Conference on Mathematical and Statistical Methods for Actuarial Sciences and Finance (MAF 2026)

Quantifying Climate Transition Risk in Portfolios using Extreme Value Theory

The transition toward a low-carbon economy introduces significant systemic risks for financial institutions, specifically “transition risks” arising from abrupt policy changes and technological shifts. Traditional mean‑variance frameworks and Gaussian‑based risk models frequently fail to capture the heavy‑tailed nature of these climate‑related financial shocks, leading to an underestimation of capital requirements.
This paper proposes a robust quantitative framework to measure the impact of climate transition risk on mixed asset portfolios (“green” vs. “brown” assets). We apply Extreme Value Theory (EVT), specifically the Peaks Over Threshold (POT) method, to model the tail behavior of asset returns under climate stress scenarios. Furthermore, we utilize Copula theory (focusing on Archimedean and Vine copulas) to capture the non‑linear tail dependence structure between carbon‑intensive sectors and sustainable indices during periods of market stress.
Our empirical analysis suggests that ignoring tail dependence leads to a significant mispricing of risk. We demonstrate that the proposed EVT‑Copula approach provides more accurate estimates of Value‑at‑Risk (VaR) and Expected Shortfall (ES) compared to historical simulation or normal distribution assumptions. The findings have direct implications for actuarial risk management, strategic asset allocation, and compliance with evolving regulatory frameworks such as Solvency II and IFRS S2 regarding climate‑related disclosures.
Keywords: Climate Finance, Extreme Value Theory (EVT), Tail Risk, Copulas, Transition Risk, Solvency II.
JEL Classification:
• C46: Semiparametric and Nonparametric Methods: Specific Distributions (Heavy tails).
• G11: Portfolio Choice; Investment Decisions.
• G32: Financing Policy; Financial Risk and Risk Management.
• Q54: Climate; Natural Disasters and Their Management.
Selected References
[1] Battiston, S., Mandel, A., Monasterolo, I., Schütze, F., & Visentin, G. (2017). A climate stress-test of the financial system. Nature Climate Change, 7(4), 283–288.
[2] Bolton, P., Despres, M., Pereira da Silva, L. A., Samama, F., & Svartzman, R. (2020). The Green Swan: Central banking and financial stability in the age of climate change. Bank for International Settlements (BIS).
[3] Embrechts, P., Klüppelberg, C., & Mikosch, T. (1997). Modelling Extremal Events for Insurance and Finance. Springer Berlin Heidelberg.
[4] McNeil, A. J., Frey, R., & Embrechts, P. (2015). Quantitative Risk Management: Concepts, Techniques and Tools (2nd ed.). Princeton University Press.

Climate Change Risk and Innovative Finance Insurance Mechanisms

Climate change risk is increasingly associated with a growing frequency and severity of extreme weather events, causing the unsustainable increase of insurance premiums. This reinforces the need for innovative risk transfer mechanisms and mitigation strategies. A notable trend is the shift from traditional ex‑post compensation models to hybrid insurance mechanisms, such as resilience bonds, which combine insurance protection with premium discounts to encourage the adoption of risk reduction measures. Resilience bonds belong to the broader category of environmental impact bonds and are designed to transfer risk to financial markets. These instruments provide investors with a risk premium if a risk reduction project is successfully implemented, whereas they only reimburse a fraction of the investment in the event of a catastrophe, which in our work is defined by exceeding a loss threshold M. Unlike catastrophe bonds, resilience bonds not only provide financial protection against extreme events but also support risk mitigation measures.
In our framework, governmental authorities can refund insurance companies for the policy losses caused by discounts to virtuous policyholders (i.e., firms adhering in each period to green technology) and/or offer guarantees for the regulatory capital required to insurers. In detail, we imagine the dynamics to be defined by the action taken at discrete times t by policyholders, insurers and financial investors. Specifically, policyholders decide whether to adopt the green technology in the next period, comparing the costs they have to bear with the granted premium discount; insurers fix the entity of the discount, which is proportional to the amount saved in the cat bond interest payment if a risk reduction has occurred; investors, finally, ask for a resilience bond interest rate depending on the probability of a catastrophe occurring in the following period.
After discussing the mathematical functions describing the above choices, we study the global dynamics of the consequent three‑dimensional system, taking place in a suitable trapezoidal prism. We find that there always exists a virtuous (optimal) equilibrium, corresponding to the full achievement of the green transition, with a minimum resilience bond interest rate paid and a maximum discount applied to the totality of the firms adhering to the green transition project. Moreover, there may exist a generically even number of equilibria inside the prism: we consider the cases when such equilibria are zero or two. If the latter case occurs, the prism is divided by an invariant surface into two regions. In one of them, containing the internal equilibria, all the trajectories converge to some sub‑optimal attractor, where the full transition is not achieved and the interest rate is not minimum. Vice versa, in the so‑called green transition region, we prove that all the trajectories converge to the virtuous equilibrium. In this way, policymakers would have a very convenient means to check if the initial conditions are such as to guarantee the success of the resilience bond project for a green transition.
In particular, we illustrate the possible outcomes of our model taking as a reference the case of flood risk mitigation in Italy. Specifically, we conduct a sensitivity analysis, observing that, at the starting point of the resilience bond project, the variables given by the existing green policyholders ratio and the current bond interest rate are not determined by the insurers, who instead can decide the initial and the maximum intensities of discount. Hence, numerical simulations show how the shape and size of the green transition region are modified by such choices.

Explaining Return Sensitivity to Monetary Variables in ESG and Traditional Indices: Evidence from Germany, Spain, France and Italy

The rapid growth of sustainable finance has intensified academic interest in whether equity indices that incorporate environmental, social and governance (ESG) criteria respond differently to macro‑financial dynamics than conventional benchmarks do. Despite this growing attention, there is still limited and often inconclusive empirical evidence on how ESG and non‑ESG indices react to monetary policy variables, particularly nominal and real interest rates and inflation expectations. Against a backdrop of heightened volatility caused by the effects of the Coronavirus (Covid‑19) crisis, the inflationary cycle in Europe and the European Central Bank’s move towards normalising monetary policy, this study provides a comprehensive cross‑country analysis of how different equity markets react to these key macroeconomic drivers.
The objective of this analysis is to examine whether the integration of Environmental, Social and Governance (ESG) criteria affects the responsiveness of stock markets in Germany, Spain, France and Italy to changes in nominal and real interest rates, as well as inflation expectations. Furthermore, the analysis investigates whether these sensitivities differ depending on market conditions (i.e. bearish, normal or bullish) and whether there are differences between the pre‑ and post‑Covid‑19 periods, taking into account the subsequent period of economic disruption and policy adjustments.
The study uses weekly data from May 2017 to November 2025 (445 observations) to consider MSCI conventional and ESG indices, complemented by interest rate data from the European Central Bank and inflation expectations derived from the EU breakeven inflation rate. After testing for stationarity, all models are estimated using Quantile Regression (QR) across the 0.1–0.9 quantiles. This methodology is particularly well‑suited to financial return series, which often exhibit non‑normality, heavy tails and asymmetry. It also allows for a richer characterisation of the impact of macro‑financial variables on returns across the entire distribution of outcomes.
The results show that the performance of the European market (both ESG and non‑ESG) is the dominant driver of national equity returns in all countries, quantiles and periods. This confirms the strong financial integration across the euro area. In contrast, the influence of monetary variables is weaker and more inconsistent. Real interest rates display the greatest explanatory power, particularly in conventional indices and in the lower quantiles, suggesting greater sensitivity in adverse market conditions. Nominal interest rates have limited but occasionally significant effects, while inflation expectations only matter in extreme quantiles, typically in high‑stress or high‑growth scenarios.
A key contribution of the study is the consistent evidence showing that ESG indices are less sensitive to monetary policy shocks than conventional indices. ESG indices exhibit fewer significant coefficients, smaller effect magnitudes and more stable responses across quantiles, which suggests that they are more resilient. These findings are consistent with literature that highlights ESG portfolios as tools for risk mitigation and long‑term stability. Sensitivity patterns also differ across countries: Germany and Spain exhibit greater heterogeneity, whereas France and Italy present more uniform relationships across quantiles and index types.
Overall, this research shows that the interaction between macroeconomic variables and equity markets is inherently asymmetric and varies depending on market conditions and the country in question. The findings emphasise the importance of quantile‑based methodologies in capturing non‑linearities and tail behaviour, and highlight how integrating Environmental, Social and Governance (ESG) factors into investment decisions may provide an added layer of resilience to shocks associated with monetary policy and inflation dynamics.
Keywords
ESG investing; Interest rates; Inflation expectations; Quantile regression; European stock markets; Sustainable finance; Market sensitivity; Financial integration; Monetary policy; Risk asymmetry
Acknowledgements and Funding Statement
This work was supported by the Spanish Ministerio de Ciencia e Innovación (PID2021‑128829NB‑100) funded by MCIN/AEI/10.13039/501100011033 and by “ERDF A way of making Europe”, as well as the Spanish Junta de Comunidades de Castilla‑La Mancha (SBPLY/21/180501/000086) and the Spanish Universidad de Castilla‑La Mancha (2025‑GRIN‑38406), both of which were co‑financed with ERDF funds.

Key Determinants of Reverse Mortgage Adoption in Spain: Evidence from PLS-SEM and Necessary Condition Analysis

This work investigates the behavioural determinants shaping the intention to adopt reverse mortgages (RMs) in Spain, drawing on an extended version of the Unified Theory of Acceptance and Use of Technology (UTAUT). The framework incorporates both core UTAUT constructs (performance expectancy (PE), effort expectancy (EE), and social influence (SI)) and contextual variables relevant to older adults’ financial behaviour, namely bequest motive (BM), sense of place attachment (SPA), and perceived risk (Rk). By combining Partial Least Squares Structural Equation Modelling (PLS-SEM) with Necessary Condition Analysis (NCA), the study provides a dual analytical perspective that distinguishes between factors that are sufficient to increase RM acceptance and those that constitute indispensable preconditions for it.
The PLS-SEM results reveal that three variables (PE, SI, and BM) play a statistically significant role in explaining the intention to use (IU) RMs. PE emerges as a strong positive predictor, underscoring the importance of individuals’ beliefs about the benefits and financial value of RMs. SI also contributes positively, suggesting that social pressure can facilitate acceptance. In contrast, BM exerts a negative effect, indicating that strong preferences for leaving an inheritance act as a barrier to the adoption of RMs. Neither EE, SPA, nor Rk show significant explanatory power within the sufficiency‑oriented PLS-SEM framework.
The NCA results complement these findings by identifying PE as a robust necessary condition for high levels of IU. Without a favourable evaluation of the product’s perceived usefulness, it is not possible to achieve strong adoption intentions, regardless of the levels of other behavioural drivers. EE, although non‑significant in the PLS-SEM model, appears as a necessary but weak condition. This implies that RMs must meet a minimal threshold of perceived ease of use; however, surpassing this threshold does not translate into additional gains in acceptance. SI, while relevant in the sufficiency perspective, does not qualify as a strictly required condition for high IU. Similarly, SPA and Rk are not identified as necessary constraints within the NCA framework. BM acts as a significant barrier in the PLS-SEM analysis, although its bottleneck effect is only visible at very high levels of IU.
Overall, this study enhances the understanding of the key behavioural determinants underlying the acceptance of RMs in Spain. By integrating sufficiency and necessity analyses, it offers nuanced insights for policymakers, financial institutions, and practitioners seeking to promote informed, transparent, and responsible diffusion of RMs among older adults.

Gender as a tool for diversification

This study investigates the impact of managerial gender on portfolio risk, based on the assumption that women are more risk averse than their male counterparts. The research focuses on a key question: Does this risk aversion manifest itself in the stock marketperformance of firms that are majority-led by women? To answer this question, we constructed two distinct portfolios based on thegender of top managers: one composed of predominantly female- led companies and one composed of exclusively male-led companies. The selection of companies was based on MSCI’s annual global gender diversity reports, and the analysis was conducted over a sample period from 2018 to 2022. Portfolio risk was assessed using the value-at-risk (VaR) metric, with dynamic optimisation procedures applied through an AR(1)-EGARCH(1,1) model to minimise variance. Our results reveal that portfolios composed of predominantly female-managed companies exhibit lower risk, as measured by VaR, compared to portfolios of male-only managed companies.

Value-for-Money for Life Policyholders: An Actuarial KPIs Framework

The value created for the individual by a life insurance contract has received regulatory and market attention in recent years. In particular, the POG (Product Oversight and Governance) discipline in Europe requires insurers to compare the price in relation to benefits, including in the latter not only their monetary amount, but also their qualitative features, as well as taking into account the individual’s needs and expectations.
Regulation focuses, in particular, on the investment opportunity offered by a life insurance contract; for instance, EIOPA has provided detailed directions for the Value‑for‑Money of Insurance‑Based Investment Products (the so‑called IBIPs), in which the protection against lifetime events plays a minor role in respect of the investment component. In this way, the suggested assessments are very close to those suited to measuring the performance of financial investments, with limited attention paid to the implications of the mutuality process.
When considering life insurance products primarily aimed at providing protection against either early death (i.e., term insurance) or longevity (i.e., annuities), the mutuality component cannot be disregarded in the assessment of the value to the individual. It is necessary, in this case, to use actuarial metrics, whose understanding requires education that may not be part of individuals’ common cultural background or general knowledge. It is worth making an effort to identify metrics that favour a pragmatic and (as much as possible) intuitive interpretation. Moreover, these metrics should enable individuals to more clearly understand their risk exposure and the value of taking out coverage.
This research focuses on the longevity risk to which an individual is exposed after retirement, a risk that is long‑term in nature and is generally poorly perceived by individuals. Our aim is to develop a range of indices, grounded in rigorous actuarial valuation, capable of highlighting various perspectives of longevity risk, on one hand, and longevity guarantees, on the other. We consider a number of post‑retirement benefits offering different longevity‑guarantee structures, including standard annuities, multi‑state annuities, longevity‑linked annuities, and tontine annuities.

Marco Corazza

Martina Nardon

Input sensitivity and relevance in Neural Network–based option pricing

In this contribution, we compare neural networks (NNs) trained on simulated data to replicate the Black–Scholes model [Black and Scholes, 1973], which serves as a benchmark, with a fully data‑driven NN approach for pricing European call options. NN‑based models offer advantages over traditional parametric approaches: they do not rely on restrictive assumptions, require no complex derivations, and exhibit high flexibility and adaptability to data patterns.
NNs have been widely applied across numerous areas of finance, and option pricing is no exception, where they have emerged as a compelling alternative pricing methodology. The closed‑form solution provided by the Black‑Scholes model is known to exhibit systematic biases when compared with observed market prices, largely due to the restrictive assumptions underlying the model. Financial data are typically non‑stationary and characterised by high dimensionality and complexity [Huang et al., 2020], exhibiting evolving autocorrelation structures and time‑varying volatility [Huang et al., 2003]. Moreover, the pronounced non‑linearity of option prices further complicates their estimation. In this context, NNs offer several distinct advantages over conventional parametric models. They do not require stringent assumptions, nor do they rely on analytically derived pricing formulas, as they are universal function approximators [Hornik et al., 1989]. As non‑parametric and adaptive models, NNs are inherently capable of responding to structural changes in data and uncovering complex, non‑obvious relationships among variables. Furthermore, their flexibility makes them particularly well suited for modelling real‑world phenomena in settings where empirical observations are available but the underlying relationships are unknown or difficult to test.
In line with these premises, this contribution pursues several interrelated objectives. First, we compare the pricing performance of a Multi‑Layer Perceptron (MLP) trained on simulated data generated under the Black–Scholes framework with that of an MLP sharing the same architecture but trained exclusively on real market data, focusing on European call options written on the S&P 500 Index in a non‑dividend‑paying setting. Second, we examine and compare the option Greeks implied by the two NN models in order to assess their ability to capture economically meaningful sensitivities. Finally, we aim to partially “open the black box” of NN‑based pricing models by conducting a relevance analysis of the input variables, with the goal of identifying the most influential factors driving option prices.

