Topics on discrete-time stochastic volatility models with applications in finance and insurance

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The pricing and hedging of financial and insurance products have been key objects of study in over two decades. Although tremendous research efforts have addressed important aspects, there are still 'puzzles' yet to be solved. This research project focuses on topics selected from the financial econometrics and mathematical finance literature.******Volatility trading has recently become almost as important as option trading, as daily volumes of volatility trading have recently become larger than daily volumes of S&P 500 option trading. Variance swap contracts are the building blocks of volatility derivatives. Although a vast majority of the mathematical finance literature examines the modelling of financial assets and pricing of volatility derivatives in continuous-time (mainly due to the tractability offered by this setup) variance swaps are in practice sampled at fixed dates and therefore, a discrete-time setting might be more appropriate. The following summarizes my future research directions. ******In the first part of this proposal, I plan to study the relationship between discrete and continuous time pricing models, by investigating the weak convergence of several popular affine and non-affine models such as the Generalized Autoregressive Conditional Heteroskedastic (GARCH) family and autoregressive Stochastic Volatility (SV) models. These results will be derived in both a univariate and multivariate setting, and applications to pricing European and American options will be discussed. One of the goals of this exercise, is to identify new pricing models and strategies which make use of the advantages of the non-affine structure, when fitting financial asset data, and the affine properties when pricing derivatives. ******In the second part of the proposal, I intend to look at novel pricing methodologies for variance swaps when the sampling is performed at discrete time points. Using the convergence results computed in the first part, I aim to derive new formulas for the discretely sampled variance swaps, when the underlying asset is modelled in continuous time, which is not possible following a direct calculation. Using real market quotes for variance swaps, I plan to construct models which fit well their term structure. For example, in the option pricing theory it is a well-known fact that adding jumps to a stochastic volatility models only helps in fitting the short-term out-of-money contracts, and therefore it will be interesting to test if that is also the case for variance swaps. I believe the proposed research plan will bring several important contributions to the modelling and pricing and hedging of financial derivatives, in particular volatility derivatives.