The Market Price of Risk for Delivery Periods: Pricing Swaps and Options in Electricity Markets
TThe Market Price of Risk for Delivery Periods:
Pricing Swaps and Options in Electricity Markets ∗ A NNIKA K EMPER [email protected] for Mathematical EconomicsBielefeld UniversityPO Box 10013133501 Bielefeld, Germany M AREN
D. S
CHMECK [email protected] for Mathematical EconomicsBielefeld UniversityPO Box 10013133501 Bielefeld, Germany A NNA K H . B ALCI [email protected] of MathematicsBielefeld UniversityPO Box 10013133501 Bielefeld, Germany
Abstract
In electricity markets, futures contracts typically function as a swap since they deliver theunderlying over a period of time. In this paper, we introduce a market price for the delivery periodsof electricity swaps, thereby opening an arbitrage-free pricing framework for derivatives based onthese contracts. Furthermore, we use a weighted geometric averaging of an artificial geometric futuresprice over the corresponding delivery period. Without any need for approximations, this averagingresults in geometric swap price dynamics. Our framework allows for including typical features asthe Samuelson effect, seasonalities, and stochastic volatility. In particular, we investigate the pricingprocedures for electricity swaps and options in line with Arismendi et al. (2016), Schneider and Tavin(2018), and Fanelli and Schmeck (2019). A numerical study highlights the differences between thesemodels depending on the delivery period.JEL
CLASSIFICATION : G130 · Q400K
EYWORDS : Electricity Swaps · Delivery Period · Market Price of Delivery Risk · Seasonality · Samuelson Effect · Stochastic Volatility · Option Pricing · Heston Model ∗ Financial support by the German Research Foundation (DFG) through the Collaborative Research Centre ‘Taminguncertainty and profiting from randomness and low regularity in analysis, stochastics and their applications’ is gratefullyacknowledged. a r X i v : . [ q -f i n . P R ] A p r . Introduction Futures contracts are the most important derivatives in electricity- and commodity markets. Dueto the non-storability of electricity, the underlying is typically delivered over a period, and the contractis therefore referred to as a swap. In electricity markets, the delivery period has an influence on pricedynamics, and Fanelli and Schmeck (2019) have provided empirical evidence indicating that impliedvolatilities of electricity options are seasonal with respect to the delivery period. In other words, thedistributional features – or the pricing measure – depend on the delivery period of the contract. Inthis paper, we introduce an arbitrage-free pricing framework that takes dependencies on the deliveryinto account. The core of our approach is the so-called market price of delivery risk , which reflectsexpectations about variations in volatility weighted over the delivery period and arises through a geometric average approach similar to that used by Kemna and Vorst (1990).In fact, the delivery period is one of the features that distinguishes electricity markets from othercommodity markets such as oil, gas, or corn. An easy way to acknowledge its existence is to usefutures price dynamics with a delivery time that represents the midpoint of the delivery period. Thisapproach has been followed, for example, by Schmeck (2016), and is advantageous as it captures thetypically observed behavior that the futures prices do not converge against the electricity spot price iftime approaches the beginning of the delivery. A possible way to model the delivery period explicitly isto average the spot price or an artificial futures price over the entire delivery time. Typically, arithmeticaveraging is used, which is the standard approach in electricity price modeling and works especiallywell for arithmetic price dynamics (see, e.g., Benth et al. (2008), and Benth et al. (2019)). However,if the underlying electricity futures are of the geometric type, the resulting dynamics are neithergeometric nor Markovian. In that case, the dynamics are approximated in line with Bjerksund et al.(2010) (see also Benth et al. (2008)).A typical feature of electricity markets is the seasonal behavior of prices. The effect is enforced throughthe rise of renewable energy, which is highly dependent on weather conditions. At present, there isa growing worldwide trend to acknowledge the need for sustainable energy production, which alsoraises the expectations of a further increasing impact of seasonal effect. Among others, Arismendiet al. (2016), Borovkova and Schmeck (2017), and Fanelli and Schmeck (2019) have addressed andmodeled seasonality in either commodity- or energy markets. Typically, a deterministic seasonal pricelevel is added to the price dynamics, but the dynamics can also exhibit seasonal behavior. Fanelli andSchmeck (2019) distinguish between seasonalities in the trading day and seasonalities in the deliveryperiod . Arismendi et al. (2016) suggest the use of a seasonal stochastic volatility model for commodityfutures. As in the Heston model, stochastic volatility follows a square-root process, but with a seasonalmean-reversion level. Indeed, a volatility smile can also be observed in electricity option markets (seeFigure 1) such that a stochastic volatility model seems appropriate.Finally, a well-known feature in electricity- and commodity markets is the Samuelson effect (see2 igure 1
The implied non-accumulated volatility surface with respect to strikes from 18 to 38 overthe last trading month in September 2016 for a European option on the Phelix DE/AU BaseloadMonth futures at the European Energy Exchange (EEX) delivering in October 2016.
Samuelson (1965)), which implies that futures close to delivery are much more volatile than are thosewhose expiration date lies far off. This effect can be observed in the implied volatility of electricityoptions, especially far out and in the money (see also Figure 1 and Kiesel et al. (2009) ). The effectis typically included in any electricity futures price dynamics. Schneider and Tavin (2018) includesuch a term-structure effect within the framework of stochastic volatility modeling. Schmeck (2016)investigates analytically the impact of the Samuelson effect on option pricing.In this paper, we suggest modeling the delivery period explicitly through a geometric averaging approach for electricity futures prices of the geometric type, in line with Kemna and Vorst (1990)and Bjerksund et al. (2010). This approach leads directly to Markovian and geometric swap pricedynamics. Indeed, the geometric averaging of futures prices coincides with the arithmetic procedureapplied to logarithmic futures prices. In line with the literature, we base the averaging procedure onan artificial futures contract that is a martingale under a pricing measure Q . In our framework, theresulting swap price dynamics are not a martingale under Q due to a drift term in the dynamics thatis characterized by the variance of the weighted delivery and is used to define the market price ofdelivery risk and an equivalent martingale measure (cid:101) Q for the swap price. (cid:101) Q can thus be used as apricing measure for derivatives on the swap. We characterize the market price of delivery risk for theSamuelson effect, and for seasonalities in the trading day and in the delivery period following Schneiderand Tavin (2018), Arismendi et al. (2016), and Fanelli and Schmeck (2019), respectively.3or option pricing, we consider a general stochastic volatility model that is inter alia feasible formean-reverting square-root volatility processes in line with the models used by Arismendi et al. (2016)and Schneider and Tavin (2018). The volatility structure is rich enough to include the categories ofseasonalities and the Samuelson effect. Both models share the feature that their commodity futuresprices are based on an affine stochastic volatility structure. Indeed, the averaging procedure of thefutures price model as well as the change of measure preserve the affine model structure of the artificialfutures price dynamics.In this paper, we focus on the pricing of a single swap contract. As mentioned above, the pricingmeasure depends on this particular contract, and it thus cannot be used for pricing derivatives onanother swap contract with a different delivery period. Nevertheless, several swap contracts are usuallyalso tradable, such that arbitrage possibilities must be excluded. Furthermore, overlapping deliveryperiods are tradable as a quarter and the corresponding three months. We address how to tackle theseissues in Chapter 4.The paper is organized as follows: Chapter 2 presents the geometric averaging approach andintroduces the market price of delivery based on a general stochastic volatility model. In order toillustrate the averaging procedure, we discuss the method based on the models created by Arismendi etal. (2016), Schneider and Tavin (2018), and Fanelli and Schmeck (2019) in Chapter 3. In Chapter 4,we address how to exclude arbitrage opportunities that might appear when there are several, possiblyoverlapping swap contracts traded on the market. Option pricing is discussed in Chapter 5. In addition,all adjusted commodity market models are investigated numerically. Finally, Chapter 6 presents ourconclusions.
