arXiv subjects

Andrea Molent

Publications and source records attributed to Andrea Molent.

17 recordsLinked to original sources

arXiv2026

Adaptive singular-point method for pricing and hedging surrenderable equity-linked contracts

We propose a deterministic numerical method for pricing and hedging surrenderable equity-linked life-insurance contracts with periodic premiums and fund contributions, maturity and death guarantees, and Bermudan surrender under correlated stochastic volatility and stochastic interest rates. The main computational challenge is the non-recombining accumulated fund, which couples with multiple stochastic factors and early exercise. Our key idea is to avoid a full multidimensional fund lattice: variance and interest-rate factors are discretized on recombining lattices, while at each factor node the contract value is represented as an adaptive one-dimensional function of the fund. Periodic contributions then act as translations of the fund argument, whereas surrender is handled directly through a backward obstacle condition. Piecewise-cubic representations propagate payoff and exercise singularities and are compressed by continuous pruning criteria that control the representation error. We establish weak convergence of the financial chains, convergence of the adaptive valuation under vanishing representation error, and Delta consistency on regular fund regions. For the strict binomial scheme, additional regularity yields first-order weak accuracy and a Talay-Tubaro expansion supporting Richardson extrapolation. Numerical experiments show compact representations, favorable cost--accuracy, and close agreement with independent Monte Carlo and cross-fitted least-squares Monte Carlo benchmarks. Hedging results further show that contracts with similar values can generate materially different exposures to equity, volatility, and interest-rate risk.

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arXiv2026

Neural Calibration of a Complete Market Model

We propose a neural calibration method to construct a recombining binomial tree directly from a set of given option prices. Rather than estimating a continuous option pricing function or a local volatility surface as an intermediate object, a neural network is used to deform a benchmark lattice. This leads to a discrete pricing model which is guaranteed to be arbitrage-free, complete, easy to interpret, and can be used directly for pricing and to find replicating trading strategies. Calibration is formulated as a penalized optimization problem that combines a repricing error with an admissibility penalty, and an optional spatial regularization term based on implied local volatilities. Numerical experiments on synthetic and SPX market data show that the proposed approach yields accurate repricing and is very competitive when compared to recently proposed other neural calibration methods. It preserves the computational advantages of lattice-based valuation and hedging. In particular, the calibrated tree can be reused to price contracts that allow early exercise, and could even be calibrated directly with American option prices.

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arXiv2026

Market-Informed Valuation of GMMB Riders with Surrender Options under a Heston Stochastic-Local Volatility Model

We develop a market-informed valuation framework for guaranteed minimum maturity benefit (GMMB) riders with rational surrender under the Heston stochastic-local volatility (SLV) model. The guarantee is written on the fee-deducted account value and is considered both in its terminal-only form and in the presence of early surrender rights. The Heston SLV specification combines stochastic volatility with a leverage function calibrated to a prescribed local-volatility surface. The leverage surface is obtained through a forward Markovian-projection equation so that, at the model level, the SLV dynamics are constrained to the same one-dimensional marginals as the corresponding local-volatility (LV) model. The latter is used only as a one-factor benchmark, allowing us to isolate the effect of stochastic volatility on continuation values and surrender decisions while preserving the same option-calibrated local-volatility target. We derive the associated backward pricing equations and propose a hybrid tree/finite-difference algorithm for the SLV model with a calibrated leverage function. Synthetic experiments and a market-informed case study show that SLV and LV valuations are numerically close for terminal-only guarantees, as expected from the common marginal target, whereas materially larger differences can arise once surrender is allowed. These differences are reflected in guarantee values, fair insurance fees and volatility-dependent surrender regions. The results indicate that matching one-date marginals implied by vanilla-option prices does not eliminate model risk for insurance liabilities whose value depends on conditional continuation dynamics and endogenous surrender decisions.

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arXiv2026

Valuation of GLWB-LTC Annuities with L\'evy Equity Dynamics, Stochastic Interest Rates and Health-State Transitions

This paper develops a valuation framework for guaranteed lifetime withdrawal benefit (GLWB) contracts with long-term care (LTC) features when the reference fund follows exponential Levy dynamics and the short rate follows the Hull-White model. The contract combines financial guarantees, longevity protection, health-contingent LTC payments, and surrender optionality, requiring the joint treatment of jump risk, stochastic discounting, and disability risk. The numerical method couples a recombining Hull-White trinomial tree with an implicit-explicit (IMEX) finite difference scheme. The framework incorporates a seven-state health model, annual fees, LTC payments, guaranteed withdrawals, and bang-bang policyholder actions, and is benchmarked against Monte Carlo simulation. Numerical results show that the hybrid tree-IMEX method delivers stable long-maturity prices consistent with simulation benchmarks. They also show that Levy equity dynamics and stochastic interest rates have a material impact on fair fees and surrender incentives, and affect the decomposition of contract value. The findings highlight the importance of modelling financial tail risk and interest-rate risk jointly when pricing long-term insurance guarantees with LTC-contingent benefits.

