1 Introduction

In many countries electricity is traded on exchanges, with prices determined by supply and demand. Such markets usually are quite liquid and in many regards comparable with financial markets. However, still there are unique frictions, not existent on financial markets (or other commodity markets): in particular, electricity is produced from fuels but cannot be converted back to fuels. Electricity also cannot be stored in large quantities at the time being. Furthermore, all kinds of restrictions on physical fuel storage and generation capacity are relevant for the production process. Finally, produced and used electric power has to be balanced immediately in an electrical network, because deviations may lead to damaged equipment or even breakdown of the net.

Still, the notions of arbitrage and market completeness—cornerstones of modern finance—can be applied also to electricity markets. Basically, a market is arbitrage free if riskless profits are not possible and it is complete if any relevant payoff can be replicated from the basic traded securities (“underlyings”), contracts or commodities, traded at this market. Moreover, a financial market is arbitrage free if and only if there exists an equivalent (local) martingale measure, such that all basic securities can be priced by taking expectation of their future discounted prices with respect to this measure, i.e. the discounted prices are martingales under the equivalent measure. An arbitrage free financial market is complete if and only if there is a unique martingale measure. As a consequence, on complete markets, every contingent claim is attainable by dynamic hedging, and financial derivatives can be priced by calculating the expected discounted value of the derivatives payoff with respect to the unique martingale measure. On incomplete markets there are claims that can not be replicated by the basic traded assets. Still, under absence of arbitrage there exist equivalent martingale measures, but uniqueness does not hold any more.

Because of the discussed frictions, electricity markets are not complete. Some submarkets for electricity, in particular futures markets, are organized like financial markets. However, the delivery profiles of traded futures usually cannot fully replicate physically traded delivery profile (although hedging with futures contracts is important in practice, see e.g. Deng et al. 2001).

The question remains, whether electricity markets or related market models, used by decision makers, are arbitrage free. This is not only a theoretical question, but is important both for valuation of contracts and also for taking model based management decisions. Valuation of contracts usually has to be based at least on the assumption that arbitrage is not possible. Moreover in a decision making context, facing a market with arbitrage possibilities decision makers clearly would like to exploit this. However, in reality for most market participants markets usually are arbitrage free (or arbitrage possibilities are unexploitable because of costs or other frictions). If the management of a firm then uses a market model that is not arbitrage free (which could e.g. happen by using flawed price models or by discretizing continuous state price models in a wrong way), there is a danger that the resulting decisions try to exploit arbitrage possibilities that are present in the model but not in reality. This is especially the case if problematic price models are used to parametrize optimization models, see e.g. Geyer et al. (2010) for a discussion in the context of tree-based stochastic optimization in finance. When this happens, high expectations are deceived, and it is likely that the implemented decisions lead into a completely wrong direction.

In Kovacevic (2018), no-arbitrage conditions for an electricity market with electricity generation from fuel and fuel storage were derived analytically, and the results were used to find valuation and pricing formulas for electricity delivery contracts. The present work goes back one step and aims at a deeper discussion of arbitrage properties in a more general setup. This includes several generating units with different production efficiency and takes into account the effects of storage costs for fuel. In a first step the results are based on duality theory for cone-constrained optimization in Banach spaces and it turns out that existence of a martingale measure (as in finance) has to be replaced by more complex requirements.

The next question then is, how restrictive the no-arbitrage conditions are in a more concrete—model based—setup. In particular, when estimating parameter values of an econometric model from data, one may ask whether the no-arbitrage conditions put any restrictions on the parameters (and hence the estimation). In the present work these questions are analyzed for price processes with (potentially nonlinear) vector-autoregressive structure. Moreover, the consequences for the valuation of electricity delivery contracts and for the construction of scenario trees for stochastic optimization are discussed.

The work is organized as follows: Sect. 2 uses a basic optimization problem to derive and analyze no-arbitrage conditions for a model with spot prices for fuel and electricity when electricity can be produced with given efficiency. Several generating units as well as storage costs are considered. A necessary and a sufficient condition are derived. In Sect. 3 we analyze the problem of testing the assumption of no-arbitrage in such a market based on the necessary, respectively the sufficient condition. Section 4 gives an outlook of applications of the obtained results to the valuation of delivery contracts and to tree construction for stochastic optimization. Finally, Sect. 5 concludes the work.

2 No-arbitrage conditions for an electricity market with production and storage

In the present section, conditions for absence of arbitrage between fuel and electricity prices when it is possible to produce electrical energy from fuel, are derived. The analysis starts with describing a basic discrete time, general state space framework (in Banach spaces). In the present section the word “market” may refer as well to a market in the economical sense as to a market model (e.g. an econometric price model). Insofar economical markets are considered, it is assumed that a potential producer acts as a price taker at both markets, fuel and electricity. The resulting formulation does not aim at modeling an economic market equilibrium. Instead, as usual in financial economics, the simpler question which properties of price processes and (in the case of electricity production) production equipment allow or prohibit arbitrage is in the forefront. In particular, the presented approach does not state, whether, how or after which time span arbitrage possibilities are removed (when detected). In a production context arbitrage (when observed) may persist some time, because adjusting the production capacities (which finally would lead to price adjustment) is expensive and takes time. This is the reason why in the present work arbitrage is defined with respect to production efficiencies, which takes into account that some producers with very efficient generators might be able to use arbitrage possibilities while this is not possible with smaller efficiency.

Subsequently, the analysis is based on a definition of arbitrage which is a slight modification of the standard approach from finance but is adapted to dealing with electricity production from fuel, including the handling of fuel storage. It is then possible to use an optimization problem as an equivalent test tool for implementing the rather unwieldy original definition. In the main part of this section, duality theory in Banach spaces is used to derive convenient conditions in terms of stochastic discount factors, which are formulated directly as properties of fuel and electricity prices and can be compared to standard financial results.

The formulation uses discrete points in time \(t\in \{0,1,2,\dots ,T\}\). At these times t all relevant information is observed and decisions are taken. Time 0 represents the begin (“here and now”) and T denotes the end of the planning horizon. For simplicity constant time increments, e.g. hours, days or weeks, are used. In order to analyze arbitrage properties, a stochastic process \(X_{t}^{f}(\omega )\) [currency units/MWh] of fuel prices and a stochastic process \(X_{t}^{e}(\omega )\) [in currency units/MWh] of electricity prices are considered. Both price processes are defined on a filtered probability space \(\mathfrak {Y=}\left( \Omega ,\mathcal {\mathcal {F}},\mathfrak {F}=\left\{ \mathcal {F}_{t}\right\} _{t\ge 0},\mathbb {P}\right) \) in discrete time \(t=0,1,\dots ,T\). Each \(\sigma \)-algebra \(\mathcal {F}_{t}\) represents the information available at time t. At the beginning, the \(\sigma \)-algebra \(\mathcal {F}_{0}\) is the trivial \(\sigma \)-algebra, i.e. \(\mathcal {F}_{0}=\left\{ \emptyset ,\Omega \right\} \). The filtration \(\mathfrak {F}\) may be generated by the price processes, but this is not a necessary requirement. To simplify the notation the sets \(\mathcal {T}=\{0,1,\dots ,T\}\), \(\mathcal {T}_{0}=\{0,1,\dots ,T-1\}\), \(\mathcal {T}_{1}=\{1,\dots ,T\}\) and \(\mathcal {T}_{1}^{T-1}=\left\{ 1,\dots ,T-1\right\} \) are used in the following. As in reality, fuel prices are assumed to be almost surely nonnegative while electricity prices may be negative with positive probability.

Both, the energy content of the fuel and electrical energy are measured in MWh (energy content). Immediately before taking decisions at time t, the producer owns a cash position \(c_{t}\) with associated interest rate \(r\ge 0\) (per period) and an amount of fuel \(s_{t}\) [MWh]. Instead of the interest rate r, often the compounding factor \(R=(1+r)\) is used. At each point in time t the producer decides first the amount \(z_{t}\) [MWh] of fuel traded on the fuel market at price \(X_{t}^{f}\). This trade happens at (or immediately after) time t. Positive values of \(z_{t}\) indicate that an amount of fuel is bought, negative values indicate selling of fuel. Electricity is produced by generators \(i\in \{1,2,\dots ,I\}.\) Generator i produces with efficiency \(\eta _{i}\) and the set of efficiencies is denoted by \(\eta =\left( \eta _{1},\dots ,\eta _{I}\right) \). The amounts \(y_{it}\) [MWh] of electricity produced with generators i over period \([t,t+1]\) is planned in advance at time t. It is sold at time \(t+1\) at price \(X_{t+1}^{e}\), immediately before calculating the new cash positionFootnote 1 Hence the amount of fuel burned for producing electricity is given by \(\sum _{i}\eta _{i}^{-1}y_{it}\) [MWh]. Electricity production \(y_{t}\) and fuel storage \(s_{t}\) are almost surely nonnegative.

