Pricing Reliability Options under different electricity prices' regimes
Luisa Andreis, Maria Flora, Fulvio Fontini, Tiziano Vargiolu
PPricing Reliability Options under different electricity prices’regimes
Luisa Andreis ∗ , Maria Flora † , Fulvio Fontini ‡ , Tiziano Vargiolu § September 13, 2019
Abstract
Reliability Options are capacity remuneration mechanisms aimed at enhancing securityof supply in electricity systems. They can be framed as call options on electricity sold bypower producers to System Operators. This paper provides a comprehensive mathematicaltreatment of Reliability Options. Their value is first derived by means of closed-form pricingformulae, which are obtained under several assumptions about the dynamics of electricityprices and strike prices. Then, the value of the Reliability Option is simulated under a real-market calibration, using data of the Italian power market. We finally perform sensitivityanalyses to highlight the impact of the level and volatility of both power and strike price, ofthe mean reversion speeds and of the correlation coefficient on the Reliability Options’ value.
Keywords:
Pricing; reliability option; option value; electricity markets. ∗ WIAS-Weierstrass Institute, Berlin; [email protected] † Department of Economics, University of Verona; maria.fl[email protected] ‡ Department of Economics and Management ”Marco Fanno”, University of Padua, and Interdepartmental Centrefor Energy Economics and Technology ”Giorgio Levi-Cases”, University of Padua; [email protected] § Department of Mathematics ”Tullio Levi Civita”, University of Padua and Interdepartmental Centre for EnergyEconomics and Technology “Giorgio Levi-Cases”, University of Padua; [email protected] a r X i v : . [ q -f i n . P R ] S e p Introduction
Capacity Remuneration Mechanisms (CRM) have been implemented in several electricity marketsworldwide, in order to remunerate explicitly power capacity. Among these, the Reliability Option(RO) mechanism recognizes the option nature of the investments in power capacity and creates amarket for such an option. ROs, firstly proposed in [6, 35], have been implemented in Colombia(Firm Energy Obligations [13]), in NE-ISO (Forward Capacity Market [18]) and in Ireland [27, 28,29] and are about to be implemented in Italy [23, 32, 33, 34]. They are tools to commercialize,through a financial product, the possibility, given by generation capacity, of providing security ofsupply by producing electricity. They give their holder, i.e. the System Operator (SO), whichacquires them in a competitive setting, the right to call the generation capacity to produce power,and to receive the positive difference between the electricity price that effectively occurs in themarket and a pre-defined price. Such a pre-defined price corresponds to the strike price of theoption, and it is set to represent the value that power has at the specific level for which load isnot shed, i.e., it is the highest system marginal price compatible with load provision with no loadshedding.In this paper, we evaluate ROs following the financial approach , which requires to identifythe stochastic property of the asset under evaluation, and to assume that, in a complete market,a continuous hedging between the financial derivative and the underlying asset is possible. At afirst glance, this assumption seems quite hard to be met in the electricity sector, given that theunderlying asset of the option is electricity, which is not a storable good . However, derivatives canbe and are indeed written on several underlying assets that are not liquidly traded, such as interestrates or temperatures (see e.g. [7, Chapter 15]). What is needed for the application of risk-neutralpricing based on hedging is the existence of liquid assets that are traded and that correlate withthe underlying of the derivative, such as forwards. The seminal paper [5] has questioned suchan assumption, considering the relationship between derivative (future) prices and spot prices inmarkets with limited liquidity and risk averse agents. However, we believe that the assumptionof limited liquidity was more justified at the beginning of the liberalization process of the powermarket, while this concern is less justified now, after several years of functioning of liberalizedelectricity markets. This approach is shared by other scholars, who have evaluated exotic optionson electricity, such as spark-spread options (options on the differential between power prices andthe heat content of the fuel, [15, 20]), Asian options (options written on average prices, [11]) oroptions which are implicit in demand response mechanisms, [31].We formulate different possible assumptions on the dynamics of the stochastic processes onwhich the RO depends, and estimate the relative RO value. ROs are complex options on powersupply which can have different maturities and can be exercised several times at different, andpossibly random, strike prices. Therefore, we provide a comprehensive mathematical treatment ofall their aspects.Though many authors, as seen above, have evaluated various exotic options on electricity, tothe best of our knowledge our paper is the first one to evaluate ROs under different assumptionson the electricity price process. Several models for electricity prices have been proposed in theliterature (see e.g. [3, 11, 19, 20, 25, 36] and the book [2] for a presentation and critical discussionof various models), and it would not be feasible to present RO pricing formulae for each one ofthese. For this reason, we choose a set of simple and significant ones, and present semi-explicitpricing formulae that have clear economic interpretations. We first start from the simplest possible See [14], ch. 22 and 23, for an introduction and an analysis of Capacity Remuneration Mechanisms. This is a standard approach to price financial derivatives, see for instance [21]. At least as long as storage of electric energy by means of conversion into a different form of energy, such askinetic energy of water in power dams or as chemical energy in batteries, is limited because of its cost or for technicalreasons. [8] evaluates, through a Monte Carlo approach, a contract composed of a portfolio of 4344 call options onhourly prices, all with the same strike price. It corresponds to a discrete-time version of the option we consider inProposition 3.1. We start by desribing in general what ROs are. These contracts are sold in an auction, typicallyonce a year, and they aim to deliver electricity with a given T -length period in advance (leadtime), for a pre-defined period of delivery, which has length ( T − T ). The rules of the ROspecify that the capacity provider, the subject who sells the option, must commit to deliver acertain capacity to the subject buying the option, in general the SO. Such a commitment is madeeffective by prescribing that the seller must offer in the market an amount of electricity equal tothe committed capacity and return any positive difference between the reference market price anda previously set strike price K . Each RO contract scheme specifies what the reference market is. Ina first approximation, the reference market can be a convex combination of different markets, suchas the day-ahead and the balancing or real-time ones. In practice, different RO schemes can havedifferent reference markets. For instance, in Ireland, exclusively the day-ahead market is taken asa reference, while in NE-ISO it is the real-time one. If we call P the day-ahead market price and P ( b ) the price in the balancing market (or in the real-time market), we can define the referencemarket price R as the following convex combination R = λP + (1 − λ ) P ( b ) , where, as said, λ ∈ [0 ,
1] depends on the country: λ = 0 for ISO New England; λ = 1 for Colombiaand Ireland; λ ∈ [0 ,
1] in the case of Italy (see [23] for a description of the forthcoming Italianmarket).The strike price is in general determined by taking into account the variable costs of thereference peak technology, that is, the dispatchable technology that would be included in theoptimal generation mix with the lowest unitary investment cost. In actual RO markets, the rulefor the strike price is communicated to potential sellers of ROs before the auction takes place. Thus,in some implementations it can be treated as a deterministic and constant parameter. However,it is also possible that the strike price changes over time during the life span of the RO. This is apossibility envisaged, for instance, in the forthcoming Italian RO scheme, where it is establishedthat the rule linking the strike price to a reference marginal technology is set before the auction,but the marginal cost of such a technology is computed every given period (a month) during thelife span of the RO. This implies that the strike price can also be conceived as a stochastic process. See [23] and [32, 33, 34].
