Abstract
This thesis consists of an introductory chapter and five papers. The papers are each concerning questions within the topics life insurance, optimal stopping or the interplay between these. Each paper is presented in a chapter,
and thus each of the chapters are self-contained and may be read alone. Below, I give a brief overview of the results of each of the chapters. A more thorough overview is presented in Chapter 1.
In Chapter 2 we consider a general geometric Lévy process and solve the non-linear optimal stopping problem of maximizing the variance at the stopping time. For solving this problem we solve an auxiliary quadratic optimal stopping problem. We show that the solution to maximizing variance depends on whether randomized stopping times are included in the set of stopping times we maximize over. For some problems the inclusion of randomized stopping times increase the value function and for some it does not. Even when the value function is not affected by inclusion of randomized stopping times, a solution may be easier to identify when they are.
In Chapter 3 we consider the non-linear optimal stopping problem of maximizing the mean minus a positive constant times the variance at the stopping time. First we solve the problem for spectrally negative geometric
Lévy process. We derive both static and dynamic solutions which are excess boundary stopping times. Afterwards we solve the problem for a Cramér-Lundberg process with exponential upwards jumps. We derive a statically optimal stopping time which is a hitting time of an interval, and we derive a dynamically optimal stopping time which is an excess boundary stopping time. Finally, we derive optimal stopping times to the optimal stopping problem of minimizing the variance conditioned on a lower bound on the
mean.
In Chapter 4 we consider the American put in a Black-Scholes market. We suggest a model for irrational exercises. We model the exercise by a stochastic intensity which depends on the profitability. Our model contains a single parameter which express how strongly the exercise intensity is affected by the profitability. This parameter we denote the rationality parameter. We give sufficient conditions and a probabilistic proof that when the rationality parameter increases to infinity the corresponding prices converge to to classical arbitrage-free price. We conclude the chapter with partial differential equations for valuation under irrational exercise, and we discuss relations to the penalty method.
In Chapter 5 is related to Chapter 4, but in Chapter 5 we consider modelling the time of surrender in a classical life insurance model. We suggest a model where the probability of surrender at any time depends on the profitability. We measure the profitability as the difference between the value of the insurance contract and the surrender value. The value of the insurance contract may be determined as a solution to a differential equation
much similar to the Thiele differential equation. As in Chapter 4 the model contains a rationality parameter which express how strongly the surrender probability is affected by the profitability. Again we derive a probabilistic proof of the intuitive convergence result that when the rationality parameter increases to infinity, the value of the life-insurance contract converge to the value corresponding to if the policyholder surrendered at the optimal time.
In Chapter 6 we add stochastic retirement to a classical finite state life insurance model. We do this by splitting the active state in a premium paying state and a retired state. We derive formulas for scaling the benefits
reasonably according to the time of retirement. Then we show how to calculate the reserves and expected cash. Afterwards we describe a way to add to the model that policyholders might change their benefit structure upon
retirement. We determine formulas for calculating reserves and cash flows in this model too. Finally, we conclude with a numerical investigation of the implication stochastic retirement has on reserves and cash flows.
and thus each of the chapters are self-contained and may be read alone. Below, I give a brief overview of the results of each of the chapters. A more thorough overview is presented in Chapter 1.
In Chapter 2 we consider a general geometric Lévy process and solve the non-linear optimal stopping problem of maximizing the variance at the stopping time. For solving this problem we solve an auxiliary quadratic optimal stopping problem. We show that the solution to maximizing variance depends on whether randomized stopping times are included in the set of stopping times we maximize over. For some problems the inclusion of randomized stopping times increase the value function and for some it does not. Even when the value function is not affected by inclusion of randomized stopping times, a solution may be easier to identify when they are.
In Chapter 3 we consider the non-linear optimal stopping problem of maximizing the mean minus a positive constant times the variance at the stopping time. First we solve the problem for spectrally negative geometric
Lévy process. We derive both static and dynamic solutions which are excess boundary stopping times. Afterwards we solve the problem for a Cramér-Lundberg process with exponential upwards jumps. We derive a statically optimal stopping time which is a hitting time of an interval, and we derive a dynamically optimal stopping time which is an excess boundary stopping time. Finally, we derive optimal stopping times to the optimal stopping problem of minimizing the variance conditioned on a lower bound on the
mean.
In Chapter 4 we consider the American put in a Black-Scholes market. We suggest a model for irrational exercises. We model the exercise by a stochastic intensity which depends on the profitability. Our model contains a single parameter which express how strongly the exercise intensity is affected by the profitability. This parameter we denote the rationality parameter. We give sufficient conditions and a probabilistic proof that when the rationality parameter increases to infinity the corresponding prices converge to to classical arbitrage-free price. We conclude the chapter with partial differential equations for valuation under irrational exercise, and we discuss relations to the penalty method.
In Chapter 5 is related to Chapter 4, but in Chapter 5 we consider modelling the time of surrender in a classical life insurance model. We suggest a model where the probability of surrender at any time depends on the profitability. We measure the profitability as the difference between the value of the insurance contract and the surrender value. The value of the insurance contract may be determined as a solution to a differential equation
much similar to the Thiele differential equation. As in Chapter 4 the model contains a rationality parameter which express how strongly the surrender probability is affected by the profitability. Again we derive a probabilistic proof of the intuitive convergence result that when the rationality parameter increases to infinity, the value of the life-insurance contract converge to the value corresponding to if the policyholder surrendered at the optimal time.
In Chapter 6 we add stochastic retirement to a classical finite state life insurance model. We do this by splitting the active state in a premium paying state and a retired state. We derive formulas for scaling the benefits
reasonably according to the time of retirement. Then we show how to calculate the reserves and expected cash. Afterwards we describe a way to add to the model that policyholders might change their benefit structure upon
retirement. We determine formulas for calculating reserves and cash flows in this model too. Finally, we conclude with a numerical investigation of the implication stochastic retirement has on reserves and cash flows.
| Originalsprog | Engelsk |
|---|
| Forlag | Department of Mathematical Sciences, Faculty of Science, University of Copenhagen |
|---|---|
| Antal sider | 142 |
| ISBN (Trykt) | 978-87-7078-961-5 |
| Status | Udgivet - 2015 |