Introduction

According to [26], no economical development is possible without the availability of energy. Accepting this reality, several governments consider electricity as one of their main priorities. For example in France, electricity is recognized by the law as a basic necessity. Its price is determined by regulated tariffs made by the government [4]. Several changes have been operated in electricity sector. Regulation and deregulation are the main mechanisms which caused these changes observed in electricity sector. The aims were to create a competitive economical environment in which the producers, the investors and the large part of consumers would get their satisfaction.

The introduction of deregulation induced many consequences. One main consequence was the high variation in price which encouraged the development of a new breed of financial products in electricity markets. These new products may help cover both physical and financial risks on the new market. Therefore, there has been an important research effort devoted to electricity price modeling for derivative pricing. Due to the non-storable nature of electricity, the challenge of the researchers was the development of a completely satisfying methodology that would help to obtain realistic and robust models. Two standard approaches have often been used to handle this problem in the literature. The first consists in modeling directly the forward curve dynamics and deduces the spot price [2, 5]. The second approach starts from a spot price model to derive future prices as the expectation of the spot price under a risk-neutral probability. Relevant contributions have been made by [8, 20] in pricing energy derivatives and electricity. They took into account seasonality and mean reversion. However, their model did not take into account the huge and non-negligible observed spikes in the market. Further [3, 6] were among the first to consider price spikes using jump-diffusion models. Similar works were done in [10, 14, 15, 24, 25, 30]. Regular increase in electricity prices and crises observed in the unregulated market are a point of focus in the media and raised the question of regulation. Moreover, direct link between the price of electricity and the national strategy of poverty reduction motivated governments to limit electricity prices, so that it can be more accessible. This leads to the reintroduction of price regulation by most governments. For instance, [19] was the first to propose the price-cap regulation to British government. Several works were done to study effect and impact of regulation in electricity network and the wholesale electricity market [12, 16].

The main contribution of this paper is, using the rate of increase given by the price-cap, to construct a spot price model in regulated electricity market. In addition to well-known specific features of electricity such as mean-reverting and spikes, our proposed model captures important characteristics of price-cap regulation: inflation rate and efficiency rate. Furthermore, we compute explicitly the forward price in this electricity model.

The rest of this paper is structured as follows: In section two, we present a review of some recent electricity price models. Section three deals with the formulation of our model. Moreover, in section three, we discuss the mean-reversion feature of our model and further compare it with [3] model. In section five, we present some numerical simulations of our model to illustrate our theoretical results and support our discussion.

Review of some recent electricity models

Several models on electricity price dynamics have been proposed in the literature, among which the jump-diffusion model. Merton [23] was among the first to work on this class of models. His first model was developed to describe the dynamic return on equity. This model was progressively extended by [3, 8, 20, 24].

Schwartz [8] considered spot prices as a stochastic process with two components represented by

$$\begin{aligned} P_t=f(t)+X_t;\; t\in [0,\infty ); \end{aligned}$$
(1)

where f is a deterministic differentiable function and \(X_t\) is the stochastic component satisfying

$$\begin{aligned} \hbox {d}X_t=-\alpha X_t\hbox {d}t+\sigma \hbox {d}W_t, \end{aligned}$$
(2)

with \(\alpha >0\) representing the speed of mean reversion, \(X(0)=x_0\) being the initial condition and W being a standard Brownian motion. Applying Itô’s formula on (2), they obtained the following dynamic for spot price

$$\begin{aligned} \hbox {d}P_t=\alpha (a(t)-P_t)\hbox {d}t+\sigma \hbox {d}W_t, \end{aligned}$$
(3)

where

$$\begin{aligned} a(t)=\frac{1}{\alpha }f'(t)+f(t). \end{aligned}$$

In the same paper, they also considered \(\log\) of spot prices, i.e., \(\ln P_t= f(t) + Y_t\) where \(Y_t\) follows process (2). In this case, they obtained

$$\begin{aligned} \hbox {d}P_t=\alpha (b(t)-\ln P_t)P_t\hbox {d}t+\sigma \hbox {d}W_t, \end{aligned}$$
(4)

where

$$\begin{aligned} b(t)=\frac{1}{\alpha }\left( \frac{\sigma ^2}{2}+f'(t)\right) +f(t). \end{aligned}$$