The producer can store fuel, e.g. by using the facilities of an oil & gas storage service company (above ground reservoirs, pipe storage or underground storage like salt caverns), which causes costs. In the present work, a very simple cost model is considered, namely storage costs \(\Psi _{t}\) (payable at time t) that are proportional to the stored amount, in particular the specification

$$\begin{aligned} \Psi _{t}=\psi \frac{s_{t}+s_{t-1}}{2}, \end{aligned}$$

is used, where \(\psi \ge 0\) is the related cost factor [currency units/MWh]. This additional simplification can be interpreted as an assumption that storage is filled (respectively emptied) uniformly over any period \([t-1,t]\).

Startup costs are neglected in the present work (or rather it is assumed that they are approximately independent from the implemented production plan). Even if such assumptions may be easily violated in practice, it is clear that if arbitrage is not possible in a model without startup costs then arbitrage is also not possible when arbitrage costs are added.

In the rest of the section, to simplify notation, all equations and inequalities involving random variables are assumed to hold almost surely. It is assumed that fuel prices and electricity prices are essentially bounded, i.e. \(X_{t}^{f},X_{t}^{e}\in L^{\infty }\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \). Later on, the notation \(X_{[t]}^{f},X_{[t]}^{e}\) will be used in order to denote relevant price histories up to time t. The decision processes \(y_{t}\) and \(z_{t}\) and the decision processes \(c_{t}\) and \(s_{t}\) are considered as real-valued random processes defined on \(\mathfrak {Y}\) and it is assumed that they are integrable, i.e. \(y_{t},z_{t},c_{t},s_{t}\in L^{1}\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \). In particular, they are also adapted to the filtration \(\mathfrak {F}\), so decisions at time t are based only on information available at (up to) this time. Because \(\mathcal {F}_{0}\) is the trivial \(\sigma \)-algebra, the starting values \(c_{0},s_{0},y_{\text {0}},z_{0}\) are deterministic decisions. For the subsequent optimization problem this setup allows to use Lagrange multipliers from \(L^{\infty }\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \), which can be identified with the dual space of \(L^{1}\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \).

Based on this setup it is possible to adapt the standard concepts of a self financing strategy in the following way:

Definition 2.1

A strategy \(\left\{ y_{t},z_{t}\right\} _{t\ge 0}\) with a cash position \(c_{t}\) and fuel storage \(s_{t}\), where \(y_{t}\ge 0\) and \(s_{t}\ge 0\), is self financing if the following conditions hold almost surely for all \(t\in \mathcal {T}_{1}\):

$$\begin{aligned} c_{t}&=\left( c_{t-1}-z_{t-1}X_{t-1}^{f}\right) R+X_{t}^{e}\sum _{i=1}^{I}y_{it-1}-\psi \frac{s_{t}+s_{t-1}}{2}, \end{aligned}$$
(2.1)
$$\begin{aligned} s_{t}&=s_{t-1}-\sum _{i=1}^{I}y_{i\,t-1}\eta _{i}^{-1}+z_{t-1}. \end{aligned}$$
(2.2)

At any time t the asset value of a strategy is given by

$$\begin{aligned} V_{t}^{\eta }=c_{t}+X_{t}^{f}\cdot s_{t} \end{aligned}$$

The first equation models a cash position (an interest paying account) which changes when fuel is bought, electricity is sold or storage costs are paid. The second equation is related to a fuel storage, which is reduced when electricity is generated and which is filled up by buying fuel on the market.

Using the notion of a self financing strategy, the classical definition of arbitrage (see e.g. Björk 2009 definitions 2.14, 2.15) can be adapted to the current context. As already discussed, arbitrage is defined with respect to production efficiencies.

Definition 2.2

An \(\eta \)-arbitrage for a market \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \) is a self financing strategy \(\left\{ y_{t},z_{t}\right\} _{t\ge 0}\) with

$$\begin{aligned} V_{0}^{\eta }&\le 0, \end{aligned}$$
(2.3)
$$\begin{aligned} \mathbb {P}\left( V_{T}^{\eta }\ge 0\right)&=1. \end{aligned}$$
(2.4)
$$\begin{aligned} \mathbb {P}\left( V_{T}^{\eta }>0\right)&>0. \end{aligned}$$
(2.5)

A market \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \) is called \(\eta \)-arbitrage free, if no \(\eta \)-arbitrage exists.

This definition is hard to implement directly, especially because of the third condition. However, based on the definition, it is possible to formulate the following optimization problem which can be used to detect arbitrage strategies in the described setup.

$$\begin{aligned} \max _{y,z,c,s}&\mathbb {E}^{\mathbb {P}}\left[ c_{T}+X_{T}^{f}s_{T}\right] \end{aligned}$$
(2.6)
$$\begin{aligned} \text {subject to:}\nonumber \\ \left( t\in \mathcal {T}_{1}\right) :\;&c_{t}=\left( c_{t-1}-z_{t-1}X_{t-1}^{f}\right) R+X_{t}^{e}\sum _{i=1}^{I}y_{it-1}-\psi \frac{s_{t}+s_{t-1}}{2} \end{aligned}$$
(2.7)
$$\begin{aligned} \left( t\in \mathcal {T}_{1}\right) :\;&s_{t}=s_{t-1}-\sum _{i=1}^{I}\eta _{i}^{-1}y_{i\,t-1}+z_{t-1} \end{aligned}$$
(2.8)
$$\begin{aligned}&c_{0}+X_{0}^{f}s_{0}\le 0 \end{aligned}$$
(2.9)
$$\begin{aligned}&c_{T}+X_{T}^{f}s_{T}\ge 0 \end{aligned}$$
(2.10)
$$\begin{aligned} \left( t\in \mathcal {T}\;\right) :\;&s_{t}\ge 0 \end{aligned}$$
(2.11)
$$\begin{aligned} \left( t\in \mathcal {T}_{0}\right) :\;&y_{t}\ge 0 \end{aligned}$$
(2.12)

The first two constraints (2.7)–(2.8) are the self financing equations of Definition 2.1. Equations (2.9)–(2.10) are identical with the first two arbitrage conditions of Definition 2.2. Furthermore, (2.11)–(2.12) formulates the physical restrictions that fuel storage and electricity production cannot be negative. Finally, it can be seen that the expected end value of the strategy—the objective function (2.6) of the optimization problem—is unbounded if and only if the strategy value [nonnegative by (2.10)] can become positive. This follows from the fact that the feasible set is a pointed cone, and feasible solutions can be scaled by positive factors ad libitum. Altogether, optimization problem (2.6)–(2.12) is unbounded if and only if there is an \(\eta -\)arbitrage between fuel and electricity price. Note that there always exists a solution because setting all decision variables to zero is feasible.

The test problem has the form of a simple production planning problem, where the expected end value is maximized. However, this is only a consequence of Definitions 2.1 and 2.2. Its aim is not production planning, but detecting inconsistencies (arbitrage) between a fuel price and the electricity price. The focus is on one fuel, however if several fuels are relevant, then the same test can be applied to all of them separately.

Note also that Problem (2.6)–(2.12) is formulated without upper bounds on storage and production (which would be important when considering planning, or contract valuation problems). They are not necessary in the present pure test context because of the cone property of the feasible set: If the optimization problem is reformulated with upper bounds on storage and production and get a positive optimal value, then this means that the solution can be scaled such that it leads to infinity in the problem without bounds. In the same manner, unboundedness of (2.6)–(2.12) means that there is a solution that leads to a positive end value with positive probability. Such a solution can be scaled, such that boundaries on storage and production are fulfilled.

Let \(\eta _{max}=\max \left\{ \eta _{1},\dots ,\eta _{I}\right\} \) be the efficiency of the most efficient generating unit. Then, using duality in Banach spaces, it is possible to derive the following equivalent conditions:

Proposition 2.3

A market \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \) is \(\eta \)-arbitrage free in the described setup if and only if there exist adapted stochastic processes \(\left\{ \xi _{t},\lambda _{t}\right\} \) with the following properties:

  1. A1:

    \(\xi _{t},\lambda _{t}\in L^{\infty }(\Omega ,\mathcal {F}_{t},\mathbb {P})\) for each \(t\in \mathcal {T}_{1}\).