3e shall first derive the RO value starting with the simplest case, and then increase the level ofcomplexity, to derive a general representation of the value of the RO.
The mathematical modeling of the general RO is quite complex, as many auctions and prices areinvolved. We simplify it by defining a mathematical model for the case when the reference priceis simply the day-ahead price P , i.e. λ = 1, as it is in the Colombian or the Irish CRM. In thisway, only one state variable is needed for the reference market price R , and it is indeed P .We start by computing the fair price of a RO, written only on the reference price P and basedon a generation capacity, i.e., for a power plant that is already in place. As said at the beginning ofthis section, the RO is sold in an auction at a certain time, but it becomes active in a subsequenttime period. Let us denote by t = 0 the auction time and by [ T , T ], with T >
0, the timeperiod when capacity has to be committed. It is assumed that the power plant will be productiveat least until T . The idea of pricing the RO is to compute the expected operational profits attime t = 0 (auction time) of the power plant over the period [ T , T ], both in the case when thecapacity provider enters a RO scheme, and in the case it does not. The difference between thesetwo operational profits will be the fair price of the RO.We work on a filtered probability space (Ω , F , {F t } t ≥ , Q ) such that the probability measure Q is the risk-neutral pricing measure used by the market, and the day-ahead electricity price P = ( P t ) t ≥ is a Q -semimartingale. We consider the simple case of a thermal plant, with totalcapacity Q > that converts a fuel, for example oil, gas or coal, into electricity. The cost C = ( C t ) t ≥ of running the thermal plant summarizes the fuel price, CO price, operational andother costs. The power plant sells the electricity at time t ≥ b t is less than or equal to P t . We adopt the usual simplifications, continuous timeinstead of hourly granularity and no ramping penalties/constraints. The plant can decide its bidprocess b = ( b t ) t ≥ to maximize its revenues.We first evaluate the expected operational profits of the power plant over [ T , T ] in the casewhen a RO scheme is not in place, This is the value of the power plant V ( T , T ) at t = 0 and itdepends on the power plant’s income over [ T , T ]. It can be defined as V ( T , T ) = sup b ∈B E Q (cid:34) (cid:90) T T e − rt Q b t ≤ P t ( P t − C t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) , (1)where B is the set of adapted processes on [ T , T ], r is the instantaneous risk-free rate of returnand E Q is the expectation with respect to Q . Remark . In this setting, we assume that the investor is risk-neutral. Although here we are notevaluating financial assets, but rather incomes coming from industrial activity, this is in line with allthe related literature (see e.g. [15, 24, 31]), and is justified by the following financial argument. Theunderlying assets P and C could be in principle not storable, or even not traded in some markets.However, even in such a situation, the risk-neutral evaluation in Eq. (1) can be applied as long asone can find hedging instruments that can be storable and liquidly traded, and that are correlatedwith P and C : for the mathematical derivation of such a result, see e.g. [7, Chapter 15] for vanillaproducts like call and put options (as we will end up to have), and [9, Remark 3.6] for structured Moreover, we do not consider congestion in the transmission network, and therefore we implicitly assume thatthe market for ROs have the same size of the electricity market, namely, that there are no differences between thepricing zones of the electricity and the capacity markets. Q is to be interpreted as the available capacity of a power plant, as described by [22]. In the real-world examplesof ROs, available capacity is computed by measuring the average availability of a power plant over a given timespan (usually a year) and derating the nominal capacity accordingly (as suggested in the academic literature by [12],and in practical market implementations in [32, 33] for the Italian scheme, and in [30] for Ireland). As an example,consider a 100MW plant with a maintenance period of one month per year. Its capacity factor is equal to 0.91; thisfigure can be used to de-rate the relevant capacity of the plant for the RO, which would amount to 91MW. . Here, we indeed have such suitable hedginginstruments, i.e. forward contracts on power and fuel (for P and C , respectively), which are liquidlytraded on financial markets, as they are basically equivalent to any other financial asset up to fewdays before physical delivery. When physical delivery approaches, in order to maintain the hedgingposition it is sufficient to liquidate the position on the maturing future(s) and open an equivalentnew one on another future with a physical delivery farther in time. This is a standard practice inenergy markets, called “rolled-over portfolios”, see e.g. [1, 17] for two applications.Going back to Eq. (1), it is optimal to choose b such that b t ≤ P t = 1 if and only if P t > C t , i.e.the optimal bidding process is b t = C t ∀ t ∈ [ T , T ]. Thus, the final payoff for a thermal plant is V ( T , T ) = E Q (cid:34) Q (cid:90) T T e − rt ( P t − C t ) + dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) . We now consider the case when the thermal plant writes a RO with strike price K = ( K t ) t ≥ . Theplant must now pay back ( P t − K t ) + . Therefore, the value V ro ( T , T ) of the thermal plant witha RO scheme in place is V ro ( T , T ) = sup b ∈B E Q (cid:34) (cid:90) T T e − rt Q ( b t ≤ P t ( P t − C t ) − ( P t − K t ) + ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) . The bidding strategy b t = C t is again optimal for all t ∈ [ T , T ]. Thus, V ro ( T , T ) = V ( T , T ) − E Q (cid:34) (cid:90) T T e − rt Q ( P t − K t ) + dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) . In a risk-neutral world, the value RO ( T , T ) of a RO written on the time interval [ T , T ] shouldmake the investor indifferent between having the original plant without the RO, and having itwith the RO written on it plus the price of the option, i.e. V ( T , T ) = V ro ( T , T ) + RO ( T , T ) . Therefore, the final result is RO ( T , T ) = V ( T , T ) − V ro ( T , T )= E Q (cid:34) (cid:90) T T e − rt Q ( P t − K t ) + dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) (2)Thus, the value of a reliability option issued by a thermal plant is equivalent to the price of aninsurance contract against price peaks. Interestingly enough, notice that the operating strategy ofthe power plants does not change. In electricity markets, it is well known that perfectly competitivemarkets without CRMs, the so called energy only markets, provide enough incentives to investment,and the same is true for optimally designed CRMs, since the latter simply anticipate ex ante thesupermarginal profits that investors would gain in energy only markets. In other words, theamount of remuneration of capacity accruing from perfectly competitive markets for CRMs equalsthe expected discounted value of the supermarginal profits gained in electricity markets; in a worldwithout market failures, the two levels coincide (see [14, Chapter 22]). This is confirmed in ourframework: without market power, the value of operating the plant is independent of the form ofremuneration of power production, i.e., if revenues accrue ex-ante from the CRM or ex-post fromselling electricity in the market. This is exactly the same argument used to evaluate derivative assets written on non-tradable quantities likeinterest rates, temperature, etc. Pricing of Reliability Options
