A probabilistic numerical method for optimal multiple switching problem in high dimension 

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A structural risk neutral model for pricing and hedging electricity derivatives

This first part of the thesis deals with an electricity structural model suitable for pricing and partially hedging power derivatives. It led to the publication of the article [1]. The classical approach in financial mathematics in order to study a given derivative product is to first propose a model for the dynamics of the underlying asset, from which the pricing and the hedging strategy can be deduced. This is the so-called reduced-form approach.
In this way, several authors tried to propose models that aim at taking into account the idiosyncrasies of the price of electricity, while remaining practical for pricing and hedging purposes (cf. [15] for example). One of these distinguishing features is that electricity is non storable 1, which implies that electricity production has to be adjusted in real time to the exact level of electricity demand.
There exist several types of power plants, based either on renewable energies (hydro, wind, . . .) or fossil energies (coal, gas, . . .). Consequently, it is clear that the composition of the electric fleet as well as the price formation mechanism will significantly impact the price of electricity. Against this backdrop, on the opposite side from reduced-form models, the stacking models try to take into account this mechanism. These models require a detailed modeling of the electricity demand, of the prices of fuels, and of all the available power plants. A global optimization then leads to the price of electricity, which is the smallest price that allows to satisfy the level of demand, making an optimal use of the available production assets (cf. [61] for example). This approach allows for a great modeling precision (it is possible to take into account many details, like the dynamic production constraints of thermal plants for example, cf. [80]). However, its main drawback is that it is very heavy and clumsy to implement, and is ill-suited to the study of derivative products.
Halfway between these two extremes, the class of structural models tries to take account in a simplified manner of the price formation mechanism, all the while making use of the classical mathematical finance tools from the reduced-form approach. In particular, this intermediary approach is very well suited to the pricing of multi-asset options that include the electricity as well as other energies like gas for example, as it allows to take into account the fine dependence structure between these variables. A survey on this class of structural models is available in [30]. The starting point of this chapter is the structural model by marginal cost developed [3], that we recall below. Consider a given power market, which includes n types of power plants. For each i = 1, . . . ,n let:
• Si t be the price of the fuel used by this type of plant (if it is based on a renewable energy, then Si t © 0).
• hi be its heat rate, that we assume to be constant (it is such that hiSi t is expressed in e/MWh). • Ci t be the power generation capacity from this type of plants (in MW).

A probabilistic numerical method for optimal multiple switching problem in high dimension

The second part of this thesis deals with the numerical solution of stochastic control problems in high dimension, more precisely of optimal switching problems. It led to the article [2], currently under review for publication. First, here is a definition of this specific class of problems. We consider the following elements:
• X = (Xt)tØ0 is a stochastic process taking values in Rd, starting from x0 oe Rd at time t = 0.
• I– = (I– t )tØ0 is a piecewise constant process taking values in RdÕ , starting from i0 oe RdÕ at time t = 0. More precisely, I– is supposed to take its values into a finite subset Iq = {i1, . . . , iq} of RdÕ. This process I– is controlled over time by a strategy –.
• – is an impulse control defined by a sequence (·n, ÿn)noeN of increasing stopping times ·n Ø 0 and Iq-valued F·n≠measurable random variables. The controlled processed I– can be deduced from this sequence as follows:  I– t = ÿn when t oe [·n, ·n+1[ .
• Among the possible strategies –, we only consider those that belong to an admissible set A. Broadly speaking, it means that we only consider the strategies such that ·n æ +OE a.s. when næOE (i.e. accumulation points are excluded).
• f : R ◊ Rd ◊ RdÕ æ R and k : R ◊ RdÕ ◊ RdÕ æ R two measurable functions. Then, the stochastic problem that we consider is the following: uncontrolled state variable X, as well as upon the controlled variable I–. The goal is thus to adjust adequately the strategy over time. However, every modification of the strategy generates a cost given by the function k. This cost depends on the values of the control immediately before and after the move. The problem (2.2.1) is called optimal switching because th controlled process I– takes its values within a finite discrete set. To be more precise, some usual regularity assumptions are required in order for the problem (2.2.1) to be well defined (cf. Section 4.2.2).
First of all, here is why we were interested in the study of this specific problem. In the first chapter, we built a structural model for the price of electricity, which is able to model properly the time dependence between the prices of electricity and other energies. In particular, this model allows to price spread options properly (between electricity and another energy), and thus, by extension, to value, in a real option framework, a given power plant. Using this tool, we wanted to know if it was possible to detect the best possible investments in power plants over time. (Which type of plant to build? How much? And when?) We will see later that this kind of investment problem can indeed be expressed as an optimal switching problem of the form (2.2.1) in high dimension (d + dÕ ∫ 3). This is why we tried to build a numerical method able to solve in practice the problem (2.2.1) in high dimension.
Now, let us detail how to proceed to solve the problem (2.2.1). If the strategy is allowed to change only on a fixed time grid # = {t0 = 0 < t1 < .. . < tN = T}, then, using the dynamic programming principle, the discretized value function v! satisfies the following backward induction:
v! (tn, x, i) = max joeIq {E (tn, x, j) ≠ k (tn, i, j)} .