  2. A2:

    \(\xi _{t}>0\)

  3. A3:

    \(R^{t+1}\,\mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}|\mathcal {F}_{t}\right] =R^{t}\xi _{t}\) for \(t=1,\dots ,T-1\), and \(R\,\mathbb {E}^{\mathbb {P}}\left[ \xi _{1}\right] =1\)

  4. A4:

    \(\mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}X_{t+1}^{e}|\mathcal {F}_{t}\right] \le \eta _{max}^{-1}\xi _{t}X_{t}^{f}\) for \(t\in \mathcal {T}_{0}\)

  5. A5:

    \(\mathbb {E}^{\mathbb {P}}\left[ \lambda _{t+1}|\mathcal {F}_{t}\right] =\xi _{t}\cdot X_{t}^{f}\) for \(t\in \mathcal {T}_{0}\text { and }\mathbb {E}^{\mathbb {P}}\left[ \lambda _{1}\right] =X_{0}^{f}\)

  6. A6:

    \(\xi _{t}\cdot \left[ X_{t}^{f}-\frac{\psi }{2}\left( 1+\frac{1}{R}\right) \right] \le \lambda _{t}\) for \(t\in \mathcal {T}_{1}^{T-1}\text { and }\xi _{T}\left[ X_{T}^{f}-\frac{\psi }{2}\right] \le \lambda _{T}\)

Proof

The proof of is rather long and is moved to the “Appendix” in order to enhance readability. \(\square \)

Remark 2.4

Given Proposition 2.3 one might also refer to \(\eta _{max}\)-arbitrage instead of \(\eta \)-arbitrage.

From A1)–A6) it can be seen that the process \(\xi \)—closely related to the process of Lagrange multiplicators of the cash equation (2.7)—plays the role of positive stochastic discount factors or state price deflators (see e.g. Cochrane 2005 section 1.2 and Back 2010 section 2.2) which fulfill the martingale property A3). In particular it is easy to show \(\mathbb {E}\left[ \xi _{t}\right] =\frac{1}{R^{t}}\). In fact, the proposition states that there is no arbitrage if and only if there exists a stochastic discount factors with the properties A4)–A6). This is similar to the situation in finance where absence of arbitrage also can be linked to the existence of positive discount factors. However, in standard financial applications the typical condition is that all discounted market prices are martingales. Condition A4) then aims at consistency between fuel and electricity prices and states that the conditional expected proceeds from selling of one MWh electricity should not be larger than the costs for producing it, if everything is discounted properly. Conditions A5)–A6) are harder to interpret in financial terms because of the additional process \(\lambda \), which is derived from the shadow costs of storage.

It is possible however, to state an easy interpretable necessary condition and an easy interpretable sufficient condition for absence of arbitrage, without mentioning the process \(\lambda \).

Corollary 2.5

If a market \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \) is \(\eta \)-arbitrage free then there exists a stochastic process \(\xi \) (fulfilling properties A1–A3 of Proposition 2.3) such that the consistency requirement A4) holds together with

$$\begin{aligned} \mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}\,\left( X_{t+1}^{f}+B_{t+1}\right) |\mathcal {F}_{t}\right]&\le \xi _{t}\left( X_{t}^{f}+B_{t}\right)&\text { for }t\in \mathcal {T}_{1}^{T-1} \end{aligned}$$
(2.13)

with

$$\begin{aligned} B_{t}=\frac{\psi }{2\,\left( R-1\right) }\left( 1+\frac{1}{R}\right) \end{aligned}$$
(2.14)

for \(t\in \mathcal {T}_{1}{\setminus }\{T-1,T\}\) and

$$\begin{aligned} B_{T}=\frac{\psi }{R-1}. \end{aligned}$$
(2.15)

Proof

If a market is \(\eta _{max}\)-arbitrage free then by Proposition 2.3 there exist processes \(\xi ,\lambda \) fulfilling A1–A6. From the first inequality of A6 we have

$$\begin{aligned} \xi _{t+1}\left( X_{t+1}^{f}-\frac{\psi }{2}\left( 1+\frac{1}{R}\right) \right) \le \lambda _{t+1} \end{aligned}$$

for \(t=1,2,\dots ,T-2\). Taking conditional expectation and applying A5 and A3 leads to

$$\begin{aligned} \mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}\,\left( X_{t+1}^{f}\right) |\mathcal {F}_{t}\right] \le \xi _{t}\left( X_{t}^{f}+\frac{\psi }{2R}\left( 1+\frac{1}{R}\right) \right) . \end{aligned}$$

Now, adding \(\frac{\psi }{2R^{2}}\frac{R+1}{R-1}\xi _{t}\) on both sides of the inequality and again applying A3 gives the first case of (2.13). In similar manner the second case can be obtained by starting with the second equation of A6, taking conditional expectations and then using A5, A3. Here the resulting inequality is expanded by \(\frac{\psi }{R(R-1)}\xi _{T-1}\)\(\square \)

Regarding fuel prices, condition (2.13) of this corollary is already close to the standard financial case: a discount factor process \(\xi _{t}\) is required such that the modified fuel price \(X_{t+1}^{f}+B_{t}\) is a supermartingale, if discounted by \(\xi _{t}\). The terms \(B_{t}\) account for storage costs, respectively the cost of carry. In the absence of storage costs (i.e. \(\psi =0\)) the fuel price itself must be a supermartingale, if properly discounted. Still, the electricity price is restricted only indirectly via the consistency requirement A4).

An interpretable sufficient condition for absence of arbitrage can be formulated in the following way.

Corollary 2.6

Consider a market with prices \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \). If there is a process \(\xi \) (fulfilling properties A1–A3 of Proposition 2.3) such that the consistency requirement A4) holds together with

$$\begin{aligned} \mathbb {E}^{\mathbb {Q}}\left[ \xi _{t+1}X_{t+1}^{f}|\mathcal {F}_{t}\right] =\xi _{t}X_{t}^{f} \end{aligned}$$
(2.16)

(i.e. the discounted fuel pieces are a martingale) then the market is \(\eta \)-arbitrage free.

Proof

Set \(\lambda _{t}=\xi _{t}X_{t}^{f}\). This choice fulfills A6 because \(\frac{\psi }{2}\left( 1+\frac{1}{R}\right) \ge 0\). Substituting \(\lambda _{t+1}\) for \(\xi _{t+1}X_{t+1}^{f}\) at the left side of (2.16) leads to A5. Because A1–A4 hold already by assumption, Proposition 2.3 implies absence of \(\eta _{max}\)-arbitrage. \(\square \)

If there exists a process of stochastic discount factors such that already the discounted fuel price is a martingale (the typical financial requirement) and the consistency requirement holds, then absence of arbitrage follows.

3 Implications of the Arbitrage conditions

The preceding section gave a complete characterization of arbitrage in a general framework. The question arises what the obtained results mean in a more concrete context when statistical models are used to describe the price processes. Can arbitrage easily arise, such that one has to use restrictions when estimating model parameters? Many models for fuel and electricity prices have been proposed in literature, see e.g. Nowotarski and Weron (2018) for a recent overview of electricity price modeling. The present work therefore can only be a starting point and the analysis therefore is restricted to one possible class of models, namely a class of (potentially nonlinear) vector-autoregressive econometric models. This class of models for electricity and fuel prices can be described by

$$\begin{aligned} X_{t+1}^{e}-E_{t+1}&=\phi ^{e}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta )+\varepsilon _{t+1}^{e} \end{aligned}$$
(3.1)
$$\begin{aligned} X_{t+1}^{f}-F_{t+1}&=\phi ^{f}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta )+\varepsilon _{t+1}^{f}. \end{aligned}$$
(3.2)

Here \(E_{t}\) and \(F_{t}\) are given and denote a given deterministic process, e.g. market expectations, or an observed forward price-curve, or can be even zero. The considered model then is formulated relative to these processes. The notation \(X_{[t]}^{i}=(X_{t}^{i},\,X_{t-1}^{i},\dots ,X_{t-p}^{i})^{\prime },\text {where }\,i\in \left\{ e,f\right\} \) represents (for some \(p\in \mathbb {N}\)) a price history up to time t (this could also be a suitable selection of past prices with certain lags). Negative \(t<0\) denote observations that have been made before the actual planning horizon \(\left\{ 0,\dots ,T\right\} \). The functions \(\phi ^{i}\) are measurable and bounded and model the one step expectation. They depend on past price differences from the reference values \(E_{t},F_{t}\) and are parametrized by some model parameter vector \(\theta \). Note that the short notation \(\phi _{t}^{i}(\theta )=\phi ^{i}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta )\) is used in the following.

In the simplest case such a model would be just linearly vector-autoregressive, e.g.

$$\begin{aligned} X_{t+1}^{e}-E_{t+1}&=\theta _{1}^{\prime }(X_{[t]}^{e}-E_{[t]})+\varepsilon _{t+1}^{e} \end{aligned}$$
(3.3)
$$\begin{aligned} X_{t+1}^{f}-F_{t+1}&=\theta _{2}^{\prime }(\,X_{[t]}^{f}-F_{[t]})+\varepsilon _{t+1}^{f}, \end{aligned}$$
(3.4)

where \(\theta _{1},\theta _{2}\) are parameter vectors of dimension p. Other linear specifications might use suitable differences between lagged variables. Clearly, nonlinear models may be computational demanding.

