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Numerical methods by optimal quantization in finance

Abstract : This thesis is divided into four parts that can be read independently. In this manuscript, we make some contributions to the theoretical study and financial applications of optimal quantization. In the first part, we recall the theoretical foundations of optimal quantization as well as the classical numerical methods to build optimal quantizers. The second part focuses on the problem of numerical integration in dimension 1. This problem arises when one wishes to numerically compute expectations, such as the valuation of derivatives in finance that are expressed as the expectation of a function of a single financial asset. We recall the existing strong and weak error results and extend the results of order 2 convergence rate to other function classes with less regularity. In a second step, we present a weak error development result in one dimension and a second development in a higher dimension when the chosen quantizer is a product quantizer. In the third part, we look at a first numerical application. We introduce a stationary Heston model in which the initial condition of volatility, instead of being deterministic as in the standard model, is assumed to be randomly distributed with the stationary distribution of the CIR EDS governing volatility. This variant of the original Heston model produces for European options on short maturities a steeper smile of implied volatility than the standard model. We then develop a product recursive quantization-based numerical method for the valuation of Bermudan options and barriers. The fourth and last part deals with a second numerical application, the pricing of Bermudan exchange rate options in a 3 factor model, i.e. where the exchange rate, domestic and foreign interest rates are stochastic. These products are known in the markets as PRDC (Power Reverse Dual Currency). We propose two schemes to evaluate this type of options, both based on optimal product quantization and establish a priori error estimates.
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Submitted on : Thursday, September 2, 2021 - 12:00:30 PM
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  • HAL Id : tel-03112849, version 2

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Thibaut Montes. Numerical methods by optimal quantization in finance. Probability [math.PR]. Sorbonne Université, 2020. English. ⟨NNT : 2020SORUS156⟩. ⟨tel-03112849v2⟩

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