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arXiv2025

Leveraging Machine Learning for High-Dimensional Option Pricing within the Uncertain Volatility Model

This paper explores the application of Machine Learning techniques for pricing high-dimensional options within the framework of the Uncertain Volatility Model (UVM). The UVM is a robust framework that accounts for the inherent unpredictability of market volatility by setting upper and lower bounds on volatility and the correlation among underlying assets. By integrating advanced Machine Learning algorithms, we aim to enhance the accuracy and efficiency of option pricing under the UVM, especially when the option price depends on a large number of variables, such as in basket or path-dependent options. In this paper, we consider two approaches based on Machine Learning. The first one, termed GTU, evolves backward in time, dynamically selecting at each time step the most expensive volatility and correlation for each market state. Specifically, it identifies the particular values of volatility and correlation that maximize the expected option value at the next time step, and therefore, an optimization problem must be solved. This is achieved through the use of Gaussian Process regression, the computation of expectations via a single step of a multidimensional tree and the Sequential Quadratic Programming optimization algorithm. The second approach, referred to as NNU, leverages neural networks and frames pricing in the UVM as a control problem. Specifically, we train a neural network to determine the most adverse volatility and correlation for each simulated market state, generated via random simulations. The option price is then obtained through Monte Carlo simulations, which are performed using the values for the uncertain parameters provided by the neural network. The numerical results demonstrate that the proposed approaches can significantly improve the precision of option pricing particularly in high-dimensional contexts.

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arXiv2025

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.

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arXiv2024

Enhancing Valuation of Variable Annuities in Lévy Models with Stochastic Interest Rate

This paper extends the valuation and optimal surrender framework for variable annuities with guaranteed minimum benefits in a Lévy equity market environment by incorporating a stochastic interest rate described by the Hull-White model. This approach frames a more dynamic and realistic financial setting compared to previous literature. We exploit a robust valuation mechanism employing a hybrid numerical method that merges tree methods for interest rate modeling with finite difference techniques for the underlying asset price. This method is particularly effective for addressing the complexities of variable annuities, where periodic fees and mortality risks are significant factors. Our findings reveal the influence of stochastic interest rates on the strategic decision-making process concerning the surrender of these financial instruments. Through comprehensive numerical experiments, and by comparing our results with those obtained through the Longstaff-Schwartz Monte Carlo method, we illustrate how our refined model can guide insurers in designing contracts that equitably balance the interests of both parties. This is particularly relevant in discouraging premature surrenders while adapting to the realistic fluctuations of financial markets. Lastly, a comparative statics analysis with varying interest rate parameters underscores the impact of interest rates on the cost of the optimal surrender strategy, emphasizing the importance of accurately modeling stochastic interest rates.

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arXiv2023

Backward Hedging for American Options with Transaction Costs

In this article, we introduce an algorithm called Backward Hedging, designed for hedging European and American options while considering transaction costs. The optimal strategy is determined by minimizing an appropriate loss function, which is based on either a risk measure or the mean squared error of the hedging strategy at maturity. The proposed algorithm moves backward in time, determining, for each time-step and different market states, the optimal hedging strategy that minimizes the loss function at the time the option is exercised, by assuming that the strategy used in the future for hedging the liability is the one determined at the previous steps of the algorithm. The approach avoids machine learning and instead relies on classic optimization techniques, Monte Carlo simulations, and interpolations on a grid. Comparisons with the Deep Hedging algorithm in various numerical experiments showcase the efficiency and accuracy of the proposed method.