Finally, it is assumed that—given the past—the error terms \(\varepsilon _{t+1}^{i}\) follow a distribution described by some joint conditional distribution function G. The distribution function respects the assumption that fuel prices \(X_{t+1}^{f}\) are nonnegative. Moreover, the error terms are characterized by

$$\begin{aligned} \mathbb {E}\left[ \varepsilon _{t+1}^{i}|\mathcal {F}_{t}\right] =0 \end{aligned}$$
(3.5)

and

$$\begin{aligned} \Sigma :=Cov\left[ \varepsilon _{t+1}^{e},\varepsilon _{t+1}^{f}|\mathcal {F}_{t}\right] =\left[ \begin{array}{cc} \sigma _{e}^{2} &{}\quad \rho \sigma _{e}\sigma _{f}\\ \rho \sigma _{e}\sigma _{f} &{}\quad \sigma _{f}^{2} \end{array}\right] \in \mathbb {R}^{2} \end{aligned}$$
(3.6)

Although the concrete joint distribution function of the error terms might also be parametrized by additional model parameters, only the covariance matrix and its components will be relevant in the following. The further parameters of the market model, i.e. R and \(\eta _{max}\) are given externally.

3.1 Implications of the necessary condition for absence of arbitrage

If one aims at testing for arbitrage in an electricity market, it is a natural approach to assume absence of arbitrage as the zero hypothesis. This is in line with the fact that according to economic theory it is hard to achieve arbitrage, i.e. riskless profits. If a person claims to know the secret of how to achieve extraordinary profits, usually it is wise not to believe this too fast.

A sensible way for constructing a test then would be to use a necessary condition like Corollary 2.5 and try to reject it. Therefore, the consequences of conditions A1–A4 and inequality (2.13) are analyzed to find out under which circumstances they are fulfilled. Here the first question is when the conditions are fulfilled for given parameter values. The answer allows to analyze whether the no-arbitrage conditions imply any restrictions that have to be observed when estimating unknown parameters. It will turn out that the answer is positive only in very exceptional cases.

Recall that \(R^{t}\xi _{t}\) has to be is a martingale according to condition A3). Therefore, it is possible to write

$$\begin{aligned} R^{t+1}\xi _{t+1}=R^{t}\xi _{t}+R^{t+1}u_{t+1}, \end{aligned}$$
(3.7)

where the stochastic process \(u_{t}\) is a martingale difference sequence. In order to ensure that \(\xi _{t+1}\) is essentially bounded (respectively \(\xi _{t+1}\in L^{\infty }\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \) as requested by A1), we assume that \(u_{t+1}\) is essentially bounded (respectively \(u_{t+1}\in L^{\infty }\left( \Omega ,\mathcal {F}_{t},\mathbb {P}\right) \)), as well. Note that because of the requirement \(\xi _{t+1}>0\) the sequence \(u_{t+1}\) a lower bound for \(u_{t+1}\)is given by \(u_{t+1}>-\frac{1}{R}\xi _{t}\). This means e.g. that the martingale difference sequence cannot be a sequence of i.i.d. random variables.

In the current context with (3.1)–(3.7) Corollary (2.5) can be reformulated in the following way.

Corollary 3.1

Let the price processes \(\left\{ X_{t}^{e},X_{t}^{f}\right\} \) be specified by model (3.1)–(3.2). If the model is \(\eta \)-arbitrage free then there exists a stochastic process \(\xi \) (fulfilling properties A1–A3 of Proposition 2.3) together with a related martingale difference sequence \(u_{t}\) [obeying (3.7)] such that

$$\begin{aligned} \frac{1}{R}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) +\mathbb {E}\left[ \frac{u_{t+1}}{\xi _{t}}\varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] \le \eta _{max}^{-1}X_{t}^{f}\text { for }t\in \mathcal {T}_{0} \end{aligned}$$
(3.8)

holds together with

$$\begin{aligned} \frac{1}{R}\left( F_{t+1}+\phi _{t}^{f}(\theta )\right) +\mathbb {E}\left[ \frac{u_{t+1}}{\xi _{t}}\varepsilon _{t+1}^{f}|\mathcal {F}_{t}\right] \le X_{t}^{f}+B_{t}-\frac{B_{t+1}}{R} \end{aligned}$$
(3.9)

Proof

Under the assumptions Corollary 2.5 is fulfilled. Plugging the model Definition (3.1) of \(X_{t+1}^{e}\) into A4) one gets

$$\begin{aligned} \left( E_{t+1}+\phi _{t}^{e}(\theta )\right) \cdot \mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}|\mathcal {F}_{t}\right] +\mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}\varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] \le \eta _{max}^{-1}\xi _{t}X_{t}^{f}. \end{aligned}$$

A3 and (3.7) now can be used to get

$$\begin{aligned} \frac{1}{R}\xi _{t}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) +\frac{1}{R}\xi _{t}\mathbb {E}^{\mathbb {P}}\left[ \varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] +\mathbb {E}^{\mathbb {P}}\left[ u_{t+1}\varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] \le \eta _{max}^{-1}\xi _{t}X_{t}^{f}. \end{aligned}$$

The first expectation here is zero by the model definition, see (3.5). Then, dividing by \(\xi _{t}\) (recall A2) leads to (3.8), the first statement of the corollary.

The same arguments, applied to (2.13), lead to the second statement (3.9). \(\square \)

Defining now a process

$$\begin{aligned} v_{t}=\frac{u_{t+1}}{\xi _{t}}, \end{aligned}$$
(3.10)

it can be seen that \(\mathbb {E}\left[ v_{t+1}\varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] =Cov\left[ v_{t+1}\varepsilon _{t+1}^{e}|\mathcal {F}_{t}\right] =\rho _{et}\cdot \sigma _{e}\cdot \sigma _{vt}\), where \(\rho _{et}\) is the conditional correlation between \(v_{t+1}\) and \(\varepsilon _{t+1}^{e}\), where \(\sigma _{e}\) denotes the (constant) standard deviation of \(\varepsilon _{t}^{e}\) [see (3.6)] and where \(\sigma _{vt}\) is the conditional standard deviation of \(v_{t+1}.\) In the same manner the relation \(\mathbb {E}\left[ v_{t+1}\varepsilon _{t+1}^{f}\right] =\rho _{ft}\cdot \sigma _{f}\cdot \sigma _{vt}\) is obtained, where \(\rho _{ft}\) is the conditional correlation between \(v_{t+1}\) and \(\varepsilon _{t+1}^{f}\), and \(\sigma _{f}\) denotes the (constant) standard deviation of \(\varepsilon _{t}^{f}\) [see (3.6)]. It is important to keep in mind that inequalities (3.8)–(3.9) hold almost surely at each point in time.

The two equations of Corollary 3.1 now can be written as

$$\begin{aligned} \rho _{et}\cdot \sigma _{e}\cdot \sigma _{vt}\le G_{t} \end{aligned}$$
(3.11)

and

$$\begin{aligned} \rho _{ft}\cdot \sigma _{f}\cdot \sigma _{vt}\le H_{t} \end{aligned}$$
(3.12)

with

$$\begin{aligned} G_{t}=\eta _{max}^{-1}X_{t}^{f}-\frac{1}{R}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) \end{aligned}$$

and

$$\begin{aligned} H_{t}=X_{t}^{f}-\frac{1}{R}\left( F_{t+1}+\phi _{t}^{f}(\theta )\right) +B_{t}-\frac{B_{t+1}}{R}. \end{aligned}$$

Note that \(-G_{t}\) can be considered as a discounted version of the “spark spread” (where the actual electricity price is replaced by the discounted expectation of the one step ahead electricity price under model 3.1) and \(H_{t}\) is the difference between fuel price and the discounted expectation of the one step ahead fuel price.

Is it possible now to choose \(\rho _{et},\sigma _{vt}\text { and }\rho _{ft}\) to fulfill the inequalities when \(\sigma _{e}\) and \(\sigma _{f}\) (and also \(\theta \) and \(R,\,\eta \)) are given parameters? First \(\sigma _{vt}\) must be nonnegative and can be chosen as large as necessary, as shown by the following corollary.

Corollary 3.2

Given any real numbers \(M\ge 0\), \(-1\le \rho _{1}\le 1\), \(-1\le \rho _{2}\le 1\) it is always possible to define a random variable \(v_{t+1}\)such that

$$\begin{aligned}&\mathbb {E}\left[ v_{t+1}|\mathcal {F}_{t}\right] =0, \end{aligned}$$
(3.13)
$$\begin{aligned}&v_{t+1}>-\frac{1}{R} \end{aligned}$$
(3.14)

and

$$\begin{aligned} \sigma _{vt}=M. \end{aligned}$$
(3.15)

Moreover, a joint distribution for \(v_{t+1},\,\varepsilon _{t+1}^{e},\,\varepsilon _{t+1}^{f}\) can be defined such that the covariance structure (3.6) is kept, while

$$\begin{aligned} \rho _{et}=\rho _{1}\text { and }\rho _{ft}=\rho _{2}. \end{aligned}$$
(3.16)

is ensured.