This model captures mean-reverting feature which is one of the main characteristics of electricity, but the model has failed to take into account the spikes which can occur in electricity markets. Cartea and Figueroa [3] in the same context of deregulation markets extended (4) by adding a jump term and obtained a mean-reverting and jump-diffusion model. They supposed that the spot price process is in the form \(\ln S_t= g(t) + Y_t\), where g is a seasonal deterministic function and that \(Y_t\) follows a stochastic process given by

$$\begin{aligned} \hbox {d}Y_t=-\alpha Y_t\hbox {d}t+\sigma \hbox {d}W_t + \ln J\hbox {d}q_t. \end{aligned}$$
(5)

Using Itô’s formula and equation (5), [3] obtained the following model:

$$\begin{aligned} \hbox {d}S_t = \alpha (\rho (t) - \ln S_t)S_t\hbox {d}t + \sigma (t) S_t \hbox {d}W_t + S_t (J-1)\hbox {d}q_t, \end{aligned}$$
(6)

where

$$\begin{aligned} \rho (t) = \frac{1}{\alpha }\left( g'(t) + \frac{1}{2}\sigma ^2(t)\right) + g(t), \end{aligned}$$

J is the proportional size of jump and \(q_t\) is the Poisson process. Hence, on contrary to [3, 8] in their model considered the non-constant volatility, jump and deterministic part of spot price as a seasonal function of time.

Model derivation and the main result

Our model is partly inspired from the electricity price-cap regulation proposed by Littlechild [19] that we recall as follow.

Price-cap market regulation

The price-cap regulation is an economical principle which aims to establish an incentive scheme for the regulated market. A key objective is to enable companies to maximize the well-being while seeking to maximize their own interests, see [1]. Its principle is to cap the market price. The main components of the price cap include the efficiency factor (G), for transferring the gains to consumers through the reduction of costs; the inflation rate (I), which drives the price changes; the exogenous factors such as customer portion of earnings' sharing (E), service quality penalties (H) and flow-through and uncontrollable costs, if any (F). ENMAX [11] proposed price-cap formula:

$$\begin{aligned} \frac{P_i-P_{i-1}}{P_{i-1}}=I_i-G_i + \left( \frac{-E_i-HS_i+F_i}{P_{i-1}}\right) , \end{aligned}$$
(7)

where \(P_i\) represents the current year’s price and \(P_{i-1}\) preceding year price. Later, we would be inspired by the economic formula (7) to model the drift of the model.

Spot price modeling procedure

The daily (resp. weekly and monthly) change in price is the difference between today’s price (resp. this week’s price and this month’s price) and yesterday’s price (resp. last week’s price and last month’s price). In general, one denotes a change over a given time period dt by \(\hbox {d}S_t\). For a daily change, we therefore have \(\hbox {d}t=\frac{1}{365}\), \(\hbox {d}t=\frac{1}{52}\) for weekly change and \(\hbox {d}t=\frac{1}{12}\) for a monthly change. The change in price \(\hbox {d}S_t\) over a given time period dt is the sum of two components: the “drift” term and the stochastic (or “random”) term, that is,

$$\begin{aligned} \hbox {d}S= \text {drift term} + \text {stochastic term}. \end{aligned}$$

The drift term represents the portion of the movement in the spot price S, which we expect to see with certainty. This term is proportional to the time period over which the change in the price is measured. That is,

$$\begin{aligned} \text {drift}\propto \hbox {d}t. \end{aligned}$$

The stochastic term represents the portion of the change that is random and cannot be predicted. This term is proportional to the increment \(\hbox {d}W_t\) of standard Brownian motion, which is normally distributed with mean zero and variance \(\hbox {d}t\) (see [28, pp. 377–380]. That is,

$$\begin{aligned} {\text {stochastic term}}\propto \hbox {d}W_t,\;\; \hbox {d}W_t\sim {\mathscr {N}}\,(0,\hbox {d}t). \end{aligned}$$

The main result

Before stating the following theorem, let us recall that a càdlàg stochastic process is the right continuous with left-limit stochastic process.

