Keynotes

Title: Computational Finance: Correlation, Volatility, and Markets

Summary:

Financial data, by nature, are interrelated and should be analyzed using multivariate methods. Many models exist for the joint analysis of multiple financial instruments. Early models often assumed constant behavior between the financial instruments over the analysis period. But today, time-varying covariance models are a key component of financial time series analysis leading to a deeper understanding of changing market conditions.

Specifically, modeling the daily volatility (variance or standard deviation) of a stock or asset return is an important step in estimating how much risk a particular asset carries. However, the variance is not directly observable from a time series since there is only one observation at each time point. The 1982 seminal paper of Nobel Laureate Robert Engle introduced the Autoregressive Conditional Heteroscedasticity (ARCH) model allowing analysts a path to obtain such an estimate. Over the last half-century, an explosion in strategies for modeling heteroscedasticity in time series, or the changing variance of the process, has been made. One overarching model is the generalization to the (G)ARCH model, which includes lagged components of squared innovations of the time series in the model.

Another key feature one wants to capture when modeling the volatility across financial markets is the correlation across and within asset classes. The multivariate Generalized Autoregressive Conditional Heteroscedasticity (MGARCH) model is used for studying the relationships between the volatilities and co-volatilities of multiple stocks.  An excellent survey on MGARCH models (with over 1000 citations) is provided by Bauwens in 2006.

Challenges of MGARCH modeling include the fact that the covariance matrix must be positive definite at every time point, as well as the obvious curse of dimensionality leading to a large number of parameters for even simplistic model formulations. A very popular MGARCH model from the early era that solves these issues is the constant conditional correlation (CCC) model, which decomposes the conditional covariances of the stock returns into conditional correlations and conditional standard deviations. The basic premise of the CCC model is that the conditional correlation matrix between returns is constant over time, but the univariate conditional standard deviations change over time. This changing structure is captured using time series models for the conditional standard deviations.

Of further importance is the recognition that markets structurally change over time and that any model or system must adapt to this change.  Dynamic conditional correlation (DCC) models adapt to market conditions. Variants of the DCC models have become one of the most popular financial time series methods in recent literature due to this need to capture changing market dynamics.  In this talk, I highlight some of the new findings for the DCC class of MGARCH models.

Another strategy to adapt to changing market dynamics is through Markov switching models where model parameterizations change with market conditions. In the univariate setting, GARCH models with varying parameters, such as the regime-switching model improve volatility forecasting. Similar improvements are seen in regime-switching models for multivariate co-volatility where copulas are used to capture the multivariate dependence.

Finally, I bring forward a hierarchical regime-switching dynamic covariance model (HRSDC) with a general discussion of estimation and how this model can capture the between and within dynamic covariance structure for a large number of stock returns potentially representing an entire market.  I will give an example of stock market returns through the late 90’s and early 2000’s, including the fall of the U.S.-based corporation ENRON leading to the 2001 global financial crisis. Further, structural change through the U.S. and global subprime crises of 2008 is also explored. We can foreshadow events such as the decline of the U.S. firm Lehmann Brothers with large lead times.

In summary, the primary motivation for this complex model characterization concerns stock portfolio diversification and early identification of stock market anomalies. The hierarchical regime-switching dynamic covariance time series model captures the changing co-movement of stocks within and between sectors as market conditions change due to market collapses and crashes or common external influences that drive economies.