## C1 - Volatility

Asset prices recorded as high-frequency data are close to continuous-time observations and thus foster statistical inference for continuous-time price models. Semi-martingales, a prominent class of continuous-time stochastic processes, comply with fundamental economic hypotheses as exclusion of arbitrage and provide flexible models allowing for stochastic volatility, jumps and leverage. Due to market microstructure of most high-frequency financial data, log-prices are not directly well fitted by semi-martingales, but should be considered within a noisy observation setup. Substantial progress on the estimation of the quadratic (co)variation, integrated (co)volatility and (co)jumps of asset prices under various assumptions on the noise process have been made. Current research focusses on the joint behaviour of assets in larger portfolios and jump behaviour during periods of financial stress.

Both, theoretically and empirically, it is still unclear how to use high-frequency data most efficiently to estimate high dimensional (quadratic) covariation matrices and how to conduct statistical tests based on these estimates. This holds not only for the covariation of the continuous (Brownian) components, the integrated covolatility, but also especially for the covariation matrix of the price jumps. Using the spectral approach of Bibinger et al (2014) one can efficiently estimate the integrated co-volatility in a noisy model. Bibinger and Winkelmann (2015) extend these results to a model with jumps. The focus is so far on low-dimensional estimates. In a pure GARCH setting one uses methods from random matrix theory to estimate large covariance matrices. In order to develop and analyse estimators for high dimensional semi-martingale models, Marcenko-Pastur-like limit theorems must be established and then shrinkage methods can be employed.

A methodological challenge in larger portfolios poses the non-synchronicity of observation times of the asset prices. In the two-dimensional setting without noise the covariation estimator by Hayashi and Yoshida was a breakthrough, overcoming the so-called Epps effect that ultra-high-frequency observations result in zero co-volatility estimation. For regular, non-synchronous observation design and moderate dimensions the problem under microstructure noise can be overcome by going over to continuous observations via linear interpolation. In larger portfolios, however, the design assumptions are often violated and other data augmentation methods should be studied, e.g. relying on the EM-algorithm or a Bayesian approach. A particular example is the estimation of the risk factor beta with respect to a much more liquid index, cf. Reiß, Todorov and Tauchen (2015) in a jump model setting. Starting from this two-dimensional situation, methods to cope with non-synchronicity need to be derived, applicable to general settings of non-synchronously observed time series, also in other application domains.

Time and space (different countries or asset classes) propagations in a global or some regional financial crisis require a more explicit and richer model structure. The popular Lévy processes have independent increments and thus do not allow for any type of serial dependence. In contrast, Hawkes process provide one promising class of models to capture the usually observed clusters of relatively large returns across markets and time. It remains an open methodological question as how to efficiently distinguish such high dimensional dependencies.

While price jumps are usually linked with specific news and a contemporaneous pricing of the new information, an increase in volatility usually reflects a period of elevated uncertainty or panic in the markets. In response to unexpected policy announcements, the asset pricing model of Pastor and Veronesi (2012) postulates a combination of jumps in prices, volatilities and correlations. It allows for the formulation of testable hypotheses to assess the expected impact of an announced new policy. Due to missing methodology, such hypotheses have not been analysed empirically. To this end, the spectral method derived by Jacod and Reiß (2014) for estimation will be employed.

**Coordination**

**Lars Winkelmann**: His main research interest is applied econometrics. His recent research concentrates on jumps in asset prices and volatility and the economic content behind such extreme events.

**Markus Reiß**: His main research interests lie in mathematical statistics and stochastic analysis. His research includes work on nonparametric statistics, statistics for stochastic processes, statistical inverse problems, stochastic differential equations and applications in finance.

**Exemplary PhD-Theses**

1. Large portfolios: random matrix theory, efficiency and algorithms

An obvious goal of PhD theses in the area of large portfolios could be to shed more light on the optimality of high dimensional covariation estimators using noisy high-frequency data. Using a convenient and mathematically tractable framework, an important contribution would be to analyse the limits of high dimensional covariation matrix estimates for growing dimensions. Generalizing the Marcenko-Pastur law of the i.i.d. case, the covariation matrix estimator in the semimartingale setting shall be analysed, which poses intriguing mathematical questions and might result in surprising features of the model and/or its estimator. Using shrinkage of the matrix estimator, more stable results for larger portfolios can be obtained. These may be then applied to determine risk measures as considered in B02 to study portfolio variations.

