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Theoretical and numerical study of problems nonlinear in the sense of McKean in finance

Abstract : This thesis is dedicated to the theoretical and numerical study of two problems which are nonlinear in the sense of McKean in finance. In the first part, we study the calibration of a local and stochastic volatility model taking into account the prices of European vanilla options observed in the market. This problem can be rewritten as a stochastic differential equation (SDE) nonlinear in the sense of McKean, due to the presence in the diffusion coefficient of a conditional expectation of the stochastic volatility factor computed w.r.t. the solution to the SDE. We obtain existence in the particular case where the stochastic volatility factor is a jump process with a finite number of states. Moreover, we obtain weak convergence at order 1 for the Euler scheme discretizing in time the SDE nonlinear in the sense of McKean for general stochastic volatility factors. In the industry, Guyon and Henry Labordere proposed in [JGPHL] an efficient calibration procedure which consists in approximating the conditional expectation using a kernel estimator such as the Nadaraya-Watson one. We also introduce a numerical half-step scheme and study the the associated particle system that we compare with the algorithm presented in [JGPHL]. In the second part of the thesis, we tackle the pricing of derivatives with initial margin requirements, a recent problem that appeared along with new regulation since the 2008 financial crisis. This problem can be modelled by an anticipative backward stochastic differential equation (BSDE) with dependence in the law of the solution in the driver. We show that the equation is well posed and propose an approximation of its solution by standard linear BSDEs when the liquidation duration in case of default is small. Finally, we show that the computation of the solutions to the standard BSDEs can be improved thanks to the multilevel Monte Carlo technique introduced by Giles in [G]
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Alexandre Zhou. Theoretical and numerical study of problems nonlinear in the sense of McKean in finance. Information Theory [math.IT]. Université Paris-Est, 2018. English. ⟨NNT : 2018PESC1128⟩. ⟨tel-01957638v2⟩

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