Proof

Given two positive real numbers \(0<K_{1}<\frac{1}{R}\) and \(K_{2}>0\), consider a random variable \(v_{t+1}\) that follows a mixture distribution such that with \(p=\frac{K_{2}}{K_{1}+K_{2}}\) the density of \(v_{t}\) is defined as \(f(x)=p\frac{1}{K_{1}}\imath _{[-K_{1},0]}(x)+p\frac{1}{K_{2}}\imath _{[0,K_{2}]}(x)\), where \(\imath _{A}(x)\) denotes the indicator function which equals one if \(x\in A\), and else is zero. This means that \(v_{t+1}\) follows a mixture of two uniform distributions on \([-K_{1},0]\) and \([0,K_{2}]\). If \(K_{1},\,K_{2}\) are chosen such that \(K_{1}K_{2}=12M^{2}\), then Eqs. (3.13)–(3.15) are fulfilled. Finally, given the joint distribution of \(\varepsilon _{t+1}^{e},\,\varepsilon _{t+1}^{f}\) and the marginal distribution of \(v_{t+1}\), a suitable copula function can be used to construct a joint distribution function for \(v_{t+1},\,\varepsilon _{t+1}^{e},\,\varepsilon _{t+1}^{f}\) that fulfills all requirements. \(\square \)

The first two requirements in Corollary (3.2) ensure that the process \(\xi _{t}\) is a positive martingale. Altogether the corollary ensures that it is possible to scale freely the left-hand side of inequalities (3.11)–(3.12) if \(\sigma _{e}\), respectively \(\sigma _{f}\) are positive. This means that in this case only the sign of \(\rho _{et}\) and \(\rho _{ft}\) really matters. It is preferable to choose both correlations negative, when the left-hand sides of (3.11)–(3.12) can be made arbitrarily small.

Therefore, it has to be checked, whether it is possible to select a random variable \(v_{t+1}\) which is negatively correlated to both error terms \(\varepsilon _{t}^{e}\) and \(\varepsilon _{t}^{f}\), when the errors are correlated with coefficient \(\rho \) as specified in (3.6). To answer this question, recall that in an inner product space one can define the angle \(\theta \) between two elements \(X,\,Y\) by \(\left\langle X,Y\right\rangle =\left\| X\right\| \,\left\| Y\right\| \cos (\theta )\). Considering that in the present context \(\left\| X\right\| =\sqrt{\text {Var}(X|\mathcal {F}_{t})}=\sigma _{X}\) and \(\left\langle X,Y\right\rangle =\text {Cov}\left( X,Y|\mathcal {F}_{t}\right) \) hold, it can be seen that for the angle \(\theta _{\varepsilon }\)between the errors \(\varepsilon _{t}^{e}\) and \(\varepsilon _{t}^{f}\) the relation \(\cos \left( \theta _{\varepsilon }\right) =\rho \) is valid. Moreover, for the angles \(\theta _{e}\) and \(\theta _{f}\) between \(v_{t+1}\) and the respective errors \(\varepsilon _{t}^{e}\) and \(\varepsilon _{t}^{f}\) the equations \(\cos \left( \theta _{e}\right) =\rho _{et}\) and \(\cos \left( \theta _{f}\right) =\rho _{ft}\) are obtained. By standard geometric arguments it can then be seen that it is possible to choose both (cosines) correlations negative, if and only if

$$\begin{aligned} \rho \ne -1. \end{aligned}$$
(3.17)

In fact, \(v_{t+1}\) has to be chosen such that it is at an obtuse angle with (but not orthogonal to) both error terms. This is possible if and only if the error vectors are not pointing exactly in opposite directions.

The special role of \(\rho \), the correlation between the error terms, can also be seen from the following consideration: the correlations \(\rho _{et}\) and \(\rho _{ft}\) have to be chosen such that the joint (conditional) correlation matrix for \(\varepsilon _{t+1}^{e},\,\varepsilon _{t+1}^{f}\) and \(v_{t+1}\) stays nonnegative definite. This reduces to

$$\begin{aligned} \left| \left[ \begin{array}{ccc} 1 &{}\quad \rho &{}\quad \rho _{et}\\ \rho &{}\quad 1 &{}\quad \rho _{ft}\\ \rho _{et} &{}\quad \rho _{ft} &{}\quad 1 \end{array}\right] \right| \ge 0 \end{aligned}$$

or (the first two principal minors are obviously nonnegative)

$$\begin{aligned} \rho _{et}^{2}+\rho _{ft}^{2}-2\rho \rho _{et}\rho _{ft}\le 1-\rho ^{2}. \end{aligned}$$
(3.18)

With \(\rho \) given, for \(0<\rho <1\) this inequality (in \(\rho _{et},\,\rho _{ft}\)) describes an elliptical disk. The ellipse has its center in the origin, and its main axis has positive slope if \(\rho \) is positive and it has negative slope if \(\rho \) is negative. If \(\rho =0\) then the ellipse becomes the unit circle. There are also two degenerated cases. For \(\rho =-1\) inequality (3.18) becomes equivalent to the equation of a straight line \(\rho _{ft}=-\rho _{et}\) and is valid for \(0\le \rho _{ft},\rho _{et}\le 1\). Finally, with \(\rho =+1\) inequality (3.18) becomes equivalent to the equation of a straight line \(\rho _{ft}=\rho _{et}\) and is valid for \(0\le \rho _{ft},\,\rho _{et}\le 1\). See Fig. 1 for a graphical representation of these cases.

Fig. 1
figure 1

Feasible combinations of \(\rho _{et},\rho _{ft}\) (correlation between \(v_{t+1}\) and the errors \(\varepsilon _{t+1}^{e}\), respectively \(\varepsilon _{t+1}^{f}\),), dependent on different values of \(\rho \) (the correlation between the errors)

Full size image

These results fit exactly to the above discussion. In case \(-1<\rho <+1\) it is easily possible to choose both \(\rho _{et}\) and \(\rho _{ft}\) negative (Case 2), i.e. from the lower left part of the ellipse. This is also possible for \(\rho =+1\), with the only difference that both correlations have to be chosen equal in this case. However, if \(\rho =-1\) it is not possible to choose both correlations negative, because of \(\rho _{ft}=-\rho _{et}\).

The situation is different if one of \(\sigma _{e}\) or \(\sigma _{f}\) equals zero.

If \(\sigma _{e}=0\) then inequality (3.11) can be reduced to the condition

$$\begin{aligned} \eta _{max}^{-1}X_{t}^{f}-\frac{1}{R}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) \ge 0. \end{aligned}$$
(3.19)

The left-hand side of the second inequality (3.12) can be made arbitrarily small using the same arguments as above. If \(\sigma _{f}=0\) then inequality (3.12) reduces to

$$\begin{aligned} X_{t}^{f}-\frac{1}{R}\left( F_{t+1}+\phi _{t}^{f}(\theta )\right) +B_{t}-\frac{B_{t+1}}{R}\ge 0. \end{aligned}$$
(3.20)

The left-hand side of the first inequality (3.11) again can be made arbitrarily small. Finally, if both standard deviations \(\sigma _{e}\) and \(\sigma _{f}\) are equal to zero, then absence of arbitrage is ensured if both inequalities (3.19)–(3.20) hold at all points in time t.

Altogether, with given parameters one only has to check \(\rho ,\,\sigma _{e}\) and \(\sigma _{f}\). If \(\rho \ne -1\) then the no-arbitrage hypothesis never can be rejected. If one or both of \(\sigma _{e},\,\sigma _{f}\) are zero, then the no arbitrage hypothesis must be rejected if the related equation from (3.19)–(3.20) is not fulfilled for some t, and otherwise the hypothesis is not rejected. If parameter values are not given and have to be estimated, it is possible to conclude:

Proposition 3.3

Consider a price model (3.1)–(3.2) and (3.5)–(3.6). Assume that price processes \(X_{t}^{e},X_{t}^{e}\) are observed such that for some parameter value \(\theta \) one of the deterministic cases

$$\begin{aligned}&A)\quad X_{t+1}^{e}-E_{t+1}=\phi ^{e}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta ),\\&B)\quad X_{t+1}^{f}-F_{t+1}=\phi ^{f}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta ). \end{aligned}$$

or

$$\begin{aligned} C)\quad \text {both equations }A)\text { and }B) \end{aligned}$$

hold for all points in time t. Then absence of arbitrage [inequalities (3.8)–(3.9)] cannot be rejected if for all points in time t (3.19) is fulfilled in case A), (3.20) is fulfilled in case B), respectively both (3.19)–(3.20) are fulfilled at all points in time in case C). Otherwise (3.8)–(3.9) are not valid and arbitrage is possible.

If none of these cases holds, and also it can be excluded that there is some \(\kappa >0\) such that for some parameter value \(\theta \) the equation

$$\begin{aligned} X_{t+1}^{f}-F_{t+1}-\phi _{t}^{f}(\theta )=\kappa \left( \phi _{t}^{e}(\theta )-\left( X_{t+1}^{e}-E_{t+1}\right) \right) \end{aligned}$$
(3.21)

holds for all points in time t, then absence of arbitrage [inequalities (3.8)–(3.9)] cannot be rejected.

