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Publicação
Option pricing in generalized rough Bergomi models
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Orientador(es)
Resumo(s)
The 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.
Descrição
Doutoramento em Matemática Aplicada à Economia e Gestão
Palavras-chave
rough Bergomi model VIX option pricing stochastic change of measure stochastic vol-of-vol least squares Monte Carlo
Contexto Educativo
Citação
Guerreiro, Henrique Manuel Emídio Lourenço (2024). “Option pricing in generalized rough Bergomi models”. Tese de Doutoramento. Universidade de Lisboa. Instituto Superior de Economia e Gestão.
Editora
Instituto Superior de Economia e Gestão