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arXiv2022

Computing XVA for American basket derivatives by Machine Learning techniques

Total value adjustment (XVA) is the change in value to be added to the price of a derivative to account for the bilateral default risk and the funding costs. In this paper, we compute such a premium for American basket derivatives whose payoff depends on multiple underlyings. In particular, in our model, those underlying are supposed to follow the multidimensional Black-Scholes stochastic model. In order to determine the XVA, we follow the approach introduced by Burgard and Kjaer \cite{burgard2010pde} and afterward applied by Arregui et al. \cite{arregui2017pde,arregui2019monte} for the one-dimensional American derivatives. The evaluation of the XVA for basket derivatives is particularly challenging as the presence of several underlings leads to a high-dimensional control problem. We tackle such an obstacle by resorting to Gaussian Process Regression, a machine learning technique that allows one to address the curse of dimensionality effectively. Moreover, the use of numerical techniques, such as control variates, turns out to be a powerful tool to improve the accuracy of the proposed methods. The paper includes the results of several numerical experiments that confirm the goodness of the proposed methodologies.

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arXiv2021

Moving average options: Machine Learning and Gauss-Hermite quadrature for a double non-Markovian problem

Evaluating moving average options is a tough computational challenge for the energy and commodity market as the payoff of the option depends on the prices of a certain underlying observed on a moving window so, when a long window is considered, the pricing problem becomes high dimensional. We present an efficient method for pricing Bermudan style moving average options, based on Gaussian Process Regression and Gauss-Hermite quadrature, thus named GPR-GHQ. Specifically, the proposed algorithm proceeds backward in time and, at each time-step, the continuation value is computed only in a few points by using Gauss-Hermite quadrature, and then it is learned through Gaussian Process Regression. We test the proposed approach in the Black-Scholes model, where the GPR-GHQ method is made even more efficient by exploiting the positive homogeneity of the continuation value, which allows one to reduce the problem size. Positive homogeneity is also exploited to develop a binomial Markov chain, which is able to deal efficiently with medium-long windows. Secondly, we test GPR-GHQ in the Clewlow-Strickland model, the reference framework for modeling prices of energy commodities. Finally, we consider a challenging problem which involves double non-Markovian feature, that is the rough-Bergomi model. In this case, the pricing problem is even harder since the whole history of the volatility process impacts the future distribution of the process. The manuscript includes a numerical investigation, which displays that GPR-GHQ is very accurate and it is able to handle options with a very long window, thus overcoming the problem of high dimensionality.

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arXiv2020

Taxation of a GMWB Variable Annuity in a Stochastic Interest Rate Model

Modeling taxation of Variable Annuities has been frequently neglected but accounting for it can significantly improve the explanation of the withdrawal dynamics and lead to a better modeling of the financial cost of these insurance products. The importance of including a model for taxation has first been observed by Moenig and Bauer (2016) while considering a GMWB Variable Annuity. In particular, they consider the simple Black-Scholes dynamics to describe the underlying security. Nevertheless, GMWB are long term products and thus accounting for stochastic interest rate has relevant effects on both the financial evaluation and the policy holder behavior, as observed by Goudenège et al. (2018). In this paper we investigate the outcomes of these two elements together on GMWB evaluation. To this aim, we develop a numerical framework which allows one to efficiently compute the fair value of a policy. Numerical results show that accounting for both taxation and stochastic interest rate has a determinant impact on the withdrawal strategy and on the cost of GMWB contracts. In addition, it can explain why these products are so popular with people looking for a protected form of investment for retirement.

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arXiv2019

Variance Reduction Applied to Machine Learning for Pricing Bermudan/American Options in High Dimension

In this paper we propose an efficient method to compute the price of multi-asset American options, based on Machine Learning, Monte Carlo simulations and variance reduction technique. Specifically, the options we consider are written on a basket of assets, each of them following a Black-Scholes dynamics. In the wake of Ludkovski's approach (2018), we implement here a backward dynamic programming algorithm which considers a finite number of uniformly distributed exercise dates. On these dates, the option value is computed as the maximum between the exercise value and the continuation value, which is obtained by means of Gaussian process regression technique and Monte Carlo simulations. Such a method performs well for low dimension baskets but it is not accurate for very high dimension baskets. In order to improve the dimension range, we employ the European option price as a control variate, which allows us to treat very large baskets and moreover to reduce the variance of price estimators. Numerical tests show that the proposed algorithm is fast and reliable, and it can handle also American options on very large baskets of assets, overcoming the problem of the curse of dimensionality.