2. Averaging of Futures Contracts
We consider a swap contract delivering a flow of 1 Mwh electricity during the delivery period ( τ , τ ] . At a trading day t ≤ τ , the swap price is denoted by F ( t, τ , τ ) and settled such that thecontract is entered at no cost. It can be interpreted as an average price of instantaneous delivery.Motivated by this interpretation, we consider an artificial futures contract with price F ( t, τ ) thatstands for instantaneous delivery at time τ ∈ ( τ , τ ] . Note that such a contract does not exist on themarket, but turns out to be useful for modeling purposes when considering delivery periods (see forexample Benth et al. (2019)).Consider a filtered probability space (Ω , F , ( F t ) t ∈ [0 ,τ ] , Q ) , where the filtration satisfies the usualconditions. At time t ≤ τ , the price of the futures contract follows a geometric diffusion process4volving as dF ( t, τ ) = σ ( t, τ ) F ( t, τ ) dW F ( t ) , (2.1) dσ ( t, τ ) = a ( t, τ, σ ) dt + c ( t, τ, σ ) dW σ ( t ) , (2.2)with initial conditions F (0 , τ ) = F > and σ (0 , τ ) = σ > , and where W F and W σ are correlatedstandard Brownian motions under Q . Thus, W σ = ρW F + (cid:112) − ρ W for a Brownian motion W independent of W F and ρ ∈ ( − , . We assume that both, the futures price volatility σ ( t, τ ) andthe futures price F ( t, τ ) , are F t -adapted for t ∈ [0 , τ ] , and that they satisfy suitable integrability andmeasurability conditions to ensure that (2.1) is a Q -martingale, and the solution given by F ( t, τ ) = F (0 , τ ) e (cid:82) t σ ( s,τ ) dW F ( s ) − (cid:82) t σ ( s,τ ) ds (2.3)exists (see Appendix A for details). As σ ( t, τ ) depends on both time t and delivery time τ , we allowfor volatility structures as the Samuelson effect, seasonalities in the trading day, or seasonalitiesin the delivery time. In this framework, we would like to mention the models of Arismendi et al.(2016), Schneider and Tavin (2018), as well as of Fanelli and Schmeck (2019), which are addressed inthe next chapter.Following the Heath-Jarrow-Morton approach to price futures and swaps in electricity markets,the swap price is usually defined as the arithmetric average of futures prices (see, e.g., Benth et al.(2008), Bjerksund et al. (2010), and Benth et al. (2019)): F a ( t, τ , τ ) = (cid:90) τ τ w ( u, τ , τ ) F ( t, u ) du , (2.4)for a general weight function w ( u, τ , τ ) := ˆ w ( u ) (cid:82) τ τ ˆ w ( v ) dv , for u ∈ ( τ , τ ] . (2.5)The most popular example is given by ˆ w ( u ) = 1 , such that w ( u, τ , τ ) = τ − τ . This corresponds to aone-time settlement. A continuous settlement over the time interval ( τ , τ ] is covered by ˆ w ( u ) = e − ru ,where r ≥ is the constant interest rate (see, e.g., Benth et al. (2008)). The arithmetric average of thefutures price as in (2.4) leads to tractable dynamics for the swap as long as one assumes an arithmetricstructure of the futures prices as well. This is based on the fact that arithmetic averaging is tailor-madefor absolute growth rate models. Nevertheless, if one defines the futures price as a geometric processas in (2.1), one can show that the dynamics of the swap defined through (2.4) is given by dF a ( t, τ , τ ) = σ ( t, τ ) F a ( t, τ , τ ) dW F ( t ) − (cid:90) τ τ ∂σ∂u ( t, u ) w ( τ, τ , τ ) w ( τ, τ , u ) F a ( t, τ , u ) du dW F ( t ) , τ ∈ ( τ , τ ] (see Benth et al. (2008); Chapter 6.3.1). Thus, the dynamics of the swap price isneither a geometric process nor Markovian, which makes it unhandy for further analysis. Bjerksund etal. (2010) suggest an approximation given by dF a ( t, τ , τ ) = F a ( t, τ , τ )Σ( t, τ , τ ) dW F ( t ) , (2.6)where F a (0 , τ , τ ) = F and an weighted average volatility Σ( t, τ , τ ) := (cid:90) τ τ w ( u, τ , τ ) σ ( t, u ) du . (2.7)Instead of averaging absolut price trends as in (2.4), we here suggest to focus on the averagingprocedure of relative price trends, i.e. growth rates or logarithmic prices. This leads to a geometricaveraging procedure in continuous time. In fact, the connection between exponential models andgeometric averaging seems natural: the geometric averaging of a geometric price process correspondsto an arithmetic average of logarithmic prices. Note that this approach is in line with Kemna and Vorst(1990) for pricing average asset value options on equities and also with Bjerksund et al. (2010). Thedifference of Bjerksund et al. (2010) and our approach is, that Bjerksund et al. (2010) approximatethe geometric average to receive a martingale dynamics, while we will make a change of measure.Note that the choice of pricing measures in electricity markets allows for more freedom as in othermarkets, as electricity itself is not storable, and thus no-arbitrage considerations for the spot itself arenot applicable (see Benth and Schmeck (2014)).We define the swap price as F ( t, τ , τ ) := exp (cid:18)(cid:90) τ τ w ( u, τ , τ ) log( F ( t, u )) du (cid:19) . (2.8)Assume that the volatility satisfies further integrability conditions (see Appendix A). It turns out,that the resulting swap price dynamics is a geometric process with stochastic swap price volatility Σ( t, τ , τ ) : Lemma 1.
The dynamics of the swap price defined in (2.8) under Q is given by dF ( t, τ , τ ) F ( t, τ , τ ) = 12 (cid:18) Σ ( t, τ , τ ) − (cid:90) τ τ w ( u, τ , τ ) σ ( t, u ) du (cid:19) dt + Σ( t, τ , τ ) dW F ( t ) . (2.9) Proof.
Plugging (2.3) into (2.8) and using the stochastic Fubini Theorem (see Protter (2005); Theo-rem 65) leads to F ( t, τ , τ ) = F (0 , τ , τ ) e − (cid:82) t (cid:82) τ τ w ( u,τ ,τ ) σ ( s,u ) du ds + (cid:82) t Σ( s,τ ,τ ) dW F ( s ) . (2.10)Then, (2.9) follows using Itô’s formula. 6lthough the futures price F ( t, τ ) is a martingale under the pricing measure Q , the swap price F ( t, τ , τ ) is not a Q -martingale anymore: the swap price under Q has a drift term, given by thedifference between the swap price’s variance and the weighted average of the futures price variance.We thus define a market price of risk at time t ∈ [0 , τ ] associated to the delivery period ( τ , τ ] as b ( t, τ , τ ) := 12 (cid:82) τ τ w ( u, τ , τ ) σ ( t, u ) du − Σ ( t, τ , τ )Σ( t, τ , τ ) , (2.11)where b ( t, τ , τ ) is measurable and F t -adapted as σ ( t, u ) and Σ( t, τ , τ ) are. It can be interpreted asthe trade-off between the weighted average variance of a stream of futures on the one hand and thevariance of the swap on the other hand. Since we have two independent Brownian motions, W F and W , we have a two-dimensional market price of risk b ( t, τ , τ ) = ( b ( t, τ , τ ) , b ) (cid:124) , where we choose b = 0 . The market price b ( · , τ , τ ) will enter also the dynamics of the volatility, which is driven bythe Brownian motion W σ = ρW F + (cid:112) − ρ W . Remark 1.
For a random variable U with density w ( u, τ , τ ) , we can write Σ( t, τ , τ ) = E U [ σ ( t, U )] ,b ( t, τ , τ ) = 12 V U [ σ ( t, U )] E U [ σ ( t, U )] , where E U and V U denote the expectation and variance only with respect to the random variable U .Note that σ ( t, U ) identifies the futures price volatility for a random time of delivery. Hence, the marketprice of delivery risk is the variance per unit of expectation of σ ( t, U ) . This is very similar to thewell-known coefficient of variation √ V U [ σ ( t,U )] E U [ σ ( t,U )] . We define a new pricing measure (cid:101) Q , such that F ( · , τ , τ ) is a martingale. Define the Radon-Nikodym density through Z ( t, τ , τ ) := exp (cid:26) − (cid:90) t b ( s, τ , τ ) dW F ( s ) − (cid:90) t b ( s, τ , τ ) ds (cid:27) . Assume that E Q [ Z ( τ , τ , τ )] = 1 , (2.12)which means Z ( · , τ , τ ) is indeed a martingale for the entire trading time. We will show later thatNovikov’s condition (see, e.g., Karatzas and Shreve (1991)) is fullfilled for suitable models, such that(2.12) holds true. We then define the new measure (cid:101) Q through the Radon Nikodym density d (cid:101) Q d Q := Z ( τ , τ , τ ) , ( τ , τ ] . Girsanov’s theorem states that (cid:102) W F ( t ) = W F ( t ) + (cid:90) t b ( s, τ , τ ) ds , (2.13) (cid:102) W ( t ) = W ( t ) , (2.14)are standard Brownian motions under (cid:101) Q (see, e.g., Shreve (2004)). The Brownian motion of thestochastic volatility is also affected due to the correlation structure: (cid:102) W σ ( t ) = W σ ( t ) + (cid:90) t ρb ( s, τ , τ ) ds . (2.15)A straight forward valuation leads to the following result: Proposition 1.
