of renewables, jumps are more likely downward pointing as opposed to upward pointing jumps in power systems with more nuclear, hydro, gas, and temperature dependent demand (cf. Paraschiv et al. [41] and Hinderks and Wagner [26]). To capture the renewable effect within the jump size distribution, we assume for simplicity a negative jump size distribution captured by the Dirac measure assigned to a jump size of -1. Under Assumptions 4and 5in Appendix A, the swap price dynamics under the physical measure Pbased on geometric averaging evolve according to Lemma 4.2 as dF(t,τ 1,τ 2) F(t−,τ 1,τ 2)=σdWP t+ReE[η(t,U)]z−1 NP(dt,dz) +μ−κln F(t,τ 1,τ 2)+1 2σ+ψP(E[η(t,U)])dt, (4.36) where the swap’s jump coefficient for a constant settlement type function w(u,τ 1,τ 2)= 1 τ2−τ1is given by E[η(t,U)]=1−e−(τ2−τ1) (τ2−τ1) η(t,τ 1). (4.37) Note that the Assumptions 4and 5are satisfied, since μ,κ,andσare constant, η(t,u)is deterministic and finite for t∈[0,τ 1]and u∈[τ1,τ 2]for a finite trading horizon τ1<∞, and the second moment of the negative exponential distribution as well as the moment generating function exist. To turn to the swap’s risk neutral measure Q, the swap’s market price of volatility and jump risk are given according to Definition 4.1 by P Q 1(t,τ 1,τ 2):= μ−κln F(t,τ 1,τ 2)+1 2σ2 σ,(4.38) P Q 2(t,τ 1,τ 2):= 1+E[η(t,U)] λ− JReE[η(t,U)]z−1G(dz).(4.39) Thinking in the spirit of the short-term long-term framework, the Gaussian part of the dynamics stands for long term evolutions. Thus, here 1captures effects connected to the incorporation of the delivery period that have a long term character. On the other hand, the Jump component stands for short-term evolutions due to imbalances in supply and demand. 2captures delivery dependend effects connected to expectations on e.g. negative spikes in the underlying. Under Assumption 6, the swap price process Fis a martingale under the risk neutral measure Qevolving as dF(t,τ 1,τ 2) F(t−,τ 1,τ 2)=σdW Q t+ReE[η(t,U)]z−1 N Q(dt,dz), (4.40) where W Q tand N Q(dt,dz)are as in Equations (4.22)and(4.23). Note that the Assumption 6 (i) is satisfied whenever 0≥2e−E[η(t,u)]+E[η(t,u)]−2,(4.41) 123
Mathematics and Financial Economics (2025) 19:293–327 323 +s 0 σνν(t)dB Qn(t)2 () ≤4v2 0+E Qnsup s∈[0,τ1]s 0 κνθνdt2 +sup s∈[0,τ1]s 0 (κν+δν1[0,ˆτn](t))ν(t)dt2 +sup s∈[0,τ1]s 0 σνν(t)dB Qn t2 () ≤4v2 0+4E Qnτ1 0 κνθνdt2 +4E Qnτ1 0 (κν+δν1[0,ˆτn](t))ν(t)dt2 +4E Qnτ1 0 σνν(t)dB Qn t2 () ≤4v2 0+4τ1τ1 0 κ2 νθ2 νdt +4τ1(κν+|δν|)2τ1 0 E Qnsup s∈[0,t] ν(s)2dt +4σ2 νE Qnτ1 0 ν(t)dt, where the first equality represents the integral version of ν. Inequality () results from the Cauchy-Schwartz inequality to the sum