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Study of numerical methods for partial hedging and switching problems with costs uncertainty

Abstract : In this thesis, we give some contributions to the theoretical and numerical study to some stochastic optimal control problems, and their applications to financial mathematics and risk management. These applications are related to weak pricing and hedging of financial products and to regulation issues. We develop numerical methods in order to compute efficiently these quantities, when no closed formulae are available. In the first part, we study the partial hedging of European claims, and in particular the quantile hedging problem. In Chapter 3, when the market is linear, we show that it is possible to solve explicitly the optimal control problem, and that its value is a function of the law of the asset price at maturity. This formulation gives a new and simple proof for explicit formulae for the quantile hedging price of vanilla options in the Black & Scholes model. In addition, we compute a convergence rate for the approximation of the control problem by one with bounded controls. In Chapter 4, we consider a non-linear market, and we approximate the control problem by one where a control is admissible only if it is piecewise constant over time. We show the convergence when the time discretization parameter goes to 0. When the log-price follows a Brownian diffusion, we couple this time discretization with a spatial discretization using finite differences, and we show the convergence towards the value function. Some numerical applications are given, showing that the scheme converges, and that the theoretical conditions on the discretization parameters are also necessary in practice. The second part of this thesis is dedicated to the approximation of risk measures on the distribution of the future balance sheet of a bank or an insurance company. This problem comes from Solvency II, the insurance companies regulation. In a Gaussian model with stochastic rates, the company sells a financial product and hedges itself discretely in time. We develop a numerical method allowing efficient approximation, in moderate dimension, of the unknown distribution. We show that spectral risk measures, computed on the approximated distribution, converge to the risk measures computed on the unknown distribution. We give numerical applications showing the efficiency of the method, in comparison with the Nested Simulations approach. The third part is dedicated to the study of some switching problems with costs uncertainty. We introduce a new family of stochastic optimal control problems, generalising usual switching problems. We show, under classical hypotheses on the driver of the backward equation and under positive costs, that the value of these control problems is given by the Y component of the solution of some backward equations with oblique reflections. We deduce the uniqueness of the solutions to these equations, assuming that there exists a solution. We then show that the precedent equations actually admits solutions, in the case we call “uncontrolled” and “irreducible”, without assuming that the costs are positive. We give a geometric characterisation of the domain, with respect to the costs.
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Submitted on : Friday, December 6, 2019 - 10:58:14 AM
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Cyril Bénézet. Study of numerical methods for partial hedging and switching problems with costs uncertainty. Probability [math.PR]. Université de Paris, 2019. English. ⟨tel-02396757⟩

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