The swap price F ( t, τ , τ ) defined in (2.8) is a martingale under (cid:101) Q . The swap priceand volatility dynamics are given by dF ( t, τ , τ ) F ( t, τ , τ ) =Σ( t, τ , τ ) d (cid:102) W F ( t ) , (2.16) dσ ( t, τ ) = ( a ( t, τ, σ ) − ρb ( t, τ , τ ) c ( t, τ, σ )) dt + c ( t, τ, σ ) d (cid:102) W σ ( t ) , (2.17) where Σ( t, τ , τ ) is defined in (2.7) . Note that the stochastic volatility process σ ( t, τ ) also depends on the delivery interval, which wedrop for notational convenience. As the swap price F ( t, τ , τ ) is a martingale under the equivalentmeasure (cid:101) Q , we can use it to price options on the swap. Nevertheless, (cid:101) Q depends on the particulardelivery period of the swap and cannot be used to price options on swaps on other delivery periods.We address this issue in Chapter 4.1.We would like to compare the approximated swap price F a ( t, τ , τ ) under Q following Bjerksundet al. (2010) with the swap price F ( t, τ , τ ) under (cid:101) Q as defined in (2.8) assuming that both have astochastic volatility based on (2.2). The swap price dynamics have the same form, the difference is inthe drift term of the stochastic volatility. If the volatility is deterministic as in the setting of Bjerksundet al. (2010), the distribution of F a ( t, τ , τ ) under Q and the distribution of F ( t, τ , τ ) under (cid:101) Q arethe same. For the swap prices both under the same measure we have the following result. Lemma 2.
For the swap prices F a ( t, τ , τ ) and F ( t, τ , τ ) , both under Q , it holds that F ( t, τ , τ ) − F a ( t, τ , τ ) = F a ( t, τ , τ ) (cid:104) e (cid:82) t V U [ σ ( s,U )] ds − (cid:105) ≥ . Proof.
From (2.6), we know that F a ( t, τ , τ ) = F e − (cid:82) t Σ ( s,τ ,τ ) ds + (cid:82) t Σ( s,τ ,τ ) dW F ( s ) . F a ( t, τ , τ ) = F ( t, τ , τ ) e − (cid:82) t V U [ σ ( s,U )] ds (2.18)and the result follows. The expression in squared brackets is strictly positive as it is the case for thevariance.Note that in (2.18) V U [ σ ( s, U )] can be interpreted as discount rate.
3. Electricity Swap Price Models
In this chapter, we transform three commodity market models from the recent literature intoelectricity swap models using the geometric averaging procedure presented in Chapter 2. That is,we examine the influence of seasonality in the mean-reversion level of the (stochastic) volatilityfollowing Arismendi et al. (2016), the impact of the Samuelson effect in line with Schneider and Tavin(2018), as well as the seasonal dependence on the delivery time following Fanelli and Schmeck (2019).Moreover, we investigate the corresponding swap and market prices numerically. In Chapter 5, wethen adress option pricing for these three models.
Arismendi et al. (2016) consider a generalized Heston model, where the mean-reversion rate ofthe stochastic volatility is seasonal. That is, they suggest a futures price dynamics of the form dF ( t, τ ) = (cid:112) ν ( t ) F ( t, τ ) dW F ( t ) , (3.1) dν ( t ) = κ ( θ ( t ) − ν ( t )) dt + σ (cid:112) ν ( t ) dW σ ( t ) , (3.2)where W σ and W F are defined as before under Q . The stochastic volatility ν ( t ) is given by a Cox-Ingersoll-Ross process with a time-dependent level. The Feller condition κθ min > σ needs tobe satisfied with θ min := min t ∈ [0 ,τ ] θ ( t ) in order to receive a strictly positive solution. If the mean-reversion level θ ( t ) is in particular of exponential sinusoidal form, that is θ ( t ) = αe β sin(2 π ( t + γ )) , for α, β > , γ ∈ [0 , , then θ min = αe − β . In the framework of Chapter 2, the futures price volatility isgiven by σ ( t, τ ) = (cid:112) ν ( t ) . The corresponding swap price dynamics under the Q evolve as dF ( t, τ , τ ) = (cid:112) ν ( t ) F ( t, τ , τ ) dW F ( t ) , (3.3) dν ( t ) = κ ( θ ( t ) − ν ( t )) dt + σ (cid:112) ν ( t ) dW σ ( t ) . (3.4)9 Stochastic Volatility V o l a t ili t y Time κ =0.6 κ =3 κ =10Mean-Reversion Level Swap Price Evolution S w ap P r i c e Time
Figure 2
Stochastic volatility for different choices of mean-reversion speed and the correspondingmean-reversion level (left). Swap prices based on the stochastic volatilities (right). For the choice ofparameters, see Table 2 in Chapter 3.4.
Typical trajectories of the volatility and swap prices are illustrated in Figure 2. As the futures pricevolatility does not depend on the delivery time τ , the resulting volatility of the swap is given by thefutures price volatility Σ( t, τ , τ ) = (cid:112) ν ( t ) , (3.5)for all choices of weight functions w ( · , τ , τ ) . Then, the market price of the delivery period is alsozero, that is b ( t, τ , τ ) = 0 , (3.6)for all t ∈ [0 , τ ] and we arrive directly at swap price dynamics of martingale form. Since the modelis not linked to the delivery time, the pricing measures for the futures and swap contract coincide, asthe dynamics do. In Figure 2, we illustate the model for different speed of mean-reversion parametersof the volatility process. The higher the parameter κ , the closer the seasonal mean-reversion level isreached by the stochastic volatility, and the higher the stochastic volatility oscillates. This affects theswap price evolution as well. Schneider and Tavin (2018) include the so-called Samuelson effect within the framework ofa futures price model under stochastic volatility. The Samuelson effect describes the empiricalobservation that the variations of futures increase the closer the expiration date is reached (seealso Samuelson (1965)). Typically this is captured with an exponential alteration in the volatility of theform e − λ ( τ − t ) , for λ > . For t → τ , the term converges to and the full volatility enters the dynamics.10f the time to maturity increases, that is for τ − t → ∞ , the volatility decreases. While Schneider andTavin (2018) base their model on a multi-dimensional setting, we here focus on the one-dimensionalcase following dF ( t, τ ) = e − λ ( τ − t ) (cid:112) ν ( t ) F ( t, τ ) dW F ( t ) , (3.7) dν ( t ) = κ ( θ − ν ( t )) dt + σ (cid:112) ν ( t ) dW σ ( t ) . (3.8)This approach includes a term-structure in the volatiliy process of the form σ ( t, τ ) = e − λ ( τ − t ) (cid:112) ν ( t ) .Applying the geometric averaging method as in (2.8) with weight function ˆ w ( u ) = 1 , the volatility ofthe swap is Σ( t, τ , τ ) = d ( τ − τ ) e − λ ( τ − t ) (cid:112) ν ( t ) , (3.9)and the new swap martingale measure (cid:101) Q is defined via the market price of risk b ( t, τ , τ ) = d ( τ − τ ) e − λ ( τ − t ) (cid:112) ν ( t ) , (3.10)where d ( x ) = 1 − e − λx λx and d ( x ) = 12 (cid:18)
12 (1 + e − λx ) − d ( x ) (cid:19) . (3.11)The volatility and the market price of risk factorize into three parts: a constant d ( τ − τ ) dependingonly on the length of the delivery period, the Samuelson effect counting the time to maturity at τ , andthe stochastic volatility (cid:112) ν ( t ) . The Samuelson effect enters both swap price dynamics and marketprice of risk through the term e − λ ( τ − t ) . Σ and b become small if we are far away from maturity, andincreases exponentially if we approach the maturity of the option. The swap price dynamics under (cid:101) Q are given by dF ( t, τ , τ ) =Σ( t, τ , τ ) F ( t, τ , τ ) d (cid:102) W F ( t ) , (3.12) dν ( t ) = (cid:0) κθ − (cid:2) κ + ρσd ( τ − τ ) e − λ ( τ − t ) (cid:3) ν ( t ) (cid:1) dt + σ (cid:112) ν ( t ) d (cid:102) W σ ( t ) . (3.13)We observe that the drift of the dynamics of ν ( t ) is now altered by the market price of risk, whichagain depends on the delivery period. The speed of mean reversion is now given by κ + ρσd ( τ − τ ) e − λ ( τ − t ) ≥ κ , if ρ ≥ ,< κ , if ρ < . Market Price of Delivery Risk M a r k e t P r i c e Time λ =1.5 λ =3.5 λ =5.5 Swap Price Evolution P r i c e Time λ =1.5 λ =3.5 λ =5.5 Figure 3