and an application of the triangle inequality. We apply Doob’s inequality to all expectations in Inequality (). In Inequality (), we apply the Cauchy-Schwartz inequality to the first and second integral and apply Ito’s isometry to the last summand. We finish with the stochastic Fubini to the second integral while making the integrand even bigger. Note that for the last summand, we have E Qnτ1 0ν(t)dt≤˜cν⊥⊥ n since we can find explicit expressions in Cont and Tankov [16] (cf. Chapter 15). Setting cν:= 4v2 0+16τ2 1κ2 νθ2 ν+16σ2 ν˜cν, then, by Gronwall, we receive E Qnsups∈[0,τ1]|ν(s)|2≤ cνe16(κν+|δν|)2τ2 1=: c2⊥⊥ n. Next, we show that |ln F|2is uniformly integrable: E Qnsup s∈[0,τ1]|ln F(s,τ 1,τ 2)|2 =E Qnsup s∈[0,τ1]ln F(0,τ 1,τ 2)+s 01−1[0,ˆτn](t)(E[μ(t,U)]−κ(t)ln F(t,τ 1,τ 2))dt −s 0 1 2E[σ(t,U)]2ν(t)1[0,ˆτn](t)dt +s 0E[σ(t,U)]ν(t)dW Qn t+s 0E[η(t,U)]d J Qn t −s 0E[η(t,U)]1−E[η(t,U)]RzP(dz) ReE[η(t,U)]z−1P(dz)1[0,ˆτn](t)R zP(dz)dt2 () ≤7ln F(0,τ 1,τ 2)2+E Qnsup s∈[0,τ1]s 01−1[0,ˆτn](t)E[μ(t,U)]dt2 123
324 Mathematics and Financial Economics (2025) 19:293–327 +E Qnsup s∈[0,τ1]s 0 1 2E[σ(t,U)]2ν(t)1[0,ˆτn](t)dt2 +E Qnsup s∈[0,τ1]s 01−1[0,ˆτn](t)κ(t)ln F(t,τ 1,τ 2)dt2 +E Qnsup s∈[0,τ1]s 0E[σ(t,U)]ν(t)dW Qn t2+E Qnsup s∈[0,τ1]s 0E[η(t,U)]d J Qn t2 +E Qn⎡ ⎣sup s∈[0,τ1]s 0E[η(t,U)]1−E[η(t,U)]RzP(dz) ReE[η(t,U)]z−1P(dz)1[0,ˆτn](t)R zP(dz)dt2⎤ ⎦ () ≤7ln F(0,τ 1,τ 2)2+4E Qnτ1 01−1[0,ˆτn](t)E[μ(t,U)]dt2 +4E Qnτ1 0 1 2E[σ(t,U)]2ν(t)1[0,ˆτn](t)dt2 +4E Qnτ1 01−1[0,ˆτn](t)κ(t)ln F(t,τ 1,τ 2)dt2 +4E Qnτ1 0E[σ(t,U)]ν(t)dW Qn t2+4E Qnτ1 0E[η(t,U)]d J Qn t2 +4E Qn⎡ ⎣τ1 0E[η(t,U)]1−E[η(t,U)]RzP(dz) ReE[η(t,U)]z−1P(dz)1[0,ˆτn](t)R zP(dz)dt2⎤ ⎦ () ≤7ln F(0,τ 1,τ 2)2+4τ1τ1 0E[μ(t,U)]2dt +4τ1 0E[σ(t,U)]4dt E Qnτ1 0ν(t)2dt+4τ1 0κ(t)2dt E Qnτ1 0ln F(t,τ 1,τ 2)2dt +4E Qnτ1 0E[σ(t,U)]ν(t)dW Qn t2+4E Qnτ1 0E[η(t,U)]d J Qn t2 +4τ1 0E[η(t,U)]2dtE Qn ⎡ ⎢ ⎣τ1 0⎛ ⎜ ⎝1−E[η(t,U)]2RzP(dz)2 ReE[η(t,U)]z−1P(dz)2⎞ ⎟ ⎠R zP(dz)2 dt⎤ ⎥ ⎦ ≤7ln F(0,τ 1,τ 2)2+4τ1τ1 0E[μ(t,U)]2dt +4c2τ1τ1 0E[σ(t,U)]4dt +4τ1 0κ(t)2dt E Qn τ1 0sup s∈[0,t] ln F(s,τ 1,τ 2)2dt +4*τ1 0E[σ(t,U)]4dt√τ1c2+4τ1 0E[η(t,U)]2R z2 Qn(dz)dt +4τ1 0E[η(t,U)]2dt τ1 0⎛ ⎜ ⎝R zP(dz)2 +E[η(t,U)]2RzP(dz)4 ReE[η(t,U)]z−1P(dz)2⎞ ⎟ ⎠dt, =: cY+28 τ1 0κ(t)2dt E Qnτ1 0sup s∈[0,t] ln F(s,τ 1,τ 2)2dt. 123
Mathematics and Financial Economics (2025) 19:293–327 325 The first equality represents the integral version of ln F. Inequality () results from the Cauchy-Schwartz inequality to the sum and an application of the triangle inequality. We apply Doob’s inequality to all expectations in Inequality (). In Inequality (), we apply the Cauchy-Schwartz inequality to the first three integrals. We finish with Itô-Lévy