Proof

Cases \(A),\,B)\) and C) refer to cases where either \(\sigma _{e}\) or \(\sigma _{f}\) are equal to zero in the given model: If there exists a parameter value \(\theta \) such that A),B) or C) are fulfilled this shows that the prices follow exactly a model with some price variances equal to zero. The correct conditions for validity of (3.8)–(3.9) already have been discussed above.

If Eq. (3.21) is not fulfilled this means that \(\rho \ne -1\), because \(\rho =-1\) is equivalent to \(\varepsilon _{t+1}^{e}=-\frac{1}{\kappa }\varepsilon _{t+1}^{f}\) (for some \(\kappa >0\)). In this case it was already shown that (3.8)–(3.9) can be made valid for arbitrary parameter value \(\theta \). \(\square \)

In a context with given data and parameters to be estimated, therefore it is usually no problem to estimate the parameters without restrictions, e.g. using unrestricted maximum likelihood. It is highly unlikely in a real world application that one of the deterministic equations in Proposition (3.3) holds for observed price data and therefore absence of arbitrage usually cannot be rejected.

3.2 Implications of the sufficient condition for absence of arbitrage

So far the main result is that for a large class of possible models the necessary conditions A4) and (2.13) for absence of arbitrage can be rejected only in rare, even unrealistic cases. In the following, similar arguments are used to to analyze the implications of the sufficient condition A4) and (2.16) in Corollary 2.6. Again the class of autoregressive models specified in (3.1)–(3.2) and (3.5)–(3.6) is considered. The process \(\xi \) is rewritten by (3.7). In particular the processes u and v, defined in (3.10), have the same properties as before. Then the sufficient conditions A4 and (2.16) can be reduced to

$$\begin{aligned} \rho _{et}\cdot \sigma _{e}\cdot \sigma _{vt}\le G_{t} \end{aligned}$$
(3.22)

and

$$\begin{aligned} \rho _{ft}\cdot \sigma _{f}\cdot \sigma _{vt}=H_{t} \end{aligned}$$
(3.23)

with

$$\begin{aligned} G_{t}=\eta _{max}^{-1}X_{t}^{f}-\frac{1}{R}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) \end{aligned}$$

and

$$\begin{aligned} H_{t}=X_{t}^{f}-\frac{1}{R}\left( F_{t+1}+\phi _{t}^{f}(\theta )\right) . \end{aligned}$$

Surprisingly, like with necessary conditions, the sufficient conditions (3.22)–(3.23) do not put severe restrictions on the model parameters.

Proposition 3.4

For the autoregressive model (3.1)–(3.2) and (3.5)–(3.6) absence of arbitrage is ensured in any of the following cases:

  1. 1.

    \(\sigma _{e}=0,\,\sigma _{f}>0\) and for all t

    $$\begin{aligned} \eta _{max}^{-1}X_{t}^{f}-\frac{1}{R}\left( E_{t+1}+\phi _{t}^{e}(\theta )\right) \ge 0.\quad \end{aligned}$$
    (3.24)
  2. 2.

    \(\sigma _{e}>0,\,\sigma _{f}=0\) and for all t

    $$\begin{aligned} X_{t}^{f}-\frac{1}{R}\left( F_{t+1}+\phi _{t}^{f}(\theta )\right) =0\quad \end{aligned}$$
    (3.25)
  3. 3.

    Else, either \(|\rho |<1\), or both (3.25) and (3.24) hold.

Proof

The “trivial” cases of zero variance are considered first. If \(\sigma _{e}=0\) and \(\sigma _{f}>0\) then it is sufficient for absence of arbitrage that the almost sure conditions (3.22), hence (3.22), hold at each point in time. This is true because one is free to choose a conditional correlation \(\rho _{ft}\) and a standard deviation \(\sigma _{vt}\) such that (3.23) is also fulfilled. In similar manner, if \(\sigma _{f}=0\) and \(\sigma _{e}>0\) then conditions (3.23), hence (3.25), must hold at any point in time to ensure absence of arbitrage. If both conditional variances are zero, then both of Eqs. (3.24) and (3.25) must hold simultaneously at any point in time to imply absence of arbitrage.

Let now \(\sigma _{e}>0\) and \(\sigma _{f}>0\). In each scenario \(\omega \) where \(H_{t}(\omega )=0\), it is possible to choose \(\rho _{ft}(\omega )=0\) to ensure (3.23). This allows to take \(\rho _{ft}\) and \(\sigma _{vt}\) such that also (3.22) stays valid. From Eq. (3.18) it can be seen that the choice \(\rho _{ft}(\omega )=0\) fully works as long as \(|\rho |\ne 1\). If \(|\rho |=1\) then (3.18) implies that \(\rho _{et}(\omega )=0\) and therefore (3.25).

Finally, consider the case \(H_{t}\ne 0\). Then (3.23) leads to

$$\begin{aligned} \sigma _{vt}=\frac{H_{t}}{\rho _{ft}\sigma _{f}}. \end{aligned}$$

Note that because of \(\sigma _{e}>0\) and \(\sigma _{f}>0\), the conditional correlation \(\rho _{ft}\) must have the same sign as \(H_{t}\) (and is not equal to zero because of the considered case \(H_{t}\ne 0\)). Plugging \(\sigma _{vt}\) into (3.22) leads to

$$\begin{aligned} \frac{\sigma _{e}}{\sigma _{f}}\frac{H_{t}}{\rho _{ft}}\rho _{et}\le G_{t}. \end{aligned}$$
(3.26)

The multiplicator of \(\rho _{et}\) on the left-hand side here is positive. If \(G_{t}\ge 0\) then it suffices to choose some nonpositive \(\rho _{et}\) [and some \(\rho _{ft}\) that fulfills equation (3.26)]. In order to account also for the case \(G_{t}<0\), \(\rho _{et}\) has to be chosen negative and the sign of \(\rho _{ft}\) depends on the sign of \(H_{t}\) as discussed above. Taking into account the correlation \(\rho \) and Eq. (3.18) (see also Fig. 1), such a choice of signs is always possible unless either \(\rho =-1\) or \(\rho =+1\). The absolute values of \(\rho _{ft},\rho _{et}\) can be chosen such that the left-hand side of (3.26) becomes small enough. \(\square \)

Using the model specification (3.1)–(3.2) and (3.2) it is possible to reformulate Proposition (3.4) in the following way:

Corollary 3.5

Assume that price processes \(X_{t}^{e},X_{t}^{e}\) are observed such that for some parameter value \(\theta \) one of the deterministic cases

$$\begin{aligned}&A)\quad X_{t+1}^{e}-E_{t+1}=\phi ^{e}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta _{1}),\\&B)\quad X_{t+1}^{f}-F_{t+1},=\phi ^{f}(X_{[t]}^{e}-E_{[t]},\,X_{[t]}^{f}-F_{[t]};\theta _{2}). \end{aligned}$$

hold for all points in time t. Then absence of arbitrage [sufficient conditions (3.22)–(3.23)] can be guaranteed if (3.24) holds in case A), respectively (3.25) holds in case B). If both, A) and B) hold for some parameter value \(\theta \) then absence of arbitrage is implied if (3.24) and (3.25) hold simultaneously.

If none of A), B) holds, still (3.24) together with (3.25) implies absence of arbitrage. Moreover, arbitrage can also be excluded if there is no \(\kappa \in \mathbb {R}\) such that for some parameter value \(\theta \) the equation

$$\begin{aligned} X_{t+1}^{f}-F_{t+1}-\phi _{t}^{f}(\theta )=\kappa \left( \phi _{t}^{e}(\theta )-\left( X_{t+1}^{e}-E_{t+1}\right) \right) \end{aligned}$$
(3.27)

holds for all points in time.

Proof

This follows immediately from Proposition (3.4). Regarding cases A), B) the error terms \((\varepsilon _{t}^{e},\varepsilon _{t}^{f})\) are almost surely equal to zero, if and only if their expectation and variance is zero. Moreover, a correlation \(\rho \) equal to 1 or \(-1\) occurs if and only if \(\varepsilon _{t}^{e}=\tau \varepsilon _{t}^{f}\) for some \(\tau \in {\mathbb {R}.}\)\(\square \)

We see now that for the vector-autoregressive model the sufficient conditions for are only slightly stronger than the necessary conditions. The requirement \(|\rho |<1\) is not very severe, and one can expect that unrestricted parameter estimation leads to an arbitrage free price model in most practical situations.

4 Two applications of arbitrage conditions

In the following two direct applications of the arbitrage results got so far are analyzed. First physical delivery contracts and their valuation is considered. Here the absence of arbitrage has the important technical consequence that in this case valuation procedures based on stochastic discount factors can be obtained. In the second discussion it turns out that the cases with zero variance are important in tree-based stochastic optimization, which is a way to deal with the valuation problem, but also for taking other decisions in the energy management context.