Equation (2) already allows us to produce model-free no-arbitrage bounds on the price of theRO. No-arbitrage bounds have been derived by [15] for analogous contracts, yet in a differentsetting. In fact, in [15], it is assumed that a continuum of forward contracts is traded, both forelectricity and for the relevant fuel (whose spot price here is K ), which deliver at any given date t . However, many energy markets do not satisfy this assumption, and especially forward contractson electricity, which guarantee the delivery of power over a period (e.g., [ T , T ]), rather than on asingle date t . Even if this does not jeopardize the evaluation mechanism developed in Section 2, asthese forwards written on a period are liquid assets that can be used for hedging, the no-arbitragebounds available in [15] cannot be directly applied to our framework, but must be modified. Noticethat these model-free bounds do not require any assumption on the electricity price apart from P being bounded from below by a constant price floor − P ∗ , with P ∗ ≥
0. This is consistent withthose electricity markets in which negative prices are allowed with a lower bound (as for instancein the German and French markets).We start from the identity ( P t − K t ) + = ( K t − P t ) + + P t − K t . Since 0 ≤ ( K t − P t ) + ≤ K t + P ∗ , we have P t − K t ≤ ( P t − K t ) + ≤ P t + P ∗ . By multiplying the inequalities by e − rt , integrating and taking the expectation, we have that Q E Q (cid:34) (cid:90) T T e − rt ( P t − K t ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) ≤ RO ( T , T ) ≤ Q E Q (cid:34) (cid:90) T T e − rt ( P t + P ∗ ) dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) . The right-hand side represents the forward price of delivering the quantity Q of electricity over theperiod [ T , T ] with an additional constant QP ∗ e − rT − e − rT r , depending on the price floor. Welabel F P (0; T , T ) := E Q (cid:34) (cid:90) T T e − rt P t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) the (unitary) forward price. Then, since RO ( T , T ) ≥
0, when K t ≡ K , i.e. with fixed strike, wecan rewrite the no-arbitrage relation above as Q (cid:18) F P (0; T , T ) − K e − rT − e − rT r (cid:19) + ≤ RO ( T , T ) ≤ QF P (0; T , T )+ QP ∗ e − rT − e − rT r . (3)Thus, the value of a reliability option written on a total capacity Q over the period [ T , T ] liesbetween the intrinsic value of Q call options on the forward F P (0; T , T ) and the modified strike K e − rT − e − rT r , and Q forwards F P (0; T , T ) adjusted by an additional constant proportional tothe price floor P ∗ .Conversely, when K follows itself a stochastic process, we define F K (0; T , T ) := E Q (cid:34) (cid:90) T T e − rt K t dt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) F (cid:35) , and obtain Q ( F P (0; T , T ) − F K (0; T , T )) + ≤ RO ( T , T ) ≤ QF P (0; T , T ) + QP ∗ e − rT − e − rT r . (4) this is alternatively referred to as flow forward or swap , see e.g.[2]. K , the upper bound is unaffected. On the other hand,the lower bound is now the intrinsic value of Q exchange options on the forward F P (0; T , T ) forthe forward F K (0; T , T ).The advantage of these no-arbitrage bounds lies in the fact that, even though there is not aforward contract traded on the market for the total period [ T , T ], this period is usually a multipleof calendar years, whose contracts are commonly traded. For example, in the Italian RO design,the period [ T , T ] starts on January, 1 of year Y and lasts until December, 31 of year Y + 2: thus, F P (0; T , T ) ends up simply being the sum of the three calendar products for the years Y , Y + 1and Y + 2. In the case when the stochastic strike K is indexed with some marginal technologydetermined in advance (e.g. combined cycle gas turbines), analogous forward contracts possiblyexist for the corresponding fuel (gas in this case).The no-arbitrage bounds above are model-free, in the sense that they hold for any no-arbitragemodel that we specify in the following for the dynamics of P , and possibly of K , the only assumptionneeded being the existence of a price floor for P . However, to evaluate the RO as a financialcontract, it is necessary to specify the stochastic process modeling electricity prices. The electricityprice shows peculiarities that make it difficult to model, such as strong seasonality and mean-reversion. For this reason, several processes have been adopted to reproduce the price dynamics.In what follows, we provide semi-explicit formulae to price a RO over [ T , T ] under different pricedynamics. Note that the price models generally used to evaluate options do not allow for negativeprices. We will use models of this kind in the subsequent sections, while allowing for negativeprices in Section 3.6 below. Let us start with the simplest assumption, i.e. that the price of electricity P evolves as a GBM,and that the option’s strike price K is a fixed deterministic value. We stress that the former isan assumption that we already know is unreasonable, in the sense that it cannot be assumed toprovide a realistic representation of the electricity price dynamics. However, it is the simplestpossible assumption that is used to derive explicit pricing formulae for call options. Thus, we treatit as a first simplified approach to help us presenting the main features of the model. In this case,the price P , under the risk-neutral measure Q , is assumed to be the solution of the following SDE: dP t = rP t dt + σP t dB t , (5)where B is a one-dimensional Q -Brownian motion and r is the instantaneous risk-free rate ofreturn.The price of a RO in this case is equivalent to the time integral over the interval [ T , T ] ofa European call option with strike price K and maturity ranging in [ T , T ]. In the followingproposition, we provide a semi-explicit formula to price the RO, under the assumptions above. Proposition 3.1.
