Gaussian Volterra processes as models of electricity markets

Introduction

Soon after the liberalization of electricity markets in Europe, I. Simonsen in presented empirical evidence that electricity prices demonstrate stylized facts (most notably, antipersistence and self-similarity) which are well explained by fractional Brownian motion (fBm). This was confirmed in several successive empirical studies (see e.g. and references therein). However, the wide application of fBm in energy markets, as well as in general financial markets, was hindered by the fact that fBm is not a semimartingale, thus (roughly speaking) it produces the possibility of arbitrage (see e.g. the wide discussions on this aspect in [4, Chapter 7] and [11, Chapter 5]). Partial remedies to this could be the introduction of transaction costs, or the use of mixed-fBm models; however, both of these models are comparatively difficult to consider analytically.

The aim of this paper is to show that it is indeed possible to use fBm, and even more general Gaussian Volterra processes, in electricity markets without introducing arbitrage. We also provide an application of our model to portfolio optimization and option pricing problem.

In order to illustrate our approach, we start with the following toy model. Assume that the spot price of electricity is of the form $S_{t}=\varphi(t)+\bar{S}_{t}$, where $\varphi$ is a deterministic seasonality function and $\bar{S}$ evolves according to Langevin equation as

In turn, this implies that the correlation of future increments of $B^{H}$ with the past trajectory is positive (i.e., $B^{H}$ is persistent) when $H>1/2$ and negative (i.e., $B^{H}$ is antipersistent) when $H<1/2$. Moreover, for $H>1/2$, the decay of this dependence as the time intervals grow apart is slow, and we talk about long-range dependence (long memory). For $H<1/2$, this decay is fast, and referred by short-range dependence (short memory). The Hurst index is also a measure for the roughness of the paths of fBm: the larger the Hurst index, the smoother the paths. Other basic properties captured by $H$ are stationarity of increments. that is (B^{H}_{t+h}-B_{h}^{H})_{t\geq 0}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tinyd}}}{=}}(B_{t})_{t\geq 0}, $h>0$ and self-similarity, that is (a^{-H}B^{H}_{at})_{t\geq 0}\mathrel{\overset{\makebox[0.0pt]{\mbox{\tinyd}}}{=}}(B_{t}^{H})_{t\geq 0}, $a>0$. It can be shown that $B^{H}$ is the unique centered $H$-self-similar Gaussian process with stationary increments. We underline, moreover, that if $H\in(0,\frac{1}{2})\cup(\frac{1}{2},1)$ the fBm is neither a Markov process nor a semimartingale (as most Gaussian-Volterra processes). . These properties are transferred also to the process $\bar{S}$, and consequently to the spot price $S$.

The paper is organized as follows. In Section 2, we first introduce the model in its full generality, with the spot price driven by $m$ Gaussian-Volterra processes. Then, we provide sufficient and necessary conditions for the absence of arbitrage, as well as for the completeness of the market. Section 3 states the optimization portfolio problem in terms of utility maximisation problem for an agent who invests in an electricity market; we present explicit solution to this problem for the case of Constant Relative Risk Aversion (CRRA) utility functions. In Section 4, we present the problem of pricing and hedging for classical vanilla options, i.e. calls and puts, written on the forward contracts, as well as the pricing of the so-called Reliability Options. Finally, in Section 5, we give some examples of Gaussian Volterra processes which can be used to model electricity spot prices. We show analytically that, when these kind of processes drives the spot prices, under conditions ensuring the absence of arbitrage, market completeness is satisfied. We point out that in this section there are mathematical results that are interesting on their own, such as Proposition 21 and Lemma 22, where we provide a novel representation of Ornstein-Uhlenbeck (OU) processes driven by Gaussian Volterra processes.

Acknowledgements

This work was initiated while the first author was visiting the University of Padova in July and August 2022 under the framework of ”UniPD Rescue Fund”. The first author is also supported by The Swedish Foundation for Strategic Research, grant Nr. UKR22- 0017 and by Japan Science and Technology Agency CREST, project reference number JPMJCR2115. The second author is supported by the European Union - FSE-REACT-EU, PON Research and Innovation 2014-2020 DM1062 / 2021. The third author is supported by the project funded by the EuropeanUnion – NextGenerationEU under the National Recovery and Resilience Plan (NRRP), Mission 4 Component 2 Investment 1.1 - Call PRIN 2022 No. 104 of February 2, 2022 of Italian Ministry of University and Research; Project 2022BEMMLZ (subject area: PE - Physical Sciences and Engineering) ”Stochastic control and games and the role of information”. The authors wish to thank for fruitful discussions Fabio Antonelli, Marco Mastrogiovanni, Wolfgang Runggaldier, Anton Yurchenko-Tytarenko.