Theorem 1

Suppose that the spot price\(S_t\)is acàdlàgprocess in a complete filtered probability space\(\left( \varOmega ,{\mathscr {F}},({\mathscr {F}}_{t})_{0\le t\le T},{\mathbb {P}}\right)\)where\(({\mathscr {F}}_{t})_t\)is a natural filtration of\(S_t\). Assume the following conditions:

  1. (i)

    for a small time interval \(\Delta t\), the change in the electricity price is proportional to \(\Delta t\),

  2. (ii)

    the inflation rate I always differs from the efficiency factor G,

  3. (iii)

    the stock prices jumps from the previous value \(S_{t^-}\) to a next value \(JS_{t^-}\) where J is the proportional size of the random jump assumed log-normally distributed, i.e., \(\ln J\sim {\mathscr {N}}(m_J,\sigma _J^2)\) with \({\mathbb {E}}[J]=1\),

  4. (iv)

    the change before and after the jumps is driven by increments \(\hbox {d}q_t\) of a Poisson process \(q_t\) defined by

    $$\begin{aligned} \hbox {d}q_t = \left\{ \begin{array}{ll} 1,&{}\quad \text{ with } \text{ probability }\,\, \ell \hbox {d}t \\ 0, &{}\quad \text{ with } \text{ probability }\,\, 1-\ell \hbox {d}t, \end{array} \right. \end{aligned}$$

    where\(\ell\)is the intensity or frequency of the process.

Then, the price-cap principle (7) yields the stochastic differential equation (SDE) below

$$\begin{aligned} \hbox {d}S_t = -\alpha (t)(\gamma (t)-S_t)\hbox {d}t + \sigma (t)S_t\hbox {d}W_t + (J-1)S_t\hbox {d}q_t, \end{aligned}$$
(8)

where\(W_t\)is the standard Brownian motion and the coefficients involved are deterministic functions of time denoted as such:\(\sigma (t)\)is the volatility,\(\beta (t):=E(t)+H(t)-F(t)\)defines the exogenous factors,\(\alpha (t):= I(t)-G(t)\), and \(\gamma (t):=\beta (t)/\alpha (t)\).

Proof

For a small time interval \(\Delta t\), the change in the electricity price is proportional to \(\Delta t\) and the expected change is by (7) therefore we have

$$\begin{aligned} \Delta S= \left[ S_t\left( I(t)-G(t)\right) -E(t)-H(t)+F(t)\right] \Delta t, \end{aligned}$$
(9)

where \(\Delta S_t=S_{t+\Delta t}-S_t\). For \(\Delta t\rightarrow 0,\) we obtain

$$\begin{aligned} \hbox {d}S_t= \left[ S_t\left( I(t)-G(t)\right) -E(t)-H(t)+F(t)\right] \hbox {d}t. \end{aligned}$$
(10)

To take into account market volatility in the model, the stochastic, or random, contribution to the change in the spot price is represented by \(\sigma (t)S_t\hbox {d}W_t\) (see [28, pp. 103–104]). Hence, we obtain the following SDE

$$\begin{aligned} \hbox {d}S_t= \left[ S_t\left( I(t)-G(t)\right) -\beta (t)\right] \hbox {d}t+ \sigma (t)S_t\hbox {d}W_t. \end{aligned}$$
(11)

Next, to capture the market shocks we add the jump term in (11) using [3] idea as follows. We suppose that the stock prices jump from the previous value \(S_{t^-}\) to a next value \(JS_{t^-}\) where J is the proportional size of the random jump assumed log-normally distributed such that \({\mathbb {E}}(J)=1\) this assumption is motivated by the fact that under regulation we want that the risk of the market shocks fluctuates around unit. Next, we assume that the term \((J-1)S_{t^-}\), which give the change before and after the jumps, is driven by increments \(\hbox {d}q_t\) of a Poisson process. Hence, from equation (11), setting \(\alpha (t):= I(t)-G(t)\), \(\gamma (t):=\beta (t)/\alpha (t)\), we finally obtain the SDE (8). \(\square\)

Mean-reversion condition

A mean-reverting process has a drift term that brings the variable being pulled back to some equilibrium. This feature is captured by one stochastic differential equation if the following definition is verified.

Definition 1

(Condition\((A_3)\)of [22]) Consider a jump-diffusion process \(Y_t\) with a differentiable drift function \(\mu (.)\).

