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Theses

Stratégies de couverture presque optimale : théorie et applications

Abstract : Abstract: This thesis has 8 chapters. The chapter 1 is an introduction to the issues encountered in the energy market : low frequency trading, high transaction costs, spread option pricing. The chapter 2 studies the hedging error convergence of a call option in the Bachelier model, for proportional transaction costs (Leland-Lott's model) and when the intervention frequency becomes infinite. We prove that this error is bounded by a random variable proportional to the convergence rate. However, the proof of convergence in probability requires some restrictive regularities on the sensitivities. The following chapters avoid these difficulties by studying the almost sure convergence. The chapter 3 develop new tools for the almost sure convergence. These results have many consequences on the control path by path of martingales and of their quadratic variations, as their increments between two general stopping times. These convergence results are well-known to be difficult to demonstrate without any information on the laws. Moreover, we apply these results to the almost sure minimization of the renormalized quadratic variation of the hedging error for a general payoff (multidimensional setting, Asian and Lookback option) for a broad class of trading dates. A lower bound for our criterion is found and an optimal sequence of stopping times is described, which is given by hitting times of random ellipsoids, depending only on the option gamma. The chapter 4 studies the hedging error convergence of an option with convex payoff (dimension 1) taking into account Leland-Lott's transaction costs. We decompose the error into a martingale part and a negligible part, then we minimize the quadratic variation of this martingale on a class of hitting times for Deltas satisfying some non-linear EDP on the second derivative. Moreover, we find a minimizing sequence of hitting times. Numerical tests illustrate our approach w.r.t. a series of strategies from the literature. The chapter 5 extends the chapter 3 by considering a discrete variation functional of order Y and Z for two Ito processes Y and Z; the minimization is on a broad class of stopping times. Lower bound and minimizing sequence are obtained. A numerical study on the transaction costs is done. The chapter 6 studies the Euler discretization of a multidimensional process X, controlled by a semi-martingale Y . We lessen some quadratic criterion on the error scheme over the discretization time grid. We find a lower bound and an optimal grid, independent of the observable data. The chapter 7 gives a Limit Central Theorem for the discretization of stochastic integrals on hitting times of any adapted ellipsoids. The asymptotic correlation is a consequence of sharp limits involving solutions to Dirichlet's problem. In the chapter 8, we are interested to expansion formulas for spread options in local volatility model. The originality of our approach is to keep the martingale property for the approximation and to exploit the call payoff structure. Numerical tests show the relevance of our approach.
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Submitted on : Wednesday, February 13, 2013 - 4:10:01 PM
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Nicolas Landon. Stratégies de couverture presque optimale : théorie et applications. Finance quantitative [q-fin.CP]. Ecole Polytechnique X, 2013. Français. ⟨pastel-00788067⟩

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