Let the reference market price P follow the dynamics (5) . The price of areliability option over the time interval [ T , T ] with fixed strike price K ≥ is given by thefollowing formula: RO ( T , T ) = (cid:90) T T Q (cid:2) P N ( d ( K, P , t )) − e − rt KN ( d ( K, P , t )) (cid:3) dt , (6) where N is the cumulative distribution function (CDF) of a standard Gaussian random variableand d ( K, P , t ) : = 1 σ √ t (cid:20) ln (cid:18) P K (cid:19) + (cid:18) r + σ (cid:19) t (cid:21) ,d ( K, P , t ) : = d ( K, P , t ) − σ √ t . RO ( T , T ) can be defined asthe time integral of a family of call options with the same underlying and strike price, indexedby their maturity in [ T , T ]. Thus, it provides a formula that can be applied to compute thevalue of the RO, once the parameters upon which the call depends on have been set; namely, therisk-free interest rate r , the starting price P and the electricity price volatility σ . A first step to increase the level of complexity consists in modeling the strike price as a stochasticprocess. Recall that, in ROs, the strike price is the marginal cost of the marginal technology.Complex RO schemes can allow it to change over time, according to a predefined rule. For instance,it can be assumed that the strike price is given by the fuel cost of a predefined marginal technology,such as Combined Cycle Gas Turbines. In such a way, the strike price will be linked to a referencefuel price. Alternatively, it can be established that the reference price changes at fixed regulardates according to a given indexing formula, for example monthly, and stays constant in each ofthese sub periods. Both cases imply that the strike price is a stochastic process. Thus, a firstextension of the model defined in Section 3.2 is to model K and P as two (possibly correlated)geometric Brownian motions. This means that the prices ( K t , P t ) t ≥ follow a risk-neutral dynamicsof the following type: (cid:26) dK t = ( r − q k ) K t dt + σ k K t dB t ,dP t = ( r − q p ) P t dt + σ p P t dB t , (7)where ( B , B ) are correlated Q -Brownian motions, with correlation ρ ∈ [ − , P would be more volatile than the strike price K . Finally, it is also possible that the strike price isnegatively correlated with the electricity price, depending on how the strike price is defined andon what reference basket it is linked to. However, this possibility is rather unlikely, for the reasonsmentioned above.The following proposition provides the value of the RO with two GBMs: Proposition 3.2.
Let the reference market price P and the RO strike price K follow the dynam-ics (7) . Then the price of a reliability option over the time interval [ T , T ] is given by RO ( T , T ) = (cid:90) T T (cid:0) P e − q p t N ( a ( K , P , t )) − K e − q k t N ( a ( K , P , t )) (cid:1) dt , (8) where N is the CDF of a standard normal random variable, and a ( K , P , t ) : = ln (cid:16) P K (cid:17) + ( q p − q k ) tσ √ t + 12 σ √ t ,a ( K , P , t ) : = a ( K , P , t ) − σ √ t ,σ : = (cid:113) σ k + σ p − ρσ k σ p = (cid:113) ( σ k − σ p ) + 2(1 − ρ ) σ k σ p . Interestingly enough, this result solves also a problem firstly posed in [24], in the framework of firms’ evaluations. As mentioned, this is going to be the case of the future Italian RO scheme.
8n contrast to Proposition 3.1, in Proposition 3.2 we used the Margrabe formula with dividends(see, for instance, [10]), instead of the Black-Scholes one. Here, the RO ( T , T ) value is equal to thetime integral of a family of options to exchange the (random) electricity price P with the (random)strike price K , again indexed by their maturity. As usual in the Margrabe formula, the relevantvolatility is σ , that can be interpreted as the volatility of the ratio P/K (i.e., of the electricityprice expressed in units of the strike price), which is decreasing with respect to the correlation ρ .In particular, for ρ → σ → | σ k − σ p | . In this case, when also σ k = σ p , the volatility vanishes, and the valueof the option is determined just by its intrinsic value. Instead, for ρ → − σ → σ k + σ p , i.e., thevolatility is maximized. However, we stress that this latter case is rather unlikely for the case ofRO, as typically a stochastic strike price K is defined in terms of quantities related to electricitygeneration (as e.g. the marginal price of the marginal technology, or some related market index),so that we should expect a positive correlation. As mentioned, a GBM does not capture typical stylized facts of electricity prices, namely seasonalityand mean-reversion. A natural extension is thus to price the RO when the dynamics of thereference price reflects the aforementioned stylized facts. In particular, we model the log-spotprice of electricity as a mean-reverting process encoding different types of seasonality by meansof a time-dependent function. This approach has been widely adopted in energy markets, see forinstance [2] and references therein. We first assume a deterministic strike price. In the next section,we shall remove this assumption.We define the function describing seasonality trends for all t ≥
0, as µ ( t ) = α + (cid:88) i =1 β i month i ( t ) + (cid:88) i =1 δ i day i ( t ) + (cid:88) i =1 γ i hour i ( t ) , (9)where month i ( t ), day i ( t ) and hour i ( t ) are dummies for month, day of week and hour, used tocapture different types of seasonality. Specifically, we assume that day can take 4 values: ‘Friday’,‘Weekend’, ‘Monday’, and ‘other working day’. This captures the differences between working daysand weekend as well as possible first- or end-of-the-working-week effect.We then consider the day-ahead price P as P t = e µ ( t ) e X t , (10)where X t , under the risk-neutral measure Q , is the solution of the SDE dX t = − λX t dt + σdW t , (11)where W is a one dimensional Q -Brownian motion, σ stands for the volatility and λ > Proposition 3.3.
