Almost sure optimal stopping times : theory and applications.

Summary This thesis has 8 chapters. Chapter 1 is an introduction to the problems encountered in energy markets: low intervention frequency, high transaction costs, and spread option pricing. Chapter 2 studies the convergence of the hedging error of a call option in the Bachelier model, for proportional transaction costs (Leland-Lott model) and when the intervention frequency becomes infinite. It is shown that this error is bounded by a random variable proportional to the transaction rate. However, proofs of convergence in probability require regularities on sensitivities that are quite restrictive in practice. The following chapters circumvent these obstacles by studying almost certain convergences. Chapter 3 first develops new tools for almost sure convergence. These results have many consequences on the almost safe control of martingales and their quadratic variation, as well as their increments between two general stopping times. These trajectory convergence results are known to be difficult to obtain without information about the laws. In the following, we apply these results to the almost certain minimization of the renormalized quadratic variation of the hedging error of a general payoff option (multidimensional framework, Asian payoff, lookback) over a large class of intervention times. A lower bound on our criterion is found and a minimizing sequence of optimal stopping times is exhibited: these are random ellipsoid reaching times, depending on the gamma of the option. Chapter 4 studies the convergence of the hedging error of a convex payoff option (dimension 1) taking into account Leland-Lott transaction costs. We decompose the hedging error into a martingale part and a negligible part, and then minimize the quadratic variation of this martingale over a class of general attainment times for Deltas satisfying a certain nonlinear PDE on the second derivatives. We also exhibit a sequence of stopping times reaching this bound. Numerical tests illustrate our approach against a series of known strategies in the literature. Chapter 5 extends Chapter 3 by considering a functional of discrete variations of order Y and Z of two real-valued Itô processes Y and Z, the minimization being over a large class of stopping times used to compute the discrete variations. Lower bound and minimizing sequence are obtained. A numerical study on the transaction costs is done. Chapter 6 studies the Euler discretization of a multidimensional process X driven by an Itô semi-martingale Y. We minimize on the discretization grid times a quadratic criterion on the error of the scheme. We find a lower bound and an optimal grid, depending only on the observable data. Chapter 7 gives a central limit theorem for stochastic integral discretizations on any adapted ellipsoidal reach time grids. The limit correlation is a consequence of fine asymptotics on Dirichlet problems. In Chapter 8, we focus on expansion formulas for spread options, for models with local volatility. The key to the approach is to retain the martingale property of the arithmetic mean and to exploit the payoff call structure. Numerical tests show the relevance of the approach.