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  • Graphic Stochastics
Applied Stochastics


Project managment: Prof. Dr. Markus Bibinger

Employees: Malon Jansen, Michael Sonntag

Project period: 2018-2022

Funding institution: DFG

Funding code: 403176476

Project description: Volatility refers to the magnitude of random fluctuations in a system described by a stochastic differential equation. In recent years, considerable attention has been devoted to stochastic volatility processes. Their path properties determine, for instance, optimal volatility estimation methods, forecasting techniques and the volatility persistence. In this project, we develop statistical theory to infer path properties of volatility. Recently, there have been important contributions to advance estimation and testing procedures on path properties of stochastic processes from discrete observations. On the other hand, even though the path properties of volatility are of particular interest for applications, there is so far no groundwork on a statistical analysis. The key difference and difficulty is that volatility is latent and thus cannot be directly observed. Based on statistics involving pre-estimated volatility instead, Moritz Jirak and Markus Bibinger started to work in this new direction in a first article focusing on a change-point analysis. Exploiting recent contributions regarding volatility estimation from high-frequency data and inference on path properties from direct observations, we aim to establish a novel theory with optimal test and estimation approaches for path properties of the latent volatility. This is pursued for three related yet different observation models of interest. While one model assumes discrete observations of a semimartingale, the two others incorporate regular or non-regular observational noise. Our theory shall also explore the limitations of identifiability. While currently conflicting models for volatility processes are put forward in the literature, our methods aim to provide evidence which models are suitable. Our nonparametric statistical analysis reveals, moreover, if path properties are stable over time or changing. Particular interest in this work is motivated by statistics for financial markets. Volatility is the prevailing concept to describe market risk in price evolutions. Reliable volatility estimates are thus key ingredients for risk management and risk prediction. Our main focus is on intra-day high-frequency data at highest available recording frequencies. For many liquid financial instruments huge numbers of observations are nowadays available – usually several observations per second. Modeling and analysing such high-frequency financial data becomes more and more important as much volume, currently almost 70%, is attributed to high-frequency trading. At the same time, specific market frictions need to be taken into account, inducing noisy observations.

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