Volatility of prices of financial assets

When it comes to analyze a financial time series, volatility modelling plays an important role. As an example, the variance of financial returns often displays a dependence on the second order moments and heavy-peaked and tailed distributions. In order to take into account for this phenomenon, known at least from the work o f [22] and [14], econometric models of changing volatility have been introduced, such as the Autoregressive Conditional Heteroskedasticity (ARCH) model by Engle, see [13]. The idea behind the ARCH model is to make volatility dependent on the variability of past observations. Taylor, in [26], studied an alternative formulation in which volatility was driven by unobserved components, and has come to be known as the Stochastic Volatility (SV) model. Both the ARCH and the SV models, covered in Section 2, have been intensively studied in the past decades, together with more or less sophisticated estimation approaches, see [25], as well as concerning concrete applications, see, e.g., [9], and references therein. Parallel to the study o f discrete-time econometric models for financial time series, more precisely in the early 1970’s, the world of option pricing experienced a great contribution given by the work of Fischer Black and Myron Scholes. The Black-Scholes (BS) model, see [4], assumes that the price of the underlying asset of an option contract follows a geometric Brownian motion. Latter type of approach has been also used within the framework of interest rate dynamics, see, e.g., [6], and references therein. One of the most successful extensions has been the continuous-time Stochastic Volatility (SV) model, introduced with the work of Hull and White, see, [19]. A major contribution was successively due to Heston in [18], indeed he developed a model which led to a quasi-closed form expression for European option prices. Differently from the BS model, the volatility is not longer considered constant, but it is allowed to vary trough time in a stochastic way. In Section 3 we will start from a sub-class of SV models, which is the one of Local Volatility (LV), being characterized by a deterministic time-varying volatility, and then we will consider the general SV case, providing information about the pricing equation as made, e.g., in [5] or, from a point of view more centred towards applications, in [12], and references therein.