Guerra, JoãoGrossinho, Maria do RosárioGuerreiro, Henrique Manuel Emídio Lourenço2024-11-292024-11-292024Guerreiro, Henrique Manuel Emídio Lourenço (2024). “Option pricing in generalized rough Bergomi models”. Tese de Doutoramento. Universidade de Lisboa. Instituto Superior de Economia e Gestão.http://hdl.handle.net/10400.5/95798Doutoramento em Matemática Aplicada à Economia e GestãoThe present thesis concerns generalizations of the rough Bergomi model which are able to t observed VIX implied volatility smiles. First, we propose a new stochastic change of measure based on a fractional Ornstein-Uhlenbeck process with a regime switching long term mean. We solve the relevant fractional stochastic di erencial equation and obtain a semi-closed formula for the forward variance curve. Moreover, we employ two variance reduction methods which substantially reduce the computational cost of simulation and pricing. Then, we consider stochastic Volterra models, where the variance follows a truncated Brownian semi-stationary process with stochastic volatility (of volatility). We device a least squares Monte Carlo method which does not require running regressions on an in nite dimensional predictor variable. In general, this would be the case for non-Markovian models. This least squares Monte Carlo method constitutes a new way for pricing VIX options in a setting where volatility and vol-of-vol are not independent. We provide numerical experiments which attest to the accuracy and e ciency gain in the numerical methods we propose. Moreover, we compare the outputs of both generalizations of the rough Bergomi to market data. The models prove to be able to reproduce key characteristics of both SP500 and VIX option markets. Finally, we discuss a possible framework for a (pseudo) rough vol-of-vol through a multi-factor Markovian approximation of the vol-of-vol process. We identify a key martingale condition which may allow to express the VIX in terms of the solution of a certain Riccati ordinary di erencial equation. We derive this equation and provide su cient conditions for the existence of solutions. We also provide some partial results regarding the martingale condition. In particular, we verify a local martingale condition.engrough Bergomi modelVIX option pricingstochastic change of measurestochastic vol-of-volleast squares Monte Carlodoctoral thesis