On pricing, hedging and trading in financial management

Abstract

In this thesis, efforts are devoted to mathematically model and address practical problems arising in credit risk management and portfolio management which include modeling credit risk, pricing and hedging credit derivatives, and dynamic optimal trading strategies in financial markets.

A generalized reduced-form intensity-based credit model with a hidden Markov process is developed for credit risk management. The intensities of defaults are determined by the hidden economic states which are governed by a Markov chain, as well as the past defaults. The model is applicable to a wide class of default intensities with various forms of dependent structures. For the hidden Markov process, a filtering method is proposed for extracting the underlying state given the observation processes. Closed-form formulas for the joint distribution of multiple default times are derived without imposing stringent assumptions on the form of default intensities. In addition, applications in credit risk management are also discussed. Based on this model, three explicit practical default models are established. The parameters in these three default intensity models are further estimated statistically using the Expectation-Maximization (EM) algorithm with real market data. The results demonstrate that these models are able to capture the hidden features and simulate credit default risks which are critical in risk management and the extracted hidden economic states are consistent with the real market data. Credit Default Swap (CDS), an important credit derivative, under these models can be priced efficiently using explicit formulas.

Copula models are also studied to capture the dependent structure of defaults in pricing and hedging credit derivatives. The thesis establishes three different classes of models, which are sophisticated and coherent with instruments implemented in practice. These models are applied to basket CDSs valuation conditional on survivorship information under Gaussian Copula model. The idea and method can be extended to general Copula models. Based on these models, both the Delta hedge and the Delta-Gamma hedge are considered. The price movements in credit derivatives with respect to shifts in correlations, one of the key parameters in the Gaussian Copula model, are intended to be hedged. The hedging method is to hedge the basket CDSs by using underlying single name CDSs as the hedging instruments. Hedging efficiency and effectiveness could be tested with a developed criteria measure.

Optimal pairs trading strategies under dynamic Mean-Variance (MV) are studied in this thesis. The strategy is implemented by choosing a pair of securities whose prices move together historically. The Ornstein-Uhlenbeck (OU) process has been adopted for modeling the mean-reverting process of the price difference of a pair of securities. The MV portfolio optimization problem is then formulated and solved dynamically by utilizing the purely-buy-and-sell-securities strategy. This optimization problem is extended to a quadratic optimization problem when optimally allocating money in both pairs of securities and riskless asset is considered. Time-consistent and closed-form solutions to these MV pairs trading problems are derived. A calibration method is also introduced for estimating model parameters.