If

$$\begin{aligned} \limsup \limits _{\mid Y_t\mid \rightarrow \infty }\frac{\mid Y_t + \mu (Y_t)\mid }{\mid Y_t\mid }< 1, \end{aligned}$$

then \(Y_t\) is mean-reverting.

From this definition, we have the following proposition

Proposition 1

The jump-diffusion model (8) is mean-reverting.

Proof

It is straightforward and is based on the fact that from an economic point of view, \(\beta (t)\) is bounded on [0, T] and we have \(\mid 1 + \alpha (t)\mid <1\) for all \(t\in [0,T]\). \(\square\)

Regulated electricity forward price

Computation of regulated electricity forward price

The price at time t of the forward expiring at time T (i.e., \({\mathbf {F}}(t,T)\)) is obtained as the expected value of the spot price under an equivalent \({\mathbb {Q}}\)-martingale measure, conditional on the information set available up to time t, precisely

$$\begin{aligned} {\mathbf {F}}(t,T)= {\mathbb {E}}_{t}^{{\mathbb {Q}}}\left[ S_T\right] . \end{aligned}$$

where \({\mathbb {E}}_{t}^{{\mathbb {Q}}}\) represents the conditional expectation knowing a natural filtration of \(S_t\) under the risk-neutral probability Q. To incorporate the non-opportunity of arbitrage in the model, we use the same approach as in [20] and [3], which consists of incorporating a market price of risk in the drift, to obtain

$$\begin{aligned} {\widehat{\gamma }}(t)=\gamma (t)-\lambda \frac{\sigma (t)}{\alpha (t)}S_t, \end{aligned}$$
(12)

where \(\lambda\) denotes the market price of risk per unit risk linked to the state variable \(S_t\). This market price of risk to be calibrated from market information pins down the choice of one particular martingale measure. Recall that when a market subjected to that measure, the opportunity of arbitrage is theoretically excluded in this market. Hence, under this equivalent martingale measure SDE (8) becomes

$$\begin{aligned} \hbox {d}S_t = -\alpha (t)\left( {\widehat{\gamma }}(t)-S_t\right) \hbox {d}t+ \sigma (t) S_t\hbox {d}{\widehat{W}}_t + (J-1)S_t\hbox {d}q_t; \end{aligned}$$
(13)

substituting (12) in (13), we obtain

$$\begin{aligned} \hbox {d}S_t &= -\alpha (t)\left( \gamma (t)-\left( 1 + \lambda \frac{\sigma (t)}{\alpha (t)}\right) S_t\right) \hbox {d}t+ \sigma (t) S_t\hbox {d}{\widehat{W}}_t \nonumber \\ &\quad + (J-1)S_t\hbox {d}q_t, \end{aligned}$$
(14)

where \(\hbox {d}{\widehat{W}}\) is the increment of a Brownian motion in the \({\mathbb {Q}}\)-martingale measure specified by the choice of \(\lambda\).

The next addresses the forward price computations.

Proposition 2

Assume that J, the increments of\(q_t\)and\(W_t,\)are independent. Under the risk-neutral or martingale measure\({\mathbb {Q}}\)and Novikov hypothesis, i.e.,\({\mathbb {E}}\left[ \hbox {e}^{\frac{1}{2}\int _{0}^{t}\sigma (s)^2\hbox {d}s}\right] <\infty\), electricity forward price under regulated market is given by

$$\begin{aligned} {\mathbf {F}}(t,T) &= S_t\hbox {e}^{\int _{t}^{T}(\alpha (s)+\lambda \sigma (s))\hbox {d}s}-\int _{t}^{T}\beta (s)\hbox {e}^{\int _{s}^{T}(\alpha (u)+\lambda \sigma (u))\hbox {d}u}\hbox {d}s. \end{aligned}$$
(15)

Before proving Proposition 2, let us first prove the following lemmas.

Lemma 1

The solution of equation (14) is the process \((S_t,0\le t\le T)\)defined by

$$\begin{aligned} S_t=Z_t\left( S_0-\int _{0}^{t}\beta (s)Z_{s}^{-1}\hbox {d}s\right) , \end{aligned}$$

where\(Z_t=\hbox {e}^{\left( \int _{0}^{t}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{0}^{t}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{0}^{t}\ln J\hbox {d}q_s\right) }\).