Market prices of delivery risk (left) and swap prices under (cid:101) Q (right) both for differentvalues for λ . The mean reversion level is given by κκ + ρσd ( τ − τ ) e − λ ( τ − t ) θ ≤ θ , if ρ ≥ ,> θ , if ρ < . For a positive correlation between swap price- and volatility dynamcis, the speed of mean reversionincreases and the level of mean reversion decreases and vice versa for a negative correlation. If κ > σ max { λ ( τ − τ ) } , Novikov’s condition is satisfied such that the measure change is welldefined and F ( t, τ , τ ) is indeed a true martingale under (cid:101) Q (see Appendix B). Using the notation ofRemark 1, we can write Σ( t, τ , τ ) = E [ e − λ ( U − τ ) ] e − λ ( τ − t ) (cid:112) ν ( t ) ,b ( t, τ , τ ) = 12 V [ e − λ ( U − τ ) ] E [ e − λ ( U − τ ) ] e − λ ( τ − t ) (cid:112) ν ( t ) , for a random variable U ∼ U [ τ , τ ] . The impact of the Samuelson effect on the market price of risk aswell as the swap price dynamics is illustrated in Figure 3. The parameters are chosen as in Table 2 (seeChapter 5.2.4).The exponential behavior of the market price becomes more pronounced the higherthe Samuelson parameter λ . At terminal time, it is equal to d ( ) , which depends by definition on λ (see Equation (3.11) and Table 1). Moreover, we clearly observe the Samuelson effect within the swapprice evolution. The higher the Samuelson parameter, the smaller the variance of the Samuelson effect(see Table 1), and thus the smaller the swap’s variance (see Figure 3). However, the closer we reachthe expiration date, the higher the swap price volatility.12 able 1 Expectation, variance and market price of delivery risk for different Samuelson parameters. d ( ) = E (cid:2) e − λ ( U − τ ) (cid:3) V (cid:2) e − λ ( U − τ ) (cid:3) d ( ) λ = 1 . . . . λ = 3 . . . . λ = 5 . . . . Fanelli and Schmeck (2019) show that the implied volatilities of electricity options depend onthe delivery period in a seasonal fashion. Incorporating this idea into a stochastic volatility framework,we start with the following futures price dynamics under Q : dF ( t, τ ) = s ( τ ) (cid:112) ν ( t ) F ( t, τ ) dW F ( t ) , (3.14) dv ( t ) = κ ( θ − ν ( t )) dt + σ (cid:112) ν ( t ) dW σ ( t ) . (3.15)Here, s ( τ ) models the seasonal dependence on the delivery in τ . Deriving the swap price model as inChapter 2, again with the choice of ˆ w ( u ) = 1 , the swap price volatility is given by Σ( t, τ , τ ) = S ( τ , τ ) (cid:112) ν ( t ) . (3.16)Moreover, the swap’s pricing measure (cid:101) Q is defined via the market price of risk b ( t, τ , τ ) = S ( τ , τ ) (cid:112) ν ( t ) , (3.17)where S ( τ , τ ) = 1 τ − τ (cid:90) τ τ s ( u ) du (3.18)and S ( τ , τ ) = 12 (cid:32) τ − τ (cid:82) τ τ s ( u ) du − S ( τ , τ ) S ( τ , τ ) (cid:33) . (3.19)Here, S ( τ , τ ) describes the average seasonality in the volatility during the delivery period, and S ( τ , τ ) the relative trade-off between the average squared seasonaltity (resulting from the averagevariance of a stream of futures) and the squared average seasonality (e.g. the variance part of the13 Seasonality Function and Expectation
Delivery Time s( τ )S ( τ , τ ) Variance of s(U) and Deterministic Market Price
Delivery Time S ( τ , τ )V[s(U)] Figure 4 s ( τ ) and S ( τ , τ ) for different delivery periods over one year (left). S ( τ , τ ) and thevariance of s ( τ ) with respect to the delivery time (right). average seasonality). The swap price dynamics under (cid:101) Q then follow dF ( t, τ , τ ) = S ( τ , τ ) (cid:112) ν ( t ) F ( t, τ , τ ) d (cid:102) W F ( t ) , (3.20) dν ( t ) = ( κθ − [ κ + σρS ( τ , τ )] ν ( t )) + σ (cid:112) ν ( t ) d (cid:102) W σ ( t ) . (3.21)A possible choice for the seasonality function is s ( τ ) = a + b cos(2 π ( τ + c )) , where a > b > and c ∈ [0 , to ensure that the volatility stays positive. In this case, Novikov’s condition is satisfied if κ > α σ , such that the measure change is well defined and F is indeed a true martingale under (cid:101) Q (see Appendix B). In the setting of Remark 1, we have Σ( t, τ , τ ) = E [ s ( U )] (cid:112) ν ( t ) ,b ( t, τ , τ ) = 12 V [ s ( U )] E [ s ( U )] (cid:112) ν ( t ) , for a uniformly distributed random variable U ∼ U [ τ , τ ] . Having option pricing in view, we wouldlike to mention that we again preserve the affine structure of the model, that is as (log( F ( t, τ )) , ν ( t )) isaffine in the volatility, so is (log( F ( t, τ , τ )) , ν ( t )) after applying the averaging procedure of Chapter2. In Figure 4 the deterministic part of the swaps volatility S ( t, τ , τ ) is plotted as well as thedeterministic part of the market price of risk S ( t, τ , τ ) . The parameters can be found in Table 2.While S ( t, τ , τ ) is the hightest in the winter and the lowest in the summer, S ( t, τ , τ ) has twopeaks in spring and autumn when the changes in s ( u ) are the biggest.14 able 2 Parameters for the simulations.
Joint Parameters F ν τ τ ρ r κ σ θ
30 0 . .
75 0 . − . .
01 3 0 . . Seasonality in Trading Days Samuelson Seasonality in the Delivery α β γ λ a b c . . . . . Swap Price Volatility V o l a t ili t y Time
Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Swap Prices P r i c e Time
Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Figure 5
The swap price volatility Σ( t, τ , τ ) for each example (left) and swap prices F ( t, τ , τ ) for each example (right). The trajectories are based on the parameters in Table 2.3.4. Comparison of the Models For our simulation study, we applied the Euler-Maruyama procedure to the swap process andthe drift-implicit Milstein procedure to the volatility process. The parameters used in this chapter aresummarized in Table 2. For each model, they fulfill the Feller-condition to ensure that the stochasticvolatility stays strictly positive as well as the Novikov condition such that the measure change is welldefined (see Chapter 3.1–3.3 for details). Note that the initial swap price volatility
Σ(0 , τ , τ ) mightbe different for each model even if the initial value ν of the Cox-Ingersoll-Ross process is always thesame. For the parameters in Table 2, the initial swap price volatilities of the two seasonality modelsare equal since S ( , ) ≈ (see Figure 4).In Figure 5, we have plotted the evolution of the stochastic volatility as well as the swap pricesof all three considered models both under the measure (cid:101) Q . For better comparison, we use the sameBrownian increments for each model. Our time scale reflects 9 month starting from January withdelivery in October. In the red trajectory, we can clearly observe the Samuelson effect, which diminishes15he volatility at the beginning and pushes in the end towards d ( ) (cid:112) ν ( τ ) . Moreover, the swap pricevolatility with seasonality in the delivery time is oscillating around the swap price volatility of the firstexample. This is caused by the choice of parameters since S ( τ , τ ) ≈ and can be observed both inthe swap price volatility and in the swap price trajectory.