Let the reference market price P follow the dynamics (9) – (10) – (11) . Then theprice of a reliability option over the time interval [ T , T ] with fixed strike price K ≥ is given by RO ( T , T ) = Q (cid:90) T T e − rt [ f (0 , t ) N ( d ( K, P , t )) − KN ( d ( K, P , t ))] dt , (12)9 here N is the CDF of a normal random variable, P = e µ (0)+ X and f (0 , t ) : = E [ P t |F ] = exp (cid:18) µ ( t ) + e − λt + 12 V ar ( t ) (cid:19) ,V ar ( t ) : = σ λ (1 − e − λt ) ,d , ( K, P , t ) : = 1 (cid:112) V ar ( t ) log f ( t, T ) K ± (cid:112) V ar ( t ) , where, by abuse of notation we mean that the definition of d ( K, P , t ) involves the + sign and thedefinition of d ( K, P , t ) involves the − sign.Remark . Equation (12) is a generalization of Equation (6): in fact, if we let µ ( t ) := ( r − q p − σ ) t and λ →
0, then we reobtain at the limit the model of the previous section. In fact, we havethat m t ≡ X , V ar ( t ) → σ t , e − rt f (0 , t ) → e ( r − q p ) t + X , and d ( K, P , t ) → σ √ t (cid:18) X + ( r − q p ) t − σ t − ln K (cid:19) = 1 σ √ t ln e X +( r − q p ) t K − σ √ t . Thus, the pricing formula in Equation (12) collapses into that of Equation (6).
As a natural extension of the model in Section 3.4, we now consider the case when the strike K is a mean-reverting process (with seasonality) as well. The dynamics of the state variables thenbecomes (cid:26) P t = e µ ( t ) e X t ,K t = e ν ( t ) e Y t . (13)Here, µ is given by (9) and ν is a seasonality function for K of the same form, while the processes X and Y are solution to (cid:26) dX t = − λ x X t dt + σ x dW t ,dY t = − λ y Y t dt + σ y dW t , (14)where ( W , W ) are correlated Q -Brownian motions, with correlation ρ ∈ [ − , Proposition 3.4.
Let the reference market price P and the RO strike price K follow the dynam-ics (13) ; then the price of a reliability option over the time interval [ T , T ] is given by RO ( T , T ) = Q (cid:90) T T e − rt ( f P (0 , t ) N ( d ( K , P , t )) − f K (0 , t ) N ( d ( K , P , t ))) dt , (15) where N is the CDF of a normal random variable, P = e µ (0)+ X , K = e ν (0)+ Y and f P (0 , t ) : = E [ P t |F ] = exp (cid:18) µ ( t ) + e − λ x t + σ x λ x (1 − e − λ x t ) (cid:19) , (16) f K (0 , t ) : = E [ K t |F ] = exp (cid:32) ν ( t ) + e − λ y t ) + σ y λ y (1 − e − λ y t ) (cid:33) , (17) d , ( K , P , t ) : = 1 (cid:113) V ar ( t ) log f P (0 , t ) f K (0 , t ) ± (cid:113) V ar ( t ) , (18) V ar ( t ) : = σ x − e − λ x t λ x + σ y − e − λ y t λ y − ρσ x σ y − e − ( λ x + λ y ) t λ x + λ y . (19)10his result is a similar to that of Proposition 3.3 in the same sense as Proposition 3.2 is similarto Proposition 3.1: here RO ( T , T ) can be again defined as the time integral of a family of optionsto exchange the electricity price P with the strike price K . Here too, the relevant volatility is V ar ( t ), which can again be interpreted as the volatility of the ratio P/K (i.e., the electricity priceexpressed in units of the strike price: this is made explicit in the proof in the Appendix), which isagain decreasing with respect to the correlation ρ . In particular, for ρ → λ x = λ y =: λ (i.e. when the two mean-reversion speeds are the same), we have V ar ( t ) → − e − λt λ ( σ x − σ y ) . In this case, when σ x = σ y ,the volatility vanishes, and the value of the option is given just by its intrinsic value. Instead, inthe unlikely case (see the discussion at the end of Section 3.3) when ρ → − λ x = λ y =: λ ,we have V ar ( t ) → − e − λt λ ( σ x + σ y ) , i.e., the volatility is maximized. In principle, it is possible to allow for negative power prices, since we know this is a possibilityin energy markets, see [16] and references therein. An analogous extension can be also envisagedfor strike prices, especially when these are linked to power prices. A possible approach to modelnegative prices is to set negative values − P ∗ and − K ∗ , for certain P ∗ , K ∗ ≥
0, as price floors for P and K , respectively, and to consider the following shifted dynamics (cid:26) P t = (cid:0) e µ ( t ) e X t − P ∗ (cid:1) ,K t = (cid:0) e ν ( t ) e Y t − K ∗ (cid:1) . (20)where µ and ν are again seasonality functions for P and K and the processes X and Y are solutionof Equation (14), in analogy with the previous section.By setting C := P ∗ − K ∗ , one can prove that the price of the reliability option is now given bythe following expression: RO ( T , T ) = Q (cid:90) T T e − rt E Q (cid:104) ( e µ ( t ) e X t − e ν ( t ) e Y t − C ) + (cid:12)(cid:12)(cid:12) F (cid:105) dt . (21)The above formula is the time integral of a family of spread options with a fixed strike price C andindexed by their expiration date in [ T , T ]. Therefore, considering dynamics of type (20) relatesthe problem of pricing a Reliability Option to the problem of pricing a spread option (see [10] fora survey of classical frameworks and methods for spread options). Unfortunately, a general closedformula for the pricing of spread options is not available. However, since the RO is in principle aquite illiquid product, one can use a numerical method to price it in this general case, for exampleMonte Carlo. In this section we simulate the value of the RO under realistic assumptions on the parametervalues. To do so, we fit the parameters of the electricity price dynamics to a real market, usingdata of the Italian market. For simplicity, we consider day-ahead prices only, and use the weightedaverage of Italian zonal prices, called PUN (
Prezzo Unico Nazionale ), ranging from January 1 toDecember 31, 2016.As previously explained, we used dummies to capture monthly, daily and hourly seasonality,as defined in Eq. (9). We chose ‘January’, ‘Friday’ and ‘hour 1’ as reference groups, against whichthe comparisons are made. Figure 1 shows the calibrated seasonality function, plotted against thehistorical PUN data. Furthermore, we considered an annual risk-free rate r = 0 .