4.1 Valuation of delivery contracts

Consider now a producer who participates on a market for physical electricity delivery contracts. While exchanges nowadays frequently use financial delivery, physical delivery is still in use. Moreover, off-exchange contracts are about physical delivery quite often. When deciding whether to enter a contract or how to valuate an already agreed contract, because the market is incomplete as discussed above, there is not a unique discount factor process \(\xi \) which gives an estimate of market values or prices in advance without taking into account individual preferences. Still the valuation problem can be analyzed from the viewpoint of the producer who may ask questions like: “What is the smallest up-front payment (alternatively: delivery price) such that all contractual obligations (deliveries) can be fulfilled and the (random) end value is acceptable?” This does not lead to market values or prices, but if e.g. market delivery prices are larger than the producers smallest possible price, then he will contract, otherwise he will not agree or at least he will have to take additional risk.

Building on arbitrage free models, it is possible to derive pricing and valuation formulas (super-hedging and acceptability prices) for delivery contracts with random delivery patterns under several types of assumptions. This was the focus in Kovacevic (2018), see also Kovacevic and Pflug (2013) where the topic was touched on the first time. This approach builds on and generalizes the ideas in King (2002), Flåm (2008), Pennanen (2011a, b) and Pennanen (2012), where financial markets instead of electricity markets are considered. In any case it turns out that the smallest value or price such that a contract does not lead to an unfavorable outcome can be expressed in terms of expectations w.r.t. equivalent measures, respectively stochastic discount factors.

In a simplified framework, the producer again is able to generate electricity with some efficiency \(\eta \) but also has the obligation to deliver amounts \(D_{t}\) [MWh] of electric energy at a fixed price of K per MWh for periods \(\left[ t,t+1\right] \)—which defines at delivery contract. The stochastic demand \(D_{t}\) is adapted to the same filtration \(\left\{ \mathcal {F}_{t}\right\} \) as the other processes considered so far, which contains all relevant information. Among others, \(D_{t}\) can be a deterministic delivery pattern, a pattern that depends on the fuel and/or the electricity prices, as well as a pattern that depends on other relevant variables like the observed temperature.

The producer cannot neglect that fuel storage and production capacity is restricted. So \(S>0\) will denote the upper bound on storage and \(P_{t}\) is an \(\left\{ \mathcal {F}_{t}\right\} \)-adapted process of upper bounds on the production of a generator with efficiency \(\eta \). The producer has to allocate the produced electricity between the market and the contractual obligations. Finally, the producer is also allowed to buy electricity from the market to meet obligations. The amount of electricity sold at the market is given by \(y_{t-1}-D_{t-1}\). If this difference is negative, an amount of energy is bought.

Out of the various cases analyzed in Kovacevic (2018), only the so called superhedging value is considered in the present work, i.e. the answer to the following question: “What is the minimum initial asset value or upfront-payment \(V_{0}=c_{0}+s_{0}X_{0}^{f}\) such that the producer is able to fulfill all contractual obligations and the asset value at the end of the planning horizon, i.e. \(V_{T}=c_{T}+s_{T}X_{T}^{f}\) is almost surely non negative.

The superhedging value can be calculated as the optimal value of the following optimization problem, where the objective is to minimize the asset value or upfront payment at the beginning.

$$\begin{aligned} V_{0}^{*}(K,D,\eta )=\min _{y,z,c,s}&c_{0}+X_{0}^{f}s_{0}\nonumber \\ \text {subject to}\nonumber \\ \left( t\in \mathcal {T}_{1}\right) :\;&c_{t}=\left( c_{t-1}-z_{t-1}X_{t-1}^{f}\right) R+y_{t-1}X_{t}^{e}-D_{t-1}X_{t}^{e}+KD_{t-1}\nonumber \\ \left( t\in \mathcal {T}_{1}\right) :\;&s_{t}=s_{t-1}-y_{t-1}\eta ^{-1}+z_{t-1}\nonumber \\&c_{T}+X_{T}^{f}s_{T}\ge 0\nonumber \\ \left( t\in \mathcal {T}\right) :\;&0\le s_{t}\le S\nonumber \\ \left( t\in \mathcal {T}_{0}\right) :\;&0\le y_{t}\le P_{t}. \end{aligned}$$
(4.1)

All assumptions on the involved processes made in Sect. 2 are kept also in the current subsection. In addition it is assumed that S is a real number, \(P_{t}\in L^{\infty }(\Omega ,\mathcal {F}_{t},\mathbb {P})\), and \(D_{t}\in L^{1}(\Omega ,\mathcal {F}_{t},\mathbb {P}).\)

The dual problem, related to (4.1) then is given by.

Notice that the Lagrange dual of the valuation problem (4.1) is given by

$$\begin{aligned} U_{0}^{*}(K,D,\eta )=\max _{\xi ,\lambda ,\mu ,\nu }&\sum _{t=0}^{T-1}\mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}\left( X_{t+1}^{e}-K\right) D_{t}\right] -\sum _{t=0}^{T-1}\mathbb {E}^{\mathbb {P}}\left[ \mu _{t}P_{t}\right] -S\sum _{t=0}^{T}\mathbb {E}^{\mathbb {P}}\left[ \nu _{t}\right] \nonumber \\ \text {subject to}\nonumber \\ A1)&-A3)\text { and }A5)\nonumber \\ A4')&\mathbb {E}^{\mathbb {P}}\left[ \xi _{t+1}X_{t+1}^{e}|\mathcal {F}_{t}\right] \le \eta ^{-1}\xi _{t}X_{t}^{f}+\mu _{t}\text { for }t\in \mathcal {T}_{0}\nonumber \\ A6')&\xi _{t}\cdot \left[ X_{t}^{f}-\frac{\psi }{2} \left( 1+\frac{1}{R}\right) \right] \le \lambda _{t}+\nu _{t} \text { for }t\in \mathcal {T}_{1}^{T-1}\nonumber \\&\quad \text { and }\xi _{T}\left[ X_{T}^{f}-\frac{\psi }{2}\right] \le \lambda _{T}+\nu _{T}\nonumber \\ \left( t\in \mathcal {T}\right) :\;&\mu _{t}\ge 0,\quad \nu _{t}\ge 0, \end{aligned}$$
(4.2)

where \(\xi _{t},\lambda _{t},\mu _{t},\)\(\nu _{t}\in L^{\infty }(\Omega ,\mathcal {F}_{t},\mathbb {P})\).

If the optimal values \(V_{0}^{*}(K,D,\eta ),\,U_{0}^{*}(K,D,\eta )\) are equal, this reformulation shows that the superhedging value can be interpreted as a (modified) expected present value of the opportunity costs for selling parts of the production according to the contract and not at the market. The opportunity costs here are discounted by stochastic discount factors which fulfill conditions similar to the no-arbitrage conditions A1)–A6). Expected present value and constraints are modified by effects related to the upper bounds on production and storage.

As shown in Kovacevic (2018), a sufficient condition for equality of the primal and dual optimal values \(V_{0}^{*}(K,D,\eta )=U_{0}^{*}(K,D,\eta \) (strong duality) is that the market is arbitrage free (which implies that the interior of the feasible set is nonempty). In the light of the previous subsection when using vector autoregressive models an arbitrage free model of the form (3.3)–(3.6) can be ensured by just using unrestricted estimation and the valuation model (4.1), respectively its dual version (4.2) can be applied likewise.

4.2 Arbitrage free price models in tree-based multistage stochastic optimization

As already pointed out in the introduction, arbitrage free price models are important for any kind of optimization problem based on estimated prices. This comprises pricing problems as above, but also e.g. planning of electricity generation. So far a class of statistical models was considered, and it was shown that they are arbitrage free under quite general conditions. However, the used statistical model is not the only possible reason for arbitrage in an optimization problem. Often the statistical model is not used in a direct way, but by implementing a suitable approximation like e.g. a discretization of the original model. Then it must be ensured that arbitrage is not induced by limitations of the data structure representing the discretized model.

An important framework for solving decision problems is tree-based multistage stochastic optimization (see e.g. Pflug and Pichler 2014), which is based on distributions on finite state spaces. This is achieved by replacing a decision problem that is initially formulated on a continuous state space with a reformulation on a “tractable” finite state space. Here, scenario trees are the tools to model the discretized processes, their distributional properties and also the information flow.