Italian Default Rates Forecasting: Enhancing RR+–LSTM Through K-PCA and Rank-Correlation Design Choices

The default rate dynamics observed within a given economy are commonly interpreted through the concept of the business cycle, which accounts for the fluctuating behavior of the economy, featuring alternating expansion and contraction phases. The majority of models developed to describe a business cycle use macroeconomic and financial variables as risk drivers to forecast the timing and severity of the next recession phase—namely, the next massive increase in an economy’s default rates. Conversely, the RecessionRisk+ (RR+) model, a reduced‑form latent‑factor model proposed in 2024, posits that the existence of a business cycle is sufficient to predict recession phases within a simple autoregressive framework, thereby considering only the default rate time series.
The predictivity of the model, applied to forecasting industry‑wise Italian default rates, has been dramatically improved in 2025 by coupling it with artificial neural network (ANN) models, with the most accurate results obtained using the well‑known Long Short‑Term Memory (LSTM) networks. As is often the case in problems involving ANNs, the choice of the ANN model and its hyperparameters is made empirically, without any proof that the identified architecture is optimal, regardless of the chosen optimality criterion.
This talk presents the progress made to date in determining the most effective ANN configuration to enhance the predictive performance of RR+ when applied to Italian default rates. In particular, the introduction of rank‑correlation criteria in the feature‑selection layer and the adoption of Kernel Principal Component Analysis (K‑PCA) in the pre‑processing layer—replacing the PCA and auto‑encoder techniques considered in previous work on this topic—improve the average forecast accuracy across Italian industries, easing the application of the proposed framework to practical problems.

A proposal for a quantitative definition of indirect algorithmic discrimination in insurance pricing

The progressive automation of insurance risk pricing, with the application of machine learning methodologies, particularly in the P&C sector, is raising awareness of the potential discrimination of risk subsets by insurance premium calculation algorithms. In this regard, it is important to distinguish the discriminatory nature of certain variables (gender, race, religion, etc.) that may be excluded from consideration by law. However, attention must be paid to the possibility that the effects of these variables may be indirectly included in the calculation due to their interaction with other variables used in pricing.
The tools for analyzing these indirect effects are multiple regression (ANOVA method) and in‑depth analysis using the Shapley index, which in this work will be applied to real insurance datasets. Specifically, regarding the use of the Shapley index to measure indirect interaction effects, reference is made to the papers of Borgonovo et al. (2019, 2024) and the references contained therein.
It will be highlighted how the role of dependence between the variables used in pricing plays a crucial role in the potential manifestation of very distant price levels, and how smoothing mechanisms can or should be created to reduce these gaps by implementing principles of solidarity between risk groups that form the basis of insurance risk management. By measuring the solidarity triggered by a pricing scheme, a possible definition of an indirect discrimination measure will be proposed, based on the results obtainable through the aforementioned statistical tools, which could potentially have applicability in regulatory frameworks.

Main References
Borgonovo, E., & Rabitti, G. (2019). A Shapley‑Owen index for interaction quantification. SIAM/ASA Journal on Uncertainty Quantification, 7(3), 1060–1075.
Borgonovo, E., Plischke, E., & Rabitti, G. (2024). The many Shapley values for explainable artificial intelligence: A sensitivity analysis perspective. European Journal of Operational Research, 318(3), 911–926.

Investigating expected lifetime utility under the Life Care Reverse Mortgage insurance scheme

This paper investigates Life Care Reverse Mortgage (LCRMs), a bundled insurance product that integrates the lifetime annuity provided by a Reverse Mortgage with long‑term care (LTC) coverage. This product is designed to address the joint challenges of longevity risk and health deterioration faced by elderly homeowners. By linking housing equity release to health‑contingent benefits, LCRMs provide state‑dependent cash flows that increase upon the borrower’s transition from functional ability to functional dependence. The analysis adopts the borrower’s perspective and examines how access to an LCRM contract affects consumption behavior and welfare over the retirement life‑cycle.
The contract is modelled in discrete time within a multi‑state health framework, in which the borrower may become functionally dependent or deceased. Installments are determined at inception through an actuarially fair pricing condition that accounts for longevity risk, health transitions, interest‑rate and house‑price dynamics, and the embedded no‑negative‑equity guarantee. The valuation framework jointly models stochastic house prices and interest rates, while mortality and morbidity dynamics are calibrated using demographic projections for the Italian population.
To assess the economic implications for retirees, the paper develops a stochastic life‑cycle model in which a retired homeowner maximizes expected lifetime utility from consumption and bequest under constant relative risk aversion preferences. The borrower faces uncertainty related to survival, health status, medical and housing‑maintenance expenditures, and financial‑market conditions. Within this framework, the LCRM delivers a stream of payments while the borrower remains healthy and a higher, health‑contingent benefit upon the onset of functional dependence. The optimization problem is solved via dynamic programming, allowing consumption and wealth dynamics to adjust endogenously to health and asset evolution.
An important component of the analysis concerns the welfare evaluation of LCRMs through the borrower’s willingness to pay (WTP). Welfare gains are assessed by comparing the maximum expected lifetime utility achieved when the LCRM is available with that obtained in a counterfactual scenario in which the borrower relies solely on liquid wealth and housing without access to the contract. WTP is defined as the additional initial liquid wealth required to offset the utility loss associated with the absence of the LCRM, thereby providing a monetary measure of the value of health‑contingent housing‑equity release within a life‑cycle setting. This approach facilitates an economically interpretable assessment of the trade‑offs faced by retirees. The analysis further examines how WTP responds to variations in preference parameters, such as risk aversion and bequest motives, as well as to alternative mortality scenarios.
These findings contribute to the actuarial literature on housing‑based retirement products and highlight the relevance of integrating health‑contingent features into decumulation strategies in ageing societies.

Optimal Cancellation of Policies with Subsidies

We consider an optimal stopping problem for an insurer deciding when to cancel insurance policies in the presence of regulatory subsidies and liquidation costs, under incomplete information about portfolio profitability. The observable value of the policy book follows an arithmetic Brownian motion whose drift depends on an unobserved parameter governing the underlying risk profile. The insurer updates its beliefs dynamically through filtering based on observed performance.
Building on Gerber’s optimal cancellation model, we incorporate structural regulatory benefits that may be granted during adverse market conditions, as well as a one‑time exit cost upon liquidation. We characterize the resulting value function and optimal stopping boundary as functions of the posterior belief and subsidy structure, and examine how regulatory incentives and learning interact to delay or accelerate market exit. The framework provides quantitative insight into regulatory interventions, capital allocation, and optimal exit decisions in insurance markets subject to uncertainty.
This is joint work with Albert Cohen (Michigan State University).

Asymptotics for aggregated interdependent multivariate subexponential claims with general investment returns

This paper investigates asymptotic estimates for the entrance probability of the
discounted aggregate claim vector from a multivariate renewal risk model into some rare set. We provide asymptotic results for the entrance probability on both finite and infinite time horizons under various assumptions regarding the stochastic price process of the investment portfolio, the distribution class of claim vectors, and the dependence structure among the claim vectors. We note that the main results extend beyond the class of multivariate regular variation. Furthermore, we introduce two dependence structures to model the dependence among the claim vectors. The immediate consequence of the main results is the asymptotic estimates of the ruin probabilities on finite and infinite time horizons.

Forecasting Mortality-at-Risk using Autoregressive with Conditional Heteroscedasticity and its Integration with Support Vector Regression

Mortality risk is a major concern for insurance industries, particularly life insurance and pension funds, due to the potential impact of declining mortality rates on longevity risk and capital adequacy. Accurate modeling and forecasting of mortality dynamics are therefore essential for risk assessment and reserve planning. This paper considers the forecasting of Mortality-at-Risk (MaR), a tail risk measure designed to quantify the maximum tolerable fall in mortality rates over a fixed future horizon at a given confidence level.
Mortality dynamics are modeled using autoregressive specifications applied to changes in log mortality rates, allowing for temporal dependence in the conditional mean. To capture timevarying uncertainty in mortality fluctuations, conditional heteroscedasticity is incorporated into the innovation process. Specifically, volatility dynamics are accommodated using Autoregressive Conditional Heteroscedastic (ARCH) models as well as stochastic volatility autoregressive (SVAR) models. The ARCH specification assumes observable volatility driven by past shocks, while the SVAR specification introduces latent volatility evolving according to its own stochastic process. These alternative formulations allow for a comparative assessment of observable and unobservable volatility effects in mortality modeling.
Within this framework, MaR is formulated using a Value-at-Risk–type approach based on the conditional or unconditional distribution of future mortality changes, depending on the underlying volatility specification. MaR forecasts are obtained through two approaches. The first is an estimative forecast, constructed by substituting estimated model parameters into the MaR expression. The second is an improved forecast, which applies a correction to account for finitesample coverage distortions arising from parameter estimation. The resulting MaR calculations differ substantially across ARCH and SVAR models due to their distinct volatility structures and distributional assumptions.
In addition, a hybrid Autoregressive–Support Vector Regression (AR–SVR) specification is examined for mortality rate forecasting. In this setting, the autoregressive component captures linear temporal dependence, while the SVR component accommodates potential nonlinear and heteroscedastic patterns that may not be fully captured by parametric time-series models.
An empirical application based on United States mortality data illustrates the behavior of estimative and improved MaR forecasts under alternative volatility specifications. The proposed modeling framework is intended for the assessment of tail mortality risk in actuarial applications, particularly in life insurance and pension fund modeling.
Keywords: Autoregressive, Life insurance, Pension fund, Risk measures, Time series forecasting, Stochastic volatility

Core-periphery analysis of risk dependence among cryptocurrencies

The paper aims at analysing the structure of dependence of risk among 221 cryptocurrencies, whose values and volumes are available through YahooFinance, starting from January 1st, 2022 until November 21st, 2025. A preliminary analysis has been performed for detecting time breakpoints. The tests Yao’s Change Point, Variance‑based change‑point, and the Binary Segmentation have been compared. The latter test gives the most stable and chronologically ordered breakpoints. According to its result, the time frame has been split into four parts, and the subsequent analysis of tail risk has been performed on each of them.
Models of tail risk have been calibrated on the dataset (historical VaR, parametric normal, EWMA, AEWMA). For each selected time interval, the data in the first part of the interval have been used as the historical dataset, and the computation of VaR has been performed through a rolling window until the last day of the selected time interval. We perform a comparison among the models through VaR backtesting tests: the Violation Indicator Test, Kupiec’s Unconditional Coverage Test, Christoffersen’s Independence Test, and the Conditional Coverage Test. The EWMA model is the best performer, so it has been used as the basis for the computation of CoVaR.
Then, a network is set up: each node represents a cryptocurrency, and the weight of the links is given by the value of their CoVaR. Since the network is directed and weighted, we apply a core–periphery algorithm suitable for directed weighted networks. At each selected time, the core highlights the cryptocurrencies that are strongly dependent, in contrast to those in the periphery, and allows for a visual representation. Specific dates are selected to show the evolution and permanence of the nodes either in the core or the periphery. The analysis points out the different roles of the cryptocurrencies in the overall risk landscape.

References
Alexander, C., & Dakos, M. (2023). Assessing the accuracy of exponentially weighted moving average models for Value‑at‑Risk and Expected Shortfall of crypto portfolios. Quantitative Finance, 23(3), 393–427. https://doi.org/10.1080/14697688.2022.2159505
Chen, Y., Bilokon, P., Hales, C., & Kerr, L. (2025). Real‑Time VaR Calculations for Crypto Derivatives in kdb+/q. The Journal of FinTech. https://doi.org/10.1142/S270510992550004X
Christoffersen, P. (1998). Evaluating interval forecasts. International Economic Review, 39(4), 841–862.
Elliott, A., Chiu, A., Bazzi, M., Reinert, G., & Cucuringu, M. (2020). Core–Periphery structure in directed networks. Proc. R. Soc. A.
Killick, R., Fearnhead, P., & Eckley, I. A. (2012). Optimal detection of changepoints with a linear computational cost. Journal of the American Statistical Association, 107(500), 1590–1598.
Kupiec, P. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73–84.
Nguyen, A. P. N., Crane, M., & Bezbradica, M. (2023). Cryptocurrency Volatility Index: An efficient way to predict the future CVI. In L. Longo & R. O’Reilly (Eds.), Artificial Intelligence and Cognitive Science (AICS 2022), Communications in Computer and Information Science (Vol. 1662). Springer, Cham. https://doi.org/10.1007/978-3-031-26438-2_28
Scott, A., & Knott, M. (1974). A cluster analysis method for grouping means in the analysis of variance. Biometrics, 30, 507–512.
Smith, M. (2023). ITNr: Analysis of the International Trade Network (R package version 0.7.0). CRAN. https://cran.r-project.org/web/packages/ITNr
Yao, Y.-C. (1988). Estimating the number of change‑points via Schwarz’ criterion. Statistics & Probability Letters, 6(3), 181–189.

A simple metric for measuring contagion-based name concentration risk in credit portfolios

Financial institutions managing credit portfolios face the risk of substantial aggregate losses due to name concentration, where a small number of obligors constitute a significant portion of the total exposure. This risk is typically managed by identifying and limiting large exposures to single obligors. The Herfindahl–Hirschmann Index (HHI) is commonly used for measuring name concentration, but it accounts only for direct concentration risk through obligor‑level exposures [see e.g. Lütkebohmert, 2008].
Crucially, however, the HHI does not capture indirect concentration risk that arises due to default contagion effects, where the default of one obligor causes a direct increase in the default risk of others. This propagation of default contagion is empirically observed in various dependence structures, ranging from simple parent–subsidiary relationships and hierarchical government structures to complex supply chain networks. The latter is increasingly relevant for financial institutions given the recent rise in supply chain disruptions driven by geopolitical instability, trade wars, and volatile energy prices. Importantly, even an obligor with a small direct exposure can constitute material concentration risk if it contributes strongly to default contagion. For this reason, the identification and quantification of contagion‑based concentration risk have received increased attention from regulatory authorities [European Banking Authority, 2022].
In this talk, we showcase that network‑based default contagion can significantly impact tail‑based risk measures, such as Value‑at‑Risk and Expected Shortfall. To address this, we introduce a new metric that generalizes the HHI for direct name concentration by additionally measuring indirect contagion‑based concentration risk. Intuitively, the metric depends on the exposure sizes of obligors, the quantity of network connections, and the strength of those connections. We show that this generalization has an intuitive theoretical motivation and that it preserves attractive properties. Furthermore, we highlight that the metric can be used to determine useful “rule‑of‑thumb” materiality thresholds. We validate this approach by comparing the simple metric against a more complex structural contagion model [Schiphorst et al., 2025], demonstrating that it provides a useful approximation for various credit portfolios and network structures. Overall, the generalized HHI metric offers a simple tool for measuring concentration risk, allowing risk managers to easily identify vulnerable (sub)portfolios and to quantify risk‑sensitive exposure limits.

Resilience of Spatially Dependent Risk Pools

Pooling is a central technique underpinning the management of insurance risks. Pooling promotes stability and diversification of a portfolio of risks. Yet, this technique becomes less effective, and can even fail, in the presence of dependence. This presentation considers spatial dependence, a special type of relationship that is motivated by climate-related risks including hurricanes, earthquakes, and so forth. Our interest is in the “resilience” of pools that we think of as the ability to anticipate, prepare for, respond to, and recover from the impacts of spatially dependent outcomes while maintaining the essential functions and structure of a pool.
This presentation will frame the problem based on first principles, beginning with a broad set of strategies that one could use to shape an appropriate pool. Next, we review data models underpinning such strategies, focusing on spatial-temporal statistical models that are most useful in this insurance context. Finally, we demonstrate how to utilize constrained optimization tools that deliver the least risky portfolio given a budget constraint, using this tool to show how dependence affects both underwriting and other requirements to enter the pool as well as risk transfer opportunities to manage the pool’s stability.

LEDecomp. Life Expectancy Decomposition R package

Differences in life expectancy across populations are often analyzed using age-specific decomposition methods, yet these approaches remain scattered and inconsistently applied. This paper introduces LEdecomp, a user-friendly tool available as an R package, designed to standardize decomposition analyses by age group and cause of death. We implement widely used decomposition techniques and extend them to incorporate sensitivity analyses and cause-specific mortality analysis. Our package collects the principal decomposition methods in the literature, providing a systematic comparison and highlighting their key properties. Empirical illustrations from the US population between 2010 and 2020 demonstrate how these approaches explain disparities and temporal changes in life expectancy. All methods are fully operational within LEdecomp, enabling researchers and practitioners to perform robust analyses without advanced programming skills. By integrating decomposition analysis into a single R package (LEdecomp), it becomes easier to carry out evidence-based interventions aimed at reducing health disparities and improving population longevity.
Keywords: Life expectancy; Mortality; Decomposition; Demography.