4. Further Arbitrage Considerations
So far, we have considered a market with one swap contract. Nevertheless, in electricity markets,typically more than one swap is traded at the same time. For example, at the EEX, the next 9 months,11 quarters and 6 years are available. In Chapter 4.1, we address the issue of arbitrage in a marketconsisting of N monthly delivering swaps and then discuss a market with overlapping delivery periodsin Chapter 4.2. N Swaps
In this chapter, we consider a market with N swap contracts having subsequent monthly deliveryperiods ( τ m , τ m +1 ] for m = 1 , . . . , N . According to the First Fundamental Theorem of Asset Pricing,the market is arbitrage-free if there exists a measure (cid:101) Q under which all swap contracts are martingales.In a market with N assets, N Brownian motions are needed such that a market price of risk exists (see,e.g., Shreve (2004)). Therefore, we add another factor for each contract and the underlying futuresprice dynamics are given by dF ( t, τ ) = F ( t, τ ) N (cid:88) j =1 σ j ( t, τ ) dW Fj ( t ) , F (0 , τ ) = F > , (4.1)where W Fj , for j = 1 , . . . , N , are independent standard Brownian motions under Q . As in Chapter 2,we define the swap price with delivery period ( τ m , τ m +1 ] , for m = 1 , . . . , N via geometric averaging F ( t, τ m , τ m +1 ) := exp (cid:18)(cid:90) τ m +1 τ m w ( u, τ m , τ m +1 ) log( F ( t, u )) du (cid:19) . The resulting swap price dynamics for the monthly delivery period ( τ m , τ m +1 ] , m = 1 , . . . , N aregiven by dF ( t, τ m , τ m +1 ) F ( t, τ m , τ m +1 ) = 12 N (cid:88) j =1 (cid:18) Σ j ( t, τ m , τ m +1 ) − (cid:90) τ m +1 τ m w ( u, τ m , τ m +1 ) σ j ( t, u ) du (cid:19) dt + N (cid:88) j =1 (cid:90) τ m +1 τ m w ( u, τ m , τ m +1 ) σ j ( s, u ) du dW Fj ( t ) . (4.2)16 earMonth 1 Month 2 Month 3 Quarter 2 Quarter 3 Quarter 4Month 4 Month 5 Month 6 Figure 6
The cascading procedure of overlapping electricity swap contracts.
Then the standard theory for multidimensional markets (see, e.g., Shreve (2004)) leads to the marketprice of risk equations and a risk-neutral probability measure.
In electricity markets, it is possible to trade into overlapping delivery periods. For example, theswap contract on the next quarter of the year is available as well as the three swaps on the correspondingmonths. Also here, arbitrage has to be excluded: It should not matter if the electricity is bought via aquarterly contract or the corresponding three underlying monthly contracts.One has to find a pricing measure under which all contracts, the monthly and the quarterly ones,are martingales. If we would price an overlapping contract using the geometric averaging procedure,we would have that F overl ( t, τ , τ N +1 ) = e (cid:82) τN +1 τ w ( u,τ ,τ N +1 ) log( F ( t,u )) du = N (cid:89) m =1 F ( t, τ m , τ m +1 ) w m , where w m = (cid:82) τm +1 τm ˆ w ( u ) du (cid:82) τN +1 τ ˆ w ( u ) du . The price of the quarterly swap would be the product of the monthlycontracts. This might create arbitrage opportunities: In general, the product of martingales is not amartingale anymore. In this framework, the so-called cascading process of overlapping contracts offersa solution. The cascading process describes the division of an overlapping contract into its buildingblocks. A swap contract delivering over a quarter is transformed into its corresponding monthly swapcontracts at its maturity (see Figure 6). Analogously, the price of a yearly swap contract is convertedinto the first 3 monthly contracts and the subsequent 3 quarterly contracts. Each quarterly contract willbe cascaded later. The monthly contracts thus play the role of building blocks for overlapping contractsand are also called atomic contracts. Consequently, the quarterly and yearly swap contracts can beseen derivatives on the monthly contracts, and we propose to price them as such. If we have found apricing measure under which all atomic swap prices are martingales, then F overl is also a martingalesince the sum of (cid:101) Q -martingales stays a martingale under (cid:101) Q .17 . Electricity Options We consider a European call option with strike price
K > and exercise time T < τ writtenon an electricity swap contract delivering in ( τ , τ ] . In Chapter 2, we have determined an equivalentmeasure (cid:101) Q , such that the swap price F ( · , τ , τ ) is a martingale. Hence, (cid:101) Q can be used as pricingmeasure for derivatives on the swap. In general, (cid:101) Q depends on τ and τ since it includes a riskpremium for the delivery period as discussed in Remark 1. Hence, the pricing measure is tailor-madefor this particular contract. Motivated by the market models considered in Chapter 3, we stick to a general factorizingvolatility structure Σ( t, τ , τ ) = S ( t, τ , τ ) (cid:112) ν ( t ) , where S ( t, τ , τ ) = E [ s ( t, U )] (5.1)identifies averaged seasonalities and term-structure effects for a random variable U with density w ( u, τ , τ ) (see also Remark 1). We assume that s ( t, u ) is positive and bounded by R , so that theswap price model dF ( t, τ , τ ) = S ( t, τ , τ ) (cid:112) ν ( t ) F ( t, τ , τ ) d (cid:102) W F ( t ) , (5.2) dν ( t ) = ( κθ ( t ) − ( κ + σρξ ( t, τ , τ )) ν ( t )) dt + σ (cid:112) ν ( t ) d (cid:102) W σ ( t ) . (5.3)is a (cid:101) Q -martingale if κ > σ R (see Appendix B). The market price of delivery risk is given by b ( t, τ , τ ) = ξ ( t, τ , τ ) (cid:112) ν ( t ) , where ξ ( t, τ , τ ) = 12 V [ s ( t, U )] E [ s ( t, U )] . (5.4)The price of the corresponding electricity call option at time t ∈ [0 , T ] is given by the risk-neutralvaluation formula C ( t, τ , τ ) = E ˜ Q (cid:2) e − r ( T − t ) ( F ( T, τ , τ ) − K ) + |F t (cid:3) , (5.5)see, e.g., Shreve (2004). The dynamics of the logarithmic swap price X ( t ) := log( F ( t, τ , τ )) aregiven by dX ( t ) = − S ( t, τ , τ ) ν ( t ) dt + S ( t, τ , τ ) (cid:112) ν ( t ) d (cid:102) W F ( t ) . (5.6)We skip the dependencies on the delivery period for notational convenience. We thus have the following18esult: Theorem 1.
The electricity option price at time t ≤ T with strike price K > is given by C ( t, τ , τ ) = e − r ( T − t ) ( e x (1 − Q ( t, x, ν ; log( K ))) − K (1 − Q ( t, x, ν ; log( K )))) , (5.7) where the probabilities of exercising the option are given by − Q k ( t, x, ν ; log( K )) = 12 + 1 π (cid:90) ∞ Re (cid:32) e − iφ log( K ) ˆ Q k ( t, x, ν ; φ ) iφ (cid:33) dφ , k = 1 , . (5.8) The characteristic functions ˆ Q k ( t, x, ν ; φ ) are given by ˆ Q k ( t, x, ν ; φ ) = e Ψ k ( t,T,φ )+ ν Ψ k ( t,T,φ )+ iφx , k = 1 , , (5.9) where Ψ k ( t, T, φ ) and Ψ k ( t, T, φ ) solve the following system of differential equations ∂ Ψ k ∂t = − σ Ψ k + ( β k ( t, τ , τ ) − ρσS ( t, τ , τ ) iφ ) Ψ k + (cid:18) φ − α k iφ (cid:19) S ( t, τ , τ ) , (5.10) ∂ Ψ k ∂t = − Ψ k κθ ( t ) , (5.11) for α = , α = − , β ( t, τ , τ ) = κ + σρ ( ξ ( t, τ , τ ) − S ( t, τ , τ )) , β ( t, τ , τ ) = κ + σρξ ( t, τ , τ ) . The proof follows the Heston procedure and can be found in Appendix C. There exists a uniquesolution to each Riccati equation (see Appendix D) and thus also for Ψ and Ψ . Then, the character-istic functions in (5.9) are uniquely determined. The related put option price can be determined by the Put-Call-Parity .The value of this result depends strongly on the tractability of the Riccati equations (5.10). Inthe classical Heston model, all coefficients of the Riccati equations (5.10) are constant, so that onecan find an analytical solution. For time-dependent coefficients, it is not clear that an analyticalsolution or closed-form expression exists. In Arismendi et al. (2016), the mean reversion level θ ( t ) of the stochastic volatility process is seasonal, but as θ ( t ) does not appear in the Riccati equation,an analytical solution exists. Schneider and Tavin (2018) include the Samuelson effect such that thefutures price dynamics have time-dependent coefficients. Nevertheless, the volatility process hasconstant coefficients. The Samuelson term appears in the Riccati equations and Schneider and Tavin(2018) are able to give a solution depending on Kummer functions. In our framework, the Samuelsoneffect appears in the drift of the stochastic volatility via the market price of delivery risk, making theRiccati equations more complicated (see Chapter 5.2.2).19 .2. The Effect of Seasonalities and Samuelson on the Swaps’ Riccati Equation In this chapter, we state the differential equations (5.10) and (5.11) for each model, and discusshow to solve them. Furthermore, we compare the corresponding option prices numerically. Let usmention, that for the classical Riccati equation setting, which corresponds to the Heston model withconstant coefficients, the exact formula for the solution can be found easily. In general, the equationhas to be solved numerically. We discuss this situation in the subsequent sections.
