Due to the 2007-2008 financial crisis, risk management has been playing a central role for those who operate in both the financial and life markets. Pertaining the latter, agents must confront themselves with some peculiar forms of risk, such as the Longevity risk and the Lapse risk.
The aim of the present research project is to give new insights to the management of the aforementioned types of risk, adopting a quantitative approach, which takes into account different standpoints. On the one hand, by selecting suitable mortality forecasting models that produce a reliable measurement of Longevity risk, the research aims to analyze optimal hedging strategies, mixing internal and external strategies, also in the presence of lapse risk. On the other hand, the research aims to investigate why, and in what extent, the introduction of suitable financial instruments (typically derivatives) might be used to reduce and mitigate the exposure to such a form of uncertainty.
The research project contains several innovative proposals compared to the current literature.
Firstly, we would aim to fill the existing gap between the class of stochastic processes employed to evaluate the real-world dynamics of mortality and the ones used for risk-neutral pricing of LLS. More precisely, generalizing the intensity-based term-structure models borrowed from the credit risk framework, we will examine a general class of affine stochastic processes for the force of mortality with both stochastic volatility and jumps to take into account some important stylized facts justified by empirical evidence. Indeed, the current literature pose a different and parallel use of stochastic process for describing mortality depending on the evaluation purpose. In particular, there is not a uniform treatment of mortality dynamics in the two contexts although they are considered the same class of models.
In addition, the lack of a stochastic volatility factor provides inadequate empirical results for parameters estimation, see e.g. Andersen, Lund (1997) and references therein. Moreover, we intend to enlarge the model including the presence of (simultaneous) jumps in the state variables dynamics. The rationale behind such a further generalization is due to the empirical evidence, which shows that mortality jumps are fundamental to explain sudden mortality improvements, see e.g. Chen, Cox (2007). To these scopes, we will consider Heston-type models.
The main advantage of using the aforementioned models in the LR modeling and management lies in the possibility to exploit well-established and standardized pricing techniques for LLSD evaluation and, at the same time, describe a more complex real-world evolution of mortality.
Recently, the evolution of financial instruments market has led investors to focus on several hedging tools, in order to capture the risk diversification. For this reason, we also investigate a stochastic version of the well-known Lee-Carter model, studying how to build a risk-neutral measure, which verifies two features: on the one hand, we may connect the continuous-time Lee-Carter model and the stochastic behavior of mortality rate previously introduced. On the other hand, we obtain the unique equivalent probability measure able to adjust the observed values with respect to the LR.
Once the theoretical framework has been defined, we will proceed to calibrate the model, to obtain suitable parameters estimates. We will also test our model by generating different scenarios, related to just as many kinds of LLSD. We resort to simulation techniques to implement this step. The numerical approach will range between standard Monte Carlo routines and some specific extensions of lattice-based methodologies, see e.g. Costabile et al. (2012), according to the type of financial instrument taken into account. The use of lattice methods is justified in terms of bigger efficiency, better convergence rate and reduced computational time, compared to other numerical methods.
Regard to the hedging strategies, we would build an optimal combination of internal and external hedging strategies comparing the risk mitigation performance in presence or not of the surrender activity. The current literature on hedging strategies is developed assuming implicitly that there is not surrender risk. The literature concerning the valuation of the surrender option is not very abundant, and most of the papers on this subject rely on numerical approaches. The contribution of this research project is at least twofold. On the one hand, we aim at providing an analytical valuation of the surrender option as well as a detailed study of the optimal surrender boundary. On the other hand, we aim at evaluating, for the first time in literature, how, and in what extent, the policyholder behavior can impact on hedging strategies formulation and performance oriented to mitigate LR. The hedging strategies optimality will be analyzed in terms of immunization approach and eventually minimizing a pertinent risk measure.
Given the new perspective in which the LR has been placed, the study of optimal portfolio allocation and optimal consumption is also of particular interest, see e.g. Menoncin (2008). The results produced in the aforementioned paper are valid for both complete and incomplete markets, when the instantaneous riskless interest rate evolves according to a Gaussian process. We intend to extend this framework to the case of stochastic volatility mortality rate model with co-jumps, using the results provided in Oliva, Renò (2018) in a financial context.