Statistical Modeling of Financial Extremes and Volatility Dynamics

Overview

The occurrence of extreme events is a defining characteristic of financial markets. Although infrequent, these "black swan" events carry disproportionate impact. This thesis aims to address key challenges in financial tail dependence, causal inference in extremes, and rough volatility modeling by proposing novel statistical frameworks.

First, to assess market efficiency under extreme market conditions, this thesis proposes the Efficient Tail Hypothesis (ETH). By constructing a novel measure of Directional Tail Dependence (DTD), we quantify the asymmetric dependence of asset returns in the joint tail within the framework of Extreme Value Theory. Empirical studies indicate that the Chinese futures market rejects the ETH, revealing the existence of predictable dependence structures under extreme conditions, which provides new early warning signals for risk management.

Second, to identify the transmission mechanisms of systemic risk in extreme scenarios, we establish the Extreme Structural Causal Model (XSCM). Utilizing transformed-linear algebra and the Partial Tail Correlation Coefficient, this model addresses the limitations of traditional causal discovery algorithms when applied to heavy-tailed and extreme data. We prove that the XSCM satisfies the tail causal Markov and faithfulness conditions, enabling the learning of Directed Acyclic Graphs (DAGs) for extremes from observational data via separation-based algorithms. This method is successfully applied to Danube river flow data and high-frequency analysis of the Chinese futures market.

Finally, addressing the issues of roughness and computational complexity in volatility modeling, we propose the rMat\'ern-V model. This model specifies log-volatility as the solution to a Matérn Stochastic Differential Equation (SDE) and utilizes rational approximations to convert it into a Markovian Gaussian Random Field with a sparse precision matrix. This framework not only simultaneously captures short-term rough behavior and long-term memory in volatility but also significantly reduces the computational cost of Bayesian inference and Monte Carlo simulations, achieving simulation-based option pricing with linear computational complexity.

In summary, by integrating Extreme Value Theory, causal inference, and stochastic volatility modeling, this thesis provides comprehensive statistical tools for understanding and modeling extreme risks and volatility dynamics in financial markets.