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Numerical analysis of the stochastic volatility jump diffusion models: case of options pricing

Abstract : In the modern economic world, the options contracts are used because they allow to hedge against the vagaries and risks refers to fluctuations in the prices of the underlying assets. The determination of the price of these contracts is of great importance for investors.We are interested in problems of options pricing, actually the European and Quanto options on a financial asset. The price of that asset is modeled by a multi-dimentional jump diffusion with stochastic volatility. Otherwise, the first model considers the volatility as a continuous process and the second model considers it as a jump process. Finally in the 3rd model, the underlying asset is without jump and volatility follows a model CEV without jump. This model allow better to take into account some phenomena observed in the markets. We develop numerical methods that determine the values of prices for these options. We first write the model as an integro-differential stochastic equations system "EIDS", of which we study existence and unicity of solutions. Then we relate the resolution of PIDE to the computation of the option value. This link, which is based on the notion of infinitesimal generators, allows us to use different numerical methods. We therefore introduce the variational equation associated with the PIDE, and drawing on the work of Zhang [106], we show that it admits a unique solution in a weights Sobolev space We focus on the numerical approximation of the price of the option, by treating the problem in a bounded domain. We use the finite elements method of type (P1), and the scheme of Euler-Maruyama, for this serve, on the one hand the finite differences method in time, and on the other hand the method of Monte Carlo and the Quasi Monte Carlo method. For this last method we use of Halton sequences to improve the speed of convergence.We present a comparative study of the different numerical results in many different cases in order to investigate the performance and effectiveness of the used methods.
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Submitted on : Tuesday, September 28, 2021 - 10:13:53 AM
Last modification on : Saturday, October 9, 2021 - 3:21:36 AM

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  • HAL Id : tel-03356473, version 1

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Abdelilah Jraifi. Numerical analysis of the stochastic volatility jump diffusion models: case of options pricing. Statistics [math.ST]. Université de Valenciennes et du Hainaut-Cambrésis, 2014. English. ⟨NNT : 2014VALE0002⟩. ⟨tel-03356473⟩

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