Semi-Markov Models for Insurance and Option Rewards

Abstract: This thesis presents studies of semi-Markov models for insurance and option rewards. The thesis consists of the introduction and six papers. The introduction presents the results of the thesis in an informal way.In paper A, a general semi-Markov reward model is presented. Recurrence relations for evaluation of higher moments of the reward process are given, as well as a backward semi-Markov reward processes are applied to insurance problems for the first time.In paper B, models for disability insurance given in paper A are further extended. Statistical evidences of relevance of semi-Markov setting are given. Applications to profit-risk analysis for contracts are considered.In paper C, a more detailed explanation of the algorithmic for the non-homogenous backward semi-Markov reward process is given. Two algorithmic approaches to solve the problem in an iterative manner are given. One of the algorithms is presented in a pseudo-code.In paper D, the geometrical Brownian motion with drift and volatility controlled by a semi-Markov processes is considered as a price process in option valuation. The discrete version is examined and limit theorems describing the transition from discrete to continuous time are given. Monte-Carlo algorithms are described.In paper E, a general price process represented by a two-component Markov process is considered. American options with pay-off functions, which admit power type upper bounds are studied. Both the transition characteristics of the price processes and the pay-off functions are assumed to depend on a perturbation parameter and to converge to the corresponding limits. Results about the convergence of reward functionals for American options are presented.In paper F, convergence for option rewards when the price processes are perturbed exponential Lévy type process controlled by semi-Markov indices is studied. Both European and American type options with pay-off functions which admit power type upper bounds are considered. The paper continues research started in paper D and gives a key example for paper E.