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A structural risk-neutral model for pricing and hedging power derivatives

In this chapter, we develop a structural risk-neutral model for energy market modifying along several directions the approach introduced in [3]. In particular a scarcity function is introduced to allow important deviations of the spot price from the marginal fuel price, producing price spikes. We focus on pricing and hedging electricity derivatives. The hedging instruments are forward contract on fuels and electricity. The presence of production capacities and electricity demand makes such a market incomplete. We follow a local risk minimization approach to price and hedge energy derivatives. Despite the richness of information included in the spot model, we obtain closed-form formulae for futures prices and semi-explicit formulae for spread options and European options on electricity forward contracts. An analysis of the electricity price risk premium is provided showing the contribution of demand and capacity to the futures prices. We show that when far from delivery, electricity futures behave like a basket of futures on fuels.

Model for capacity, demand and fuel prices

Let (&, P,F) be a given probability space, where P is the historical (or statistical) probability measure. E will denote the expectation operator taken with respect to P. All the subsequent processes, namely C, D and S, will be defined on this probability space. The market filtration Ft will be the natural filtration generated by all Brownian motions driving the dynamics of all such processes. We assume from the beginning that the spot interest rate r is a positive constant. For the sake of simplicity, we set storage cost and convenience yield of every fuel equal to zero.

Table of contents :

1 Introduction (en français) 
1.1 Un modèle structurel risque-neutre pour la valorisation et la couverture de produits dérivés sur l’électricité
1.2 Un algorithme probabiliste pour la résolution de problèmes de commutation optimale
en grande dimension
1.3 Un algorithme numérique pour la résolution des équations de HJB totalement non-linéaires via EDSRs à sauts négatifs
2 Introduction (in English) 
2.1 A structural risk neutral model for pricing and hedging electricity derivatives
2.2 A probabilistic numerical method for optimal multiple switching problem in high dimension
2.3 A numerical algorithm for fully nonlinear HJB equations: an approach by control randomization
3 A structural risk-neutral model for pricing and hedging power derivatives 
3.1 Introduction
3.2 Electricity spot market model
3.2.1 Spot model
3.2.2 Estimation and backtesting
3.3 Pricing and hedging
3.3.1 Model for capacity, demand and fuel prices
3.3.2 Choice of pricing measure
3.3.3 Electricity futures
3.3.4 Pricing formulae
3.3.5 Hedging derivatives
3.4 Numerical results
3.4.1 Explicit model for capacities and demand
3.4.2 Computing the Conditional Expectation of Scarcity Function
3.4.3 Pricing and Hedging
3.5 Conclusion
3.6 Appendices
3.6.1 Dataset
3.6.2 Proofs
3.6.3 Algorithms
4 A probabilistic numerical method for optimal multiple switching problem in high dimension 
4.1 Introduction
4.2 Optimal switching problem
4.2.1 Formulation
4.2.2 Assumptions
4.2.3 Outline of the solution
4.3 Numerical approximation and convergence analysis
4.3.1 Approximations
4.3.2 Convergence analysis
4.4 Complexity analysis and memory reduction
4.4.1 Complexity
4.4.2 General memory reduction method
4.5 Application to investment in electricity generation
4.5.1 Modeling
4.5.2 Numerical results
4.6 Conclusion
4.7 Appendices
4.7.1 Lp convergence speed of empirical mean
4.7.2 Positivity of cointegrated geometric Brownian motions
4.7.3 No jump measure for diffusion-based discontinuities
4.7.4 Empirical confidence intervals
4.7.5 Graphical representation of random processes
5 A numerical algorithm for fully nonlinear HJB equations via BSDEs with nonpositive jumps 
5.1 Introduction
5.2 Time discretization
5.2.1 The forward regime switching process
5.2.2 Discretely jump-constrained BSDE
5.2.3 Convergence of discretely jump-constrained BSDE
5.2.4 Approximation scheme for jump-constrained BSDE and stochastic control problem
5.3 Approximation of conditional expectations
5.3.1 Localizations
5.3.2 Projections
5.4 Applications
5.4.1 Linear Quadratic stochastic control problem
5.4.2 Uncertain volatility/correlation model
5.4.3 Comparisons with [62]
5.5 Conclusion
Bibliography 

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