Embedding the Gompertz Mortality Law within an Affine Diffusion Framework: The Long-Term Dynamics of Age-Specific Death Rates

Longevity is inherently dynamic, and understanding its temporal evolution is essential for actuarial science, longevity risk assessment, and the valuation of life‑contingent liabilities. This paper proposes a mathematical approach that embeds age‑specific mortality dynamics within an affine diffusion setting. Specifically, we model the time evolution of the logged central death rate at fixed ages using an Ornstein–Uhlenbeck (OU) process, thus capturing stochastic fluctuations together with mean‑reverting behaviour. The framework allows us to investigate how mortality levels evolve over time and how the pace of these changes contributes to long‑run patterns across ages.
Our central research question concerns the asymptotic shape of age‑specific central death rates: do their long‑run expectations remain consistent with the exponential increase with age implied by the Gompertz law? To address this question, we construct a stochastic differential equation whose drift incorporates the linear dependence on age characteristic of Gompertz mortality. By embedding the Gompertz slope directly in the drift term, we ensure that this structural age‑pattern influences both the stationary distribution and the long‑term expectation of the stochastic process. This provides a theoretically transparent way to assess whether an affine diffusion is compatible with classical mortality regularities.
Empirically, we estimate a system of OU processes simultaneously for all integer ages in the interval 50–95. This joint calibration yields age‑specific drift and diffusion coefficients and allows us to derive the stationary mean and variance for each age. Using these quantities, we obtain the implied long‑run expectation of the central death rate curve and examine its behaviour across ages. The resulting trajectory provides direct evidence on whether the exponential‑in‑age structure persists in the affine diffusion context or whether deviations emerge due to the stochastic dynamics.
Our results contribute to the understanding of long‑term mortality modelling by clarifying the compatibility between classical Gompertz patterns and modern affine stochastic representations of longevity

Single-population mortality models based on Linear Hypercubes

Exploiting the parallelism between mortality and credit risk, we propose a mortality modeling approach based on the Linear Hypercube Model of [1], originally developed for credit derivatives. This approach is particularly attractive due to its analytical tractability while always keeping mortality intensities nonnegative. It can be applied to a variety of problems, such as the explicit computation of standard actuarial quantities, the pricing of guaranteed annuity options, and the estimation of multi‑population mortality surfaces. Estimation can be performed using quasi‑maximum likelihood in conjunction with the well‑known Kalman Filter. We conduct extensive experiments to evaluate the adequacy of the proposed approach for fitting mortality surfaces.
Keywords: Mortality model; multi‑population; polynomial diffusion; life insurance; guaranteed annuity options.

References
[1] Ackerer, D., & Filipović, D. (2020). Linear credit risk models. Finance and Stochastics, 24(1), 169–214.

Tail-Risk-Aware Dynamic Allocation for Leveraged Exchange-Traded Funds

This study develops and evaluates a comprehensive framework for managing Leveraged Exchange-Traded Funds (ETFs)-specifically SPDR S&P 500 (1x), ProShares Ultra (2x), and UltraPro (3x)-under both passive buy-and-hold and dynamic risk-managed strategies. Leveraged ETFs are known to exhibit volatility decay, negative convexity, and amplified tail risk due to daily leverage resetting, making traditional passive investment approaches particularly vulnerable during periods of market stress. This paper addresses these challenges by integrating advanced return modelling, tail-risk measurement, and dynamic exposure control within a unified and empirically validated framework.
To accurately capture skewness, kurtosis, and extreme losses, the analysis employs advanced return distributions, including Skewed-t, Generalized Pareto, Gram-Charlier (GC(4)), and a modified Gram-Charlier specification (mGC(4)). A broad set of performance and risk metrics is considered, encompassing Sharpe, Sortino, and Omega ratios, as well as drawdowns, Conditional Drawdown at Risk (CDaR), Value at Risk (VaR), Expected Shortfall (ES), and Median Shortfall (MS). Strategy robustness is further assessed through extensive Monte Carlo simulations and stress testing, allowing the evaluation of performance and risk under adverse scenarios not observed historically.
The proposed dynamic strategy allocates exposure between the risky asset and a risk-free asset using a disciplined, rule-based mechanism. Risk information is synthesized from historical volatility, the VIX, and Median Shortfall using Principal Component Analysis (PCA) to construct a synthetic tail-risk regime indicator. Exposure is defined as an inverse linear function of combined risk and is subject to conservative operational constraints, including exposure caps, transaction cost modelling, and a firewall mechanism that reduces exposure during severe drawdowns or extreme systemic risk regimes. This adaptive design aims to preserve capital during stress periods while maintaining participation in favourable market conditions.
Empirical results show that the dynamic strategy consistently outperforms passive management across all leverage levels and distributional assumptions. The dynamic approach significantly reduces volatility, drawdowns, tail-risk measures, and leverage-induced negative convexity, often achieving risk profiles comparable to unleveraged ETFs. Risk-adjusted performance improves markedly, with statistically significant gains in Sharpe, Sortino, and Omega ratios, and positive Jensen’s alpha ranging between 2.5% and 7%. Formal Jobson-Korkie and Lo-Wang tests confirm the statistical significance of these improvements. From an extreme-risk perspective, the dynamic strategy reduces VaR, ES, and MS exceptions by 70% to 100% and lowers effective beta to the range of 0.27-0.39, resulting in approximately half the annualized volatility of passive leveraged strategies.
This study contributes to the literature by accurately modelling tail risk in leveraged ETFs using advanced return distributions and by integrating multiple complementary risk measures into a coherent dynamic allocation rule. The results demonstrate the consistent superiority of dynamic over passive management across leverage levels and market regimes, while providing a practical framework that mitigates negative convexity and complies with regulatory and prudential risk constraints. The results have clear implications for investors, fund managers, and regulators, and suggest that dynamic risk management is essential for efficient and sustainable investment in leveraged equity products.
Keywords: Leveraged ETFs; Dynamic Portfolio Management; Volatility Targeting; Tail Risk; EGARCH models; Risk-Adjusted Performance

Mapping Risk in Italian Real Estate Market: A Feature-Based Clustering Approach

Assessing real estate risks and vulnerabilities may provide significant support to financial institutions and policymakers, as housing market imbalances may amplify macroeconomic shocks, affect credit conditions, and pose threats to financial stability. However, these risks necessitate tailored monitoring approaches, as real estate dynamics are inherently heterogeneous across space and strongly influenced by local economic, demographic, and institutional factors. Ignoring such geographical heterogeneity may lead to an underestimation of vulnerabilities and to ineffective risk management strategies.
In this work, we provide a deeper understanding of the Italian real estate market at the municipal level, offering a fine‑grained perspective. In particular, we explore the riskiness of the Italian market with the aim of identifying homogeneous and stable sub‑markets, so as to provide stakeholders—such as banks, regulators, and investors—with valuable support in terms of risk management performance and portfolio allocation. To present a comprehensive outlook, we apply the k‑means clustering method to multiple features derived from historical Italian municipalities’ house price time series. These features capture different dimensions of market dynamics, including price volatility, persistence, and tail risk behavior. This approach enables us to uncover distinct geographical patterns and to group municipalities with similar risk profiles.
Furthermore, we conduct a comparative analysis that tracks municipality transitions across clusters when alternative feature sets are considered, enabling the identification of potential regime‑switching areas. Our results shed light on the possible instability of features‑based classification, especially when dealing with geographically distributed variables, and underscore the need for multi‑metric robustness in real estate risk assessment.
Overall, our methodology advances traditional housing market analysis by integrating tail risk measures on price levels with cluster mapping, providing a more nuanced and operational framework for monitoring and managing real estate risk. The results enable the categorization of the obtained clusters by analyzing their levels of risk and assigning a rating based on the different risk indicators considered.

Set-valued expectiles as risk measures for multi-currency portfolios

Expectile regions–like depth regions in general–capture the idea of centrality of multivariate distributions. If an order relation is present for the values of random vectors and a decision maker is interested in dominant/best points with respect to this order, centrality is not a useful concept. Therefore, cone expectile sets are introduced which depend on a vector preorder generated by a convex cone. This provides a way of describing and clustering a multivariate distribution/data cloud with respect to an order relation. Fundamental properties of cone expectiles are established including dual representations of both expectile regions and cone expectile sets. It is shown that set-valued sublinear risk measures can be constructed from cone expectile sets in the same way as in the univariate case. Inverse functions of cone expectiles are defined which should be considered as ranking functions related to the initial order relation rather than as depth functions.
References
[1] Artzner P, Delbaen F, Eber JM, Heath D, Coherent measures of risk, Mathematical Finance 9(3), 203-228, 1999
[2] Bellini F, Klar B, Müller A, Rosazza Gianin E, Generalized quantiles as risk measures, Insurance: Mathematics and Economics 54, 41-48, 2014
[3] Cascos I, Ochoa M, Expectile depth: Theory and computation for bivariate data sets, J. Multivar. Anal. 184, 104757, 2021
[4] Hamel AH, Ha TKL, Set-valued expectiles for ordered data analysis, Journal of Multivariate Analysis 208, 105425, 2025
[5] Giorgi E, McNeil AJ, On the computation of multivariate scenario sets for the skew-t and generalized hyperbolic families, Computational Statistics & Data Analysis, 100, 205-220, 2016

A Parametric Quantile-based approach to Premium estimation in count data insurance models

The measurement of the risk premium is fundamental to actuarial pricing; traditionally, this involves decomposing aggregate claims into frequency and severity to obtain a pure premium, which is later adjusted with a safety loading to account for random fluctuations and ensure profitability.
Recent actuarial literature has increasingly embraced quantile‑based premium principles, applying quantile regression to estimate conditional thresholds—mostly within the domain of continuous claim severity. Compared to traditional mean‑based frameworks, these approaches better capture tail behavior and offer a more stable basis for pricing in the presence of skewed or heavy‑tailed loss distributions.
This study moves the actuarial focus from claim severity to claim frequency by proposing a quantile‑based premium principle that operates directly on the number of claims. Quantile regression has traditionally been used to model severity, but several important insurance contexts are fundamentally driven by frequency risk. A prominent example is daily disability benefit insurance, where the benefit is typically structured as a contractually fixed monetary unit multiplied by a discrete quantity—such as the number of days or weeks of inability to work. As a result, a discrete variable becomes the main source of uncertainty.
Extending quantile regression to count data raises methodological issues that have hindered widespread adoption. The discrete nature of count variables makes quantiles difficult to define in a coherent way, due to the presence of jumps in the cumulative distribution function. As a consequence, the classical quantile regression framework, which relies on continuity assumptions, cannot be straightforwardly applied. To overcome the discreteness of the response variable, an artificial degree of smoothness is typically introduced through “jittering,” a technique that adds small random noise to the data to make it approximately continuous. While this allows standard quantile regression algorithms to run, it introduces unwanted randomness, complicates inference, and reduces the interpretability and stability of the resulting estimates. Due to these limitations, quantile regression for count data has seen limited application in actuarial practice.
These limitations highlight the need for quantile models that respect the discrete nature of count data without relying on ad‑hoc transformations. In this direction, we adopt a parametric quantile regression framework for count variables, originally proposed by [1]. This approach defines quantiles directly within a specified discrete distribution, ensuring that the conditional quantile function remains coherent with the underlying probabilistic structure of the data. By modeling the conditional distribution parametrically, the method yields smooth and interpretable quantile curves and avoids the conceptual and practical drawbacks of jittering (see also [2]).
To demonstrate the practical relevance of the approach, we analyze real data from daily disability benefit insurance. Modeling conditional quantiles for different policyholder profiles allows us to characterize not only average outcomes but also extreme claim behaviors. This quantile‑based description feeds directly into the pricing process, enabling the construction of premiums that incorporate both expected and tail risks.
In summary, this study adapts the quantile premium principle to frequency‑driven insurance contracts through a parametric quantile regression model for count data. The method provides a coherent way to incorporate tail risks and policyholder heterogeneity into pricing; its structure naturally supports future extensions, including dynamic models, multivariate counts, and links with capital or solvency metrics.

References
[1] P. Frumento, N. Salvati. Parametric modeling of quantile regression coefficient functions with count data. Statistical Methods & Applications, 30 (2021), 1237–1258.
[2] V. Carcaiso, L. Grilli. Quantile regression for count data: jittering versus regression coefficients modelling in the analysis of credits earned by university students after remote teaching. Statistical Methods & Applications, 32 (2023), 1061–1082.

Modelling Insurance Claim Frequency and Severity with Ridge, Lasso, and Elastic Net Regularization

The increasing sophistication of predictive models in automobile insurance pricing is essential for insurers seeking technical excellence in a highly competitive market. To remain competitive and offer fair and accurate tariffs, companies must continuously invest in updating their modeling techniques and adopting more advanced algorithms. For decades, Generalized Linear Models (GLMs) have been the industry standard due to their solid predictive performance, statistical interpretability, and suitability for regulatory environments. However, the likelihood‑based optimization procedures commonly used in GLMs assign full credibility to the training data, which may impair their ability to generalize effectively to unseen observations. Moreover, the multicollinearity commonly present in insurance datasets for pricing models inflates the variance of the coefficients.
The main objective of this study is to explore regularization techniques, also referred to as penalized models, to improve pricing models in automobile insurance. In particular, Ridge, Lasso, and Elastic Net regularization techniques are investigated as mechanisms to alleviate overfitting—reducing the excessive dependence on training data inherent in classical GLM estimation—and addressing multicollinearity. These approaches aim to preserve the interpretability of GLMs, necessary in a highly regulated insurance sector, while offering a modelling framework that is more suited to insurance pricing data.
The study examines the application of Ridge, Lasso, and Elastic Net regularization within Poisson and Gamma GLMs for modeling claim frequency and severity using simulated datasets and a sample of a real insurance company. To this end, an efficient framework is developed that allows fitting a grid of regularized models in a single step, utilizing both Python and R. The framework enables the user to specify a set of L1 ratios and variables, while automatically selecting the optimal path of penalty hyperparameters (λ) and fitting the models accordingly. This research includes the interpretation of a set of Python‑based visualizations designed to extend the graphical outputs currently available in closed‑source software for GLM analysis.
These plots are used to pursue three goals. The first involves selecting and transforming the original variables via groupings, polygonizations, and interactions to maximize their effectiveness in the model through careful management of the bias–overfitting trade‑off. For this purpose, the isolated effect of the transformed variable in the model is compared with its effect had the variable not been transformed. The second goal is to evaluate the predictive performance of the proposed algorithm. In this process, the predictions generated by the model are averaged within each category of the displayed variable and compared with the corresponding observed averages. Predictions from a second model are then incorporated to achieve the third goal, which is the comparison of the models. Other graphical solutions and key performance indicators are proposed to support the last two objectives.
Leveraging these capabilities, the study demonstrates how introducing Ridge, Lasso, and Elastic Net regularization to the unregularized models—using exactly the same variables—improves predictive performance while maintaining interpretability.

Partially Factorized Variational Inference for claim counts

Overdispersion, which occurs when the variability of claim counts exceeds that predicted by a Poisson model, is typically linked to unobserved heterogeneity across policyholders, reflecting latent differences in underlying risk not captured by observed covariates. It can be naturally accommodated by standard Bayesian mixed Poisson models by introducing latent random effects that induce extra‑Poisson variation, providing a coherent probabilistic representation of risk heterogeneity. While these hierarchical models are highly flexible and capable of representing complex data structures, posterior inference using Markov Chain Monte Carlo (MCMC) methods can become very computationally demanding, particularly when applied to high‑dimensional datasets with a large number of covariates. These computational demands often limit the practical applicability of fully Bayesian approaches in large‑scale modeling, as the required time can make such analyses infeasible.
To address these challenges, we develop Variational Inference (VI) alternatives for a sequence of Poisson mixture models, with particular focus on the Poisson–Gamma (Negative Binomial), Poisson–Lognormal, and Poisson–Inverse Gaussian specifications. We implement both a standard Mean‑Field VI (MFVI) approach and a Partially Factorized VI (PFVI) framework. MFVI assumes full independence among all model parameters, allowing for extremely fast posterior approximation but often at the expense of both accuracy and the ability to capture posterior dependencies inherent in hierarchical models.
Our primary contribution lies in the PFVI formulation, which relaxes the full‑independence assumption by preserving dependence structures within carefully selected parameter groups. This structured approximation more faithfully represents the hierarchical nature of mixed Poisson models while retaining the scalability and efficiency advantages of VI. PFVI enables better uncertainty quantification by capturing posterior correlations between the key model parameters, bridging the gap between computational tractability and inferential fidelity.
We conduct multiple simulation studies to evaluate both the posterior approximation accuracy and the computational efficiency of the proposed methods as the complexity increases, using MCMC as a benchmark to evaluate the results. Finally, the proposed methods are illustrated through a real‑world case study using an open‑access insurance claims dataset.