01 and, in thepricing models where the only stochastic variable is the electricity price, we considered K = 40 (cid:164) /MWh. According to the scheme to be implemented in Italy, the pricing of the RO starts 4 yearsfrom now, and the option has a maturity of 3 years ( T = 4 , T = 7).11he starting point X is taken equal to 0. Table 1 reports the estimated parameters for eachdifferent model, while Table 2 shows the estimated seasonality parameters. Figure 1:
Seasonality function in (9) (solid red line, upper panel) calibrated on historical 2016 PUNelectricity data (solid blue line, upper panel) and residuals (bottom panel).GBM 1-OU 2-OUˆ σ λ - 294.84 294.84 Table 1:
Estimated yearly parameters ˆ σ and ˆ λ for each pricing model (electricity price following a Geo-metric Brownian motion (GBM), electricity price following a mean-reverting Ornstein-Uhlenbeck process(1-OU), correlated electricity and strike prices following mean-reverting Ornstein-Uhlenbeck processes (2-OU)). stimate S.E. pValue Estimate S.E. pValueIntercept 3 .
79 0 .
01 0 hour − .
13 0 .
01 0 month − .
22 0 .
01 0 hour − .
01 0 .
01 0 . month − .
27 0 .
01 0 hour . .
01 0 month − .
36 0 .
01 0 hour .
18 0 .
01 0 month − .
28 0 .
01 0 hour .
16 0 .
01 0 month − .
23 0 .
01 0 hour .
12 0 .
01 0 month − .
07 0 .
01 0 hour .
07 0 .
01 0 month − .
21 0 .
01 0 hour .
01 0 . month − .
07 0 .
01 0 hour − .
05 0 .
01 0 month .
14 0 .
01 0 hour − .
02 0 .
01 0 . month .
23 0 .
01 0 hour .
04 0 .
01 0 month .
21 0 .
01 0 hour .
09 0 .
01 0Monday − .
01 0 .
01 0 . hour .
15 0 .
01 0Weekend − .
14 0 .
01 0 hour .
22 0 .
01 0Working day 0 .
02 0 .
01 0 hour .
28 0 .
01 0 hour − .
08 0 .
01 0 hour .
27 0 .
01 0 hour − .
15 0 .
01 0 hour . .
01 0 hour − .
18 0 .
01 0 hour .
12 0 .
01 0 hour − .
18 0 .
01 0 hour .
03 0 .
01 0 . Table 2:
Linear regression estimates, standard errors and p-values obtained using the specification in (9).The base group categories for each dummy variable are month , friday and hour . As mentioned, real electricity prices do not follow GBMs. Therefore, in the simulation, we startfrom the model defined in Section 3.4.
We simulate the value of the RO using the Monte Carlo methodology. Specifically, we computethe RO value using 10,000 simulations of the price path of the underlying.Figure 3 shows the comparative statics for different ranges for the parameters σ and λ andstrike price K . As expected, the higher the strike price, the lower the value of the reliability optionfor each value of σ (left panel). On the other hand, both the left and right panels show that, when σ increases, the RO value rises as well. Moreover, when λ is low, the relative increase in the ROvalue is high (right panel). This is consistent with the fact that a low λ allows fluctuations of theunderlying that are far from the long term mean to be more persistent. We simulate now the value of the RO using the model described in Section 3.5, again by means ofa Monte Carlo method (again using 10,000 runs). We start from a given correlation coefficient, setat ρ = 0 .
