Modelli a volatilità non-semimartingala per l'asset pricing

The semimartingale property is fundamental in the asset pricing theory. This special condition characterizes the stochastic integration in the Ito framework and constitutes the foundation of modern mathematical finance. On the other hand, many natural and social phenomena require a different representation. In this thesis we introduce a new non-semimartigale model that can be used to describe volatility or interest rate. In Chapter 1 we present the basic tools to understand the fractal structure of the process. Indeed, the definition of fractional integral plays an essential role in order to manage different results of this treatment. In Chapter 2 we describe a non-semimartingale volatility model known as Rough Volatility. In this section, in a completely informal way, we examine also the illusory antipersistence effect and we point out that it is impossible to know whether volatility is rough or smooth. In Chapter 3 we discuss the properties of our model that extends the fractional Ornstein-Uhlenbeck (used in Rough Volatility Framework) and generalises the Extended Vasicek (used in Bond Pricing). We call our model Extended Fractional Vasicek. We prove, using the pathwise integration in Riemann-Stieltjes sense, that the process defined is the unique solution of a specific stochastic differential equation driven by fractional Brownian motion. The Extended Fractional Vasicek process is neither Markov processes nor semimartingales, but it is gaussian. So, if we can compute expected value and covariance we know everything about the process. Therefore, we determine two equivalent forms of its autocovariance function: Reduced Form (RF) and Extended Fractional Form (EFF). We also study the long run behavior of expected value and the asymptotic variance. Finally we discuss a new characterization of a stationary stochastic process.