Proof

To solve equation (14), we consider a process Z, solution of the following equation

$$\begin{aligned} \hbox {d}Z_t &= Z_t\left( \alpha (t)\left( 1 + \lambda \frac{\sigma (t)}{\alpha (t)}\right) \hbox {d}t+ \sigma (t)\hbox {d}{\widehat{W}}_t + (J-1)\hbox {d}q_t\right) \\ &\quad \text {and}\,\,\,Z_0=1. \end{aligned}$$

Applying Itô formula with jumps stated in 7.10 [9], we obtain

\(Z_t=Z_0\hbox {e}^{\left( \int _{0}^{t}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{0}^{t}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{0}^{t}\ln J\hbox {d}q_s\right) }\).

Now, let us set \(f(S_t,Z_t)=\frac{S_t}{Z_t}\). By applying Itô formula with jumps one more, we obtain

$$\begin{aligned} \frac{S_t}{Z_t} &= \frac{S_0}{Z_0} +\int _{0}^{t}\frac{1}{Z_s}\left[ ((\lambda (s)\sigma (s) +\alpha (s))\hbox {d}s+\sigma (s)\hbox {d}{\widehat{W}}_s\right. \nonumber \\ &\quad \left. +(J-1)\hbox {d}q_s)S_s-\beta (s)\hbox {d}s\right] \nonumber \\ &\quad -\int _{0}^{t}\frac{S_s}{(Z_s)^2}Z_s((\lambda (s)\sigma (s)+\alpha (s))\hbox {d}s\nonumber \\ &\quad+\sigma (s)\hbox {d}{\widehat{W}}_s+(J-1)\hbox {d}q_s)\nonumber \\ &\quad+\frac{1}{2}\left( \int _{0}^{t}2\frac{S_s}{(Z_s)^3}(\sigma (s)Z_s)^2\hbox {d}s -\frac{2}{(Z_s)^2}\sigma (s)^2S_sZ_s\hbox {d}s\right) . \end{aligned}$$
(16)

The development of (16) leads to

$$\begin{aligned} \frac{S_t}{Z_t} &= \frac{S_0}{Z_0}+\int _{0}^{t}\frac{S_s}{Z_s}((\lambda (s) \sigma (s)+\alpha (s))\hbox {d}s+\sigma (s)\hbox {d}{\widehat{W}}_s\nonumber \\ &\quad+(J-1)\hbox {d}q_s)-\int _{0}^{t}\frac{1}{Z_s}\beta (s)\hbox {d}s\nonumber \\ &\quad -\int _{0}^{t}\frac{S_s}{Z_s}((\lambda (s)\sigma (s)+\alpha (s))\hbox {d}s\nonumber \\ &\quad +\sigma (s)\hbox {d}{\widehat{W}}_s+(J-1)\hbox {d}q_s)\nonumber \\ &\quad +\int _{0}^{t}\frac{S_s}{Z_s}\sigma (s)^2\hbox {d}s-\frac{S_s}{Z_s}\sigma (s)^2\hbox {d}s. \end{aligned}$$
(17)

By observing that the sum of the first and fourth term of equation (17) is equal to zero and since \(Z_0=1\) we obtain

$$\begin{aligned} \frac{S_t}{Z_t}= S_0 - \int _{0}^{t}Z_{s}^{-1}\beta (s)\hbox {d}s. \end{aligned}$$

Finally, we obtain

$$\begin{aligned} S_t= Z_tS_0 - Z_t\int _{0}^{t}Z_{s}^{-1}\beta (s)\hbox {d}s. \end{aligned}$$

This ends the proof. \(\square\)

Furthermore, the solution of (14) at T starting at t is given by

$$\begin{aligned} S_{T} = \frac{Z_T}{Z_t}S_t - Z_T\int _{t}^{T}\alpha (s)\mu (s)Z_{s}^{-1}\hbox {d}s, \end{aligned}$$
(18)

where \(Z_T= Z_t\hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{t}^{T}\ln J\hbox {d}q_s\right) }\).

Lemma 2

IfJis a log-normal distributed process with\({\mathbb {E}}[J]=1\)andqa Poisson process, then\({\mathbb {E}}_{t}^{{\mathbb {Q}}}[\hbox {e}^{\int _{t}^{T}\ln J_s\hbox {d}q_s}]=1\).