The model

We now fix a final horizon $\bar{T}<+\infty$, and assume that the spot price of electricity at each time $t\in[0,\bar{T}]$ is a $n$-factor process, i.e. driven by $n$ Gaussian Volterra processes, see, e.g. , as

This allows the process $S$ to be the sum of several Gaussian components, e.g. the sum of a Brownian motion and an Ornstein-Uhlenbeck (OU) process as in , or a variant of this model driven by fractional Brownian motions and/or fractional OU processes, possibly with different Hurst exponents, and so on. As one can see, the toy model presented in the Introduction corresponds to a fractional OU process. More details about these models, together with details about the Volterra kernels suited for this, can be found in Section 5.

We assume that the spot price of electricity $S=(S_{t})_{t\in[0,\bar{T}]}$ is not traded in the market, but the traded products are instead $m$ forward contracts with maturities $0<T_{1}<\ldots<T_{m}\leq\bar{T}$, and we assume that the general dynamics of the $j$th forward price $F(\cdot,T_{j})$ with maturity $T_{j}$, $j=1,\ldots,m$, evolves as

In the following section, we will discuss the conditions for absence of arbitrage and completeness in our market. In doing this, we follow closely the framework present in , which is especially suited to the general dynamics presented in Equation (7) whereas other similar sources are more specialized in assuming that asset prices are strictly positive, see e.g. . As it is well known, these notions are related to that of portfolio. As we already pointed out, in electricity markets, the spot price $S$ is not traded (and, if a proxy for that is quoted, it cannot be stored without transaction costs, that in this case means some energy loss), while forward contracts with maturities $T_{j}$, $j=1,\ldots,m$ are. Thus, we assume that an agent can build a financial portfolio by investing at time $t$ the quantity $\Delta_{t}^{j}$ in the forward contract with maturity $T_{j}$, $j=1,\ldots,m$. To keep things simple, we assume that our agent is interested in trading in this market only while all these forward contracts are still traded: since each forward contract is defined only until its maturity $T_{j}$, the global portfolio position $\Delta_{t}:=(\Delta^{1}_{t},\ldots,\Delta^{m}_{t})$ is well-defined only for $t\in[0,T_{1}]$, being $T_{1}$ the first maturity of our forwards: thus, we assume that our agent trades in the market only until a final time $T$, with $T\leq T_{1}$. As a consequence, we will give all the relevant definitions about portfolio, arbitrage and completeness relative to the final horizon $T$.

Starting from the discussion above, we assume that an agent builds a financial portfolio by investing at time $t\in[0,T]$, with $T\leq T_{1}$, the quantity $\Delta_{t}^{0}$ in the riskless asset, and the quantity $\Delta_{t}^{j}$ in the forward contract with maturity $T_{j}$, $j=1,\ldots,m$. This results in a portfolio value defined as $X_{t}^{\Delta}:={\Delta_{t}^{0}+}\sum_{j=1}^{n}\Delta^{j}_{t}F(t,T_{j})$. We call this portfolio self-financing if the dynamics of its value is

In order to rule out the possibility of arbitrage, in the next section we present the classical criterion of existence of an equivalent martingale measure.

Thus, we have a pricing relation between the spot and forward price, that leads to an arbitrage-free pricing dynamics for the forward price.

Now, we provide a necessary and sufficient condition for the absence of arbitrage in the market.

The proof of this proposition is quite standard and follows in part that of [14, Theorem 12.1.8]. For the readers’ convenience, we present a version adapted to our case in the Appendix.

Equation (15) states that $\mu$ and $\sigma$, which in principle are defined only for their second argument equal to the observed maturities $T_{j}$, $j=1,\ldots,m$, are instead defined structurally as functions of the vector of Volterra kernels $K$ (and $\theta$), which instead are defined for all times, also different from $T_{j}$. For this reason, if Equation (15) is true, we can extend the definition of $\mu$ and $\sigma$ for $T$ possibly different from $T_{j}$, $j=1,\ldots,n$, as

2 Completeness

where $I_{m}$ is the identity matrix of order $m$.

Let us note that, by virtue of the previous theorem, if the market is complete then the function $\theta_{t}$ satisfying (17) is unique. Indeed, the only solution is given by

The existence of the left inverse for which equation (20) holds is equivalent to the injectivity of $\bar{K}(t)$ for a.a. $t\leq T_{1}$. Thus, $rank(\bar{K}(t))=n$ for a.a. $t\leq T_{1}$, in particular, $m\geq n$.