In the setting of (3.3) and (3.4), option pricing has been treated by Arismendi et al. (2016). Recall,that Ψ k ( t, T, φ ) and Ψ k ( t, T, φ ) solve the following system of differential equations d Ψ k dt ( t, T, φ ) = − σ Ψ k ( t, T, φ ) + ( β k − ρσiφ ) Ψ k ( t, T, φ ) + 12 φ − α k iφ ,d Ψ k dt ( t, T, φ ) = − κθ ( t )Ψ k ( t, T, φ ) , for α = , α = − , β = κ − σρ , and β = κ . Since all coefficients of the Riccati equations (thefirst equation of the system) are constant, the solutions can be calculated by Ψ k ( t, T, φ ) = 1 σ ( β k − σρφi − δ k ) 1 − e − δ k ( T − t ) − g k e − δ k ( T − t ) , k = 1 , , where δ k := (cid:114) ( β k − σρφi ) + 2 σ ( 12 φ − α k φi ) ,g k := β k − σρφi − δ k β k − σρφi + δ k . Finally, numerical integration leads to the solution of Ψ ( t, T, φ ) and Ψ ( t, T, φ ) (see Chapter 5.2.4).In Figure 7, we illustrate the option prices for different speed of mean-reversion parameters( κ = 0 . , κ = 3 , and κ = 10 ) over the entire time horizon based on the parameters in Table 2 and 3.The calculations are conducted by using the analytical solution for the Riccati equations. The solutionsfor Ψ k are attained using the Runge-Kutta method. We here require a relative and absolute toleranceof e − . As proposed by Arismendi et al. (2016), we apply a trapezoidal integration scheme to obtainthe integral values for each strike which are mandatory to determine the corresponding probabilities − Q and − Q as in (5.8) and thus the related option price for the considered strike at a specificpoint in time (see (5.7)). As a result, we can observe decreasing call option prices over time. Moreover,the lower the speed of mean-reversion, the higher is the option price except for the first trading days.The closer we reach the expiration date, the smaller is the difference between each option price for a20 Time O p t i on P r i c e Option Prices with Seasonality in the Trading Day for K=28 over Time κ =0.6 κ =3 κ =10 Figure 7
Option prices with seasonality in the trading day over the whole time horizon for a fixedstrike K = 28 . fixed strike price. In the setting of Chapter 3.2, the resulting dynamics under (cid:101) Q are given by (3.12) and (3.13).Under the measure (cid:101) Q , the Samuelson effect appears in the drift term of the stochastic volatility. Ψ k ( t, T, φ ) and Ψ k ( t, T, φ ) for k = 1 , solve the following two systems of differential equa-tions: d Ψ dt ( t, T, φ ) = − σ Ψ ( t, T, φ )+ (cid:16) κ + σρ (cid:104) d ( τ − τ ) − d ( τ − τ )(1 + iφ ) (cid:105) e − λ ( τ − t ) (cid:17) Ψ ( t, T, φ )+ 12 d ( τ − τ ) ( φ − iφ ) e − λ ( τ − t ) ,d Ψ dt ( t, T, φ ) = − κθ Ψ ( t, T, φ ) , and d Ψ dt ( t, T, φ ) = − σ Ψ ( t, T, φ )+ (cid:16) κ + σρ (cid:104) d ( τ − τ ) − iφd ( τ − τ ) (cid:105) e − λ ( τ − t ) (cid:17) Ψ ( t, T, φ )+ 12 d ( τ − τ ) ( φ + iφ ) e − λ ( τ − t ) ,d Ψ dt ( t, T, φ ) = − κθ Ψ ( t, T, φ ) , where d ( x ) and d ( x ) are defined in (3.11). Compared to Schneider and Tavin (2018), the Samuelsoneffect appears additionally in front of Ψ and Ψ . This leads to the setting of time-dependent21oefficients in the Riccati-type equations. The explicit solution is expressed in terms of hypergeometricexpressions.Figure 8 illustrates option prices with respect to seven strike prices based on the default parametersintroduced in Table 2 and 3. We use the Runge-Kutta method with adaptive step size to solve bothsystems of differential equations. The trapezoidal integration of the integrands with respect to φ leadsto the cumulative distributions Q and Q for each strike price K . An application of Equation (5.7)gives the corresponding option prices for each strike. In fact, we approximate the analytic expressionsfor Ψ , Ψ , Ψ , and Ψ since two coefficients in the Riccati equation include the Samuelson effect.For models with time-dependent θ, σ, and ρ , the method by Benhamou et al. (2010) can be appliedwith the help of a volatility of variance expansion using the Lewis representation. For time-dependentcoefficients of piece-wise constant structure, one can also use the model of Mikhailov and Nögel(2004). However, in general, the solution has to be found numerically. Most of them concern theone dimensional case, for example, standard second order finite difference methods, see Tavella andRandall (2000). More recently, results include stochastic volatility with high-order compact finitedifference schemes such as Crank–Nicolson scheme, see Düring et al (2014).In order to investigate the impact of the Samuelson effect, we set the parameter λ to . , . ,and . . As a result, we observe higher option prices for smaller Samuelson parameters, which aredecreasing in increasing strike prices. The differences are especially large for at the money strikes.With increasing time to maturity, these differences become even larger. To add, the option pricesbecome more affeced the closer we reach the expiration date (see Figure 8 (right)).
27 28 29 30 31 32 3300.511.522.533.5
Option Prices with Samuelson Effect O p t i on P r i c e Strike λ =1.5 λ =3.5 λ =5.5 Option Prices with Samuelson Effect for K=28 over Time O p t i on P r i c e Time λ =1.5 λ =3.5 λ =5.5 Figure 8
Option prices with Samuelson effect 10 days before maturity (left) and over the entire timehorizon for a fixed strike K = 28 (right). .2.3. Delivery-Dependent Seasonality Finally, we consider the resulting option prices corresponding to Chapter 3.3. Ψ k ( t, T, φ ) and Ψ k ( t, T, φ ) for k = 1 , solve the following two systems of differential equations: d Ψ dt ( t, T, φ ) = − σ Ψ ( t, T, φ ) + (cid:16) κ + σρ (cid:104) S ( τ , τ ) − S ( τ , τ )(1 + iφ ) (cid:105)(cid:17) Ψ ( t, T, φ )+ 12 S ( τ , τ ) ( φ − iφ ) ,d Ψ dt ( t, T, φ ) = − κθ Ψ ( t, T, φ ) , and d Ψ dt ( t, T, φ ) = − σ Ψ ( t, T, φ ) + (cid:16) κ + σρ (cid:104) S ( τ , τ ) − iφS ( τ , τ ) (cid:105)(cid:17) Ψ ( t, T, φ )+ 12 S ( τ , τ ) ( φ + iφ ) ,d Ψ dt ( t, T, φ ) = − κθ Ψ ( t, T, φ ) . The differential equations can be solved analytically, while all coefficients are constant. The solutionsare given by Ψ k ( t, T, φ ) = 1 σ ( β k ( τ , τ ) − σρφi − δ k ( τ , τ )) 1 − e − δ k ( τ ,τ )( T − t ) − g k ( τ , τ ) e − δ k ( τ ,τ )( T − t ) , (5.12)where β ( τ , τ ) = κ + σρ ( S ( τ , τ ) − S ( τ , τ )) ,β ( τ , τ ) = κ + σρS ( τ , τ ) ,δ k ( τ , τ ) := (cid:114) ( β k ( τ , τ ) − σρφi ) + 2 σ ( 12 φ − α k φi ) ,g k ( τ , τ ) := β k ( τ , τ ) − σρφi − δ k ( τ , τ ) β k ( τ , τ ) − σρφi + δ k ( τ , τ ) , and Ψ k ( t, T, φ ) = − κθσ (cid:104)(cid:16) β k ( τ , τ ) − σρφi − δ k ( τ , τ ) (cid:17) ( T − t ) − (cid:18) − g k ( τ , τ ) e − δ k ( τ ,τ )( T − t ) − g k (cid:19) (cid:105) . The delivery dependent seasonality model is able to incorporate delivery dependent effects, whilebeing highly tractable and fast to implement. In Figure 9, we visualize the option prices over the lasttrading month based on the parameters in Table 2 and 3. For the calculations, we use the analytical23 .68 0.7 0.72 0.7400.511.522.533.5
Time O p t i on P r i c e Option Prices with Seasonality in the Delivery
K=29K=30K=31
Figure 9
Option prices with seasonality in the delivery over the last trading month for fixed strikes.