Proof

Firstly, we use differentiation method to compute \({\mathbb {E}}_{t}^{{\mathbb {Q}}}[\hbox {e}^{\int _{0}^{t}\ln J_s\hbox {d}q_s}]\).

Let us define \(L_t\) such that

$$\begin{aligned} L_t &\equiv \hbox {e}^{\int _{0}^{t}\ln J_s \hbox {d}q_s},\nonumber \\ &\equiv \hbox {e}^{m_t} \end{aligned}$$
(19)

where \(m_t\) is

$$\begin{aligned} m_t=\int _{0}^{t}\ln J_s\hbox {d}q_s, \end{aligned}$$

or equivalently

$$\begin{aligned} \hbox {d}m_t=\ln J_t\hbox {d}q_t. \end{aligned}$$
(20)

In order to write the dynamic followed by \(L_t\) for process define in (20) we use the generalization form of Itô’s lemma [9]. The SDE verified by \(L_t\) is

$$\begin{aligned} L_t &= L_0+\int _{0}^{t}L_s\ln J_s \hbox {d}q_s\nonumber \\ &\quad-\int _{0}^{t}L_s\ln J_s \hbox {d}q_s + \int _{0}^{t}L_s(\hbox {e}^{\ln J_s}-1)\hbox {d}q_s\nonumber \\ & = 1 + \int _{0}^{t}L_s(\hbox {e}^{\ln J_s}-1)\hbox {d}q_s. \end{aligned}$$
(21)

Then, from (21) we obtain

$$\begin{aligned} {\mathbb {E}}[L_t] &= 1+ \int _{0}^{t}{\mathbb {E}}[L_s]({\mathbb {E}}[\hbox {e}^{\ln J_s}]-1)\ell \hbox {d}s\nonumber \\ &= 1. \end{aligned}$$

Alternatively, we can remark that \(\int _{0}^{t}\ln J_s \hbox {d}q_s=\sum _{i=0}^{q_t}\ln J_s\) which is the particular case of Lévy process with the moment-generating function. Using Lévy–Khintchine representation, we have

$$\begin{aligned} {\mathbb {E}}\left[ \hbox {e}^{iu\int _{0}^{t}\ln J_s\hbox {d}q_s}\right] &= {\mathbb {E}}\left[ {\mathbb {E}}\left[ \hbox {e}^{iu\int _{0}^{t}\ln J\hbox {d}q_s}|q_t\right] \right] \nonumber \\ &= {\mathbb {E}}\left[ {\mathbb {E}}\left[ \varphi (u)^{q_t}\right] \right] \nonumber \\ &= \hbox {e}^{t\ell (\varphi (u)-1)}. \end{aligned}$$
(22)

where \(\varphi\) is the moment-generating function of the jump \(\ln J\). Evaluating (22) at \(u=-i\) leads to desired result. \(\square\)

Proof of Proposition 2

Before stating let us recall that forward price formula is given by \({\mathbf {F}}(t,T)= {\mathbb {E}}_{t}^{{\mathbb {Q}}}\left[ S_T\right]\). By substituting \(S_T\) with (18), we obtain

$$\begin{aligned} {\mathbf {F}}(t,T) &= {\mathbb {E}}_{t}^{{\mathbb {Q}}}\left[ S_T\right] \nonumber \\ &= S_t {\mathbb {E}}_{t}^{\mathbb {Q}}\left[ \hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s+\int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{t}^{T}\ln J_s\hbox {d}q_s\right) }\right] \nonumber \\ &\quad - {\mathbb {E}}_{t}^{\mathbb {Q}}\left[ Z_t\hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s)- \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{t}^{T}\ln J_s\hbox {d}q_s\right) }\right. \nonumber \\&\times \left. \int _{t}^{T}\alpha (s)\mu (s)Z_{s}^{-1}\hbox {d}s\right] . \end{aligned}$$
(23)

We first compute \({\mathbb {E}}_{t}^{\mathbb {Q}}\left[ \hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{t}^{T}\ln J_s\hbox {d}q_s\right) }\right] \equiv A\).