We require, for convenience, the injectivity of $\bar{K}(t)$ ”for a.a. $t\leq T_{1}$”. However, we can note that completeness could still hold for $t>T_{1}$, as long as the remaining forward contracts span a diffusion matrix ( that is $\bar{K}(t)$ with reduced number of rows) with rank equal to $n\leq m$. Thus, in this case, even if we have assumed, form the beginning, that we are interested in trading only until $T_{1}$, in principle, we can trade even after some contracts have reached maturity, continuing to have a complete market.

If $m=n$ the market is complete if and only if $\bar{K}(t)$ is invertible for a.a. $t\leq T_{1}$.

We underline that the forward prices $F(t,T_{j})$, $j=1,\ldots,m$, that we model here are only an approximation (or better, the building blocks) of the real contracts that are traded in electricity markets. Those last can be represented by the quantities $F(t,T_{j},T_{k})$, each one of these being the price at time $t$ of a forward contractalso called swap contracts in this framework by some authors which delivers a fixed intensity of electricity over the period $[T_{j},T_{k}]$, where $0\leq t\leq T_{j}<T_{k}\leq\bar{T}$. More precisely, according to and to the no-arbitrage condition

Assume now that $|K(T,s)-K(T^{\prime},s)|\leq C|T-T^{\prime}|^{\rho}$ for all $T,T^{\prime}\in[0,\bar{T}]$ and for a suitable $C$. Then, from Equation (22) we have that

The optimal investment

Now we want to solve the problem of an agent who can invest in the electricity market and wants to maximize the expected utility of her/his wealth at the terminal time $T\in[0,T_{1}]$. The dynamics for the portfolio wealth has already been given in Equation (8), which we can rewrite as

where $u$ is a known utility function, i.e. a real function which is non-decreasing and concave, and $\Delta$ is chosen among the admissible strategies, as defined in Section 2, with the additional requirements that $u(X_{T}^{\Delta})$ is well-defined and such that Equation (23), with initial condition $X_{0}^{\Delta}:=x>0$, has a unique strong solution $X^{\Delta}$. We thus restrict our admissible strategies to the class satisfying also the additional assumptions above, and denote this new class of admissible strategy by $\mathcal{A}$.

In the case when the utility function is well-defined only on non-negative or strictly positive wealth, as in the cases $U(x)=x^{\gamma}$, $\gamma\in(0,1)$ or $U(x)=\log x$, respectively, requiring that $u(X_{T}^{\Delta})$ is well-defined is equivalent to require that $X_{T}^{\Delta}\geq 0$ or $X_{T}^{\Delta}>0$ a.s., respectively. The first requirement simply corresponds to take the constant in the definition of admissibility being equal to zero, while the second one poses a slight restriction to this condition. In both cases, by the properties of conditional expectation, the supermartingale property for $X^{\Delta}$ implies that also $X_{t}^{\Delta}\geq 0$ or $X_{t}^{\Delta}>0$ a.s. for all $t\in[0,T]$, respectively.

As in [5, Chap. 20], we incorporate the budget constraint in Equation (25) into the Lagrange function

Since in our setting the market is complete, this problem is brought back to the formulation

i.e., we now maximize in the generic random variable $X_{T}$, remembering that, once we obtain the maximizer $X_{T}^{*}$, in order to obtain the optimal portfolio strategy $\Delta^{*}$ we use a martingale representation for the optimal portfolio $X_{T}^{*}=X_{T}^{\Delta^{*}}$. For $\lambda>0$ fixed, we solve the equation

from which [5, Proposition 20.3], the optimal terminal wealth $X_{T}^{*}$ results in

Let us now consider the case when the utility function is of Constant Relative Risk Aversion (CRRA) type, i.e. $u(x):=\frac{1}{\gamma}x^{\gamma}$, with $\gamma<1$, $\gamma\neq 0$, or $u(x)=\log x$ (which is often seen conventionally as ”the case $\gamma=0$”, see e.g. the discussion in [5, Chapter 20.7]). Then, in order for $\Delta$ to be an admissible strategy, we must impose that $X^{\Delta}_{T}\geq 0$ in the case $\gamma\in(0,1)$ and $X^{\Delta}_{T}>0$ in all the other cases. Moreover, for all $\gamma<1$ (including the case ”$\gamma=0$”, which is the case of a log utility function) we have that $u^{\prime}(x)=x^{\gamma-1}$, thus $I(y)=y^{\frac{1}{\gamma-1}}$.