Pflug and Pichler (2014), 1.4. (for an alternative formulation see e.g. Alonso-Ayuso et al. 2009) describes an approach where the original time oriented formulation involving time indices is replaced by a node oriented formulation: consider a finite probability space \(\Omega =(\omega _{1},\dots ,\omega _{KS})\), representing S scenario-paths. Any stochastic process defined on this sample space can be represented as a finite tree with node set \({\mathcal N}=\{0,1,\dots ,N\}\). The levels of the tree correspond to the decision stages. Let \({\mathcal N}_{t}\) be the set of nodes at level t, for \(t=0,\dots ,T\). The final level \({\mathcal N}_{T}\) contains the S leaves of the tree, each of which can be identified with a scenario paths: \({\mathcal N}_{T}=\Omega =(\omega _{1},\dots ,\omega _{S})\). The tree structure represents the filtration of the process and can be defined (as a data structure) by stating the (unique) predecessor node \(n_{-}\) for each node n. There is a unique root node, by convention denoted with 0, which represents the present. By construction there is a one-to-one relation between any node n and an assigned pair \((\omega ,t\)), which means that each node relates to the state of the system at time t in sample path \(\omega \) and vice versa.

The price processes \(X^{e},X^{f}\) are represented w.r.t. the nodes of the tree, i.e. \(X_{n}^{e},X_{n}^{f}\) for some \(n\in {\mathcal N}\) is used instead of \(X_{t}^{e}(\omega ),X_{t}^{f}(\omega )\) for some \(t\in \mathcal {T}\). In similar manner the decision processes x, c, s, z, y are related to the nodes: So far \(s_{t}(\omega )\) denoted the amount of fuel stored at time t in state \(\omega \). In the discretized model, \(x_{n}\) denote the value of produced energy planned at node n, which can be identified with a point in time t and a scenario \(\omega \). Almost sure constraints then are obtained by formulating the same constraint for all nodes of a stage \(\mathcal {N}_{t}\). Moreover, constraints between points in time can be rewritten with node indices instead of time indices, using the predecessor relation \(n_{-}\). As an example consider the cash equation (2.1), which can be rewritten as

$$\begin{aligned} s_{n}=s_{n_{-}}-\sum _{i=1}^{I}\eta _{i}^{-1}y_{i\,n_{-}}+z_{n_{-}} \end{aligned}$$

in the node oriented formulation.

Finally probabilities \(\pi _{n}\) can be assigned to all leaf nodes \(n\in \mathcal {N}\)(T), which also implies probabilities \(\pi _{n}\) for all other nodes. The probabilities then can be used to formulate objective functions based on expectation or other probability functionals (risk or acceptability functionals).

Given an estimated price model, several methods have been proposed to construct approximating trees (with a fixed tree structure), see e.g. Dupacova et al. (2003), Heitsch and Römisch (2010), Pflug and Pichler (2015). The outcome are price values and probabilities at all nodes. In the context of the basic model (3.1)–(3.3) this means that for a node n the model equations are replaced with

$$\begin{aligned} X_{n}^{e}-E_{n}&=\phi ^{e}(X_{[n_{-}]}^{e}-E_{[n_{-}]},\,X_{[n_{-}]}^{f}-F_{[n_{-}]};\theta )+\varepsilon _{n}^{e} \end{aligned}$$
(4.3)
$$\begin{aligned} X_{n}^{f}-F_{n}&=\phi ^{f}(X_{[n_{-}]}^{e}-E_{[n_{-}]},\,X_{[n_{-}]}^{f}-F_{[n_{-}]};\theta )+\varepsilon _{n}^{f}, \end{aligned}$$
(4.4)

where \(X_{[n_{-}]}^{i}\) denotes a history of price values from predecessor values in the same paths as n and the \(\theta \) is the original estimator. On the other hand, given a node n the conditional distribution of the errors \(\varepsilon _{m}^{e},\varepsilon _{m}^{f}\) related to its successor nodes (i.e. \(m\in \left\{ k:\,n=k_{-}\right\} \)) can be described by pairs of price values and conditional probabilities (which can be derived easily from the node probabilities \(\pi \)). When \(\theta \) has been originally estimated to obtain an arbitrage free model as discussed above, also the approximating tree model stays arbitrage free under quite general circumstances.

However, when many decision stages are involved, scenario trees may contain numerous nodes. Decomposition techniques and other algorithmic tools exist in literature to reduce the resulting computational effort. However, often it is necessary to reduce the density of scenario trees: the number of node-successors might be small, some nodes even might have a single successor. In such nodes the conditional variances of the errors become zero, i.e. the distributions degenerate as discussed in the previous sections. It is not enough then to define the prices in these single successor nodes by

$$\begin{aligned} X_{n}^{e}-E_{n}&=\phi ^{e}(X_{[n_{-}]}^{e}-E_{[n_{-}]},\,X_{[n_{-}]}^{f}-F_{[n_{-}]};\theta ) \end{aligned}$$
(4.5)
$$\begin{aligned} X_{n}^{f}-F_{n}&=\phi ^{f}(X_{[n_{-}]}^{e}-E_{[n_{-}]},\,X_{[n_{-}]}^{f}-F_{[n_{-}]};\theta ), \end{aligned}$$
(4.6)

as usually done. These expected prices may lead to arbitrage because they pretend perfect foresight. According to the previous results, this can be avoided by prices fulfilling (3.24)–(3.25). These conditions have priority and one might use e.g. prices that are as close as possible to the conditional expectations as possible (in some distance) but still fulfill the no-arbitrage conditions.

Note that in financial applications one also aims at using arbitrage free prices in planning applications like portfolio optimization or asset-liability management, to exclude unrealistic gains. In Geyer et al. (2010) it was claimed that if M assets and N successor nodes for some node in the tree are considered and \(N<M\), then arbitrage is always possible. Following Duffie (2001), if \(\pi \) denotes the vector of prices (dimension M) in this node and D is the \(M\times N\) matrix of asset prices in the N successor nodes, then absence of arbitrage is equivalent to the existence of some vector \(\xi \in \mathbb {R}^{N}\), \(\xi >0\) (stochastic discount factor) such that

$$\begin{aligned} D\cdot \xi =\pi . \end{aligned}$$
(4.7)

While the claim in Geyer et al. (2010) is not fully correct,Footnote 2 it is often true in practical cases. For \(N<M\), arbitrage can only be excluded if the price vector \(\pi \) (dimension M) lies in the N-dimensional cone of positive linear combinations of the column vectors of D, which is a strong requirement. For one successor node (\(N=1\)) and two prices (\(M=2\)), as in the discussion above, this criterion means that the payoff D must be a (positive) multiple of the price vector \(\pi \) to avoid arbitrage. As a special case: when D is defined by expected prices and the prices are martingales under the physical measure (which is very unlikely), this would ensure the absence of arbitrage. When prices are not martingales under the physical measure, one could enforce absence of arbitrage by replacing the expectations by the prices of the predecessor nodes, which would heavily change the system dynamics. Moreover, even if there are enough (\(N\ge M\)) successors it must be shown that \(\xi >0\) exists such that (4.7) holds to ensure that the prices are arbitrage free.

At first glance this seems to be a contradiction to the previous findings in the present work with \(M=2\)—at least in the context of vector autoregressive models, where absence of arbitrage can be excluded under normal conditions when \(N\ge M\) and even in the problematic case \(N=1\) (with variances equal to zero), absence of arbitrage can be ensured by using prices close to an initial estimate, but fulfill the inequalities (3.24)–(3.25). For fuel prices this still means that they have to martingales under the original measure, condition (3.24) however is different and much weaker than the requirement that also electricity price should be a martingale under the original measure. That absence of arbitrage is easier to achieve in the electricity production setting than in a purely financial setting can be explained by the additional frictions, present on the considered market (in particular the nonnegativity of fuel storage and irreversibility of electricity production). Still, a positive \(\xi \) is required, but the conditions (3.24)–(3.25) are much less restrictive than the pure equation system (4.7). So the frictions make it more difficult to achieve arbitrage.

5 Conclusions

In the present work a necessary and a sufficient condition for absence of arbitrage on an electricity market with generation from fuel, fuel storage and also accounted for the related costs was derived. Despite the fact that electricity markets show unique frictions, the derived conditions can be easily interpreted in a financial context. Building on these results, the main question is, how restrictive such constraints are. In particular, it is important for practical applications to know the implications for parameter estimation because for many planning and valuation problems it is important to base decisions on arbitrage free prices to avoid unrealistic outcomes.

These questions were analyzed for a class of (potentially nonlinear) vector-autoregressive prices models for fuel and electricity. It turned out that actually the derived conditions for absence of arbitrage restrict the space of parameter values only to a very slight extent. If one ignores deterministic price models, it is therefore to use unrestricted estimation approaches (like e.g. unrestricted maximum likelihood) for estimation.

Note that the fact that the considered vector-autoregressive price model has additive error terms is crucial for the proofs given in this work. While it has been shown here that it is virtually impossible to achieve arbitrage in such a setup, it remains open under which conditions arbitrage might be possible for other specifications. Therefore, the analysis of parametric models will be extended to further model classes in future work.

This work uses a simple, stylized market model. This leaves room for future research. In particular, further frictions like efficiencies dependent on the generation level and several different fuels and usage of renewable energy might be considered.