Applications of asymptotic methods in quantitative finance and insurance

This thesis deals with three essays related to studying asymptotic behavior of a portfolio tail loss, asymptotic behavior of options price and asymptotic stability of a class of jump diffusion process. In the first article that constitutes our first essay, we study an enhancement to the CreditRisk+ model termed as the 2-stage CreditRisk+. We determine under what conditions on the portfolio does the 2-stage CreditRisk+ credit risk model gives higher Value at Risk than the CreditRisk+. This entails studying rare event probability of large portfolio loss event. For the same we use technique from the theory of large deviations. In the second article, we consider an asymptotic options pricing problem in a Markov modulated regime switching market. In such market, the key model parameters are modulated by a continuous-time, finite-state, Markov chain. Such a market is incomplete and hence there exist a range of options price. For an asymptotic analysis, we consider two variations of the chain, namely, a slow chain and a fast chain. It has been observed that there exists an asymptotic option price for the slow chain case while it is been argued that such price may not exist for the fast chain case. In this article, we attempt to show why this is so by determining the range of options price for the slow chain and the fast chain. In the third and the last article, we consider a jump-diffusion process whose drift, diffusion and the jump kernel is modulated by a semi-Markov process. The semi-Markov process is a generalization over the Markov chain case since its sojourn time need not be exponentially distributed. We study the issue of asymptotic stability of this process with regards to almost sure and moment exponential sense. We study this issue here because of its motivational connection to the ruin theory in insurance. We also study the issue of stabilization and de-stabilization of a non-linear system of differential equation perturbed by a semi-Markov modulated jump diffusion process. We thereby comment on the interesting behaviour that we observe with regards to (de)-stabilization of the system of differential equation in one and in higher dimension.