Credit portfolio losses with climate change factors

We consider the problem of computing risk measures of a credit portfolio via the evaluation of the characteristic function of the loss variable. We propose a new methodology to obtain the characteristic function of the loss distribution when the dependence structure is driven by either the Gaussian or t‑copula model. This new approach relies on a quadrature method based on Shannon wavelets and the cardinal sine function. It works extremely well for the one‑factor and the multi‑factor model when, in the second case, a moderate number of risk factors are considered.
We compare our approach with some of the state‑of‑the‑art methods for performing the same task, and we obtain much better results in terms of execution time and accuracy. As quadrature methods are affected by the curse of dimensionality, we further introduce a simulation approach to evaluate the characteristic function in the case of multi‑factor models with many risk factors. The simulation is based on low‑discrepancy Monte Carlo sequences. A broad set of numerical examples illustrates the efficiency of our methodology.
We conclude our work with a real portfolio where the exposures are taken from the European Investment Bank, and we incorporate climate‑change‑related factors into the analysis. This study highlights the practical relevance of our methodology for assessing credit risk in portfolios exposed to emerging environmental challenges.
Key words: Credit risk; multi-factor models; Gaussian copula; t-copula; Climate change; Shannon
wavelets.

Modelling natural disasters – the need for time-varying loss distributions

Natural catastrophes have become more frequent and severe in recent years, posing a huge threat not only to human lives but also to property and infrastructure. Accurate models for the severity of losses caused by natural catastrophes are necessary to correctly assess risks in pricing insurance and reinsurance contracts or financial instruments such as insurance‑linked securities. In traditional frameworks, loss severities are usually described as independent and identically distributed (iid) random variables. However, with factors such as climate change and increasing urbanization in high‑risk zones, the assumption of a loss distribution not changing over time may no longer hold. As the frequency and magnitude of natural catastrophes exhibit upward trends, neglecting the change of distribution parameters over time can lead to underestimating tail risk and, consequently, the capital needed for solvency or the price of instruments.
This presentation compares two approaches to modelling loss severities. In the first approach, losses are described by a probability distribution with constant parameters, as in the traditional framework. The second approach utilizes Generalized Additive Models for Location, Scale, and Shape (GAMLSS), first introduced in [1]. Use of GAMLSS allows modelling the location and scale parameters of distributions using linear modelling techniques, where time is used as a covariate. This approach enables the probability density function to evolve over time, capturing the upward trends observed in losses caused by natural catastrophes.
We apply this methodology to a PCS index dataset consisting of natural catastrophe insured losses that occurred in the USA between 1985 and 2024. We calibrate a series of models using a variety of probability distributions such as Log‑Normal, Inverse Gaussian, and Weibull. Model selection is performed using the Akaike Information Criterion and the Bayesian Information Criterion. Next, the performance of models in VaR prediction is assessed with Kupiec and Christoffersen tests. Our results demonstrate that models incorporating time‑varying location and scale parameters provide a better fit to the historical data compared to alternatives with constant parameters.

Reference
[1] R. A. Rigby & D. M. Stasinopoulos. Generalized Additive Models for Location, Scale and Shape. Journal of the Royal Statistical Society Series C: Applied Statistics, 54(3), 2005, 507–554.

Quantifying Model Risk in Climate Change Adaptation: A Robust Vine-Copula Approach for Pricing Catastrophe Bonds

The pricing of Catastrophe (Cat) Bonds in the era of climate change faces a fundamental dilemma: the necessity to rely on historical data to model rare events, juxtaposed with the non‑stationary nature of climate signals that renders historical patterns increasingly obsolete. Standard actuarial frameworks typically assume a fixed dependence structure, often Gaussian or Student‑t copulas, between triggering variables (e.g., wind speed, flood depth, and storm surge). However, empirical evidence suggests that extreme weather events exhibit asymmetric tail dependence that traditional parametric copulas fail to capture adequately. Furthermore, the selection of a single “best‑fit” model exposes issuers and investors to significant model risk, particularly when facing Knightian uncertainty regarding the trajectory of climate pathways (e.g., RCP/SSP scenarios).

This paper addresses these challenges by proposing a novel, Robust R‑Vine Copula framework for pricing multi‑peril Cat Bonds. Unlike classical approaches that optimize a single dependence structure, we construct a robustness framework that accounts for both parameter and structural uncertainty. Specifically, we define an ambiguity set of plausible vine‑copula models centered around a nominal reference model, bounded by a statistical‑distance constraint (e.g., Wasserstein distance). Within this ambiguity set, we employ a worst‑case expectation approach (min‑max pricing) to derive no‑arbitrage prices that are resilient to model misspecification.

Methodologically, our approach involves a three‑step process: (1) Modeling marginal distributions using non‑stationary Generalized Extreme Value (GEV) theory with time‑varying covariates linked to climate indices; (2) Utilizing Regular Vine (R‑Vine) copulas to flexibly model high‑dimensional dependence structures, allowing for diverse pair‑copula families in the trees; (3) Quantifying model risk by calculating the spread between the nominal price and the worst‑case price within the ambiguity set.

We apply this framework to a dataset of European windstorm and flood triggers. The empirical results demonstrate that ignoring model uncertainty in the dependence structure leads to a systematic underpricing of the risk premium, particularly in the senior tranches of the bond. Our findings indicate that the “model risk premium,” the additional yield required to compensate for structural uncertainty, can account for up to 15–20% of the total spread in high‑emission scenarios. By explicitly quantifying this risk, our proposed model provides a rigorous tool for Solvency II capital calibration and enhances the transparency of Insurance‑Linked Securities (ILS) markets.
This research contributes to the growing literature on quantitative risk management by bridging the gap between advanced statistical dependence modeling and sustainable‑finance adaptation strategies.
Keywords: Catastrophe Bonds, Robust Vine‑Copula, Model Risk, Ambiguity Sets, Climate Adaptation, Quantitative Risk Management.

A Causal Approach to Forecasting Central Bank Decisions

We propose a causal framework to assist market participants in forecasting cumulative adjustments to the Federal Open Market Committee (FOMC) target rate by integrating market‑implied expectations from the CME FedWatch tool with a broad set of U.S. macroeconomic indicators. In this paper, we make three contributions.
First, we show the existence of a regime shift: at short horizons, market‑implied expectations fully capture the available information, making macroeconomic variables redundant, while at longer horizons, macroeconomic variables provide substantial additional predictive value beyond market expectations.
Second, we introduce a causal discovery framework that identifies horizon‑specific direct drivers reflecting the underlying mechanisms of FOMC monetary policy decisions. This approach yields parsimonious and interpretable forecasting models, mitigates overfitting, and enhances out‑of‑sample performance.
Finally, we identify and quantify horizon‑specific direct and total causal effects of inflation and market expectations on FOMC decisions, showing that standard regressions conflate these effects and are prone to confounding and mediation biases, while our causal framework delivers well‑defined and economically interpretable causal estimates.
JEL classification: C51, C53, C54, E43, E44, E47, E52, E58, G10.
Keywords: Causal inference; Causal discovery; Monetary policy; FOMC decisions; Interest rate forecasting; Market‑based expectations; FedWatch tool; Macroeconomic indicators; Stochastic discontinuities; Target rate; SOFR.

Robust Pricing of Equity-Indexed Annuities under Uncertain Volatility and Stochastic Interest Rate

In this paper, we propose a novel methodology for pricing equity‑indexed annuities featuring cliquet‑style payoff structures and early surrender risk, using advanced financial modeling techniques. Specifically, the market is modeled by an equity index that follows an uncertain volatility framework, while the dynamics of the interest rate are captured by the Hull‑White model. Due to the inherent complexity of the market dynamics under consideration, we develop a numerical algorithm that employs a tree‑based framework to discretize both the interest rate and the underlying equity index, enhanced with local volatility optimization.
The proposed algorithm is compared with a machine‑learning‑based algorithm. Extensive numerical experiments demonstrate its high effectiveness. Furthermore, the numerical framework is employed to analyze key features of the insurance contract, including the delineation of the optimal exercise region when early surrender risk is incorporated.

Characterizing the stability of intertemporal preferences through discount functions

The framework of this paper is financial mathematics and, more specifically, the field of intertemporal choice and its anomalies. In effect, when analyzing certain processes of decision‑making, the agent exhibits an inconsistent behavior which cannot be modeled by exponential discounting. One of these inconsistent behaviors is decreasing impatience, which has been studied from different points of view. In this paper, we have pointed out that the behavior of a discount function in a neighborhood of infinity is of vital importance to know the properties and characteristics of such a discount function, in particular, regularity and decreasing impatience. Thus, the objective of this paper is to present the concept of stable preference which describes the behavior of the instantaneous discount function in a neighborhood of infinity, from two points of view: from the perspective of a preference relation defined as a weak order on the set of dated rewards, and by using the discount function derived from such a preference relation.
It is shown that this concept can be used to characterize different types of non‑additivity: superadditivity, strongly and moderately decreasing impatience, decreasing impatience according to a function , decreasing impatience according to a vector , etc. Moreover, it has implications for the shape (regularity and singularity) of the discount function. In particular, we will analyze the stability of those discount functions with finite temporal domain, such as the so‑called (generalized) linear discounting. Finally, we will relate the stability of the intertemporal preference with the value of the well‑known Prelec’s convexity index.
The methodology employed in this paper consists of the methods of financial mathematics (concept of discount function, discount rate, Prelec’s index, etc.) and differential calculus, because the discount functions involved in this study will be assumed to admit continuous derivatives of all orders. The main contribution of this paper is two schemes which relate the stability of a discount function with either its non‑additivity (superadditivity / subadditivity) or its behavior in a neighborhood of infinity.
Keywords:
Discount function; intertemporal choice; non-additivity; decreasing impatience; regularity; Prelec’s index.

Validation of machine learning based scenario generators

Machine learning (ML) methods are becoming increasingly important for designing economic scenario generators for internal models. Validating data‑driven models requires different methods than validating classical, theory‑based models. We discuss two novel aspects of such validation: first, checking the multivariate distribution of risk factors, and second, detecting unwanted memorization effects.
The first task is necessary because, in ML‑based methods, dependencies are driven by data rather than derived from financial‑mathematical theory. To address this issue, we propose using an existing test from the literature. The second task is necessary because it cannot be ruled out that ML‑based models merely reproduce empirical data rather than generating new scenarios. For this second issue, we introduce a novel memorization ratio together with a thorough discussion.
We include numerical experiments based on real market data and validate a simple autoencoder‑based scenario generator.

Cross-Country Generalization Bias in Commodity Market: Do Machine Learning Models Learn the Same Way?

Despite the increasing adoption of Machine Learning (ML) models in international comparative studies, most cross‑country analyses train and evaluate models within the same national context, implicitly assuming their ability to generalize across different settings. This paper explicitly investigates the existence of a cross‑country generalization bias by assessing whether ML models trained on data from one country retain adequate predictive performance when applied to a structurally different context.
Using a harmonized dataset from Italy and Lithuania, we implement a cross‑country evaluation framework in which supervised models are trained exclusively on one country and tested out‑of‑distribution on the other. Model performance is assessed using standard predictive metrics and compared against within‑country benchmarks to quantify potential generalization gaps. In addition, we analyze the stability of feature importance and error distributions to identify systematic shifts in the patterns learned by the models.
The results highlight that model transferability is not guaranteed in the presence of cross‑country distribution shift, underscoring the need for explicit cross‑country generalization assessments in ML‑based comparative research.
Keywords: Neural Network, LSTM, Commodity, Price Prediction.

References
[1] E. Antwi, E. N. Gyamfi, K. A. Kyei, R. Gill, & A. M. Adam. Modeling and forecasting commodity futures prices: decomposition approach. IEEE Access, 10:27484–27503, 2022.
[2] M. Ashfaq, I. Khan, D. Shah, S. Ali, & M. Tahir. Predicting wheat yield using deep learning and multi‑source environmental data. Scientific Reports, 15(1):26446, July 2025. doi:10.1038/s41598‑025‑11780‑7.
[3] N. Bhanumurthy, P. Dua, & L. Kumawat. Weather shocks and agricultural commodity prices in India. Climate Change Economics, 4(03):1350011, 2013.
[4] M. R. Bhardwaj, J. Pawar, A. Bhat, Deepanshu, I. Enaganti, K. S. Sagar, & Y. Narahari. An innovative deep learning‑based approach for accurate agricultural crop price prediction. Proceedings of the 2023 IEEE 19th International Conference on Automation Science and Engineering (CASE), 1–7, 2023. doi:10.1109/CASE56687.2023.10260494.
[5] A. Vairo, G. A. Sgarro, & F. Romano. Predicting commodity prices: a neural networks approach to improve market prospects. Applied Mathematical Sciences, 18(6):289–298, 2024. doi:10.12988/ams.2024.919149.
[6] L. Doroshenko, L. Mastroeni, & A. Mazzoccoli. Wavelet and deep learning framework for predicting commodity prices under economic and financial uncertainty. Mathematics, 13(8), 2025. doi:10.3390/math13081346.
[7] Forecasting copper prices using decision learning methods. Resources Policy, 69:101859, 2020. doi:10.1016/j.resourpol.2020.101859.
[8] P. Foroutan & S. Lahmiri. Deep learning systems for forecasting the prices of crude oil and precious metals. Financial Innovation, 10(1), July 2024. doi:10.1186/s40854‑024‑00637‑z.
[9] B. Jin & X. Xu. Machine learning wholesale white wheat price index forecasts. Quality & Quantity, June 2025. doi:10.1007/s11135‑025‑02233‑4.
[10] P. Kumari, N. Harshith, & A. Ginige. Integrating recursive feature selection with automated machine learning framework for global wheat price prediction. Journal of Agriculture and Food Research, 22:102113, 2025. doi:10.1016/j.jafr.2025.102113.

Option Pricing and Risk Management via Neural Networks

The objective of this study is to investigate the potential of neural network–based models for option pricing and risk management, with particular emphasis on their ability to evaluate risk and accurately capture complex market dynamics under nonstandard and illiquid market conditions. Traditional analytical option pricing frameworks often rely on restrictive assumptions such as market completeness, continuous trading, and specific distributional properties of asset returns. These assumptions are frequently violated in real financial markets, especially in the presence of illiquidity, irregular trading activity, and structural changes in volatility dynamics. In such environments, data‑driven approaches based on neural networks provide a flexible alternative capable of learning complex nonlinear relationships directly from observed market data.
In this work, we focus on recurrent neural network (RNN) models for pricing European options using historical market observations. Recurrent architectures are particularly well suited for financial time series, as they explicitly account for temporal dependence and memory effects in asset price dynamics. Several RNN‑based models are developed and trained under different assumptions on volatility behaviour, including constant volatility, local‑volatility structures, and stochastic volatility dynamics represented by GARCH(1,1) processes. This allows us to systematically assess the robustness of neural‑network pricing models across varying volatility regimes and market conditions.
The proposed neural network approach is evaluated in comparison with the classical Black–Scholes model, which remains a benchmark in both academic research and practical applications. In addition, comparisons are conducted with subdiffusive option pricing frameworks that have previously been employed for risk evaluation in markets exhibiting anomalous diffusion and irregular trading patterns [1, 2]. Rather than focusing on the specific mathematical structure of these models, our emphasis is placed on their role in risk assessment and their ability to describe nonstandard market behaviour. This comparative analysis highlights the advantages and limitations of neural network models relative to both traditional and nonclassical pricing approaches.
A further contribution of the study is the analysis of option sensitivities—commonly referred to as Greeks—within the neural network framework. Accurate estimation of Greeks is essential for hedging strategies and risk management, yet it remains challenging in nonstandard models and data‑driven settings. We demonstrate how key sensitivities, including Delta, Gamma, and Vega, can be computed from trained RNN models using automatic differentiation and numerical techniques. The resulting Greeks are analyzed in terms of stability, interpretability, and consistency with financial intuition. The approach is motivated by recent developments in option pricing and Greeks estimation for assets with complex dependence structures, and it extends these ideas to neural‑network‑based pricing models.
Furthermore, the study is inspired by recent advances in deep‑learning approaches to option pricing based on recurrent architectures, particularly models combining Long Short‑Term Memory (LSTM) and Gated Recurrent Unit (GRU) networks [3]. Such architectures have demonstrated strong empirical performance in capturing temporal dependencies and nonlinear patterns in financial data. Building on these insights, we explore the applicability of RNN‑based models for simultaneous option pricing and risk evaluation, with a focus on their performance in markets characterized by limited liquidity and sparse trading activity.
Overall, the results of this study indicate that recurrent neural network models constitute a powerful and flexible tool for option pricing, Greeks estimation, and risk evaluation. The proposed framework offers a data‑driven alternative to classical analytical models and provides valuable insights for risk management in complex financial markets.
Keywords: Option pricing, risk evaluation, option Greeks, recurrent neural networks, deep learning, illiquid markets.