From independence between J, \(\hbox {d}q_t\) and \(\hbox {d}W_t,\) we obtain

$$\begin{aligned} A&={\mathbb {E}}_{t}^{\mathbb {Q}}\left[ \hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s\right) }\right] {\mathbb {E}}_{t}^{\mathbb {Q}}[\hbox {e}^{\int _{t}^{T}\ln J\hbox {d}q_s}]\\ & = \hbox {e}^{\int _{t}^{T}(\alpha (s)+\lambda \sigma (s))\hbox {d}s}{\mathbb {E}}_{t}^{\mathbb {Q}}[\hbox {e}^{\int _{t}^{T}\ln J\hbox {d}q_s}] \end{aligned}$$
(24)

We now compute

\({\mathbb {E}}_{t}^{\mathbb {Q}}\left[ Z_t\hbox {e}^{\left( \int _{t}^{T}(\lambda \sigma (s) + \alpha (s) - \frac{1}{2}\sigma (s)^{2})\hbox {d}s + \int _{t}^{T}\sigma (s)\hbox {d}{\widehat{W}}_s+\int _{t}^{T}\ln J\hbox {d}q_s\right) }\int _{t}^{T}\alpha (s)\gamma (s)Z_{s}^{-1}\hbox {d}s\right] \equiv A_1\).

Replacing \(Z_{s}^{-1}\) by its expression, using independence between J, \(\hbox {d}q_t\) and \(\hbox {d}W_t\) and Fubini theorem [29], we obtain

$$\begin{aligned} A_1 &= {\mathbb {E}}_{t}^{\mathbb {Q}}\left[ \int _{t}^{T}\alpha (s)\gamma (s) \hbox {e}^{\left( \int _{s}^{T}(\lambda \sigma (u) + \alpha (u) -\frac{1}{2}\sigma (u)^{2})\hbox {d}u+\int _{s}^{T}\ln J\hbox {d}q_u+\int _{s}^{T}\sigma (u)\hbox {d}{\widehat{W}}_u\right) }\hbox {d}s\right] \nonumber \\ &= \int _{t}^{T}{\mathbb {E}}^{\mathbb {Q}}_{t}[\hbox {e}^{\int _{s}^{T}\ln J\hbox {d}q_u}]\alpha (s)\gamma (s){\mathbb {E}}_{t}^{\mathbb {Q}}\left[ \hbox {e}^{\left( \int _{s}^{T}(\lambda \sigma (u) + \alpha (u) -\frac{1}{2}\sigma (u)^{2})\hbox {d}u+\int _{s}^{T}\sigma (u)\hbox {d}{\widehat{W}}_u \right) }\right] \hbox {d}s\nonumber \\ &= \int _{t}^{T}\alpha (s)\gamma (s)\hbox {e}^{\int _{s}^{T}(\alpha (u)+\lambda \sigma (u))\hbox {d}u}\hbox {d}s. \end{aligned}$$
(25)

By replacing finally (24) and (25) in (23), we obtain the forward price

$$\begin{aligned} \mathbf {F}(t,T) &= S_t\hbox {e}^{\int _{t}^{T}(\alpha (s)+\lambda \sigma (s))\hbox {d}s}-\int _{t}^{T}\beta (s)\hbox {e}^{\int _{s}^{T}(\alpha (u)+\lambda \sigma (u))\hbox {d}u}\hbox {d}s. \end{aligned}$$

This completes the proof. \(\square\)

Analytical comparison with forward price in Cartea et al. (2005)

Recall that forward price obtained in [3] is given by

$$\begin{aligned} F(t,T) &= G(T)\left( \frac{S(t)}{G(t)}\right) ^{\hbox {e}^{-\alpha (T-t)}}\hbox {e}^{\int _{t}^{T} \frac{1}{2}\sigma ^2(s)\hbox {e}^{-2\alpha (T-s)}-\lambda \sigma (s)\hbox {e}^{-\alpha (T-s)}\hbox {d}s+\int _{t}^{T}\xi (\sigma _J,\alpha ,T,s)\ell \hbox {d}s-\ell (T-t)} \end{aligned}$$

The forward price formula (15) derived in this work is an affine function of the spot price \(S_t\), Unlike in the works of [3], where they have obtained a power function of the spot price. This is what justifies the presence of fewer jumps in the forward prices. This is in line with the fact that we are in a regulating context where prices are likely to undergo less variations.

