Research
Interests & Publications
My research interests include optimal stopping and free boundary problems, optimal control, problems with partial information and filtering techniques. The studied problems are motivated by financial and actuarial applications.
Publications
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Filtering in a hazard rate change-point model and financial applications
M. Buttarazzi and C. Ceci (2026) International Journal of Theoretical and Applied Finance.Abstract
This paper develops a continuous-time filtering framework for estimating a hazard rate subject to an unobservable change-point. This framework naturally arises in both financial and insurance applications, where the default intensity of a firm or the mortality rate of an individual may experience a sudden jump at an unobservable time, representing, for instance, a shift in the firm's risk profile or a deterioration in an individual's health status. By employing a progressive enlargement of filtration, we integrate noisy observations of the hazard rate with default-related information. We characterise the filter, i.e. the conditional probability of the change-point given the information flow, as the unique strong solution to a stochastic differential equation driven by the innovation process enriched with the discontinuous component. A sensitivity analysis and a comparison of the filter's behaviour under various information structures are provided. Our framework further allows for the derivation of an explicit formula for the survival probability conditional on partial information. This result applies to the pricing of credit-sensitive financial instruments such as defaultable bonds, credit default swaps, and life insurance contracts. Finally, a numerical analysis illustrates how partial information leads to delayed adjustments in the estimation of the hazard rate and consequently to mispricing of credit-sensitive instruments when compared to a full-information setting.
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The market value of optimal annuitization and bequest motives
M. Buttarazzi and G. Stabile (2024) Mathematical and Statistical Methods for Actuarial Sciences and Finance, Springer.Abstract
Since the seminal contribution of Yaari (1965), who showed that individuals with no bequest motive should convert all their retirement wealth into annuities, a number of papers have analysed the annuitization decision under the so-called all or nothing institutional arrangement, where immediate lifetime annuities are purchased just at a one point in time. In this paper, we investigate the effect of linear bequest motives on the annuitization decision for a retired individual who maximizes the market value of future cash flows. Finally, we present numerical examples analyzing optimal annuitization under strong or weak bequest motives.
Submitted Papers
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Optimal Annuitization with stochastic mortality: Piecewise Deterministic Mortality Force
M. Buttarazzi, T. De Angelis and G. Stabile (2025)Abstract
This paper addresses the problem of determining the optimal time for an individual to convert retirement savings into a lifetime annuity. The individual invests their wealth into a dividend-paying fund that follows the dynamics of a geometric Brownian motion, exposing them to market risk. At the same time, they face an uncertain lifespan influenced by a stochastic mortality force. The latter is modelled as a piecewise deterministic Markov process (PDMP), which captures sudden and unpredictable changes in the individual's mortality force. The individual aims to maximise expected lifetime linear utility from consumption and bequest, balancing market risk and longevity risk under an irreversible, all-or-nothing annuitization decision. The problem is formulated as a three-dimensional optimal stopping problem and, by exploiting the PDMP structure, it is reduced to a sequence of nested one-dimensional problems. We solve the optimal stopping problem and find a rich structure for the optimal annuitization rule, which cover all parameter specifications. Our theoretical analysis is complemented by a numerical example illustrating the impact of a single health shock on annuitization timing, along with a sensitivity analysis of key model parameters.
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The Impact of a Health Shock on the Optimal Annuitization Time
M. Buttarazzi (2025)Abstract
In this paper, we derive explicit closed-form solutions for the value function and the associated optimal stopping boundaries in an optimal annuitization problem under a mortality shock. We consider an individual whose retirement wealth is invested in a financial fund following the dynamics of a geometric Brownian motion and has the option at any time to irreversibly convert their wealth into a life annuity. The individual faces a sudden, permanent health deterioration occurring at a random, exponentially distributed time, and the annuitization decision is modelled as an optimal stopping problem across two health states. Our analytical expressions characterise both the value function and the optimal timing of annuitization. The results provide clear economic intuition: the optimal strategy is governed by the critical interplay between the relative attractiveness of the annuity (money's worth), the financial returns from the investment fund, and bequest motives across different health states. A numerical analysis compares the optimal annuitization strategy of an individual facing a health shock against a benchmark case with constant mortality, highlighting how the likelihood and severity of a health shock significantly alter optimal annuitization behaviour.
PhD Thesis
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Optimal annuitization time under piecewise deterministic mortality force
M. Buttarazzi (2025), PhD Thesis, Sapienza University of Rome.Abstract
Purchasing an annuity is one of the most significant financial choices individuals can make for retirement. This decision marks the point where retirees convert their savings into a steady, guaranteed income stream for life, prioritizing long-term financial stability over the potential growth of other investments. The uncertainty surrounding lifespan complicates this choice, as individuals must consider how to ensure their resources last throughout their lifetime. This Thesis explores the optimal timing for an individual to purchase an annuity while facing two sources of uncertainty: fluctuations in financial markets and the random evolution of his survival probability. We consider an individual whose retirement wealth is invested in a financial fund and who may choose a random time to irreversibly convert the entire wealth into an immediate lifetime annuity. The individual's mortality force evolves according to a piecewise deterministic Markov process, where the mortality risk remains constant between random jump times, with the size of each jump governed by a probability distribution. Coupled with financial market fluctuations, this introduces a complex decision-making environment where the retiree must determine the best time to convert his wealth into an annuity. The problem is initially formulated as a three-dimensional continuous-time optimal stopping problem. However, by exploiting the particular structure of the piecewise deterministic Markov process, we are able to reduce the complexity. Specifically, we transform it into a sequence of nested one-dimensional optimal stopping problems. This transformation simplifies the mathematical analysis and enables us to derive explicit solutions. The solution to the problem hinges on analyzing two key regions: the continuation region, where the individual chooses to keep his wealth invested in the financial fund, and the stopping region, where he opts to purchase the annuity. The boundary between these regions is critical in determining the optimal timing of the annuity purchase. Initially, we focus on a simplified scenario where the force of mortality remains constant throughout the individual's life. This case allows us to derive explicit solutions and serves as a foundation to prove properties that will be generalized later for the more general model. In the general scenario, where the mortality force is modelled as a piecewise deterministic Markov process, we examine the properties of the value function and analyze the geometry of the continuation and the stopping regions. The shape of these regions varies depending on the presence of incentives or taxes on annuity purchases. Then, we investigate a special case in which the mortality force experiences a single jump at a random time. This scenario yields explicit solutions that provide insights into the financial consequences of sudden changes in mortality risk.