Table 3
Parameters for pricing options.
Parameters
T K
Iterations φ min φ max n .
75 27 , . . . ,
33 100 0 . . . solutions for Ψ , Ψ , Ψ , and Ψ . As before, numerical integration leads to the desired option pricefor each considered strike.As a result, the option prices are decreasing with an increasing strike price. Furthermore, theoption prices are decreasing over time for all strikes. In this chapter, we focus on concrete numerical examples based on the transformed models inChapter 3 and Chapter 5.2. We consider the integrands for each model as well as the resulting calloption prices. For comparative reasons, we have chosen the same parameters for all examples (seeTable 2 and 3). In order to determine both integrands for each model, we calculate the solution to thesystem of ordinary differential equations as in Chapter 5.2.1–5.2.2. We used the analytical solutionof the Riccati equations with seasonality in the trading day and in the delivery time. We get a certainintegrand depending on φ for each strike price. The possible oscillation of the integrand can have anegative influence on the numerical procedure since the standard quadrature can fail, see Rouah (2013).We can observe that both integrands are relatively smooth for each model and converge to zero around φ ≈ (see Figure 10). For the integration, we apply the standard trapezoidal rule. To be precise, wefix the upper boundary for the integrands with φ max = 100 due to the converting behavior and truncatethe lower boundary at φ min = 0 . . Plugging the integral value into (5.8) leads to the optionprices (5.7) (see Figure 11 (left)). To compare the resulting prices, we also conduct a simulation studybased on the parameters in Table 3 starting 20 days before the swap contract expires. The results can24
10 20 30 40 50 60-0.1-0.0500.050.1
Integrand for k=1 (10 Days before Maturity) I n t eg r and f o r k = φ Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Integrand for k=2 (10 Days before Maturity) I n t eg r and f o r k = φ Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Figure 10
Integrands for each model and for all strikes K = 27 , . . . , .
27 28 29 30 31 32 3301234
Option Prices (10 Days before Maturity) O p t i on P r i c e Strike
Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Option Prices for K=28 over Time O p t i on P r i c e Time
Seasonality in Trading TimeSamuelson EffectSeasonality in Delivery Time
Figure 11
Option prices for each model 10 days before maturity (left) and over the last tradingmonth for a fixed strike K = 28 (right). The parameters are based on Table 2 and 3.
25e found in Figure 11 (right). In all examples, we observe decreasing option prices for increasingstrikes. An application of the Samuelson effect leads to smaller option prices than in the first examplewith seasonality in the trading time. Seasonality in the delivery leads to higher option prices for out ofthe money strikes than in the case of the first example. Only for far in the money strikes, the optionprices are even smaller than the ones resulting from the Samuelson effect. Having the last tradingmonth in view, the option prices are decreasing in time. For the fixed strike K = 28 , the option pricesfor the Samuelson effect the smallest. In contrast, seasonalities in the delivery affect the option pricesat most so that they show the highest option prices over the last trading month.
6. Conclusion
We suggest the use of a pricing framework for swaps and options in electricity markets. Moreover,we introduced an equivalent martingale measure for the swap that explicitly depends on its deliveryperiod and can be used to price electricity options. Geometric averaging on the delivery period is thekey element here. The market price of delivery risk for an individual contract is specified by the trade-off between the variance of the swap on the one hand and the weighted average variance of a stream offutures on the other hand. We considered futures price models from the recent literature and providethe corresponding swap price models. Moreover, we investigated the effect of seasonal dependenceon the trading day, the Samuelson effect, and delivery-dependent seasonality in line with Arismendiet al. (2016), Schneider and Tavin (2018), and Fanelli and Schmeck (2019), respectively. Wheneverthe futures and thus the swap price volatility are independent of the delivery time, the market price ofdelivery risk is zero. On the other hand, typically observed characteristics of the electricity market,such as seasonalities in the delivery and term-structure effects, instead impact the market price of thedelivery risk.Moreover, we provided an outlook of our model in the case of several atomic and overlappingcontracts. For each additional atomic contract, new uncertainty occurs, and a further Brownian motionis thus needed within the futures price. The pricing procedure can be applied as before. Overlappingcontracts are treated as derivatives of the underlying atomic swap contracts, which is justified by thecascading process (see Figure 6).All examples are characterized by a volatility structure in the spirit of the Heston model. Theaffine model structure of the futures is inherited by the swaps, thereby leading us to follow the Hestonmethodology for option pricing. We investigated the option price for seasonal dependence on thetrading day, the Samuelson effect, and delivery-dependent seasonality. Whenever the deterministicvolatility part is independent of the trading time, the corresponding Riccati equations can be solvedanalytically. For the Samuelson effect, the deterministic part of the volatility is time-dependent, andwe showed that a unique solution exists. Furthermore, we provided a numerical method to solve the26iccati equations.In conclusion, this paper treats each electricity swap as a proper contract on the market andsuggests a pricing measure that is tailor-made for this particular contract, which includes acknowledgingthe existence of the delivery period. Our pricing framework allows for the evaluation of option pricesin line with the Heston method.
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A. Technical Requirements
1. For the model (2.1) and (2.2), we make the following assumtions:(a) The conditions by Yamada and Watanabe need to be satisfied (see also Karatzas and Shreve(1991); Proposition 2.13). In particular, we assume a ( t, τ, σ ) : [0 , τ ] × ( τ , τ ] × R + → R ,c ( t, τ, σ ) : [0 , τ ] × ( τ , τ ] × R + → R , are Borel-measurable functions and σ = { σ ( t, τ ) | ≤ t ≤ τ ≤ τ } is a stochasticprocess with continuous sample paths. Further, we assume • | a ( t, τ, x ) − a ( t, τ, y ) | ≤ K | x − y | for some positive constant K > with x, y ∈ R + , • | c ( t, τ, x ) − c ( t, τ, y ) | ≤ H ( | x − y | ) for x, y ∈ R + where H : [0 , ∞ ) → [0 , ∞ ) is anincreasing function with H (0) = 0 and (cid:82) (0 ,(cid:15) ) H − ( u ) du = ∞ , ∀ (cid:15) > ,which guarantees that there exists a unique strong solution for (2.2). In particular, σ ( t, τ ) is adapted to the filtration F t .(b) Next, we assume that F = { F ( t, τ ) | ≤ t ≤ τ ≤ τ } is a stochastic process withcontinous sample paths. It directly follows that σ ( t, τ ) F ( t, τ ) is process Lipschitz andthus functional Lipschitz. Then, by Protter (2005) (see Theorem 7; p. 253) Equation (2.1)admits a unique strong solution.(c) In order to attain that (2.1) is a Q -martingale, we assume that the Novikov condition (see,e.g., Karatzas and Shreve (1991); Proposition 5.12) is satisfied, that is E Q (cid:104) e (cid:82) τ σ ( t,τ ) dt (cid:105) < ∞ . (A.1)2. For the geometric weightening approach (2.8) we need to apply the stochastic Fubini Theorem(see Protter (2005); Theorem 65; Chapter IV. 6). Therefore, we assume that • ( t, u, ω ) → w ( u, τ , τ ) σ ( t, u ) is jointly progressivly measurable, • E Q (cid:104)(cid:82) τ (cid:82) τ τ w ( u, τ , τ ) σ ( t, u ) du dt (cid:105) < ∞ .29 . An Application of Girsanov’s Theorem for the Examples We want to check if Novikov’s condition is satisfied, that is E Q (cid:104) e (cid:82) τ b ( t,τ ,τ ) dt (cid:105) < ∞ (see,e.g., Karatzas and Shreve (1991)).In the case of Schneider and Tavin (2018), we can find specific upper and lower boundaries forthe deterministic part since e − λ ( τ − t ) ∈ [0 , and d ( τ − τ ) ∈ [ −
12 1 λ ( τ − τ ) , ] . Hence, E Q (cid:104) e (cid:82) τ b ( t,τ ,τ ) dt (cid:105) = E Q (cid:104) e (cid:82) τ d ( τ − τ ) e − λ ( τ − t ) ν ( t ) dt (cid:105) ≤ E Q (cid:104) e − ˜ u (cid:82) τ ν ( t ) dt (cid:105) , (B.1)where ˜ u := − max { λ ( τ − τ ) } . Following Cont and Tankov (2004) (see Chapter 15.1.2) thereexists an explicit, finite expression for the last expectation if κ + 2 σ ˜ u > .In the case of Fanelli and Schmeck (2019), we can again find specific upper and lower boundariesfor (3.17) since s ( u ) = a + b cos(2 π ( c + u )) ∈ [0 , a ] for a > b > and thus s ( u ) ≤ a s ( u ) . Inparticular, S ( τ , τ ) ∈ [ − a, a ] . Hence, E Q (cid:104) e (cid:82) τ b ( t,τ ,τ ) dt (cid:105) = E Q (cid:104) e (cid:82) τ S ( τ ,τ ) ν ( t ) dt (cid:105) ≤ E Q (cid:104) e − ˜ u (cid:82) τ ν ( t ) dt (cid:105) , (B.2)where ˜ u := − a . As before, the last expectation is limited if κ + 2 σ ˜ u > , i.e. κ > a σ .In the general case of Chapter 5, we assume that s ( t, u ) is positive and bounded. As s ( t, u ) ∈ [0 , R ] , ξ ( t, τ , τ ) ∈ [ − R, R ] . Hence, E Q (cid:104) e (cid:82) τ b ( t,τ ,τ ) dt (cid:105) = E Q (cid:104) e (cid:82) τ ξ ( t,τ ,τ ) ν ( t ) dt (cid:105) ≤ E Q (cid:104) e − ˜ u (cid:82) τ ν ( t ) dt (cid:105) , (B.3)where ˜ u := − R . As before, if κ + 2 σ ˜ u > , that is if κ > R σ , then Novikov’s condition issatisfied. C. Proof of Theorem 1Proof.