References
[1] Shchestyuk, N., Drin, S., & Tyshchenko, S. (2024). Risk evaluating for subdiffusive option price model with gamma subordinator. In Mathematical and Statistical Methods for Actuarial Sciences and Finance (MAF 2024), Le Havre, France, 286–291. Springer, Cham. https://doi.org/10.1007/978‑3‑031‑64273‑947
[2] Shchestyuk, N., & Tyshchenko, S. (2025). Subdiffusive option price model with inverse Gaussian subordinator. Modern Stochastics: Theory and Applications, 12(2), 136–152. https://doi.org/10.15559/24‑VMSTA265
[3] Zouaoui, H., & Naas, M.-N. (2023). Option pricing using deep learning approach based on LSTM–GRU neural networks: Case of the London Stock Exchange. Faculty of Economic Sciences, University of Relizane.

Transition Risk and Systemic Interconnectedness in European Banking

The transition toward a low-carbon economy has become a major source of financial risk, particularly for institutions exposed to carbon‑intensive activities and to the systemic effects of climate‑related shocks. In the banking sector, transition risk arises from policy interventions, technological change, and shifts in market preferences that may affect the value of banks’ assets and their interconnectedness within the financial system. While a growing literature has explored either climate exposure or systemic risk separately, less attention has been devoted to their joint interaction. This study contributes to this line of research by developing a network‑based framework to assess how climate transition risk propagates through the European banking sector.
The paper proposes a Transition Risk Index (TRI) that combines banks’ environmental exposure with their systemic relevance in financial risk networks. Banks are ranked annually according to their Scope 1 greenhouse gas emissions and classified into green and brown groups. Systemic relevance is then measured through network centrality indicators derived from two alternative representations of systemic risk. The first is based on ΔCoVaR, which captures tail‑risk transmission under extreme market conditions. The second relies on the spillover framework, which measures the transmission of volatility through forecast error variance decompositions in a VAR setting. These two approaches provide complementary perspectives on systemic risk, allowing us to distinguish between extreme‑risk amplification and broader connectedness effects.
Using these networks, we construct long–short portfolios that take a long position in systemically relevant brown banks and a short position in systemically relevant green banks. The daily return of this portfolio defines a Brown–Minus–Green (BMG) factor, while the cumulative return of the factor yields the Transition Risk Index. In this way, the TRI provides a dynamic indicator of how climate‑related transition risk evolves over time in the banking sector, incorporating both environmental exposure and network position.
The empirical analysis is conducted on a sample of major European banks over the period 2016–2024, using daily stock return data and annual GHG emissions. The results show that transition risk is not evenly distributed across institutions and that banks occupying more central positions in the network play a key role in amplifying climate‑related shocks. Moreover, the comparison between the two network constructions reveals important differences. The spillover‑based TRI appears smoother and less volatile, suggesting more gradual adjustments in average connectedness. By contrast, the ΔCoVaR‑based TRI displays sharper and more persistent movements, especially during periods of financial and geopolitical stress, highlighting the importance of tail‑risk channels.
To further investigate the dynamics of transition risk, we perform an event‑based analysis around major policy and geopolitical developments related to the green transition. In particular, we examine the behavior of the TRI around events such as the Russia–Ukraine war, recent EU energy regulations, and the 2024 U.S. presidential election. The evidence suggests that geopolitical shocks are associated with stronger and more persistent changes in the tail‑based network, whereas spillover‑based adjustments are more localized and transitory. This confirms that climate‑related transition risk may propagate differently depending on the dimension of systemic risk under consideration.
Overall, this study provides a novel framework for monitoring transition risk in the banking sector by integrating climate‑related indicators and financial network analysis. The results highlight the importance of considering both environmental exposure and systemic interconnectedness when assessing financial stability in the context of the low‑carbon transition. The proposed TRI may offer useful insights for regulators and supervisors seeking to identify vulnerable institutions and to monitor the evolution of climate‑related systemic risk in Europe.
Keywords: Climate Transition Risk; Banking Sector; Systemic Risk; Financial Networks; ΔCoVaR; Spillover Index; Greenhouse Gas Emissions.

A Two-Stage Automated Decision Support System for Constructing European Corporate Green Bond Portfolios

In recent years, there has been a rapid growth of green bond issues by agencies and listed companies on the European market. Furthermore, portfolio managers and institutional investors have shown an increasing interest in transparent approaches to portfolio design that can accommodate both financial and sustainability‑related information, particularly within a fixed‑income framework.
This work presents a two‑stage automated decision support system for portfolio construction in the European corporate green bond market. In the first stage, given the large and varying number of eligible portfolio constituents, we perform a preselection phase that exploits a multicriteria decision analysis (MCDA) technique. In this step, we rank green bonds according to multiple financial bond‑specific criteria, which include bond characteristics, credit quality indicators, and firm‑related measures such as market‑based risk indicators or sustainability scores. Given this methodology, we can reduce dimensionality in an economically meaningful way and ensure consistency in the portfolio size on every rebalancing window.
In the second stage, we perform the actual portfolio optimization phase, following the classical mean‑variance allocation model. In this context, since estimating a high‑dimensional covariance matrix using historical data from an in‑sample window represents a challenge, our empirical design employs a PCA‑based regularization technique inspired by some recent literature contributions.
The proposed framework is applied in an empirical setting using a dynamic investable universe of euro‑denominated agency and corporate green bonds derived from the historical constituents of the Bloomberg MSCI EUR Corporate and Agency Green Bond Index. Moreover, we construct the optimal portfolio allocation strategy by implementing a rolling‑window approach considering a four‑year investing period from the beginning of 2022 to the end of 2025. In this scheme, model parameters are estimated using a one‑year in‑sample window, and portfolio weights are rebalanced quarterly.
To conclude, this is a methodological and empirical work that offers insights into portfolio construction practices. Moreover, it illustrates how to combine multi‑criteria decision‑making techniques and standard quantitative portfolio tools in the fixed‑income green bond market, a setting that is receiving noteworthy attention nowadays.

Rough Volatility vs HAR: An High-Frequency Perspective

The availability of high‑frequency financial data has profoundly reshaped volatility modeling and forecasting. Traditional GARCH‑type models often fail to capture the strong persistence and multi‑scale structure observed in realized volatility. In this context, the Heterogeneous Autoregressive (HAR) model [2] provides a parsimonious framework by modeling realized volatility as the aggregation of components over daily, weekly, and monthly horizons, reproducing long‑memory‑like behavior in a simple linear setting. Empirical studies at high frequencies have further revealed that volatility exhibits intrinsic roughness, with dynamics resembling fractional Brownian motion and a Hurst exponent significantly below one half. This has motivated rough volatility models, which describe volatility as a non‑Markovian stochastic process with highly irregular paths. While both HAR and rough approaches aim to capture persistence across scales, they differ fundamentally in structure and interpretation, particularly in high‑frequency regimes.
Using daily realized volatility computed from 5‑minute intraday returns, the HAR model remains competitive even against rough specifications such as the Rough Fractional Stochastic Volatility (RFSV) model [3] and the rough Bergomi (rBergomi) model [1]. When forecasting performance is evaluated via the quasi‑likelihood (QLIKE) loss—a measure robust to measurement error in realized volatility [4]—HAR delivers lower loss values for major indices such as IXIC, SPX, STOXX50E, FTSE, and N225 (see Figure 1). These results indicate that, despite its reduced complexity and discrete‑time formulation, HAR provides strong empirical performance at daily horizons, motivating a careful comparison with rough models.
We first use ultra‑high‑frequency data sampled at 10‑second intervals to assess the impact of microstructure noise on the estimation of the Hurst exponent. Estimates show a clear stabilization once the sampling interval exceeds approximately two minutes (Figure 2, left panel), suggesting that finer grids induce artificial roughness, while coarser grids yield more reliable measures. Based on this, we focus on sampling frequencies between 1.5 and five minutes, as finer intervals are dominated by microstructure effects, while coarser sampling adds no forecast advantage at daily horizons.
In the empirical analysis using high‑frequency SPX data, when daily realized volatility is computed with 1.5–3‑minute returns, an RFSV‑type model improves QLIKE forecasts by up to 25–30% relative to HAR. Moreover, aggregating realized volatility to intraday horizons (e.g., 15‑minute volatility) using the same data, rough models achieve up to 98% improvement over HAR; the right panel of Figure 2 illustrates log‑volatility forecasts from HAR and rough specifications. These gains at high frequency are consistent with rough models’ ability to directly capture volatility’s intrinsic irregularity, whereas HAR approximates persistence only via coarse temporal aggregation.
Overall, based on around twenty high‑frequency global indices, our work clarifies the respective domains of relevance of HAR and rough volatility models: HAR remains competitive at daily horizons, while rough specifications deliver substantial gains when volatility is modeled and forecast at intraday and high‑frequency scales.

Bibliography
[1] C. Bayer, P. Friz, and J. Gatheral. Pricing under rough volatility. Quantitative Finance, 16(6):887–904, 2016.
[2] F. Corsi. A simple approximate long‑memory model of realized volatility. Journal of Financial Econometrics, 7(2):174–196, 2009.
[3] J. Gatheral, T. Jaisson, and M. Rosenbaum. Volatility is rough. Quantitative Finance, 18(6):933–949, 2018.
[4] A. J. Patton. Volatility forecast comparison using imperfect volatility proxies. Journal of Econometrics, 160(1):246–256, 2011.

Short-Maturity Asymptotics of Vanilla and Barrier Options under general Stochastic Volatility models with Jumps

We study the short‑time asymptotics of Vanilla options and Up‑And‑In Barrier options under a general stochastic volatility model with jumps using Malliavin Calculus in two very different ways.
For the Vanilla options, we use a decomposition formula to show that the price of European call and put options approaches a finite limit at a speed proportional to a power of the time to maturity. More particularly, we show that when the option is At‑The‑Money, the speed of convergence is of order T^{1/2}, while in the presence of jumps it is proved that in the Out‑The‑Money (OTM) and In‑The‑Money (ITM) cases the speed of convergence towards its limit is of order T. Moreover, if no jumps are involved, it can be shown that under a general stochastic volatility model the speed of convergence of the European options towards their limit in the OTM and ITM cases is of order T^{α}, where α is a parameter that depends on the roughness of the volatility process.
For the Up‑And‑In barrier options, an analysis of the probability of crossing the barrier shows that the price of the barrier option approaches zero faster than any polynomial in the time‑to‑maturity variable. To show this, we analyse the law of the supremum of the log‑price under a general stochastic volatility model in the case where the correlation factor between the log‑price and the volatility process satisfies ρ ∈ (−1, 1). The analysis of the law is performed using Malliavin Calculus, obtaining a concentration inequality and results on the law of the supremum of conditionally Gaussian processes.
The results of the talk can be found in the following articles:
• Jafari, H., Burés, O., Vives, J., and Zhao, Y. Q. (2025). Option price asymptotics under a stochastic volatility Lévy model with infinite activity jumps. International Journal of Theoretical and Applied Finance, 28(1‑2): Paper No. 2550006, 29.
• Burés, O. (2025). On the short‑time behaviour of up‑and‑in barrier options using Malliavin calculus. arXiv preprint arXiv:2510.15423.

Volatility Modeling with Rough Paths: A Signature-Based Alternative to Classical Expansions

We compare two methodologies for calibrating implied volatility surfaces: a second‑order asymptotic expansion method derived via Malliavin calculus, and a data‑driven approach based on path signatures from rough path theory. The former, developed in Alòs et al. (2015), yields efficient and accurate calibration formulas under the assumption that the asset price follows a Heston‑type stochastic volatility model. The latter models volatility as a linear functional of the signature of a primary stochastic process, enabling a flexible approximation without requiring a specific parametric form.
Our numerical experiments show that the signature‑based method achieves calibration accuracy comparable to the asymptotic approach when the true dynamics are Heston. We then test the model in a more general setting where the asset follows a rough Bergomi volatility process—a regime beyond the scope of the asymptotic expansion—and show that the signature approach continues to deliver accurate results. These findings highlight the model‑independence, robustness, and adaptability of signature‑based calibration methods in settings where volatility exhibits rough or non‑Markovian features.

A transfer learning approach for mortality forecasting

Predicting future mortality rates is fundamental to actuarial practice, yet traditional approaches such as the Lee–Carter model face well‑known limitations when historical experience is scarce or structurally incomplete, a situation frequently encountered in emerging markets, young portfolios, or jurisdictions affected by political discontinuities. In these settings, purely local calibration can lead to unstable estimates, overfitting, and poor out‑of‑sample performance, underscoring the need for methodologies that can systematically exploit information beyond the target population while preserving its specific features.
This study investigates transfer learning as a principled solution to the data‑scarcity problem by leveraging knowledge encoded in neural networks trained on data‑rich source domains to enhance forecasting performance in data‑poor target domains. Within this framework, the parameters (weights and biases) of deep learning models trained on multiple countries are interpreted as condensed representations of common mortality structures, which can then be partially reused and adapted to new populations with limited historical depth.
Two deep learning architectures are implemented and compared: Feedforward Neural Networks (FFNN) and Recurrent Neural Networks with LSTM layers. Source models are trained on mortality data for ages 0–99 from 13 European countries over the period 1950–2016, obtained from the Human Mortality Database (HMD), thereby exploiting a long and heterogeneous panel to learn generalizable mortality patterns. Transferability is then evaluated on German mortality data from 1990–2016, also from HMD, which is deliberately selected as the target domain due to its comparatively short post‑reunification history and resulting data limitations.
For each architecture, a family of target models is constructed by initializing the network with the source parameters and freezing an increasing number of initial layers, while allowing the remaining layers to be retrained on the German data. This design enables a systematic assessment of transfer depth—that is, how many layers should retain their source‑domain representation versus being specialized to the target domain—and reveals architecture‑dependent patterns in the division between general feature extraction and domain‑specific adaptation.
The empirical results demonstrate that transfer learning can substantially reduce prediction error relative to models trained solely on the target data, even without exhaustive hyperparameter optimization. In particular, FFNNs tend to benefit from freezing deeper stacks of early layers, while LSTM networks already gain markedly from transferring just the first recurrent layer, suggesting that temporal regularities in mortality are captured very efficiently at shallow depths in the recurrent setting. Overall, the investigation provides, to the best of current knowledge, the first systematic empirical evidence that transfer learning—traditionally successful in computer vision and natural language processing—extends effectively to quantitative actuarial applications.
The proposed framework offers practitioners a practical and conceptually transparent methodology to improve mortality forecasts when domain‑specific data is limited, by reusing structured information learned from richer international experience.

A Stochastic Block Model for Public Debt and International Trade Networks

The increasing complexity of the global economy represents a challenge for monitoring public debt risks (Akanbi and Sbia, 2018; Gu, 2021). We propose a twofold framework by applying a Dynamic Stochastic Block Model (SBM) to a multi‑layer network composed of extracted debt relationships and trade flows for European countries. The trade network layer is directly observable because trade edges represent the levels and growth of imports and exports between countries. In contrast, the debt/fiscal network layer is not directly observable and must be estimated from correlations in the changes in debt‑to‑GDP ratios across country pairs. A high (negative) positive correlation between two countries reflects strong (negative) synchronization in fiscal balances, while a correlation close to zero suggests no relationship between the fiscal evolution of the two economies.
Our SBM assumes a multinomial logit form to express Markov‑chain dependence through the transition matrix and accounts for the persistence of edges between the same pair of nodes across network layers and, more structurally, clustering redundancy (Baltodano and Casarin, 2022; Matias and Miele, 2017). The model allows for clusters of countries with different levels of debt synchronization that evolve over time and are subject to structural changes, suggesting an increase in systemic risk. On the other hand, the layer dependence accounts for conflicts in policy objectives such as external balance and fiscal stability. The SBM approach jointly considers the interdependence between countries and the twin‑deficit phenomenon, consistent with the composite nature of risks in economic and financial systems.
We find evidence of structural changes in the topology of the debt network during crises, shifting from assortative behavior to a core‑periphery system with high interdependence among countries’ debt. Moreover, the results suggest parameter heterogeneity. In stable periods, unemployment (real GDP per capita) differences lead to higher (lower) debt synchronization, while income inequality and financial depth reduce the similarity in debt evolution. There is no evidence of a twin‑deficit phenomenon in the period considered.