Isometry (seeØksendalandSulem[39],cf.Theorem1.17)tothelastsummandandanapplicationofthe stochastic Fubini theorem to the fourth summand (including ln F) while making the integrand even bigger. By the previous considerations, we know that E Qnτ1 0E[σ(t,U)]2ν(t)dt≤ +τ1 0E[σ(t,U)]4dtE Qn+τ1 0ν(t)2dt≤+τ1 0E[σ(t,U)]4dt√τ1c2is bounded independently of nand that τ1 0E[σ(t,U)]4dt E Qnτ1 0ν(t)2dt≤c2τ1τ1 0E[σ(t,U)]4dt is independent of n. By the choice of cY, an application of Gronwall’s inequality yields EPnsups∈[0,τ1]|ln F(s,τ 1,τ 2)|2≤cYe28τ1 0κ(t)2dt =: c3⊥⊥ n,such that we have shown, that ZP Qis indeed a true martingale. Funding Open Access funding enabled and organized by Projekt DEAL. Declarations Conflict of interest No potential competing interest was reported by the authors. Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/. References 1. Benth, F.E., Benth, J.S., and Koekebakker, S.: Stochastic modelling of electricity and related markets. Vol. 11. World Scientific Publishing Company (2008) 2. Benth,F.E., Cartea, Á., Kiesel, R.: Pricing forwardcontractsinpowermarketsbythecertaintyequivalence principle: explaining the sign of the market risk premium. J. Bank. Financ. 32(10), 2006–2021 (2008) 3. Benth, F.E., Kallsen, J., Meyer-Brandis, T.: A non-gaussian ornstein-uhlenbeck process for electricity spot price modeling and derivatives pricing. Appl. Math. Financ. 14(2), 153–169 (2007) 4. Benth, F.E., Klüppelberg, C., Müller, G., Vos, L.: Futures pricing in electricity markets based on stable CARMA spot models. Energy Econ. 44, 392–406 (2014) 5. Benth, F.E., Koekebakker, S.: Stochastic modeling of financial electricity contracts. Energy Econ. 30(3), 1116–1157 (2008) 6. Benth, F.E., Paraschiv, F.: A space-time random field model for electricity forward prices. J. Bank. Financ. 95, 203–216 (2018) 7. Benth, F.E., Piccirilli, M., Vargiolu, T.: Mean-reverting additive energy forward curves in a heath-Jarrowmorton framework. Math. Financ. Econ. 13(4), 543–577 (2019) 8. Benth, F.E., Schmeck, M.D.: Pricing and hedging options in energy markets using Black-76. J. Energy Markets 7(2), 35–69 (2014) 9. Bjerksund, P., Rasmussen, H., and Stensland, G.: Valuation and risk management in the norwegian electricity market. In: Bjørndal, E., Bjørndal, M., Pardalos, P. M., and Rönnqvist, M. (Editors) Energy, Natural Resources and Environmental Economics, pp. 167-185, (2010) 10. Borovkova, S., Permana, F.J.: Modelling Electricity prices by the potential jump-diffusion. In: Shiryaev, A.N.,Grossinho,M.R., Oliveira, P.E.,Esquível,M.L.(Editors), Stochastic Finance. Springer pp.239-263, (2006) 123
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