Hereafter, we will follow [5, Ch. 20]. Let us set

The Optimal Wealth Process. In (27), we have computed the optimal terminal wealth $X_{T}^{*}$, but it is also possible to find an explicit formula for the entire optimal wealth process $X^{*}$. In this, we follow [5, Chapter 20.5.2], but for the reader’s convenience here we write an ad-hoc derivation for our case.

The optimal wealth process $X^{*}=(X^{*}_{t})_{t\geq 0}$ is given by

By the abstract Bayes’ formula, we obtain

that looks almost like a Radon-Nikodym derivative: this leads us to define the $P$-martingale $Z^{0}$ as

Since $\theta$ is deterministic, we have that $\int_{t}^{T}\beta\theta_{s}\cdot dW_{s}$ is Gaussian and independent of ${\mathcal{F}}_{t}$, thus

and Equation (29) follows from $1-\beta=\frac{1}{1-\gamma}$. Putting all of this in Equation (30) we obtain

The Optimal Portfolio. Let us denote by $\mu_{H}(t):=-\frac{1}{2}\frac{\beta}{1-\gamma}|\theta_{t}|^{2}$ the drift of the process $H$ defined in (29), so that

The optimal portfolio process $\Delta^{*}=(\Delta^{*}_{t})_{t\geq 0}\in{\mathcal{A}}$ is given by

Proof. Starting from (28) for the optimal wealth process, by the Itô formula we have that

By comparing the diffusion part in (23) and (32), we have

System (33) admits solution (for a.a. $t\leq T<T_{1}$) given that by Theorem 5 there exists a left inverse of $\bar{K}(t)$ for a.a. $t\leq T_{1}$. Thus, taking

we obtain (33); this solution is our $\Delta_{t}^{*}$. Since $X^{*}$ appears linearly in $\Delta^{*}$, multiplied by a square integrable deterministic function of time, we also have that Equation (23) has a unique strong solution, thus $\Delta^{*}$ is admissible and optimal.

Option pricing

while the price of a put option with the same strike price $K$ and maturity $T$ is given by

We can immediately verify that, as in any other arbitrage-free market, the call-put parity holds in the following form.

If $C(t,T)$ and $P(t,T)$ denote respectively the call and put prices at time $t$, both with strike price $K$ and maturity $T$ written on the forward contract $F(\cdot,T_{j})$, with $T_{j}>T$ as in Equations (34)-(35), then we have

Proof. From Equations (34)-(35), it easily follows that

by the martingale property of $F(\cdot,T_{j})$.

By using the Bachelier formula adapted to this context (see e.g. [6, Eq. (3)]), we can also produce a closed formula for the price of call options on forwards (and consequently, by the call-put parity above, also for the price of put options).

The price of a call option at time $t$ written on the forward contract $F(\cdot,T_{j})$ with strike price $K$ and maturity $T$ with $T_{j}>T$ is

and the functions $n$ and $N$ are respectively the density and the cumulative distribution functions of a $N(0,1)$ law, i.e.

Moreover, the hedging portfolio for this call option is composed by the only asset $F(\cdot,T_{j})$ ( together with the money market account), and the quantity of this asset to be held at time $t$ is $N(d_{j}(t,T,T_{j}))$.

The results above are true for discounted prices, i.e. for a situation where the interest rate applied to prices is $r\equiv 0$. When one wants to derive explicitly the same results taking explicitly into account the presence of a risk-free short rate $r(u)$, $u\in[t,T]$, it is not difficult to see that in the case when $r$ is deterministic the call-put parity formula modifies into

The more general case when the intensity $(r_{u})_{u}$ is a stochastic process, possibly dependent on $F(\cdot,T_{j})$, can be treated with the usual change-of-numeraire techniques, see e.g. [5, Chapter 26]: Here we decided not to present this in detail so as not to further burden the reader.

