# A jump-diffusion model for pricing electricity under price-cap regulation

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## Abstract

In this paper, we derive a new jump-diffusion model for electricity spot price from the “Price-Cap” principle. Next, we show that the model has a non-classical mean-reverting linear drift. Moreover, using this model, we compute a new exact formula for the price of forward contract under an equivalent martingale measure and we compare it to Cartea et al. (Appl Math Finance 12(4):313–335, 2005) formula.

## Keywords

Model Electricity market Price-cap regulation Spot price Forward price## 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].

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

*W*being a standard Brownian motion. Applying Itô’s formula on (2), they obtained the following dynamic for spot price

*g*is a seasonal deterministic function and that \(Y_t\) follows a stochastic process given by

*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

*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:

### Spot price modeling procedure

*t*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 d

*t*is the sum of two components: the “drift” term and the stochastic (or “random”) term, that is,

### 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 a*

*càdlàg*

*process 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:*

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

- (ii)
the inflation rate

*I*always differs from the efficiency factor*G*, - (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 \({\mathbb {E}}[J]=1\), - (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*

*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*

*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 \(\mu (.)\).

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

*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

*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

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*

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*

*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*

*Z*, solution of the following equation

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

*T*starting at

*t*is given by

### **Lemma 2**

*If**J**is a log-normal distributed process with*\({\mathbb {E}}[J]=1\)*and**q**a 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 \({\mathbb {E}}_{t}^{{\mathbb {Q}}}[\hbox {e}^{\int _{0}^{t}\ln J_s\hbox {d}q_s}]\).

### *Proof of Proposition 2*

*J*, \(\hbox {d}q_t\) and \(\hbox {d}W_t,\) we obtain

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

*J*, \(\hbox {d}q_t\) and \(\hbox {d}W_t\) and Fubini theorem [29], we obtain

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

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

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.

## Notes

### Acknowledgements

We thank the African Center of Excellence in Technologies, Information and Communication (CETIC) to place in our disposal its infrastructures. This helps us to improve conditions of this work. We would like to thank anonymous referees for their helpful comments which ultimately improved the article.

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