Some illustrative curves of spot and forward price in regulated electricity market

This section deals with the numerical simulations of the forward price in order to illustrate some meaningful behaviors of the model and in comparison with the model developed in [3]. The proposed simulations also aim at highlighting the fundamental role of some particular parameters in the outcomes of the prices. For the numerical computation of spot and forward prices, we approximated the integrals using the trapezium and the Stratonovich integration methods. The parameters used in the simulations are plausible relative to those used in the literature.

Fig. 1
figure 1

Spot prices for the parameters, \(I=0.0314\); \(G=0.01\); \(E=0.05\); \(H=0.001\); \(F=0.05\); \(\ell = 2.85\); \(\sigma =0.75\); \(\sigma J=0.67\); \(S(0)=50\)

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Fig. 2
figure 2

Spot prices for the parameters, \(I=0.0314\); \(G=0.01\); \(E=0.05\); \(H=0.001\); \(F=0.05\); \(\ell = 8.85\); \(\sigma =0.75\); \(\sigma J=0.67\); \(S(0)=50\)

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Fig. 3
figure 3

Spot prices for the parameters, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001; F=0.05\); \(\ell = 1.5\); \(\sigma =\exp (-0.015t)\); \(\sigma J=0.67; S(0)=50\)

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Fig. 4
figure 4

Forward price for the parameters, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001\); \(F=0.05; \sigma =0.75\); \(\sigma J=0.67; S(0)=50\)

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Fig. 5
figure 5

Forward prices for the parameters, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001\); \(F=0.05; \sigma =\exp (-0.01t)\); \(\sigma J=0.67; S(0)=50\)

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Fig. 6
figure 6

Forward prices for the parameters, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001\); \(F=0.05; \ell = 8.5\); \(\sigma =0.75\); \(\sigma J=0.67; S(0)=50\)

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Fig. 7
figure 7

Forward prices for the parameters, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001\); \(F=0.05; \ell = 8.5\); \(\sigma =\exp (-0.01t)\); \(\sigma J=0.67; S(0)=50\)

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Figure 1 shows a simulated spot price compared to the spot price in [3] without the seasonal part. One can observe that the proposed model captures some characteristics discussed in the regulated market such as mean reversion, a property also observed in Figs. 2 and 3 confirming the theoretical results. It is further relevant to discuss that in our model, the frequency of jumps is less than in the deregulated market. Figures 4 and 5 present four different states of the evolution of the forward price process in the absence of jumps in the spot price model. Here, we observe that the forward price fluctuates around an average like the spot price. This could be justified by the fact that the forward formula obtained here is a functional of the spot price. Figure 6 obtained by introducing small jumps into the model shows that despite the jump at the beginning, the forward price latter oscillates around an equilibrium a situation which is not observed in Fig. 7 with bigger jumps. In a nut shell, these illustrations show that our model with the mean-reversion property captures the main objective of regulation principle, which is to cap prices within a given range.

Fig. 8
figure 8

Forward prices for each day for various maturities and parameter sigma, \(I=0.0314; G=0.01\); \(E=0.05; H=0.001\); \(F=0.05; \ell = 8.5\); \(\sigma =\exp (-0.01t)\); \(\sigma J=0.67; S(0)=50\)

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Fig. 9
figure 9

Forward prices for each day for various maturities and parameter sigma \(I=0.0314; G=0.1\); \(S(0)=50; \sigma J=0.67\); \(\ell =0.25\)

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Figure 8 shows that despite jumps in the prices, prices vary from a certain threshold for different maturities. We observe in Fig. 9 that when the efficiency rate factor G is more than the inflation rate factor I, forward price decreases over time. This is in accordance with the economic principle.

Conclusion

In this paper, we have proposed a new model of spot price in regulated electricity market. The principal aim was to propose the forward price in regulated electricity market using economic principle price cap. It is also motivated by the fact that incentives regulation in public utilities, especially in electricity field, becomes more prevalent. The proposed model leads to non-classical Ornstein–Uhlenbeck process due to non-constant speed of the mean reversion. The determination of the exact solution permits us to derive the explicit expression of the forward price. An important topic for further research is to use historical data to calibrate the introduced model.