5, and assume that λ K and σ K are equal to the ones estimated for the electricity priceand X = 0. In line with the PUN mean price, which is equal to 42.77 (cid:164) /MWh, K is arbitrarily13 igure 3: Sensitivity analysis of the results using a yearly σ in the range (0; 2ˆ σ ] with a strike price K inthe range [20; 60] (left panel), and a yearly σ in the range (0; 2ˆ σ ] with and a yearly λ in the range (100; 2ˆ λ ](right panel). The RO value is expressed in (cid:164) /MWh. chosen equal to 40 (cid:164) /MWh, so that, after de-seasonalizing (using the same estimated seasonalityparameters of the PUN price), we obtain Y = − . P . The upper left panel shows that the initial level of the strikeprice K has no influence on the value of the reliability option. This is due to the magnitude of theestimated λ P , and thus of λ K : a mean reversion speed as high as that estimated makes the strikeprice process return to its mean level in an amount of time negligible with respect to the maturity.This implies that the starting point of the process has no relevant impact on the RO value.The upper right panel of Figure 4 instead shows how sensitive the RO value is to changes inthe electricity price parameters λ P and σ P (and thus in turn in λ K and σ K ). Similarly to whatwe have observed before, the higher the volatility of the underlying (and, in this case, of the strikeprice), the higher the RO value. This relationship increases in proportionality as the speed of meanreversion decreases, since it takes more time to return to the mean, and thus volatility mattersmore.The impact of the correlation factor ρ is instead investigated in the bottom left panel, wherewe assess how different correlation factors in the range [ −
1; 1] affect the price of the reliabilityoption. When the two assets are perfectly correlated ( ρ = 1), the RO value is zero for all levelsof σ P . In fact, as seen in Section 3.3, the volatility is minimized and the RO can be interpretedas an integral of calls, with maturity ranging in the interval [ T , T ], being exactly at the moneyat the time of expiration, and thus having zero value. Instead, as shown, when the two processesare uncorrelated, the level of risk increases, and it reaches its maximum when they are perfectlynegatively correlated. In this case, the volatilities of the two Brownian motions sum up, increasingthe volatility of the option payoff and minimizing the risk of having the calls at the money. Finally,the bottom right panel shows that the RO price is negatively correlated with the risk free rate r :a higher r decreases the option value as it lowers the discounted cash flows.In the previous figures, the parameters for λ P and λ K , and σ P and σ K , were tied together,in the sense that λ K and σ K were always equal to, respectively, λ P and σ P . Instead, we nowinvestigate what happens when σ K equals σ P as before, but λ K changes independently from λ P .Moreover, we also investigate the effects of a variation in σ P different from that in σ K . Figure 5and 6 show the results.The left panel of Figure 5 reports the results for a variation in λ K (in the range (0; 2ˆ λ P ] andshown in log scale) independent from the value of λ P . The graph shows how K hardly affectsthe RO value, as it has an impact only when both σ K and λ K are sufficiently small. This confirmsthe result shown above that the initial condition of the parameters matters only when it takes a14 igure 4: Sensitivity analysis of the results using a yearly σ P in the range (0; 2ˆ σ P ] with an initial strike K in the range [20; 100] (upper left panel), with a yearly λ P in the range (100; 2ˆ λ P ](upper right panel),with a correlation ρ in the range [ −
1; 1] (left bottom panel) and with a yearly risk free rate r in the range[0; 0 .
2] (right bottom panel). sufficient amount of time for them (i.e., for the strike price in this case) to return to their longterm value. The right panel instead shows the sensitivity of the RO value to changes in the yearly λ K (again in the range (0; ˆ λ P ]) independent from the value of λ P , and in the correlation factor ρ (in the range [ −
1; 1]) (in this graph, σ K is always equal to σ P and they are in turn equal to ˆ σ P , λ P = ˆ λ P , and λ K is shown in log scale.). Here, the ρ value matters more when both λ K = λ P and σ K = σ P . In fact, ρ (negatively) affects the RO value only when it tends to − λ K iscloser to the value of λ P (note that, in the figure, λ K ∈ (0; ˆ λ P ], where ˆ λ P corresponds to the valueof 2 .
47 in log scale). This confirms our intuition that, when the initial value of the electricityprice and the strike price are close and the two random variables follow the same dynamics, theRO has a negligible value since it is likely that it will be always at-the-money. Conversely, if thetwo random variables are not perfectly correlated or the two variables follow different dynamics,it is unlikely that at every point in time P t and K t coincide, and this adds value to the RO.15 igure 5: Sensitivity analysis of the results using a yearly λ K in the range (0; ˆ λ P ] with an initial strikeprice K in the range [20; 100], both with a yearly σ K equal to the yearly σ P (upper left panel) and withand a scaled down yearly σ K (upper right panel), and with a correlation ρ in the range [ −
1; 1] (bottompanel) (here σ K = σ P ). The RO value is expressed in (cid:164) /MWh. Finally, Figure 6 shows the effect of a disjoint variation in the two volatilities, with a yearly σ P and σ K in the range (0; 2ˆ σ P ], for different levels of ρ (in these graphs, λ K is always equal to λ P = ˆ λ P ). When ρ ≤
0, the RO price is always increasing in the electricity price volatility σ P andin the strike price’s one σ K . This is as expected, since volatility adds value to the call options.Instead, when ρ >
0, the fact that the two processes move together can lower the aggregate risk,since the spread between the electricity price and the strike price reduces. This translates into anegative effect on the option value. The RO value is minimized when σ P = σ K . In Figure 6, panel ρ = 0 .
5, we can see that the option value is still positive; in the panel ρ = 1, the RO value becomesnull for σ P = σ K , since, as mentioned, if the two processes are perfectly positively correlated, theRO value coincides with its intrinsic value. Thus, there is a non-monotone effect of the volatilityincrease of one process, depending on the amount of volatility of the other process, and on thelevel of the correlation coefficient. The inflection is maximum when the two processes are perfectlypositively correlated. In this paper, we have studied the value of the RO from a financial perspective. The financialapproach to option pricing relies on the assumption that a risk-neutral measure exists, which isequivalent to assume that markets are complete. This is not a problem for pricing options onelectricity prices, as long as they can be written on electricity futures that can be rolled overthe delivery period of the RO. Nevertheless, such an approach does require that RO markets arecompetitive and that forward markets are liquid. Our analysis provides a benchmark value forthe RO, under the assumption that the market for the derivative is liquid enough to bring aboutcompetition. Therefore, the simplified mathematical model that we proposed can be seen as astarting point in the analysis of ROs. We obtain semi-explicit formulae for the value of the RO,under a set of different assumptions with increasing realism and complexity. We move from simpleintegrals of call options written on GBMs to correlated mean reverting processes that capture thebehavior of realistic electricity price time series, on the one hand, and complex rules for RO, onthe other. Then, we simulate the value of the Reliability Option through a real-market calibration Note that according to [6, 12] ROs are instruments that enhance competition in the electricity market. igure 6: Sensitivity analysis of the RO value to a disjoint variation in the two volatilities, with a yearly σ P and σ K in the range (0; 2ˆ σ P ] (here λ K = λ P ). In the different panels, we can see how a variation inthe correlation coefficient ρ affects the RO value: when the two processes are independent or negativelycorrelated, higher σ P and σ K result in a higher option value. However, when the correlation is positive(middle right and bottom panels), the higher the correlation, and the more the two volatilities are similar,the lower the value of the option. The RO value is expressed in (cid:164) /MWh.