However, in electricity markets other illiquid products can be found as well, usually more involved than the vanilla products seen above. A notable example is the so-called Reliability Option, which is present in several national markets (see e.g. ), and which has the peculiarity that its payoff is defined on the spot price $S$ over the time span $[T_{1},T_{2}]$, with $0\leq T_{1}<T_{2}\leq\bar{T}$. More in detail, the payoff of a Reliability Option with fixed strike price $K$ written on the time span $[T_{1},T_{2}]$ is found to be equal to

The price at time $t=0$ of the Reliability Option as defined in Equation (37) is equal to

with $\sigma(0,T,T)$ as defined in Equation (36),

Gaussian Volterra Processes to model electricity spot prices

In this section, we provide some examples of Gaussian Volterra processes which can be used to model electricity spot prices. We show analytically that, under conditions ensuring the absence of arbitrage, the completeness of the market is satisfied (see Theorem 5 and Corollary 8). While this is straightforward when the number of factors $n$ is 1, checking this for $n>1$ involves proving that the matrix $\bar{K}(t)$ is nonsingular for a.a. $t\in[0,T]$. This is generally true, and quite straightforward to prove, when the kernels $K$ have different functional form from one another, resulting in factors which belong to different classes of Gaussian processes, for example when we have a standard Ornstein-Uhlenbeck process and an Ornstein-Uhlenbeck process driven by a fractional Brownian motion (fractional Ornstein-Uhlenbeck, fOU processes), see e.g. Section 5.5Less trivial is the case when the Gaussian processes belong to the same class, for example when we have two fractional Brownian motion with different Hurst exponents. For this reason, in the following sections we check the most common cases that one could face, namely the case of Riemann-Liouville (RL) processes, and the case of a fBm, in both the cases with Hurst exponent $H>1/2$ and $H<1/2$. We also show how to formulate these results for Ornstein-Uhlenbeck (OU) processes driven by them, obtaining the original driving processes in the case when the mean-reversion speed is zero: for this reason, we first present how to treat OU processes driven by a generic Gaussian Volterra process, and then we will analyze the cases when the OU processes are driven by Gaussian-Volterra processes of the same kind as seen above, namely RL processes and fBm with $H>1/2$. We will analyze in detail the case when one has only $n=2$ factors, as all the cases with $n>2$ get more and more complicated to handle. The only exception is when we have a generic number $n$ of RL fBm, in which case market completeness is equivalent to the determinant of a generalized Vandermonde matrix being nonzero, which we check to be true.

where we search for a solution $Y=\{Y_{t},t\geq 0\}$ having Lebesgue-integrable sample paths, so that the integral $U_{t}:=\int_{0}^{t}Y_{s}ds$ in the right-hand side is well defined. Denoting $U_{t}=\int_{0}^{t}Y_{s}ds$, we can rewrite the above equation as

as the process $U$ has continuous sample paths which are a.e. differentiable with respect to the Lebesgue measure. Moreover, if $Z$ is continuous a.s., then equation (40) holds for all $t>0$. The condition of continuity of $Z$ is based on the standard Kolmogorov continuity theorem: we here follow , which specifically analyzes Gaussian Volterra processes. Let $Z=\{Z_{t},t\geq 0\}$ be a centered Gaussian process. If there exists $K>0$ and $\delta>0$ such that

then the process $Z$ has a modification that is continuous on $[0,T]$ and, moreover satisfies Hölder continuity on $[0,T]$ of any order $0<\gamma<\frac{\delta}{2}$.

In the general case of a Gaussian Volterra process $Z$ defined as in Equation (38), the condition (41) is reduced to (see e.g. )

Let the condition (41) be fulfilled. Then Equation (40) has the unique continuous solution

and Equation (39) has the unique continuous solution

which can be represented as a Gaussian-Volterra process as

Proof. If the condition (41) holds, then Equation (40) is a linear ordinary differential equation in $U$ with non-homogeneous term $Z$, which is a continuous random function (here we consider the continuous modification of $Z$). Thus, the standard theory of ordinary differential equations gives the unique continuous solution as in Equation (42), which is differentiable and has derivative given by Equation (43). This is obviously solution of Equation (39), and it is very easy to see that this equation admits a unique solution. Take in fact two solutions $Y^{1}$ and $Y^{2}$: then we have that

which, by the Gronwall lemma, has as consequence $Y^{1}_{t}-Y^{2}_{t}\equiv 0$ for all $t>0$, thus, $Y^{1}\equiv Y^{2}$, and the solution is unique. According to [17, Chapter 4, Theorem 64], we can apply the stochastic Fubini theorem to the first term in the right-hand side of (43) and get that

where $K_{Y}$ is defined as in Equation (44): this concludes the proof.

The advantage of (44) is that we immediately get it avoiding any fractional or other operators or transformation. However, if we want to reduce (44) to one term, then some additional assumptions are needed.

