Path properties of volatility: A statistical analysis
Statistics and Econometrics
Final Report Abstract
We contribute new probabilistic results and statistical methods for the analysis of stochastic processes which are of interest to describe price and volatility dynamics in financial markets. Volatility is the key quantity to model and to predict financial risk. Recently, rough fractional stochastic processes as volatility models were shown to facilitate improved risk forecasts. However, they do not account for volatility jumps. This is important, since economic shocks trigger volatility explosions, which are not adequately modelled by continuous processes. Due to the different nature of financial risk induced by jumps on the one hand and volatility on the other, the two components are separated in recent economic studies. We design statistical methods to detect jumps in a fractional continuous process and to discriminate jumps from the continuous evolution. We demonstrate the relevance of including jumps in an empirical analysis of daily volatility measures which are computed from intra-day, high-frequency prices. We derive structural results on the identifiability of path properties, as the regularity, of a latent stochastic volatility process. They shed light on the frontiers of statistical inference for volatility models. Moreover, we develop tools which allow to efficiently exploit information from intra-daily highfrequency financial data. This becomes increasingly important in the digital era with electronic markets and algorithmic trading. In fact, most volume of stock trades is nowadays attributed to high-frequency trading. Limit order books contain all information from high-frequency trading and provide huge data sets. However, it is challenging to use such data for inference due to its complexity and the market microstructure. For this purpose, we establish statistics for a model with limit order microstructure noise. In this model prices of bid and ask limit orders are discrete observations of a latent semimartingale efficient price process with additive one-sided noise. We construct test and estimation methods for volatility and jumps based on local order statistics. We establish, moreover, the estimation of an extreme value index in the difficult situation with a time-varying, stochastic boundary process. We demonstrate that our novel approach improves volatility estimation and price-jump detection. We showcase how this helps to improve risk quantification in an empirical analysis of intra-daily limit order book data. One focus for a follow-up research project is to investigate implications of the highlighted insight that the estimation uncertainty varies across different stocks. This indicates that a risk analysis for one stock, e.g., Apple, can be improved using data also from another stock, e.g., Google. Therefore, it is important to advance our research from this project to a multivariate portfolio model including strategies for optimal portfolio allocation.
Publications
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Github.com/bibinger/LOMN. R functions on testing for jumps from best ask prices in a limit order book with limit order microstructure noise
A. Ristig
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Cusum tests for changes in the Hurst exponent and volatility of fractional Brownian motion. Statistics & Probability Letters, 161, 108725.
Bibinger, Markus
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Gumbel convergence of the maximum of convoluted half-normally distributed random variables. Technical report
M. Bibinger
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Inference on the intraday spot volatility from high-frequency order prices with irregular microstructure noise. Journal of Applied Probability, 61(3), 858-885.
Bibinger, Markus