17f the parameters.The results evidence that the value of the RO moves consistently with expectations from optiontheory: a rise in the strike price lowers the RO value, which depends positively on the volatilityof the electricity price, as well as on the volatility of the strike price itself. The mean reversionspeed of the processes reduces the impact of the starting point, which was another expected result.However, when both the strike price and the electricity price are assumed to be stochastic processes,the value of the RO depends crucially on their correlation coefficient ρ . In particular, a positivecorrelation reduces the value of the RO. Moreover, there is a non-monotone impact of the volatilityof one process, depending on the level of volatility of the other process and on a positive correlation.This is important when designing the rule of the RO. For instance, if the strike price is allowed tochange with respect to a reference marginal cost, which is also believed to be the technology settingthe system marginal price at the day ahead level, the two process clearly covariate positively. Inthis case, it is very likely that a RO has a very limited value, for every possible starting valueof the state variables P and K . More in general, our results show that a careful estimate of theparameters is needed to calculate the value of the ROs. Ceteris paribus , the RO value will belower as the volatility of the electricity price decreases, the strike price increases, the speed ofmean reversion increases, the correlation of the electricity price with the strike price increases (ifthe strike price is allowed to change over time), and the two volatilities are closer. These are allfactors that need to be taken into account when designing the market for ROs and calculating theequilibrium value.
Acknowledgments . Part of this work was done while the first author had a post-doc positionunder the financial grant “Capacity markets and the evaluation of reliability options” from theInterdepartmental Centre for Energy Economics and Technology “Giorgio Levi-Cases”, Universityof Padua, which she gratefully acknowledges, and while the second author was completing her PhDat the University of Padua. The authors also wish to thank Paolo Falbo, Giorgio Ferrari, MicheleMoretto, Filippo Petroni, Marco Piccirilli and Dimitrios Zormpas for interesting discussions, andall the participants to the 2nd Conference on the Mathematics of Energy Markets in Vienna, XLIA.M.A.S.E.S. Conference in Cagliari, Energy Finance Christmas Workshop 2017 in Cracow, XIXWorkshop in Quantitative Finance in Rome, Energy Finance Italia 3 in Pescara, 3rd C.E.M.A.Conference in Rome, XLII A.M.A.S.E.S. Conference in Naples.
Appendix
A.1 Proofs of pricing formulae
Proof of Proposition 3.1.
The quantity f ( s, ω ) : = e − rs Q ( P s ( ω ) − K ) + in Equation (2) is non-negative.Then, if we set A ( K, P , s ) : = e − rs E Q (cid:104) ( P s − K ) + (cid:12)(cid:12)(cid:12) F (cid:105) , (A.1)by Tonelli’s theorem, we get RO ( T , T ) = Q (cid:90) T T A ( K, P , s ) ds . (A.2) A ( K, P , s ) is clearly the price of a European call option with strike price K and maturity s , thusEquation (6) is simply obtained with the Black and Scholes formula. Proof of Proposition 3.2.
As in the proof of Proposition 3.1, if we write A ( K , P , s ) : = e − rs E Q (cid:104) ( P s − K s ) + (cid:12)(cid:12)(cid:12) F (cid:105) , (A.3)18hen, by Tonelli’s theorem, we have RO ( T , T ) = Q (cid:90) T T A ( K , P , s ) ds . Here, A ( K, P , s ) is the price of an exchange option between the electricity price P and the strikeprice K , with maturity s , thus Equation (8) is simply obtained with the Margrabe formula withdividends (see [10]). Proof of Proposition 3.3.
As in the previous proofs, we write A ( K, P , s ) : = e − rs E Q (cid:104) ( P s − K ) + (cid:12)(cid:12)(cid:12) F (cid:105) and apply Tonelli’s theorem to obtain RO ( T , T ) = Q (cid:90) T T A ( K, P , s ) ds . We now notice that A ( K, P , s ) = e − rs E Q (cid:104) ( f ( s, s ) − K ) + (cid:12)(cid:12)(cid:12) F (cid:105) where f ( t, s ), t ∈ [0 , s ], has the dynamics df ( t, s ) = f ( t, s ) σe − λ ( s − t ) dW t The result then follows from the Black-Scholes formula with time-dependent (deterministic) volatil-ity, which enters into the formula via the integral of its square, here equal to (cid:90) s (cid:16) σe − λ ( s − t ) (cid:17) dt = σ λ (1 − e − λs ) = V ar ( s )Equation (12) follows. Proof of Proposition 3.4.
As before, we write A ( P , K , s ) : = e − rs E Q [ ( P s − K s ) + | F ], we useTonelli’s theorem and obtain RO ( T , T ) = Q (cid:90) T T A ( K, P , K , s ) ds . Now, as in the proof of Proposition 3.3, we now notice that A ( K, P , s ) = e − rs E Q (cid:104) ( f P ( s, s ) − f K ( s, s )) + (cid:12)(cid:12)(cid:12) F (cid:105) where f i ( t, s ), t ∈ [0 , s ], I = P, K , have the dynamics df P ( t, s ) = f P ( t, s ) σ x e − λ x ( s − t ) dW t ,df K ( t, s ) = f K ( t, s ) σ y e − λ y ( s − t ) dW t , The result then follows from the Margrabe formula with time-dependent (deterministic) volatilities,which now enters into the formula via the integral of the squared volatility of f p ( · , s ) /f K ( · , s ) (seee.g. [15]), here equal to (cid:90) s (cid:16) σ x e − λ x ( s − t ) + σ y e − λ y ( s − t ) − ρσ x σ y e − ( λ x + λ y )( s − t ) (cid:17) dt = V ar ( s )Equation (15) follows. 19 eferences [1] Alexander. C. Hedging the risk of an energy futures portfolio . In: Risk management in com-modity markets – from shipping to agriculturals and energy. H. Geman (ed.). Wiley Financeseries. John Wiley & Sons, 2008[2] Benth, F. E., Benth, J. ˇS. and Koekebakker, S. (2008).
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