If $K_{Z}$ is such that $K_{Z}(u,u)=0$ and is differentiable in the first variable, with $\frac{\partial K_{Z}}{\partial s}$ Lebesgue integrable in $s$, then

Proof. Under the previous assumptions, we can integrate in (44) by parts and get that

Now, let us consider some examples of Gaussian-Volterra processes.

2 Riemann-Liouville processes

The Riemann-Liouville (RL) process is defined as

where $W$ is a standard 1-dimensional Brownian motion, and normalizing factor $c_{H,U}$, by analogy with fractional integrals, is chosen as $c_{H,U}:=\frac{1}{\Gamma(H+\frac{1}{2})}$. So, the RL process is a Gaussian-Volterra process having the kernel $K_{U^{H}}(t,s):=c_{H,U}(t-s)^{H-\frac{1}{2}}$. This kernel is square-integrable in $[0,t]$ for any $H>0$, and it is well known that $U^{H}$ has continuous sample paths for all $H>0$. However, the conditions that we gave to have an OU process driven by this process with the good properties seen in Section 5.1 are valid only when $H>1/2$. In fact, we have that $K_{U^{H}}(s,s)=0$ if and only if $H>1/2$; in this case,

which is integrable in $t$ for $H>1/2$. For this reason, we will analyze OU processes driven by RL processes only in the case $H>1/2$, while we have a much more general result for RL processes. We underline that the increments of RL processes are not stationary .

As announced, in the case of RL processes, we can prove that $n$ different factors, each one with a different Hurst exponent, make the matrix $\bar{K}(t)$ nonsingular for all $t>0$. Consider $n$ independent Riemann-Liouville processes represented as in Equation (45), driven by $n$ independent standard 1-dimensional Brownian motions. We assume that the Hurst exponent $H_{i}$, $i=1,\ldots,n$, are all different; thus, without loss of generality, we can assume that $0<H_{1}<H_{2}<\ldots<H_{n}<1$.

From Corollary 8, a sufficient and necessary condition for the completeness of the market in this case is

for any $0\leq t\leq T$, where $T<T_{1}<\ldots<T_{n}$. Let us note that, as well as in the subsequent subsections, in order to assess whether the matrix (18) is invertible we can neglect the normalizing constants appearing in each line, moreover, we will consider equivalently $\bar{K}$ in (18), as well as its transpose: clearly this does not change the sign of the determinant of $\bar{K}$. Without loss of generality, we assume that $0<H_{1}<H_{2}<\ldots<H_{n}<1$. Then, by letting $\alpha_{i}:=H_{i}-\frac{1}{2}$ and $x_{j}:=T_{j}-t$, for $i,j=1,\ldots,n$, the following result is instrumental in proving the completeness.

Let us consider the determinant of the form

Let us note that this matrix has the form of the so-called unsigned exponential Vandermonde matrix. The result on the positiveness of the determinant for the signed exponential Vandermonde matrix can be found in . However, the proof for the case of the unsigned exponential matrix is not explicitly provided there, thus we report our proof in the Appendix A for the readers’ convenience.

In our case, as $x_{j}=T_{j}-t$, we have $0<T_{1}-T\leq x_{1}<\ldots<x_{n}\leq T_{n}$ for all $t\in[0,T]$, so we can see that the conditions of Theorem 23 are satisfied. Thus, also the determinant of the matrix of the processes’ kernels $\bar{K}(t)$ is positive for all $t\in[0,T]$. Therefore, in the case the electricity spot price is the sum of $n$ RL processes, we can assert that our market is complete, when the number $m$ of forward contracts is equal to $n$. Let us note that the completeness is still valid in the case $m>n$, by Theorem 5: indeed, thanks to Theorem 23, we have a minor of order $n$ different from zero, so the rank of the matrix is $n$

2.2 Ornstein-Uhlenbeck processes driven by RL processes, H>1/2𝐻12H>1/2

By Lemma 22, an OU process with mean-reversion speed $\alpha$ driven by a RL fBm with Hurst exponent $H>\frac{1}{2}$ is a Gaussian-Volterra process, with kernel given by

Consider two OU processes with the same mean-reversion speed $\alpha$, driven by two independent RL fBm with $\frac{1}{2}<H_{1}<H_{2}$, and let $0<T_{1}<T_{2}$. In this case

Obviously, $\Delta_{RL}(T_{1},T_{1})=0$, and

for a suitable $\xi\in(s,T_{1})$ to be chosen due to the Cauchy theorem about the ratio of increments of differentiable functions. Then we have

which implies that $\frac{\partial\Delta_{RL}(T_{1},T_{2})}{\partial T_{2}}>0$: this, joint to $\Delta_{RL}(T_{1},T_{1})=0$, implies that $\Delta_{RL}(T_{1},T_{2})>0$ for all $T_{2}>T_{1}$.

