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Alternative approaches by using BSDEs
Considering the large panel of methods that have been developed recently for gas storage valuation, it seems dicult to propose a new and competitive approach. As already mentioned in Paragraph 1.3.2 and in the line of Carmona and Lukovksy , our objective was to provide an alternative simulation-based numerical method using sample paths of both gas price and inventory level, to avoid inventory level discretization. In addition, we would like to derive an analysis of convergence of the method with respect to main approximation parameters.
The recently developed theory of Backward Stochastic Dierential Equations (BSDEs for short) and its application to optimal switching problems gives new possibilities going in this sense. These approaches provide alternative characterizations of the solution to optimal switching problems than the classical ones presented in Section 1.3 and the natural way to solve BSDEs are simulation-based techniques. However, this remains a relatively unexplored domain from a numerical point of view, due to the diculties raised by the practical computation of the solution to BSDEs, and more specically BSDEs related to multiple obstacle problem: that is multiple optimal stopping time, optimal switching and impulse control problems. BSDEs linked to such problems are particularly complex BSDEs: these are reected BSDEs (RBSDEs for short), see among others Bouchard and Touzi , Hu and Tang , Hamadène and Zhang , and BSDEs with constrained jumps, see Kharroubi et al.  and Elie and Kharroubi . A topic that have been a very active area of research in recent years is however the proposition of ecient numerical discrete-time schemes allowing the resolution of such BSDEs, see e.g., Bouchard and Touzi , Bouchard and Chassagneux , Bouchard and Elie , Chassagneux et. al .
And yet, it appears that only few authors have published numerical experiments on the subject. To our knowledge, only Hamadène and Jeanblanc  and Porchet  provide numerical results issued from BSDE-based computation for obstacle problems resolution: respectively for a two regimes switching problem with an one-dimensional uncontrolled forward diusion (startingstopping problem) and for the valuation by utility indierence of a coal and fuel oil-red power plant with two modes (uncontrolled two-dimensional forward process). The case with 2 regimes roughly simplies the computation of the solution as the implied two-dimensional reected BSDE can be reduced to a single BSDE with two reecting barriers (by working on the dierence value process).
Let us also notice that the Tsitsiklis and Van Roy-based numerical scheme used by Ludkovski  is equivalent to an algorithm for solving a discrete-time reected BSDE. The author handle with a 3-regimes switching problem with an uncontrolled forward diusion by solving a cascade of RBDSEs with one reecting barrier (using an iteration on the number of remaining switches). In the case of storage valuation, the main diculties raised come from the fact that it boils down to a 3-regimes switching problem whose state variable (actually, the inventory level) is controlled, constrained and degenerate (Cu admits no diusion term), see (1.2) and (1.4).
Using multi-dimensional reected BSDEs
Hu and Tang  and Elie and Kharroubi  make the link between multiple (that is jIj 3) switching problems in which the dynamics of the forward process is policy-dependent and multi-dimensional reected BSDEs, namely systems of jIj reected BSDEs with interconnected obstacles (the reection is said to be of oblique type). This follows from recent work of Hamadène and Jeanblanc  (in the special case of two regimes), Hamadène and Zhang [62, 63] and Djehiche et al.  dealing with uncontrolled forward diusions. This kind of multi-dimensional reected BSDE, in the case of an uncontrolled forward diffusion, can be solve with purely simulation-based techniques by using the numerical scheme proposed by Chassagneux et. al . The discretization error when solving the obliquely
ected BSDE on a time grid with step jj is shown to be of order jj 1 2 ; 8 > 0. A rst approach using reected BSDEs
We have rst tried to use the result from Hu and Tang . A representation of the solution of an optimal switching problem based on a multi-dimensional reected BSDE is provided under relevant assumptions, among which the forward controlled state variable X must have a dynamic like dXu t = b(Xu t ; ut)dt + (Xu t )dWt in which is invertible. The BSDE representation of the (optimal) solution is indeed based on a (optimal) policy-dependent change of measure. In our framework of storage valuation, the two-dimensional state variable (P;Cu) is degenerate since the inventory level admits no diusion term. To overcome this diculty, we had in mind to introduce a small volatility coecient > 0 (aimed at tending to 0) and transform the inventory dynamic in (1.4) as d Cu t = q( Cu t ; ut)dt + dW C t in which WC is a standard Brownian motion. Since the inventory level is required to satisfy the volumetric constraint (1.2), it seems natural to impose the same thing to Cu. It means, to make Cu intrinsically satisfy the constraint thought a modied dynamic: this constitutes a classical practical technique when solving stochastic optimization problems involving controlled and constrained forward processes. Let us thus redene d Cu t = q( Cu t ; ut)dt + dWC t dAut ; Cu 0 = c.
Using BSDEs with constrained jumps
As we have seen in the PDE formulation (1.8) or with the representation using multi-dimensional reected BSDE in previous Paragraph (1.5.1), the diculty in the derivation of a tractable BSDErepresentation for the storage valuation problem is rstly the dependence of the solution in the regime i 2 I with respect to the global solution in all possible regimes, and secondly the regimedependence of the forward process.