An important question in practice is the efficient calculation of the matrix estimator which for p assets produces a p2-dimensional quantity with a p4-dimensional Fisher information or covariance matrix for the estimator, based on usually about 104 or more data points for each asset. For moderate dimensions such as p=100 this is already computationally demanding. One way out is to start with a simple and fast pilot estimator and to use iterative gradient descent to improve on this iteratively. Another focus for a thesis could be on the non-synchronicity of observation times and different liquidity of assets. A basic question is how linear interpolation works in high dimensions and whether a variant of the EM algorithm may yield stable and reliable results.

2. Testing for breaks in high dimensional integrated covariance and co-jump matrices

Multivariate statistics provides a number of tests to distinguish between two covariance matrices. A goal of a PhD thesis is to extend existing tests, to the underlying econometric model class of high dimensional semimartingales. Change point tests for the matrix of the integrated or spot covolatility and correlation and cojumps should be developed. An automated procedure may timely detect events important for risk management and portfolio selection. A connection with text based DTM as outlined in the project group B would shed light on reasons of changes.

3. Using the limit order book to improve inference on volatilities and jumps

An important contribution could be based on the limit order book model of Bibinger, Jirak and Reiß (2016), extended to semi-martingales with jumps. The following questions should be answered: (i) How do jumps influence the volatility estimation? (ii) Is there a way to filter jumps and what are the post-filter properties of the volatility estimator? (iii) Which empirical results are obtained from traded prices versus limit-order book quotes, e.g. during a flash crash? Answers to these questions do not only require a profound understanding of the continuous semi-martingale setting, but also innovative ideas to cope with jumps. The empirical part enters unexplored territory and yields a better understanding of the functioning of the markets during crises (high jump activities).

4. Shock propagation in financial markets: Jumps in jump intensities vs. jumps in volatility

(Lars Winkelmann, Markus Reiß)

Based on Jacod and Reiß (2014), a goal of a PhD thesis is to develop statistical tests to disentangle changes of price jump intensities from volatility jumps. Different models that are able to generate jump clustering and feedback between markets should be reviewed and compared in simulations (Hawkes processes, doubly stochastic Poisson processes, vola jumps). Empirical applications of the new methods based on intraday stock and bond data include (i) an analysis of the shock propagation during the European sovereign debt crisis, (ii) assessments of impact effects of central bank and government policy announcement in the context of the Pastor and Veronesi (2012) model and (iii) a study of the implications for risk management and related high-frequency risk measures.

**References**

- Bibinger M, Hautsch N, Malec P, Reiß M (2014) Estimating the quadratic covariation matrix from noisy observations: local method of moments and efficiency, Annals of Statistics 42(4), 1312-1346, DOI:10.1214/14-AOS1224
- Bibinger M, Jirak M, Reiß M (2016) Volatility estimation under one-sided errors with applications to limit order books, Annals of Applied Probability 26(5), 2754-2790, DOI: 10.1214/15-AAP1161
- Bibinger M, Winkelmann L (2015) Econometrics of cojumps in high-frequency data with noise, Journal of Econometrics, 184(2), 361-378, DOI: http://dx.doi.org/10.1016/j.jeconom.2014.10.004
- Jacod J, Reiß M (2014) A remark on the rates of convergence for integrated volatility estimation in the presence of jumps, Annals of Statistics 42(3), 1131-1144, DOI: 10.1214/13-AOS1179
- Pastor L, Veronesi P (2012) Uncertainty about Government Policy and Stock Prices, Journal of Finance, 64, 4, 1219-1264, DOI: 10.1111/j.1540-6261.2012.01746.x
- Reiß M (2011) Asymptotic equivalence for inference on the volatility from noisy observations, Annals of Statistics 39(2), 772-802, DOI: 10.1214/10-AOS855