3 Ornstein-Uhlenbeck processes driven by fBMs, H>1/2𝐻12H>1/2

Consider now the case of fBM $B_{H}$ with Hurst index $H\in(\frac{1}{2},1)$. Then $B_{H}$ admits the compact interval representation of the form

In other words, in this case the kernel is

For $H>1/2$, this kernel is square-integrable in $[0,t]$ and is such that $B^{H}$ has continuous (even Hölder up to order $H$) sample paths. Moreover, the conditions that we gave to have an OU process driven by this process with the good properties seen in Section 5.1 are valid. In fact, following Lemma 22, we have that ${K_{{B^{H}}}(s,s)}=0$ and

which is integrable in $t$ for $H>1/2$. Thus, we can also analyze OU processes driven by fBm. After Lemma 22, an OU process with mean-reversion speed $\alpha$ lead by a fBm is a Gaussian-Volterra process with kernel

Let us consider two fOU process with the same mean-reversion speed $\alpha$ and $\frac{1}{2}<H_{1}<H_{2}$, and consider also $0<T_{1}<T_{2}$. Then, we have

for a suitable $\xi\in(s,T_{1})$ to be chosen due to the Cauchy theorem about the ratio of increments of differentiable functions. By definition of $\varphi_{H}$, we have

which implies that $\frac{\partial\Delta(T_{1},T_{2})}{\partial T_{2}}>0$: this, joint to $\Delta(T_{1},T_{1})=0$, implies that $\Delta(T_{1},T_{2})>0$ for all $T_{2}>T_{1}$.

This result ensures that for the case in which the electricity price is driven by two fOU process with $\frac{1}{2}<H_{1}<H_{2}$, the same mean-reversion speed $\alpha$ and $0<T_{1}<T_{2}$, the market is complete. Since for $\alpha=0$ the two OU processes degenerate in two fractional Brownian motions, the result holds true also for this simpler case.

4 Fractional Brownian motions, H<1/2𝐻12H<1/2

Now, let $Z=B^{H}$, a fBm with Hurst index $H\in(0,\frac{1}{2})$. In this case, we have the representation

for ease of notation in the following, we denote $K^{H}_{B}:=K_{H}$, that explicitly can be written as

Now, let $0<T<T_{1}<T_{2}$ and $0<H_{1}<H_{2}<\frac{1}{2}$. We want to investigate the behaviour of the matrix (18), that in this case writes:

specifically, we wonder if there are any points $0<s\leq T$ where $\Delta(s)$ is zero. Thus, in the following, we proceed to answer this question. Equivalently, it is sufficient to analyze the behavior of the determinant

To study the specifics of this manifold, let us calculate

Let $f(R)=\frac{K_{H_{1}}(R)}{K_{H_{2}}(R)}$. So, we are interested in the sign of the difference $f(R_{1})-f(R_{2})$ for $1<R_{1}<R_{2}$. We have

Thus, we have to study the sign of the term in the square brackets that we denote by $I(R)$. By some computations, we obtain

We can see that if $R\searrow 1$, then $I(R)\rightarrow-\infty$, if $R\rightarrow+\infty$, then $I(R)\rightarrow+\infty$. Moreover, it is easy to see that $I(R)$ strictly increases. This means that for some $R^{*}$, $f^{\prime}(R)>0$, when $1<R<R^{*}$, $f^{\prime}(R^{*})=0$, and $f^{\prime}(R)>0$ for $R>R^{*}$. That is, $R^{*}$ is a minimum point of $f$. Moreover, when $R\searrow 1$ , $f(R)\rightarrow+\infty$ and when $R\rightarrow+\infty$, $f(R)\rightarrow f_{\infty}$, where

and by the sign of the derivative of $f$, we have that there exists $R_{*}<R^{*}$, for which $f(R^{*})<f(R_{*})=f_{\infty}$.

If $\Delta(T)<0$, then for all $0<s\leq T$, $\Delta(s)<0$, on the other hand, if $\Delta(T)\geq 0$ there exists a unique $0<s\leq T$ such that $\Delta(s)=0$.

Thus, we have that the condition ensuring completeness of the market is satisfied (see Corollary 8), indeed the determinant of $\bar{K}(t)$ vanishes at most in one point.