References
Akanbi, O. A., & Sbia, R. (2018). Investigating the twin‑deficit phenomenon among oil‑exporting countries: Does oil really matter? Empirical Economics, 55(3), 1045–1064.
Baltodano López, O., & Casarin, R. (2025). A dynamic stochastic block model for multidimensional networks. arXiv Working Paper, 2209.09354.
Gu, G. W. (2021). Sovereign default, trade, and terms of trade. Macroeconomic Dynamics, 25(6), 1346–1380.
Matias, C., & Miele, V. (2017). Statistical clustering of temporal networks through a dynamic stochastic block model. Journal of the Royal Statistical Society: Series B (Statistical Methodology), 79(4), 1119–1141.

Mathematical programming models for multi-signal automated trading systems

Automated trading systems are well‑established techniques in the investment sector and can operate either as support tools or as fully autonomous agents executing market transactions without human intervention. Trading algorithms commonly rely on technical indicators computed from historical and current market data. These indicators are processed through predefined decision rules that generate trading signals for buying, selling, or staying out of the market. The reproducibility of trading rules enables the development of automated trading systems that operate according to signals on a high‑frequency basis, such as daily trading.
Such systems may consider multiple signals by assigning a weight to each, resulting in a convex combination of the signals. The action performed by the automated trading system depends on the value of this convex combination relative to two thresholds. Specifically, values above an upper threshold trigger a buy position, values below a lower threshold trigger a sell position, while values in between indicate staying out of the market.
A similar trading algorithm depends on three categories of parameters: those for indicator computation (e.g., time windows), those defining the trading rules (e.g., threshold values), and the weights for aggregating the signals. In this presentation, we propose a mathematical programming model aimed at jointly optimizing these parameter categories in multi‑signal trading systems. Particular attention is devoted to the system based on four signals, originally introduced in [CPM24]. More specifically, the signals considered in the article are generated by the exponential moving average, relative strength index, moving average convergence divergence, and Bollinger bands indicators.
An implementation of the mathematical programming model is employed to conduct several experiments on real stock prices from the Italian financial market. Both in‑sample and out‑of‑sample results are presented. In the in‑sample phase, the model is applied to historical data to identify the optimal indicator parameters, trading rules, and signal weights in order to maximize the final net capital of the investment. The resulting solutions are subsequently evaluated out‑of‑sample by applying the corresponding trading system to unseen data.
We compare the optimization results obtained through our mathematical programming model with the metaheuristic optimization proposed in [CPM24]. The results of the computational campaign are promising, although our computational times are sometimes considerably higher than the metaheuristic model in [CPM24]. The proposed mathematical programming model significantly outperforms the heuristic approach described in [CPM24]. In the out‑of‑sample phase, although the solutions obtained are less effective than in‑sample, they still yield better results than those of the heuristic solutions in general.

References
[CPM24] Marco Corazza, Claudio Pizzi, and Andrea Marchioni. A financial trading system with optimized indicator setting, trading rule definition, and signal aggregation through Particle Swarm Optimization. Computational Management Science, 21(1), 2024, p. 26.

Integrating Cause-Specific Analysis for Insights into Life Expectancy Trends

Assessing mortality dynamics, which are determined by changes in cause‑specific mortality rates, is a topic that has gained renewed focus in the current literature following the COVID‑19 pandemic. However, the issue of understanding how changes in age‑ and cause‑specific mortality rates influence overall mortality trends and period life expectancies has long been debated. For example, refer to the seminal paper [1], which analyses the impact of eliminating specific causes of death on life expectancy, as a consequence of medical improvements, and the relationship between death rates and life expectancy. Recently, among many others, [2] considers a set of log‑linear models to identify changes in mortality and life expectancy caused by trend changes to particular causes, and [3] applies a period–cohort improvement model to identify critical drivers influencing trends in US mortality.
In this framework, this study aims to gain insight into recent trends in life expectancies for countries characterised by disparate progress in mortality to understand age‑specific and cause‑specific determinants of change. To this purpose, it applies the decomposition method introduced in [4], which developed a well‑known formula to gauge the changes in life expectancy consequent to changes in age‑specific and cause‑specific components.
The analysis adopts a dual perspective. First, it offers a retrospective decomposition of observed mortality changes over the past decades, examining how shifts in age‑ and cause‑specific death rates have shaped national life expectancies. Second, it explores hypothetical counterfactual scenarios that simulate the complete elimination of selected causes of death, thereby providing a forward‑looking view of the potential longevity gains that could be achieved through targeted medical breakthroughs or public health interventions.
Empirically, the study draws on detailed cause‑of‑death data from the Human Cause‑of‑Death Database, a component of the Human Mortality Database (HCD@HMD), which offers harmonised mortality data across a wide range of countries and periods. The analysis focuses on the major cause‑of‑death categories, including neoplasms, circulatory diseases, respiratory conditions, external causes, infectious diseases, and others. For each category, we calculate the rate of cause‑specific mortality improvement and assess its impact on life expectancy and life disparity, allowing for a richer understanding of the distributional and equity implications of mortality changes.
The resulting cause‑specific decomposition and counterfactuals also provide an interpretable map of post‑COVID mortality dynamics and a structured basis for incorporating cause information into longevity risk assessment and actuarial forecasting [5]. As a natural extension, the same decomposition logic can be transferred from life expectancy to life‑contingent valuation, building on [6], to assess how cause‑ and age‑specific mortality dynamics translate into changes in annuity values and reserves.
Keywords
Cause‑of‑death mortality; Cause elimination; Life expectancy; Mortality forecasts.
Area: Actuarial Mathematics
MSC: 91G05

References
[1] N. Keyfitz. What difference would it make if cancer were eradicated? An examination of the Taeuber paradox. Demography, 14 (1977), 411–418.
[2] A. M. T. L. Yiu, T. Kleinow, & G. Streftaris. Cause‑of‑death contributions to declining mortality improvements and life expectancies using cause‑specific scenarios. North American Actuarial Journal, 28(3) (2024), 513–550.
[3] A. M. Villegas, M. Bajekal, S. Haberman, & L. Zhou. Key Drivers of Long‑Term Rates of Mortality Improvements in the United States: Period, Cohort, and Cause of Death Analysis, 1959–2016. North American Actuarial Journal, 28 (2024), 187–217.
[4] J. W. Vaupel & V. Canudas Romo. Decomposing change in life expectancy: A bouquet of formulas in honor of Nathan Keyfitz’s 90th birthday. Demography, 40 (2003), 201–216.
[5] C. Di Palo, A. M. Palazzo, & A. Pianese. COVID‑19, Cause‑Specific Mortality, and Life Expectancy Decomposition: Evidence from the USA and France. Submitted.
[6] C. Di Palo. Decomposing changes in life annuities. European Actuarial Journal, 2025. https://doi.org/10.1007/s13385‑025‑00409‑8.

Integrating Health Benefits into an NDC Pension System

Population ageing increases both pension and long-term care (LTC) expenditures, largely due to rising disability and illness at older ages. This paper proposes an extension of a notional defined contribution (NDC) pension system that incorporates health- and disability-contingent benefits throughout the retirement period. Using a continuous-time multi-state model of health, disability, and survival calibrated on data from the Health and Retirement Study, pension benefits are allowed to adjust dynamically to retirees’ current health conditions, providing enhanced protection in fragile states. We formalize four key design objectives-financial sustainability, actuarial fairness, consistency with health-related economic needs, and homogeneity of benefits-and show that they cannot generally be satisfied simultaneously. Through a numerical analysis, we quantify the trade-offs implied by alternative pension designs and demonstrate how heterogeneity in longevity can be leveraged to finance enhanced benefits for unhealthy retirees without increasing aggregate pension expenditures, while making explicit the resulting redistribution across health states.

A State-Contingent Long-Term Saving Product under Health and Financial Uncertainty

Health‑related events occurring during working life may compromise individuals’ ability to generate labor income, increasing the need for liquidity at times when regular savings are no longer manageable. In recent years, the incidence of work‑limiting health conditions has risen steadily across all age groups, highlighting the growing relevance of health‑related risks for labor income and long‑term economic outcomes. In this paper, we propose a long‑term saving product with a state‑contingent liquidity constraint, characterized by systematic accumulation during periods of full labor capacity. Resources can be accessed only when health‑related events impair income‑generating ability; otherwise, the accumulated capital remains invested until maturity.
The product is designed to address the risk of liquidity shortfall associated with adverse health conditions during working life, without relying on traditional insurance transfers or guaranteed benefits. Instead, protection arises from a contractual restriction on withdrawals, which links access to funds to objectively defined health states. The proposed product naturally fits within the framework of sustainable finance, particularly regarding the social dimension of ESG criteria. Rather than focusing on short‑term return maximization, it aims at strengthening individuals’ financial resilience during working life by providing structured access to accumulated resources in periods when adverse health conditions limit or prevent labor income generation.
From the provider’s perspective, the state‑contingent liquidity constraint represents a structural feature that enhances the stability and predictability of the funding base. Compared to fully liquid savings products such as savings accounts, the probability that assets remain invested over extended horizons is higher, while no explicit biometric risk is transferred to the issuer. This positions the product between traditional saving instruments and insurance contracts, expanding the range of available financial instruments without requiring the provision of guaranteed benefits. The main economic value for the provider arises from the increased effective duration of liabilities and the reduced short‑term uncertainty of cash outflows, which can support a more efficient asset allocation and long‑term financial planning.
In this view, we develop a quantitative framework that combines a stochastic representation of individual health dynamics with a financial market environment evolving over time, enabling the joint analysis of demographic and financial uncertainty. Individual health dynamics are described through a discrete‑time state‑based process with age‑ and duration‑dependent transitions, while the financial environment is modeled using standard stochastic techniques. The two components are assumed to be independent and are jointly simulated to generate long‑term scenarios. This structure enables the evaluation of cumulative cash flows, fund evolution, and outcome distributions over a finite horizon. The proposed framework can accommodate alternative contractual designs and payoff structures and provides a basis for assessing risk exposure, sustainability, and performance from different perspectives.

Hybrid Uncertainty in Sustainable Portfolio: Bridging Probabilistic and Possibilistic Sharpe Ratios

Climate‑related risks are now seen as financial risks, and they have motivated investors to consider sustainability in portfolio decisions. A common way to capture climate‑related exposure at the company level is through ESG scores. These scores are widely used, yet they remain imperfect because they depend on rating agencies and are updated less frequently than the availability of financial data. Their uncertainty stems not only from time‑varying information but also from credibility issues, such as inconsistent methodologies and limited confidence in the data.
To address these issues, the paper introduces a decision system for portfolio allocation that offers two distinct approaches to handling financial and sustainability uncertainties. The financial part follows the usual probabilistic setting for returns, while the ESG part is modelled through possibilistic methods using a rectangular fuzzy structure. In this setting, the use of fuzzy possibilistic descriptions for ESG ratings allows for accounting for credibility uncertainty, specifically by including minimal trust or full ignorance regarding the ratings.
We apply the framework in a Sharpe‑ratio setting and introduce a hierarchical structure in which both the usual financial Sharpe ratio and a similar sustainability version based on ESG ratings are optimised. In this formulation, two parameters become central: a parameter (γ) used to model the level of confidence placed on ESG data quality, and a second parameter (δ) that reflects investor preferences for sustainability. The contribution of the paper lies in introducing ESG information within a Sharpe‑ratio‑based framework and showing how portfolio choices change once the Sharpe ratio based on ESG is taken into account. The hierarchical structure allows preserving the classical financial objective in the first step and then exploring how ESG‑based objectives can be incorporated without discarding the financial benchmark.
The proposed model is tested on data from the European market using the EURO STOXX 100 constituents during the period 2014–2024. Our results show that increasing δ relaxes the financial Sharpe constraint and allows the portfolio to shift toward the ESG optimum. As δ increases, the financial Sharpe ratio decreases, yet the portfolio remains well diversified over a meaningful range. This suggests that financial performance can decline moderately while diversification remains intact. For large values of δ, however, diversification eventually collapses, and the solution concentrates on a few ESG‑oriented firms. This indicates that there exists an intermediate range of δ that provides a reasonable trade‑off, where financial Sharpe deterioration is limited and diversification is preserved. This trade‑off is a key feature of the hierarchical model and shows that credibility‑based ESG adjustments can be incorporated without immediately sacrificing diversification, thereby jointly satisfying both objectives.
Keywords: Hybrid Uncertainty; Fuzzy possibilistic approach; ESG Ratings; Sharpe Ratio; Portfolio Optimization.

Climate-Driven Financial Risk and Optimal Portfolio Choice with Temperature-Linked Derivatives

In recent years, climate change has been strongly affecting the economy. Temperature, extreme heat or cold waves play a key role in market dynamics and investment strategies. The impact of temperature variation on local and global economies represents a topic of fundamental importance for society. Economists have long attempted to estimate the economic costs of climate change, widely regarded as one of the largest environmental externalities (see e.g. [7]). Recent empirical studies show that temperature variability has a statistically significant effect on economic output and on the main drivers of economic growth. Temperature variations affect the economy in different ways, with special attention to total factor productivity (TFP), capital accumulation, and employment (see e.g. [3]). Poorer countries are much more affected, as their economies are more dependent on agriculture and outdoor labor. Moreover, high temperatures reduce capital and employment growth in low‑income nations, while more developed economies are also exposed through industrial channels (see e.g. [4]). In [6], the authors estimate that weather variability accounts for about 3.4% of U.S. GDP fluctuations, showing the exposure of the economy to weather factors. A growing body of evidence indicates that a share of these damages is driven by temperature changes. Future projections indicate that climate‑related damages, mostly driven by temperature dynamics, could reduce global income by about 19% by mid‑century (see e.g. [5]).
In this context, the demand for financial instruments to hedge against temperature risk is increasing. The present paper introduces an optimal investment strategy for a portfolio that includes a weather derivative indexed on temperature. We integrate temperature‑linked weather derivatives into a dynamic portfolio optimization framework, as proposed in [1].
In such a framework, temperature dynamics play a crucial role in weather derivatives and climate risk literature. One of the foundational contributions is [1], where the authors propose mean‑reverting dynamics with a deterministic long‑term mean capturing the seasonal component. The volatility is also deterministic.
The introduction of futures and option markets on weather indices in the Chicago Mercantile Exchange (CME), based on data from different cities in the United States, Europe, and Japan, is discussed in [2]. These contracts are used for hedging against weather‑related risks. In the U.S. market, the most commonly traded indices are heating degree days (HDD) and cooling degree days (CDD), calculated on monthly or seasonal intervals.
Consistent with the growing demand for climate‑risk hedging instruments, we introduce a temperature‑linked derivative in a portfolio‑choice setting to allow the investor to hedge temperature risk within a dynamic allocation framework.
We consider an economy driven by the interest rate and a temperature‑related variable. Following the financial literature, we model interest‑rate dynamics using a CIR‑type process. Along the lines of [1], we adopt a Vasicek specification for temperature dynamics to capture features such as mean reversion.
We solve an optimal portfolio allocation problem where the investor maximizes expected terminal wealth, assuming a CRRA utility function. The corresponding maximization problem is solved through standard dynamic programming techniques.
To test the soundness of our proposal, we compare the optimal investment strategy with and without weather derivatives, assessing the improvement achievable by incorporating weather‑related tools through several robust and synthetic measures.
Keywords: portfolio optimization; weather risk; weather derivatives; dynamic programming; temperature dynamics
JEL classification: C61, D81, G11, G13, Q54

References
[1] P. Alaton, B. Djehiche, and D. Stillberger. On modelling and pricing weather derivatives. Applied Mathematical Finance, 9(1):1–20, 2002.
[2] F. E. Benth and F. E. Benth. The volatility of temperature and pricing of weather derivatives. Quantitative Finance, 7:553–561, 2007.
[3] M. Henseler and I. Schumacher. The impact of weather on economic growth and its production factors. Climatic Change, 154, 2019.
[4] M. Kotz, A. Levermann, and L. Wenz. The effect of rainfall changes on economic production. Nature, 601:223–227, 2022.
[5] M. Kotz, A. Levermann, and L. Wenz. The economic commitment of climate change. Nature, 628:551–557, 2024.
[6] J. K. Lazo, P. H. Larsen, M. Lawson, and D. M. Waldman. U.S. economic sensitivity to weather variability. Bulletin of the American Meteorological Society, 2011.
[7] W. Nordhaus. Climate change: The ultimate challenge for economics. American Economic Review, 109:1991–2014, 2019.

