Last modified | 18th August 2021 |
Institution | University of Manchester |
Status | Lecturer in Financial Mathematics |
Contact | thomas.bernhardt@manchester.ac.uk |
Stochastics and its connections to Actuarial Science, Financial Mathematics and Statistics. Decumulation and investment strategies for pension funds. Long-term goals include developing new Stochastic tools beyond Itô Calculus.
Pooled annuity funds adjust the income of its members with observed mortality rates to cope with the uncertainty of rising life expectancy. However, the heterogeneity of the members negatively affects the stability of the income payments. In particular, wealth heterogeneity stands out because differences in savings can be orders of magnitude large. We analyze if every member in a given group benefits from pooling their funds together and how we can split the group so that everyone benefits.
The Root barrier of the Skorokhod embedding problem is a helpful tool to find a robust lower bound for prices of VIX call options when the terminal distribution is known. Despite its importance, there are only a few results about the properties of the barrier function in the literature. We show that the barrier function is continuous and finite at every point where the terminal distribution has no atom and its absolutely continuous part is bounded away from zero.
Tontines are well-understood under the assumption of constant market returns and a perfect pool. However the literature lacks in understanding the impact of fluctuating market returns and realized mortality rates. We analyze how many members a tontine need in order to deliver a stable income for life. We aim for practical answers that shall be used in the future for pension products.
Combining the best of drawdown and annuity, the investment returns and the longevity credits, tontines offer a great alternative to current pension products. Promoting tontines, we analyze the effect of a bequest motive on the decision to invest in a tontine. We formulate an investment problem where a pensioner chooses the percentage of wealth in the tontine, an investment strategy and their consumption rate. The investment problem is formulated such that the optimal strategy maximizes the utility of lifetime consumption and the left behind bequest. We show that, for a risk-averse investor, the percentage in the tontine is around 80% for a wide range of risk aversion and different bequest motives.
We solve a general optimal stopping problem involving generalized drift and show how the Principle of Smooth Fit is violated for specific choices of the problem data. We build on existing results from Lamberton and Zervos (2013). This is a technical analysis involving the concept of difference-of-two-convex-functions instead of twice-differentiable-functions.
The authors are part of the research project ”Minimising Longevity and Investment Risk while Optimizing Future Pension Plans”. This paper was written to familiarize the project team with the existing knowledge on decumulation strategies for pension funds. Here, highlighting important ideas and identifying promising areas for future research has been given most attention. On the other hand, topics related to implementation such as taxation or solvency or regulation has been given less attention.
In reality the flow of information arrives on a discrete time grid rather than it arrives in a continuous stream. Empirically there is no difference between models which have the same distribution at this grid. Therefore, we introduce a new class of continuous processes, the Itô semi-diffusions. These processes are modelled by a homogenous SDE between grid points, and have a prescribed distribution at grid points.
We consider the solvability of SDEs with homogeneous coefficients which are reflected in a càdlàg function. Our main idea is to use methods from Engelbert and Schmidt to show that under mild assumptions on drift and volatility the problem reduces to solve a Skorokhod-type problem. In particular, we can deal with non-Lipschitz coefficients.
We consider the problem to predict a stochastic intensity of a jump process given that we observe the occurrence of jumps in time. Under suitable integrability constraints, the literature provides already the means to deal with such problems in a general context. Our contribution is to point out that for a specific class of jump processes, the Cox-processes, these restrictions are not actually needed.
In this paper about market weights, the authors characterised polynomial jump-diffusions on the unit simplex. Here, they encountered possible measures on the complex plane with vanishing complex moments. In case of a probability measure, I suggested a proof excluding such possibilities. In the corresponding paper, see the proof of Theorem 4.3, Type 4, establishing 4.2 also in case of n=2.