5 Mixed case (OU and fOU with H>1/2𝐻12H>1/2)

As we already said at the beginning of this section, the cases when the Gaussian-Volterra processes driving the spot price $S$ belong to different classes are quite straightforward to treat. As an example, we here report a result relative to the case when we have two factors, namely a standard OU process and a fOU process with $H>1/2$.

Let us consider a standard OU process, with kernel $K_{1}(t,u):=e^{\alpha_{1}(t-u)}$, and a fOU process with $H>1/2$, with kernel

If $\alpha_{1}\leq\alpha_{2}$, then for $T_{1}<T_{2}$ we have

Now, $\Delta(T_{1},T_{2})\neq 0$ is equivalent to

which is true if $\alpha_{1}\leq\alpha_{2}$, as in this case

Conclusions

We introduce a non-Markovian model for the spot price of electricity, based on a $n$-factor Gaussian Volterra process, obtained by stochastic integrals of Volterra kernels. As is customary in electricity markets, we assume that the spot price is not traded, but forward contracts with maturities $T_{1}<T_{2}<\ldots<T_{m}$ are traded in the market. We discuss conditions for absence of arbitrage and completeness. We characterize the dynamics of forward prices when there is no arbitrage in terms of an equivalent martingale measure, finding that in that case the risk-neutral dynamics of forward prices has zero drift and diffusion coefficients equal to the Volterra kernels with a variable frozen to the forward maturity. Moreover, we find that completeness is equivalent to the invertibility (in a generalized sense) of the matrix formed by the diffusion coefficients of the traded forwards. In all the above discussion, we modeled forward prices as having instantaneous delivery, while real contracts in electricity markets deliver electricity during a time span $[T_{1},T_{2}]$: however, this is a common market practice, which we show to be justified in our framework, as the tracking error between the real asset and the one that we model goes to zero as a power of the period length $T_{2}-T_{1}$, thus this approximation is particularly robust for hourly or daily contracts.

After this, we find the optimal investment in forward markets, analyzing in detail the particular case of CRRA utility functions, for which we characterize explicitly the optimal portfolio strategy. We also present option pricing formulas: in the particular case of vanilla calls and puts written on a traded forward, we find that the price is unique also in incomplete markets and given by a version of the Bachelier formula, and we also give the hedging strategy. However, in order to have a unique price for other kind of derivatives, possibly more exotic, completeness should be assumed: as an example of this, we give the pricing formula for a particular option in electricity market, called reliability option. Finally, we analyze in detail examples of the most used Gaussian Volterra processes in our particular framework, especially concerning market completeness. The case when the Volterra kernels have different functional forms among each other is usually quite straightforward, so we just give one example of this case in Section 5.5. Instead, the more interesting case is when the Volterra kernels refer to the same kind of processes. Thanks to a novel representation of Ornstein-Uhlenbeck (OU) processes driven by Gaussian Volterra processes, we show how to treat fractional Brownian motions as a particular case of fractional OU processes. We then pass to analyze particular instances of fractional processes, starting from Riemann-Liouville (RL) processes: in this case, we find that $n$ independent RL processes generate a complete market if their Hurst exponents are all different from each other. The case of RL-driven OU processes is more involved, and we are able to prove that two such processes generate a complete market in the case when the mean-reversion speed is the same and the Hurst exponents are different. We use the same framework, i.e. two OU processes with same mean-reversion speed, also in the case of fractional Brownian motions with Hurst exponents both greater than 1/2; we prove the market completeness in a quite straightforward way, as the diffusion matrix is invertible for all times. Instead, we do not present the case of OU processes driven by fBm with Hurst exponent smaller than 1/2. We only consider two fBm driving the spot prices, with different Hurst exponents smaller than 1/2, we find that the diffusion matrix is invertible at all times but one, but the assumptions for completeness are still satisfied.

Appendix A Appendix

Assume now that there is no arbitrage, i.e. Equation (10) is true for all $j=1,\ldots,m$. This, together with Equation (12), implies

which, compared with Equation (7), gives the desired conclusion.

Conversely, now assume that Equations (14) and (15) hold, which consistently give the dynamics in Equation (7). Then, combining Equation (14) with its terminal condition and Equation (12), we obtain

We proceed by induction. Obviously, the statement is true for $n=1$. Let us assume that it is true for $n-1$. We can transform our determinant $\Delta$ as follows:

Since we have assumed that the statement holds for $n-1$ we can assert that

Now, let us consider subsequently the derivative, starting with

and it equals zero if $x_{n}=x_{1}$. Therefore,

for $x_{n}>x_{1}$, that is our case. Thus, we can proceed in the same way and get that

References