Essays in Financial and Insurance Mathematics.

Abstract

This dissertation consists of the following three parts: (i) We find the minimum probability of lifetime ruin of an investor who can invest in a market with a risky and a riskless asset. The price of the risky asset is assumed to follow a diffusion with stochastic volatility. Given the rate of consumption, we find the optimal investment strategy for the individual who wishes to minimize the probability of outliving the wealth. Techniques from stochastic optimal control are used. (ii) We extend the Heston stochastic volatility model to include state-dependent jumps in the price and the volatility, and develop a method for the exact simulation of this model. The jumps arrive with a stochastic intensity that may depend on time, price, volatility and jump counts. The jumps may have an impact on the price or the volatility, or both. The random jump size may depend on the price and volatility. The exact simulation method is based on projection and point process filtering arguments. Numerical experiments illustrate the features of the exact method. (iii) We study the properties of sovereign credit risk using Credit Default Swap (CDS) spreads for U.S. and major sovereign countries. We develop a regime-switching two-factor model that allows for both global-systemic and sovereign-specific credit shocks, and use maximum likelihood estimation to calibrate model parameters to weekly CDS data. The preliminary results suggest that there is heterogeneity across different countries with respect to their sensitivity to system risk. Furthermore, the high-volatility and low-volatility regimes behave differently with asymmetric regime-shift probabilities.