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arXiv2019

Gaussian Process Regression for Pricing Variable Annuities with Stochastic Volatility and Interest Rate

In this paper we investigate price and Greeks computation of a Guaranteed Minimum Withdrawal Benefit (GMWB) Variable Annuity (VA) when both stochastic volatility and stochastic interest rate are considered together in the Heston Hull-White model. We consider a numerical method the solves the dynamic control problem due to the computing of the optimal withdrawal. Moreover, in order to speed up the computation, we employ Gaussian Process Regression (GPR). Starting from observed prices previously computed for some known combinations of model parameters, it is possible to approximate the whole price function on a defined domain. The regression algorithm consists of algorithm training and evaluation. The first step is the most time demanding, but it needs to be performed only once, while the latter is very fast and it requires to be performed only when predicting the target function. The developed method, as well as for the calculation of prices and Greeks, can also be employed to compute the no-arbitrage fee, which is a common practice in the Variable Annuities sector. Numerical experiments show that the accuracy of the values estimated by GPR is high with very low computational cost. Finally, we stress out that the analysis is carried out for a GMWB annuity but it could be generalized to other insurance products.

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arXiv2019

Machine Learning for Pricing American Options in High-Dimensional Markovian and non-Markovian models

In this paper we propose two efficient techniques which allow one to compute the price of American basket options. In particular, we consider a basket of assets that follow a multi-dimensional Black-Scholes dynamics. The proposed techniques, called GPR Tree (GRP-Tree) and GPR Exact Integration (GPR-EI), are both based on Machine Learning, exploited together with binomial trees or with a closed formula for integration. Moreover, these two methods solve the backward dynamic programming problem considering a Bermudan approximation of the American option. On the exercise dates, the value of the option is first computed as the maximum between the exercise value and the continuation value and then approximated by means of Gaussian Process Regression. The two methods mainly differ in the approach used to compute the continuation value: a single step of binomial tree or integration according to the probability density of the process. Numerical results show that these two methods are accurate and reliable in handling American options on very large baskets of assets. Moreover we also consider the rough Bergomi model, which provides stochastic volatility with memory. Despite this model is only bidimensional, the whole history of the process impacts on the price, and handling all this information is not obvious at all. To this aim, we present how to adapt the GPR-Tree and GPR-EI methods and we focus on pricing American options in this non-Markovian framework.

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arXiv2018

Computing Credit Valuation Adjustment solving coupled PIDEs in the Bates model

Credit value adjustment (CVA) is the charge applied by financial institutions to the counterparty to cover the risk of losses on a counterpart default event. In this paper we estimate such a premium under the Bates stochastic model (Bates [4]), which considers an underlying affected by both stochastic volatility and random jumps. We propose an efficient method which improves the finite-difference Monte Carlo (FDMC) approach introduced by de Graaf et al. [11]. In particular, the method we propose consists in replacing the Monte Carlo step of the FDMC approach with a finite difference step and the whole method relies on the efficient solution of two coupled partial integro-differential equations (PIDE) which is done by employing the Hybrid Tree-Finite Difference method developed by Briani et al. [6, 7, 8]. Moreover, the direct application of the hybrid techniques in the original FDMC approach is also considered for comparison purposes. Several numerical tests prove the effectiveness and the reliability of the proposed approach when both European and American options are considered.

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arXiv2016

Pricing and Hedging GMWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

Valuing Guaranteed Minimum Withdrawal Benefit (GMWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Yang and Dai, the Black and Scholes framework seems to be inappropriate for such a long maturity products. Also Chen Vetzal and Forsyth in showed that the price of these products is very sensitive to interest rate and volatility parameters. We propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GMWB variables annuities: a hybrid tree-finite difference method and a Hybrid Monte Carlo method, an ADI finite difference scheme, and a Standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GMWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal, optimal surrender and optimal withdrawal strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.

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arXiv2015

Pricing and Hedging GLWB in the Heston and in the Black-Scholes with Stochastic Interest Rate Models

Valuing Guaranteed Lifelong Withdrawal Benefit (GLWB) has attracted significant attention from both the academic field and real world financial markets. As remarked by Forsyth and Vetzal the Black and Scholes framework seems to be inappropriate for such long maturity products. They propose to use a regime switching model. Alternatively, we propose here to use a stochastic volatility model (Heston model) and a Black Scholes model with stochastic interest rate (Hull White model). For this purpose we present four numerical methods for pricing GLWB variables annuities: a hybrid tree-finite difference method and a hybrid Monte Carlo method, an ADI finite difference scheme, and a standard Monte Carlo method. These methods are used to determine the no-arbitrage fee for the most popular versions of the GLWB contract, and to calculate the Greeks used in hedging. Both constant withdrawal and optimal withdrawal (including lapsation) strategies are considered. Numerical results are presented which demonstrate the sensitivity of the no-arbitrage fee to economic, contractual and longevity assumptions.

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