Elie and Kharroubi  make the link between RBSDEs and a new class of BSDEs with constrained jumps, previously introduced in Kharroubi et al.  in the framework of impulse control. These BSDEs with constrained jumps provide a representation for the solution of impulse control and optimal switching problems, see Remark 1.5.1. In particular, in , the family of reected BSDEs (1.18) is related to a particular member of the class of associated BSDEs with constrained jumps. The idea behind this new representation is to articially introduce a random regime I which jumps from one operating mode to another. This allows to retrieve, in the jump component of resulting one-dimensional BSDE, the required information with respect to the whole set of operating regimes.
In addition, it allows degenerate forward processes (do not require the invertibility of the volatility matrix of the forward process). According to us, this constitues the right BSDE-based formulation for numerically solving the storage valuation problem or any other multiple optimal switching problems involving degenerate and policy-dependent state variables.
Solving BSDEs with constrained jumps and link to Part II
Numerical resolution of BSDEs with constrained jumps The numerical resolution of a BSDE with constrained jumps, as for example (1.24), is a challenging problem. The main di- culty comes from the constraint, which concerns the jump component of the solution (V (j))j2I. Classical approaches by projected schemes (discretely reected backward schemes) are irrelevant, since the process K, allowing to fulll the constraint on jumps does not a priori satisfy any These kind of BSDEs might be numerically approximated by using a penalization procedure and classical backward discrete-time schemes for BSDEs with jumps, see Bouchard and Elie .
The constraint on the jump component V is introduced in the driver of the BSDE and penalized with a parameter p > 0. The solution to the penalized BSDE with jumps is known to converge to the minimal solution of the BSDE with constrained jumps, see  and . However, no convergence rate is available for such an algorithm (in particular, the convergence rate of the error due to penalization). The numerical resolution of BSDEs with constrained jumps raises various questions: The theoretical representation of the exact solution using BSDE with constrained jumps holds for a Poisson measure with any intensity measure > 0. From a numerical viewpoint, this is apparently not the case and it has to be chosen carefully, at least with respect to the time step of the discrete-time resolution grid. Even for BSDEs with jumps (without constraint), Elie  alludes to the critical role of in practical applications.
As already mentioned, no convergence rate is available for the numerical scheme using penalization described above. On one hand, in the literature of BSDEs with constraint(s), only Hamadène and Jeanblanc  provide a convergence rate of the error introduced when using a penalization procedure: for a BSDE with two reecting barriers with uncontrolled forward diusion, the authors obtain a bound on the error between the exact and the penalized solution of order p1. On the other hand, the impact of the penalization parameter p is well-known in practice: as p increases at xed discrete-time step, the driver explodes leading to numerical instabilities, see e.g., the numerical experiments of Lemor  for the resolution by penalization of a BSDE with one reecting barrier. However, to our best knowledge, there exists no explicit computation of the discretization error with respect to p in the literature.
The solving procedure described above does not provide any representation of the optimal impulse/switching strategy during the backward in time recursion. An additional numerical procedure, to be determined, is required.
Table of contents :
0.1 Une approche par EDSRs pour la résolution de problèmes de contrôle impulsionnel
0.2 Evaluation d’options sur moyenne mobile
0.3 Organisation de la thèse
0.1 A BSDE-based method for solving impulse control problems
0.2 Valuation of moving average options
0.3 Organization of the thesis
I Valuation methods of gas contracts: a review
1 Gas storage facilities
1.1 General description
1.2 A gas storage modelization
1.3 Main diculties of classical valuation methods
1.4 Review of existing methods
1.5 Alternative approaches by using BSDEs
2 Gas Swing contracts
2.1 General description
2.2 Classical formulation of the problem and practical analysis
2.3 Review of existing methods
II Solving impulse control problems by using BSDEs with jumps
2 An impulse control problem: link to BSDEs with jumps
2.1 Notations and assumptions
2.2 Link to BSDE with constrained jumps and penalization approach
2.3 Hölder property of the value function w.r.t. time maturity
3 A rate of convergence of the error due to penalization
3.1 Preliminary results
3.2 Convergence rate of the approximation by penalization
3.3 Proof of Proposition 3.2.1
4 Estimation of the discretization error
4.1 Discrete-time approximation
4.2 A rst estimate of the error due to discretization
4.3 Estimates involving the path-regularity of the continuous-time solution
4.4 Global convergence rate of the penalization approach
5 Numerical applications
5.1 A problem of optimal forest management
5.2 Swing options valuation
5.3 Valuation of gas storage facilities
6 Concluding remarks and perspectives
III Valuation methods for moving average options
2 A nite-dimensional approximation for pricing moving average options
2.1 Framework and formulation of the pricing problem
2.2 A nite-dimensional approximation of moving average options price
2.3 Uniformly-weighted moving averages
2.A Some properties of the Laguerre polynomials
3 Methods for pricing moving average options
3.1 Laguerre approximation-based numerical method
3.2 Reference methods
3.3 Some extensions
4 Numerical applications
4.1 Experiments in the Black and Scholes framework
4.2 Valuation of oil-indexed gas contracts
5 Perspectives for further research