A Scenario Simulation Model of Stock's Volatility Based on a Stationary Markovian Process

In this paper we discuss univariate statistical properties of volatility. We present a parsimonious univariate model that well reproduces two stylized facts of volatility: the power-law decay of the volatility probability density function with exponent α and the power-law decay of the autocorrelation function with exponent β. Such model also reproduces, at least qualitatively, the empirical observation than when the probability density function decays faster, then the autocorrelation decays slower. Another important feature investigated within the model is the mean First Passage Time (mFPT) Tx0 (Λ) of volatility time-series. We show that the proposed model allows to obtain the mFPT in terms of α and β. These findings are in agreement with the empirical investigations to a good extent. The model also provides useful insights on the First Passage Time distribution (FPTD), such as the fact that the FPTD decays with a power-law whose exponent is independent from the Λ threshold. These results point in the direction of (i) using the mFPT as a useful metric and (ii) using our model as a scenario simulator in the assessment procedures of the so-called volatility risk associated to derivative financial products