This review presents the set of electricity price models proposed in the literature since the opening of power markets. We focus on price models applied to financial pricing and risk management. We classify these models according to their ability to
represent the random behavior of prices and some of their characteristics. In particular, this classification helps users to choose among the most suitable models for their risk management problems.
We consider a 2-dimensional marked Hawkes process with increasing baseline intensity in order to model prices on electricity intraday markets. This model allows to represent different empirical facts such as increasing market activity, random jump si
zes but above all microstructure noise through the signature plot. This last feature is of particular importance for practitioners and has not yet been modeled on those particular markets. We provide analytic formulas for first and second moments and for the signature plot, extending the classic results of Bacry et al. (2013) in the context of Hawkes processes with random jump sizes and time dependent baseline intensity. The tractable model we propose is estimated on German data and seems to fit the data well. We also provide a result about the convergence of the price process to a Brownian motion with increasing volatility at macroscopic scales, highlighting the Samuelson effect.
We derive a model based on the structure of dependence between a Brownian motion and its reflection according to a barrier. The structure of dependence presents two states of correlation: one of comonotonicity with a positive correlation and one of c
ountermonotonicity with a negative correlation. This model of dependence between two Brownian motions $B^1$ and $B^2$ allows for the value of $mathbb{P}left(B^1_t - B^2_t geq xright)$ to be higher than $frac{1}{2}$ when $x$ is close to 0, which is not the case when the dependence is modeled by a constant correlation. It can be used for risk management and option pricing in commodity energy markets. In particular, it allows to capture the asymmetry in the distribution of the difference between electricity prices and its combustible prices.
A new method for stochastic control based on neural networks and using randomisation of discrete random variables is proposed and applied to optimal stopping time problems. The method models directly the policy and does not need the derivation of a d
ynamic programming principle nor a backward stochastic differential equation. Unlike continuous optimization where automatic differentiation is used directly, we propose a likelihood ratio method for gradient computation. Numerical tests are done on the pricing of American and swing options. The proposed algorithm succeeds in pricing high dimensional American and swing options in a reasonable computation time, which is not possible with classical algorithms.
We consider a doubly stochastic Poisson process with stochastic intensity $lambda_t =n qleft(X_tright)$ where $X$ is a continuous It^o semimartingale and $n$ is an integer. Both processes are observed continuously over a fixed period $left[0,Tright]$
. An estimation procedure is proposed in a non parametrical setting for the function $q$ on an interval $I$ where $X$ is sufficiently observed using a local polynomial estimator. A method to select the bandwidth in a non asymptotic framework is proposed, leading to an oracle inequality. If $m$ is the degree of the chosen polynomial, the accuracy of our estimator over the Holder class of order $beta$ is $n^{frac{-beta}{2beta+1}}$ if $m geq lfloor beta rfloor$ and it is optimal in the minimax sense if $m geq lfloor beta rfloor$. A parametrical test is also proposed to test if $q$ belongs to some parametrical family. Those results are applied to French temperature and electricity spot prices data where we infer the intensity of electricity spot spikes as a function of the temperature.
We propose new copulae to model the dependence between two Brownian motions and to control the distribution of their difference. Our approach is based on the copula between the Brownian motion and its reflection. We show that the class of admissible
copulae for the Brownian motions are not limited to the class of Gaussian copulae and that it also contains asymmetric copulae. These copulae allow for the survival function of the difference between two Brownian motions to have higher value in the right tail than in the Gaussian copula case. Considering two Brownian motions $B_t^1$ and $B_t^2$, the main result is that the range of possible values for $mathbb{P}left(B_t^1-B^2_t geq etaright)$ with $eta > 0$ is the same for Markovian pairs and all pairs of Brownian motions, that is $left[0,2Phileft(frac{-eta}{2sqrt{t}}right)right]$ with $Phi$ being the cumulative distribution function of a standard Gaussian random variable.