An algorithm for -constrained Gaussian copula graphical models

Gaussian graphical models provide a natural framework for learning conditional dependence networks via the sparse precision matrix, but their standard formulation relies on Gaussianity, which is often violated in practice (e.g., heavy tails, skewness, and outliers). A flexible alternative is the nonparanormal (Gaussian copula) model, where each variable is allowed an unknown monotone marginal transformation while the dependence structure is captured by an underlying latent Gaussian vector. Within this setting, the SKEPTIC approach of Liu et al. (2012) bypasses explicit estimation of the marginal transformations and instead exploits the invariance of rank-based measures: it computes pairwise Spearman’s ρ or Kendall’s τ and applies the Gaussian copula calibration to obtain a consistent estimator of the latent correlation matrix. This rank-based plug-in correlation estimator is robust to marginal misspecification and can be directly used as input for high-dimensional precision matrix estimation and graph recovery.
Building on this robust correlation estimate, we investigate a two-stage SKEPTIC–SCGlasso procedure that combines rank-based correlation estimation with SCGlasso (Fulci et al. 2026), a computationally efficient -constrained and -regularized estimator of sparse precision matrices. SCGlasso estimates the precision matrix by optimizing the Gaussian log-likelihood over the positive-definite cone, while controlling the maximum number of edges via an explicit -constraint and using an -penalty to improve numerical stability. The goal is to couple robustness in the first step with direct sparsity control in precision matrix estimation and the corresponding graph.
In simulations spanning random, cluster, hub, and scale-free topologies and different sample sizes, we compare the proposed pipeline with Glasso, SCAD, and MCP in terms of Frobenius norm error and edge selection accuracy. Overall, the proposed method achieves lower Frobenius error and better graph recovery, with higher F1 scores and fewer false positive edges on average. An application to daily returns of the French 30 industry portfolios illustrates that the resulting networks are sparse and economically plausible, and that their structure varies over time in a way that is consistent with sectoral comovement.
Keywords: Nonparanormal distribution, Gaussian copula, SKEPTIC, sparse precision matrix estimation, ℓ0-constraint, financial networks.
References
Fulci, A., Paterlini, S., Taufer, E.: An ℓ0-constrained and ℓ1-regularized estimator for graphical models. Statistics and Computing 36, 60 (2026).
Liu, H., Han, F., Yuan, M., Lafferty, J., Wasserman, L.: High-dimensional semiparametric Gaussian copula graphical models. The Annals of Statistics 40(4), 2293–2326 (2012).

Complementary aspects between hierarchical clustering and a gaussian graphical model in a crypto partition

Clustering techniques apply metrics different from correlation, and typically they reduce to the Euclidean distance in the case of quantitative variables. Keeping this in mind, the log‑returns of 16 crypto assets were examined over a period of approximately three years, leading to a final partition of 4 or 5 groups, focusing on the aggregation distances recorded in the dendrogram. This result is compared with partitions stemming from correlations—first considering the correlation matrix, then the partial correlation matrix, and finally using a Gaussian graphical model. The main outcome of these comparisons is the superposition of the partition obtained via hierarchical clustering and the one stemming from the Gaussian graphical model. The partitions considering just the correlation matrices proved to be insufficient and did not yield satisfactory results.
On the other hand, Gaussian graphical models rely on strong parametric assumptions, and they are strictly unjustified or acceptable only as approximations in most financial applications. In contrast, clustering methods belong to nonparametric statistical techniques and do not require adherence to such strong assumptions. From this perspective, the two methodologies can be viewed as complementary, and the consistency of the final partitions provides nonparametric support for the result obtained using the Gaussian graphical model.
Because of the Euclidean distance used in the clustering method, the conjecture that clusters with low inner distances on the dendrogram identify assets with equivalent risk is tested. Adopting the canonical CAPM theory, these groups were compared with the dominant market asset, namely Bitcoin. The conclusion of these tests is consistent with the hypothesis, and it opens an interpretative meaning for the cliques of the Gaussian graphical model that could not be directly inferred from the model itself.

Analysis of the determinants of skill mismatch in the labour market: a nonparametric approach

Labour market mismatch between labour demand and labour supply has become a persistent feature of many advanced economies, with important economic and social implications. Despite relatively high employment levels in several countries, firms increasingly report difficulties in filling vacancies, while parts of the workforce experience unemployment, inactivity or
underutilisation of skills. This apparent paradox reflects the multidimensional nature of mismatch,which may result from geographical frictions, skill misalignment, demographic trends, institutional factors, or imperfect information. From a quantitative perspective, measuring and analysing mismatch raises significant methodological challenges, as traditional labour market indicators, such as unemployment rates, provide only a partial picture of market imbalances and fail to capture unmet labour demand or qualitative skill gaps. Owing to its multidimensional character, the statistical analysis of mismatch naturally leads to complex inferential problems. The aim is to investigate whether the difficulties firms experience in recruitment are systematically associated with particular skill requirements, firm characteristics, sectoral contexts, or other factors. In such settings, a powerful and versatile approach is provided by the family of Combined Permutation Tests (CPT). This approach is based on performing a series of partial tests whose p-values are subsequently combined through a suitable combining function, providing a single overall test statistic that addresses the multivariate hypothesis. A key advantage over parametric procedures is that no specific multivariate distributional assumptions are required for the test statistics, nor is it necessary to specify or estimate the dependence structure among variables. The distribution-free nature of permutation tests ensures robustness and flexibility, especially when normality assumptions are violated. Dependence among the partial tests is naturally accounted for through permutations of the rows of the data matrix. Standard parametric approaches, both univariate and multivariate, rely on stringent assumptions that may only hold asymptotically. CPTs provide a sound alternative, retaining good performance under parametric assumptions and offering robustness otherwise. In the multiple testing context, controlling the family-wise error rate (FWE) plays a fundamental role in identifying the partial hypotheses driving overall significance while maintaining control of type I error. This work represents an output of the project entitled “Digital Innovation and Specialized Skills in Precision Mechanics. An Empirical Study of the Mismatch between Labor Supply and Demand in the Ferrara Area”, funded by the University of Ferrara and Chamber of Commerce, Industry, Crafts, and Agriculture of Ferrara and Ravenna in 2025.
Keywords: Multivariate Statistics, Nonparametric Inference, Combined Permutation Test, Labour Mismatch

Overcoming mortality data scarcity: a neural neural network forecasting approach

Most of the methodological developments in mortality modelling and forecasting have been based on the extrapolation of past trends. Some of the desirability criteria identified in the literature for a good mortality model relate to the ability to capture all the significant structure in the past data by means of a parsimonious number of parameters, along with the ability to provide reasonable and consistent forecast levels of uncertainty in mortality projections, compared to historical levels of variability. Recent literature has exploited neural network architectures, often combined with more traditional mortality models, to achieve these relevant targets.
In the context of the Lee–Carter (LC) mortality model, we show the advantages of using General Regression Neural Networks (GRNN) to forecast the LC stochastic time index when a limited number of historical mortality rates is available for training the models. GRNN models are parsimonious since they have only one tuning parameter to control the overall generalization. Additionally, they can detect significant features in the time series of the LC time index through fast single‑pass learning, irrespective of the sample size. Therefore, compared to other neural network models, thanks to their simple structure, GRNN models allow one to extrapolate relevant information included in the period dimension of the data, even when reliable mortality rates refer to short time frames.
As a representative case study, we use GRNN models, within the structure of the traditional LC model, applied to short time series of annual mortality rates. Deploying forecasting methods that allow capturing, from small sample sizes, the fundamental mortality dynamics and reliably projecting them into the future is useful in many respects within policymaking, at national and supranational levels. Some considerations on the empirical evidence highlight the relevance of these innovative techniques for mortality forecasting, relative to traditionally employed methodologies.
Keywords: Mortality forecasting; Lee–Carter model; neural networks.

A Stratified Lee–Carter Framework for Socio-Economic and Health-Driven Mortality Dynamics

Stochastic mortality models form the analytical backbone of actuarial science, supporting the valuation of life‑contingent liabilities, the quantification of longevity risk, and the long‑term assessment of pension and health‑care systems. The classical Lee–Carter model [Lee and Carter, 1992] remains a benchmark due to its parsimony and interpretability, yet its aggregate formulation implicitly assumes homogeneous mortality improvements across the population. A substantial body of demographic and epidemiological evidence contradicts this assumption, documenting persistent socio‑economic gradients in mortality outcomes and pronounced heterogeneity in health‑related risks, particularly those associated with cancer incidence and survival. These structural disparities motivate the development of mortality models capable of capturing heterogeneous dynamics within a unified stochastic framework. Recent extensions of stochastic mortality models incorporate observable drivers such as economic, environmental, and lifestyle variables to improve forecast interpretability and scenario analysis [Debón et al., 2024]. While valuable, such approaches typically operate at the aggregate level and do not explicitly model population stratification or health‑driven heterogeneity.
This paper introduces a hybrid stratified extension of the Lee–Carter model that integrates socio‑economic stratification, health‑status differentiation, and evolving population composition within a coherent actuarial framework. The population is partitioned into mutually exclusive strata defined by observable socio‑economic characteristics and health conditions. For each stratum, mortality is modeled through a log‑bilinear specification combining stratum‑specific age effects with two common latent factors: a general period factor capturing broad mortality improvements and a health‑related factor designed to reflect cancer‑driven mortality dynamics, motivated by recent evidence on cancer prevalence and survival [Lara and Hershman, 2026; Richards and Macdonald, 2025]. Population weights evolve exogenously over time, allowing demographic change and socio‑economic composition to directly influence aggregate mortality trajectories.
The proposed structure preserves the interpretability and transparency of the classical Lee–Carter model while introducing sufficient flexibility to accommodate heterogeneous sensitivities to mortality improvements and health shocks across strata. Identification conditions are formally established, and the aggregation properties of the model are derived. A key theoretical implication is that, even when each stratum follows a Lee–Carter‑type representation, the resulting aggregate mortality surface is generally nonlinear in the latent factors and does not admit a classical Lee–Carter decomposition. This result highlights the actuarial relevance of modeling heterogeneity explicitly rather than relying solely on aggregate mortality data.
The framework is compatible with Poisson log‑bilinear estimation, generalized singular value decomposition for initialization, and state‑space inference via Kalman filtering, extending the methodological approaches of Currie [2016]. From an actuarial perspective, the model provides a coherent basis for portfolio‑specific longevity projections, capital requirement calculations, and the pricing and reserving of life‑contingent liabilities in the presence of socio‑economic and health‑related heterogeneity. It also offers a natural extension of multi‑population coherence principles [Li and Lee, 2005] to intra‑population strata, ensuring consistent long‑term projections across socio‑economic groups.
Keywords: Stochastic mortality modeling; Lee–Carter model; socio‑economic stratification; health‑related mortality risk; longevity risk.

References
Currie, I. D. (2016). On fitting generalized linear and non‑linear models of mortality. Scandinavian Actuarial Journal, 2016(4), 356–383. doi:10.1080/03461238.2014.928230
Debón, A., Martínez‑Ruiz, F., & Montes, F. (2024). Incorporating socio‑economic and environmental covariates into stochastic mortality models. European Actuarial Journal, 14(1), 45–78. doi:10.1007/s13385‑023‑00342‑1
Lara, P. N., & Hershman, D. L. (2026). Cancer statistics, 2026: Charting a course for a national cancer research agenda. CA: A Cancer Journal for Clinicians, 76(1). doi:10.3322/caac.70043
Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87(419), 659–671.
Li, N., & Lee, R. (2005). Coherent mortality forecasts for a group of populations. Journal of the American Statistical Association, 100(470), 687–698. doi:10.1111/1467‑9965.00068
Richards, S. J., & Macdonald, A. S. (2025). On contemporary mortality models for actuarial use I: practice. British Actuarial Journal, 30, e18, 1–35.
doi:10.1017/S135732172500012.

Conditional value-at-risk under reward-penalty mechanism with applications to robust portfolio management

In this paper, we present robust portfolio selection models by incorporating a reward and penalty mechanism into portfolio management. We assume that the joint distribution of the losses of the underlying risky assets in a portfolio is uncertain but lies within a multivariate distribution set. Our goal is to identify optimal portfolio allocations by minimizing the worst-case conditional value-at-risk (CVaR) of portfolio loss under the reward and penalty mechanism and distribution uncertainty.
Our models can also be used to investigate the problem of how to balance portfolio losses
with their associated downside risk in portfolio management. The core of such a robust portfolio selection model is the worst-case CVaR. We first derive an explicit closed-form expression for the worst-case CVaR under the reward-penalty mechanism, which generalizes several existing models and results regarding the worst-case CVaR. We then apply this expression to obtain optimal portfolio allocations that minimize the worst-case CVaR under both a classical mean-covariance-based multivariate distribution set and a generalized mean-covariance-based multivariate distribution set.
Additionally, we utilize real market data to illustrate the application of the proposed models and the corresponding optimal portfolio allocations in portfolio management. Our empirical experiments show that portfolios based on the proposed models have the potential to outperform those based on several existing related models. Furthermore, the results demonstrate that incorporating downside risk into portfolio loss helps better manage risk and can achieve higher investment returns than considering either downside risk or portfolio loss alone. Moreover, our experiments reveal the trade-off between improving expected portfolio return and controlling the worst-case portfolio CVaR.

Scenario-Based Reward–Risk Ratio Optimization for Portfolios with Zero-Valued Instruments

This research addresses a critical limitation in portfolio optimization: the ill‑posed nature of reward–risk ratios when portfolios include zero‑valued instruments (ZVIs). In this context, ZVIs encompass not only derivatives whose current value may be zero—such as forwards and futures—but also instruments with numerically negligible current values. This includes, for example, deep out‑of‑the‑money European options close to maturity, whose values can become effectively zero within the Black–Scholes–Merton framework. In such cases, traditional return‑based formulations can break down because the corresponding percentage return becomes undefined or numerically unstable. We propose a unified, scenario‑based framework that overcomes this singularity by reformulating reward–risk ratio optimization—specifically for the Sharpe, Sortino, and Omega ratios—in terms of position units and profit‑and‑loss (P&L) rather than weights and returns.
In our scenario‑based setting, portfolio P&L is constructed via two components:
(i) scenario generation, where future realizations of market risk factors are simulated on a time grid, and
(ii) instrument valuation, where each instrument is mapped to scenario‑wise values (and hence scenario‑wise P&L) conditional on those simulated paths.
Together, these steps provide the P&L distribution required to evaluate reward and risk functionals. The proposed methodology is applicable to ratio problems in which the reward is an expected excess return (equivalently, expected excess P&L in unit space) and the risk is a positive, convex, and linearly homogeneous functional—a class that includes widely used measures such as the Sharpe ratio, Sortino ratio, and the Omega measure.
We show that applying the Charnes–Cooper lifting to the weight‑based formulation converts the fractional objective into a single convex program. By substituting the unit‑based representation of scenario‑wise portfolio P&L, the resulting lifted problem admits an equivalent expression in unit space, allowing the optimization to be carried out directly in terms of position units. The numerical results confirm that this scenario‑based P&L representation preserves convexity and delivers fast, stable solutions in mixed portfolios that include non‑linear instruments. Furthermore, we evaluate bisection and Dinkelbach’s methods as parametric alternatives, identifying the specific trade‑offs between convergence speed and the flexibility required to handle non‑homogeneous portfolio constraints.
Numerical experiments on mixed portfolios confirm that while all three approaches identify identical optimal allocations, the Charnes–Cooper and Dinkelbach methods significantly outperform bisection in terms of convergence speed and solver stability. Our results provide a robust computational pipeline for derivatives‑heavy portfolios, ensuring mathematical consistency where traditional weight‑based models break down.
Keywords: Reward–risk ratio optimization; Fractional programming; Derivative portfolios; Zero‑valued instruments; Scenario‑based profit‑and‑loss.

