Financial Risk Modelling and Analysis

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Mathematical and statistical modelling is used to describe and analyze complex phenomena arising in nature and human societies. Stochastic processes are a perfect instrument for modelling dynamically changing systems with uncertainty and multiple degrees of freedom. Examples of such complex systems are financial markets with millions of transactions made every day. The ability to capture the behaviour of market prices more accurately and more thoroughly is of significant importance to various financial institutions such as major Canadian banks and pension funds. The development of efficient numerical and statistical methods of mathematical finance and computational implementation of solvable stochastic volatility models are the main priorities of the proposed research program. It has the following three objectives. First, we will develop a family of multi-asset pricing models with a systemic risk component and evaluate their applications in portfolio management, derivative pricing and risk assessment. The proposed models allow for dealing with financial assets that have missing and asynchronous pricing data. New assets can be easily included in the model without recalibrating parameters for other assets. Second, we will develop structural models of credit risk based on occupation time. In these models, the time of default is defined as the first time that the occupation time of the firm's value process below a default barrier has exceeded a given threshold. Such structural models can be used to assess the liquidation risk and allow for separating default and liquidation events. Our primary focus is on the calibration of occupation time models, their applications in the pricing of financial risk instruments such as credit default swaps and contingent capital bonds. Additionally, we will evaluate the application of spectral expansions for pricing occupation time derivatives. Third, we will develop statistical models for financial news analysis and integrate them with asset price models. One objective is to uncover how financial news and other announcements can affect the stock market, whether that be changes in the price of a stock or changes in the daily volume. Additionally, we will develop a news-driven model of the stock market by combining the classification of textual financial news algorithm with asset price models. The results of this research program will help us better understand how to model financial markets and assess risks when data are incomplete, as well as how to extract useful information from unstructured business data.