Mathematical Models for Financial Data

Abstract

The first chapter studies the effects of the temporary short-selling ban on US financial stocks in 2008 by overcoming a key problem in earlier empirical works of having to use non-financial stocks as a control group since almost all financial stocks were banned. Alternatively, the control group is selected to be a synthetic portfolio of non-banned financial stocks constructed from the finance segments of large industrial companies that were not banned. With this control group, it is found that the ban leads to over-valuation of banned stocks that is highest at the beginning of the ban and steadily converges to zero at the end of the ban. To understand this dynamic, the model of Scheinkman and Xiong (2003) is solved in a finite trading horizon with zero trading cost.

The default of one bank can cause other banks to default through two channels: financial contagion in the inter-bank liability network and fire sale in the asset selling market. In the second chapter jointly written with Weinan E, a model that incorporates these two channels is developed and analyzed theoretically. An algorithm for finding the state in which both the inter-bank liability network and the market are in equilibrium is proposed and tested.

An oft-mentioned but under-studied feature of asset price bubbles is a surge of new entrants, retail investors who never invested, joining the bubble because of their friends or neighbors. The third chapter, jointly written with Harrison Hong, incorporates this viral element into an otherwise standard bubble model with forward-looking agents. Optimism spreads across households following an epidemic process and the participation rate rise as new entrants buy anticipating trending prices. Insiders or institutions gradually sell their shares, generating trading volume and moderating price growth. This model rationalizes several patterns in the data, which have been difficult to explain, including trending prices and volume peaking months before prices, participation rates, and short-selling in both stock market and housing bubbles.