On the choice of the number and types of portfolios in the cross-sectional CAPM

In the context of a recently proposed cross-sectional capital asset pricing factor model, we conduct a systematic assessment of its performance by examining various ways of constructing its two non‑market factors called RV and RS. Specifically, we consider different numbers and types of long‑short portfolios used in obtaining RV and RS among those available at the q‑factors Data Library. Based on over 10,000 two‑pass regression fits and a special version of the cross‑sectional R‑squared and block‑bootstrap confidence intervals, we identify the winning configuration as that formed by four portfolios pertaining to the groups momentum (P52W 6), value (NOP), profitability (FP 6), and intangibles (OCA), and optionally five portfolios by augmenting those four portfolios with frictions (ISFF 1). The inclusion of the innovation group is not recommended. Overall performance of the model hinges heavily on the choice of the types of portfolios.
Systematic search for the best number and types of portfolios in the cross‑sectional CAPM, using the adjusted and centered CS R‑squared as a comparison metric and q‑factors Data Library as a data source, resulted in 4–5 portfolios—P52W 6, NOP, FP 6, OCA, and optionally ISFF 1—capturing momentum, value, profitability, intangibles, and frictions anomalies, respectively. Notably, portfolios capturing investment style did not feature frequently in the winning combinations. This result is somewhat in line with Cooper et al. (2022), who suggested using five portfolios capturing size, momentum, value, quality, and beta anomalies. Also, compared to the average value of the metric of about 0.4 reported in Cooper et al. (2022), we observe predominantly low explanatory ratios, with only about 14–22% of all combinations per given L leading to one‑sided 90% BC‑percentile confidence intervals for the respective population parameter whose lower end is positive. This could be due to various reasons such as different candidate portfolios for RV‑ and RS‑factor construction, different pool of testing portfolios, different methodological choices (non‑tradability adjustment, block‑bootstrap, bias‑correction), or differing time spans, though one would expect the model to work well universally.

Risk exposure for non-life insurance in a telematics setting

Risk exposure is a core component of road accident modelling, as it weights observations by vehicle usage. Thus, their incorporation not only improves model performance but also provides crucial benefits for ratemaking, as it allows car insurance premiums to be proportional to actual usage.
Historically, two main measures have been utilized: the observation period (i.e., the contract duration) and the distance driven. However, in recent years, telematics data from On-Board Devices and smartphone applications have greatly increased the number of covariates available. Such advancements have enabled models to incorporate behavioural data, such as acceleration, and geographical data, which provide insights into driving conditions.
Against this backdrop, risk exposure can now be tracked dynamically for large data sets containing claim occurrences. This opens various research avenues, specifically regarding the impact of exposure on claim frequency. Our presentation will focus on three key aspects that impact exposure measures in a telematics setting:
The access to granular information allows data to be studied at various levels of specificity. Consequently, a balance must be struck between exposure and the granularity of covariates per observation. On the one hand, a more granular dataset provides more detailed information on telematics data but increases the number of observations. On the other hand, a less granular dataset summarizes covariate data and simplifies datasets.
Geographical telematics data has also allowed driving conditions to be tracked continuously as the driver is exposed to accident risk. It is therefore possible to examine how driving in different areas at varying exposure levels affects road accident frequency.
We also consider other distinct exposure measures that have been underdeveloped in the literature. Indeed, access to telematics data with sufficient counts of observed accidents has limited these alternative perspectives. Our talk will focus on clarifying the distinction between time-driven and mileage-based modelling.
In brief, we will present the three challenges outlined above using traditional and state-of-the-art machine learning methods to provide a broad overview of the influence of telematics data on risk exposure.

Seasonal Generalized Lagrangian Katz INAR processes

This paper proposes a new class of Integer Autoregressive (INAR) processes (Al‑Osh and Alzaid, 1987) for count data with seasonal features. We build on the general class of Generalized Lagrangian Katz INAR (GLKINAR) processes recently proposed in Baltodano et al. (2025) and extend them to a seasonal INAR specification (GLK‑SINAR). The Generalized Lagrangian Katz is a flexible distribution family, as it nests the Generalized Poisson and Negative Binomial distributions as special cases and can accommodate various data features, including over‑ and under‑dispersion, heavy tails, and skewness.
We present a seasonal specification combining the thinning operator (Weiss, 2008) and the lag operator (Bourguignon et al., 2016), and derive relevant theoretical properties, including stationarity and conditional moments. A Bayesian framework is proposed to perform inference and forecasting, and a posterior approximation procedure based on Gibbs sampling is designed. The efficiency of the procedure and the effectiveness of the inference are demonstrated using simulated data.
We consider an application to the count series of vessels following the Cape of Good Hope route. In recent decades, container vessel count data have become of interest for many applications, including monitoring climate risk, predicting dynamics in global economic activity, food security, and medical supply shortages (Brancaccio et al., 2020). The Cape of Good Hope series is particularly interesting, as cargoes take this route when draft restrictions are in place or to avoid congestion in the Suez Canal. We show that the GLK‑SINAR captures the periodic behaviour and provides better forecasts than standard GLK‑INAR models.

References
Al‑Osh, M. A., & Alzaid, A. A. (1987). First‑order integer‑valued autoregressive (INAR(1)) process. Journal of Time Series Analysis, 8, 261–275.
Baltodano, O., Bassetti, F., Carallo, G., & Casarin, R. (2025). First‑order integer‑valued autoregressive processes with Generalized Katz innovations. Econometrics and Statistics (Part B), forthcoming.
Brancaccio, G., Kalouptsidi, M., & Papageorgiou, T. (2020). Geography, transportation, and endogenous trade costs. Econometrica, 88(2), 657–691.
Bourguignon, M., Vasconcellos, K. L. P., Reisen, V. A., & Ispány, M. (2016). A Poisson INAR(1) process with a seasonal structure. Journal of Statistical Computation and Simulation, 86(2), 373–387.
Weiss, C. H. (2008). Thinning operations for modelling time series of counts – a survey. Advances in Statistical Analysis, 92, 319–341.

Bayesian estimation of Cox processes

Cox processes are popular models for risks where losses appear to form clusters randomly distributed on time. They are a generalization of compound Poisson processes, with the claim arrival intensity being a stochastic process instead of a constant. For example, catastrophes and other adverse events can be modeled by large transient peaks in the intensity process. However, in many practical applications, the claim arrival intensity is not directly observable or is imperfectly observable. Further, since in many cases peaks in the intensity are much less frequent than individual claim events, data about frequency and size of peaks is often sparse. Thus, estimation of Cox models is usually difficult.
We propose a nonparametric Bayesian approach to the estimation of such processes. A numerical scheme yields approximate calculations which can be applied in real time as new data arrives and is processed.

Effects of Traditional Reinsurance on Demographic Risk Under Solvency II

This paper analyses the role of traditional proportional reinsurance as an effective risk management instrument for demographic risk in life insurance portfolios within the Solvency II regulatory framework. While existing literature has largely focused on stochastic mortality modelling or on specialized longevity-linked reinsurance instruments, comparatively little attention has been paid to the use of standard proportional reinsurance treaties, such as quota-share and surplus, in a fully market-consistent valuation setting. This study addresses this gap by proposing a comprehensive analytical framework that integrates proportional reinsurance into both the valuation of technical provisions and the calculation of the Solvency Capital Requirement (SCR).
The analysis is developed for a homogeneous cohort of policyholders holding traditional life insurance contracts and is grounded in the market-consistent principles prescribed by Solvency II. Liabilities are valued using a cohort-based approach and replicated through valuation portfolios consistent with fair value accounting. A central contribution of the paper is the introduction and decomposition of the net-of-reinsurance Claims Development Result, which provides a transparent link between profitability and solvency over a one-year horizon. This decomposition allows the separate identification of idiosyncratic demographic risk and trend risk, and highlights how reinsurance affects both risk mitigation and the transfer of expected profits to the reinsurer.
Within this framework, proportional reinsurance is formally incorporated into the structure of cash flows, technical provisions, and solvency capital calculations. The impact of different reinsurance structures on the demographic SCR is examined, together with the corresponding effects on expected profitability. Numerical analyses illustrate how quota-share and surplus treaties can significantly reduce capital requirements, while also revealing the trade-offs associated with reinsurer pricing and profit sharing. Particular attention is paid to the role of portfolio characteristics, such as the heterogeneity of sums insured, which critically influences diversification effects and the relative effectiveness of alternative reinsurance arrangements.
The results show that proportional reinsurance can be a flexible and capital-efficient tool for managing demographic risk under Solvency II, provided that treaty design and pricing are carefully aligned with the insurer’s risk profile and business objectives. The proposed methodology is highly adaptable and applicable to a wide range of traditional life insurance products. Overall, the paper contributes to both the academic literature and actuarial practice by offering a coherent, market-consistent framework for evaluating the solvency and profitability implications of traditional reinsurance strategies, thereby supporting more informed decision-making in insurance risk management.
Main References

Bianchessi, E.; Clemente, G.P.; Della Corte, F.; Savelli, N. Effects of Traditional Reinsurance on Demographic Risk Under the Solvency II Framework. Risks 2025, 13, 203.
Wüthrich, M. V., Bühlmann, H., & Furrer, H. (2010). Market-consistent actuarial valuation. Berlin: Springer.

Systemic Risk Amplification in Pension Fund Networks: Evidence from the COVID-19 Crisis

Pension funds manage over €50 trillion globally with investment horizons spanning 20–40 years, yet systemic risk propagation in pension networks remains severely understudied compared to banking systems. This paper provides a comprehensive empirical analysis of cascade dynamics in pension fund networks, using unique data from Lithuanian pension funds covering the entire working-age population (ages 23–71) with over 70,000 daily observations spanning 2019–2025.
We document extreme systemic risk amplification during the COVID‑19 crisis: system‑wide failures exhibit 19.5–20× amplification relative to independence benchmarks, with 98–100% of funds breaching Value‑at‑Risk limits simultaneously on peak stress days despite being managed by six independent providers across eight distinct age cohorts. This synchronization persists across all analysed periods — pre‑COVID (39/40 funds failed), during‑COVID (40/40 funds), and post‑COVID (48/48 funds) — indicating structural rather than cyclical fragility.
Applying network amplification analysis to pension funds, we demonstrate that portfolio homogeneity driven by lifecycle mandates dominates apparent diversification benefits. Funds exhibit correlations of 0.88–0.96 throughout the sample period, effectively moving as a single entity. Our findings reveal three critical patterns:
(1) diversification across managers and age profiles provides negligible protection during systemic stress;
(2) amplification intensifies post‑crisis rather than diminishing, with the system becoming more interconnected; and
(3) the 10% systemic stress threshold is exceeded far more frequently in the post‑COVID period, indicating persistent elevated fragility.
These results raise urgent concerns for climate transition planning. If a temporary two‑year pandemic triggered 19.5–20× amplification and complete cascade failures, permanent climate mandates, which explicitly coordinate portfolio changes through sustainability requirements, could produce even larger systemic failures. Current pension regulation focused on individual fund solvency appears insufficient when portfolio homogeneity creates network effects that amplify shocks catastrophically.
We discuss policy implications for managing climate transition without triggering systemic cascades. Our amplification analysis demonstrates the need for macro‑prudential monitoring frameworks that quantify systemic vulnerability beyond individual fund solvency and preserve genuine portfolio diversification as pension systems transition to sustainable finance under climate mandates.
Keywords: Systemic risk, pension funds, network contagion, portfolio homogeneity, climate transition, financial stability.

When markets jump: option pricing and flash crashes in cryptocurrencies

The cryptocurrency derivatives market is distinguished by extreme volatility and relatively low liquidity when compared to traditional financial markets. These characteristics pose significant challenges for classical option pricing models that were originally developed under assumptions of continuous trading. In the first part, we concentrate on the problem of pricing vanilla options written on cryptocurrency futures contracts. The empirical analysis focuses on options on Bitcoin (BTC) and Ether (ETH) futures.
We calibrate and evaluate a range of widely used option pricing models, including the Black–Scholes model, the Merton Jump Diffusion model, the Variance Gamma model, the Kou double‑exponential jump model, the Heston stochastic volatility model, and the Bates model combining stochastic volatility with jumps. All models are calibrated to real market data, allowing for a consistent comparison of their ability to reproduce observed option prices and implied volatility surfaces. The results confirm that purely diffusion‑based models fail to capture key empirical features of cryptocurrency options, while models incorporating jumps and stochastic volatility provide a better fit.
To ensure robustness of the empirical findings, model performance is assessed across different maturities. We additionally discuss the stability of calibrated parameters over time and highlight the practical implications of model choice for pricing, hedging, and risk management in cryptocurrency derivatives markets.
The second part shifts attention to extreme market events, with a particular focus on flash crashes in cryptocurrency markets. Unlike traditional equity markets, cryptocurrency markets operate continuously, without trading halts, which complicates both the definition and detection of flash crashes. We discuss several definitions proposed in the literature and show how they can be applied to cryptocurrencies. In particular, we examine the Parisian ruin model and the Peaks‑Over‑Threshold method. We illustrate how these approaches can be employed to identify extreme downside events and quantify their severity and duration.
We conclude by highlighting the conceptual and empirical links between derivative pricing models and flash crash dynamics, arguing that information extracted from option markets can provide valuable insights into tail risk and the mechanisms underlying extreme price dislocations in cryptocurrency markets. In particular, parameters related to jumps and stochastic volatility are shown to be closely connected to the frequency and magnitude of extreme market events.

Incentive-Based Prepayment Modeling Integrated with Hull–White PDE Pricing

Mortgage‑backed securities (MBS) expose investors to significant prepayment risk, arising from borrowers’ option to refinance or prepay their mortgages when market conditions become favorable. While agency guarantees largely eliminate credit risk, accurate modeling of prepayment behavior remains essential for valuation, hedging, and risk management. This work proposes an integrated framework for modeling and pricing agency MBS by combining a Hull–White one‑factor interest rate model with a probabilistic, incentive‑driven prepayment specification.
The valuation backbone is provided by the Hull–White short‑rate model, which ensures consistency with the observed term structure of interest rates and leads to a partial differential equation (PDE) governing the value of fixed‑income instruments. For MBS valuation, the standard bond pricing PDE is augmented by coupon payments, an option‑adjusted spread, and a prepayment intensity term that acts as a time‑ and state‑dependent killing rate. The resulting PDE explicitly links interest rate dynamics with mortgage amortization behavior.
Prepayment is modeled using a generalized Vasicek framework interpreted in a probabilistic manner. Borrower prepayment is treated as a threshold event on a latent Gaussian variable driven by a systematic market factor and an idiosyncratic component. The key observable explaining refinancing incentives is the incentive ratio, defined as the ratio of the current coupon yield of a par‑priced 30‑year agency MBS to the weighted average coupon (WAC) of the mortgage pool. This ratio is transformed into a standardized market factor through an empirical quantile mapping followed by a probit transformation. Conditional prepayment probabilities are then obtained via a probit regression, producing monthly single‑month mortality (SMM) rates and annualized conditional prepayment rates (CPR). Model parameters are calibrated to historical agency data under fixed assumptions for the unconditional prepayment rate and the correlation between borrowers.
To enable forward‑looking valuation, the framework includes a projection mechanism for all prepayment inputs. The current coupon yield is forecast using a regression on medium‑term swap rates, whose future paths are generated consistently from the Hull–White model. The WAC is modeled as a slowly adjusting process that converges toward projected market mortgage rates. Combining these projections yields a forward incentive ratio, which is mapped through the calibrated probit model to obtain a projected prepayment intensity.
A central contribution of this work is the construction of a two‑dimensional prepayment intensity grid, , defined over time and short‑rate states. At each grid node, swap rates implied by the Hull–White model determine the current coupon yield, which feeds into the incentive‑based prepayment mapping. This grid is then embedded directly into a finite‑difference solver for the Hull–White PDE, ensuring full internal consistency between interest rate dynamics and prepayment behavior.
The proposed framework provides a coherent and transparent link between market rates, borrower incentives, and MBS valuation, offering a flexible foundation for further extensions such as seasoning effects, burnout, or alternative interest rate models.