Some relationships between ARCH-type and Stochastic Volatility models are investigated. New model formulations are derived through a transformation of a GARCH-M process and the name Generalized Bilinear Stochastic Volatility is suggested. Markovian-type representations are presented and estimation algorithms are proposed.

The validity of an approximation formula for European option prices under a general stochastic volatility model is proved in the light of the Edgeworth expansion for ergodic diffusions. The asymptotic expansion is around the Black-Scholes price and is uniform in bounded payoff functions. The result provides a validation of an existing singular perturbation expansion formula for the fast mean re...

Today, better numerical approximations are required for multi-dimensional SDEs to improve on the poor performance of the standard Monte Carlo integration. With this aim in mind, the material in the thesis is divided into two main categories, stochastic calculus and mathematical finance. In the former, we introduce a new scheme or discrete time approximation based on an idea of Paul Malliavin wh...

In this paper we investigate to what extent the bootstrap can be applied conditional mean models, such as regression or time series when volatility of innovations is random and possibly non-stationary. fact, many economic financial displays persistent changes possible non-stationarity. However, theory for models has focused on deterministic unconditional variance little known about performance ...

Concurrent research documents sizeable changes in the volatility of U.S. macroeconomic time series; e.g., see Kim and Nelson (1999), McConnell and Pérez-Quirós (2000), Stock and Watson (2002), and Sensier and van Dijk (2004). Most of the evidence from this literature suggests a sizeable reduction in volatility for many series; many of them used to construct business cycle indicators. With the e...