We can write C ( t, τ , τ , ) = e − r ( T − t ) E (cid:101) Q (cid:2) e X t X T ≥ log( K ) |F t (cid:3) − e − r ( T − t ) K E (cid:101) Q (cid:2) X T ≥ log( K ) |F t (cid:3) . (C.1)Due to the Markovian structure, an application of the Independence Lemma (see, e.g., Shreve (2004);cf. Lemma 2.3.4) leads to C ( t, τ , τ ) = c ( t, X ( t ) , ν ( t )) − c ( t, X ( t ) , ν ( t )) , c ( t, x, ν ) = e − r ( T − t ) e x (1 − Q ( t, x, ν ; log( K ))) , (C.2) c ( t, x, ν ) = e − r ( T − t ) K (1 − Q ( t, x, ν ; log( K ))) , (C.3)and Q ( t, x, ν ; log( K )) := ˜˜ Q (cid:2) X t,x,ν ( T ) ≤ log( K ) (cid:3) ,Q ( t, x, ν ; log( K )) := ˜ Q (cid:2) X t,x,ν ( T ) ≤ log( K ) (cid:3) , where the probability measure (cid:101)(cid:101) Q is defined by d (cid:101)(cid:101) Q d (cid:101) Q = e − (cid:82) T S ( u,τ ,τ ) ν ( u ) du + (cid:82) T S ( u,τ ,τ ) √ ν ( u ) dW F ( u ) .For k = 1 , , e − rt c k ( t, X ( t ) , ν ( t )) are martingales under (cid:101) Q . Hence, c k ( t, x, ν ) solves ∂c k ( t, x, ν ) ∂t + ( A t c k )( t, x, ν ) = rc k ( t, x, ν ) , for k = 1 , , (C.4)subject to the terminal conditions c ( T, x, ν ) = e x x ≥ log( K ) and c ( T, x, ν ) = K x ≥ log( K ) , by anapplication of the discounted Feynman Kac Theorem (see, e.g., Shreve (2004); cf. Theorem 6.4.3 andCh. 6.6). For a function f depending on x and ν , the generator of ( X, ν ) is given by ( A t f )( x, ν ) = − ∂f∂x S ( t, τ , τ ) ν + ∂f∂ν [ κθ ( t ) − ( κ + σρξ ( t, τ , τ )) ν ]+ 12 ∂ f ( ∂x ) S ( t, τ , τ ) ν + 12 ∂ f ( ∂ν ) σ ν + ∂ f∂x∂ν ρσS ( t, τ , τ ) ν . (C.5)If we plug (C.2) and (C.3) inside the partial differential equation (PDE) (C.4), we end up with ∂Q k ∂t + α k S ( t, τ , τ ) ν ∂Q k ∂x + ( κθ ( t ) − β k ( t, τ , τ ) ν ) ∂Q k ∂ν + 12 S ( t, τ , τ ) ν ∂ Q k ( ∂x ) + 12 σ ν ∂ Q k ( ∂ν ) + ρσνS ( t, τ , τ ) ∂ Q k ∂x∂ν = 0 , (C.6)for α = , α = − , β ( t, τ , τ ) = κ + σρ ( ξ ( t, τ , τ ) − S ( t, τ , τ )) , and β ( t, τ , τ ) = κ + σρξ ( t, τ , τ ) . This PDE can be solved by a martingale depending on the solutions of the dynamics dX k ( t ) = α k S ( t, τ , τ ) ν k ( t ) dt + S ( t, τ , τ ) (cid:112) ν k ( t ) d (cid:102) W F ( t ) ,dν k ( t ) = ( κθ ( t ) − β k ( t, τ , τ ) ν k ( t )) dt + σ (cid:112) ν k ( t ) d (cid:102) W σ ( t ) . Following Heston, the corresponding characteristic function solves (C.6) as well. Note, that theunderlying model structure is of affine type since the PDE is linear in ν . The characteristic function is31hus of exponential affine form (see Duffie (2010)): ˆ Q k ( t, x, ν ; φ ) = E Q k (cid:104) e iφX t,x,νk ( T ) (cid:105) = e Ψ k ( t,T,φ )+ ν Ψ k ( t,T,φ )+ iφx , k = 1 , , (C.7)for φ ∈ R , where Ψ k : [0 , T ] × [0 , τ ) × R → C and Ψ k : [0 , T ] × [0 , τ ) × R → C are time-dependentfunctions satisfying Ψ k ( T, T, φ ) = 0 and Ψ k ( T, T, φ ) = 0 at terminal time T . The last term in (C.7)is added in order to ensure the terminal condition ˆ Q k ( T, x, ν ; φ ) = e iφx . (C.8)For notational convenience, we drop the time and space indices such that Ψ k := Ψ k ( t, T, φ ) , Ψ k := Ψ k ( t, T, φ ) , and ˆ Q k := ˆ Q k ( t, x, ν ; φ ) . Plugging (C.7) into the PDEs of (C.6) for k = 1 , andrearranging terms yields ˆ Q k (cid:34) ν (cid:104) ∂ Ψ k ∂t + α k S ( t, τ , τ ) iφ − Ψ k β k ( t, τ , τ ) − S ( t, τ , τ ) φ + 12 σ Ψ k + ρσS ( t, τ , τ ) iφ Ψ k (cid:105) + ∂ Ψ k ∂t + Ψ k κθ ( t ) (cid:35) = 0 . Since ˆ Q k > for k = 1 , and ν > by definition, we apply the separation of variables argument (see Duffie (2010); cf. p. 150) to achieve the following differential equations ∂ Ψ k ∂t = − σ Ψ k + ( β k ( t, τ , τ ) − ρσS ( t, τ , τ ) iφ ) Ψ k + (cid:18) φ − α k iφ (cid:19) S ( t, τ , τ ) (C.9)of Riccati-type and ∂ Ψ k ∂t = − Ψ k κθ ( t ) , (C.10)subject to Ψ k ( T, T, φ ) = 0 and Ψ k ( T, T, φ ) = 0 for k = 1 , .An application of the Fourier inversion technique (see Gil-Pelaez (1951)) to (C.7) leads to thecumulative distribution functions Q and Q given by Q k ( t, x, ν ; log( K )) = 12 − π (cid:90) ∞ Re (cid:32) e − iφ log( K ) ˆ Q k ( t, x, ν ; φ ) iφ (cid:33) dφ , k = 1 , . (C.11)32 . On the Solutions of the Differential Equations To show that there exists a unique solution for the Riccati equations Ψ k ( t, T, φ ) for k = 1 , ,transfer them to a homogenoeus second order linear differential equation using the substitution Ψ k ( t, T, φ ) = z (cid:48) k ( t,T ) σ z k ( t,T ) (see, e.g., Poljanin and Zajcev (2018)). Rewrite the resulting differentialequation as a system of first order equations in line with Walter (1996)(cf. p. 103f.). Then, TheoremVI (see Walter (1996)) ensures that there exists a unique solution to the differential system and thusto the second order equation since all matrix elements have continuous real and imaginary parts intrading time t ∈ [0 , τ ]]