Donatien Hainaut

• PhD
• Professor at Catholic University of Louvain

We introduce the Volterra Stein-Stein model with stochastic interest rates, where both volatility and interest rates are driven by correlated Gaussian Volterra processes. This framework unifies various well-known Markovian and non-Markovian models while preserving analytical tractability for pricing and hedging financial derivatives. We derive expl...

The Amihud illiquidity measure has proven to be very popular in the economic and financial literature for measuring the illiquidity process of stocks and indices. None of the existing discrete-time illiquidity models in the literature are however adapted for reproducing peaks of illiquidity with long memory and for the management of the liquidity r...

The K-means algorithm and its variants are well-known clustering techniques. In actuarial applications, these partitioning methods can identify clusters of policies with similar attributes. The resulting partitions provide an actuarial framework for creating maps of dominant risks and unsupervised pricing grids. This research article aims to adapt...

In absence of a closed form expression such as in the Heston model, the option pricing is computationally intensive when calibrating a model to market quotes. this article proposes an alternative to standard pricing methods based on physics-inspired neural networks (PINNs). A PINN integrates principles from physics into its learning process to enha...

Guaranteed minimum accumulation benefits (GMABs) are retirement savings vehicles that protect the policyholder against downside market risk. This article proposes a valuation method for these contracts based on physics-inspired neural networks (PINNs), in the presence of multiple financial and biometric risk factors. A PINN integrates principles fr...

Classic diffusion processes fail to explain asset return volatility. Many empirical findings on asset return time series, such as heavy tails, skewness and volatility clustering, suggest decomposing the volatility of an asset’s return into two components, one caused by a Brownian motion and another by a jump process. We analyze the sensitivity of E...

This article introduces a new class of diffusive processes with rough mutually exciting jumps for modeling financial asset returns. The novel feature is that the memory of positive and negative jump processes is defined by the product of a dampening factor and a kernel involved in the construction of the rough Brownian motion. The jump processes ar...

The least squares Monte Carlo method has become a standard approach in the insurance and financial industries for evaluating a company’s exposure to market risk. However, the non-linear regression of simulated responses on risk factors poses a challenge in this procedure. This article presents a novel approach to address this issue by employing an...

Subdiffusions appear as good candidates for modeling illiquidity in financial markets. Existing subdiffusive models of asset prices are indeed able to capture the motionless periods in the quotes of thinly-traded assets. However, they fail at reproducing simultaneously the jumps and the time-varying random volatility observed in the price of these...

We propose in this paper a new framework of optimal liquidation strategies for a trader seeking to liquidate his large inventory based on a jump-dependent price impact model with propagator. This new jump-dependent price impact model allows to best reproduce the empirical direct and indirect effects of market orders on the transaction price. More p...

This article proposes a continuous time mortality model based on calendar years. Mortality rates belong to a mean-reverting random field indexed by time and age. In order to explain the improvement of life expectancies, the reversion level of mortality rates is the product of a deterministic function of age and of a decreasing jump-diffusion proces...

We have seen in Chap. 4 that self-excited processes offer a natural way to introduce contagion between shocks in financial markets. In this approach, the occurrence of a shock depends on previous ones. In the most common specification, the intensity of jumps, that is akin to the instantaneous probability of a shock, increases as soon as a jump is o...

The pricing of exotic options with a payoff involving asset prices at different times requires a model capable of explaining the covariance of underlying securities. Assuming that asset returns are ruled by a Brownian motion with drift is convenient for mathematical developments. However, this model does not replicate the time dependence observed f...

The previous chapters have initiated our journey into the world of processes that are not martingales. We introduce in this chapter a new category of processes perfectly adapted for modeling illiquidity. In emerging or in small cap markets, the number of participants is often low, and thus transactions are sparse. The time series of stock prices in...

From the 1980s to the present, many interest rate models have been proposed in the literature. They all aim to explain changes in bond or swap quotes and to replicate risks within the interest rates market. Three dominating frameworks coexist: short-term rate, forward rate, and the Libor market models. In this last approach, proposed by Brace et al...

The switching regime processes of Chap. 1 and the Heston model of Chap. 3 cannot duplicate the clustering of jumps in stock markets caused, for example, by an onslaught of bad news. Self-excited processes offer an interesting way to introduce such spillover effects into the market dynamics. In this approach, the occurrence of a shock depends on the...

There is considerable empirical evidence suggesting that the random walk model for changes in stock prices is not appropriate. One of the reasons is that this model fails to account for economic cycles because increments are independent and identically distributed. A reliable solution for modeling economic cycles consists in modulating the paramete...

The previous chapters provide empirical evidence that models with stochastic volatility outperform their deterministic counterpart. In Chap. 1, the multifractal process competes with GARCH models, whereas the Heston model of Chap. 3 achieves a better likelihood than the Black and Scholes model. On the other hand, models based on fractional Brownian...

In Chap. 10, we managed illiquidity in a Black and Scholes framework with an appropriate time change. Such an approach can be extended to jump-diffusions. Nevertheless, option pricing is a challenging task in this framework mainly because there is no analytical formula for options in the non-time-changed model. This chapter explores a new approach...

The previous chapters studied processes that depend on the convolution of a function and their past sample path. For instance, the fractional Brownian motion of Chap. 6 is proportional to \(\int _{0}^{t}\left (t-u\right )^{H-\frac {1}{2}}\mathrm {d}W_{u}\), where W u is a Brownian motion. In a similar manner, the interest rate model of Chap. 8 in t...

This chapter introduces an algorithm called particle filtering used for the inference of the most likely sample path of a hidden process driving stock prices in nested models. Particle filtering is a simulation-based method approximating the likelihood of observations. This approach allows us to fit processes for which the probability density funct...

A Rough process shares most of features of fractional Brownian motion with a small Hurst index and its sample paths exhibit a high ruggedness compared to those of a Brownian motion. This article studies a multivariate claim process in which the instantaneous probability of claim occurrences has a rough dynamic. In this setting, the claim arrival in...

The Rough Fractional Stochastic Volatility (RFSV) model of Gatheral et al. (Quant Financ 18(6):933–949, 2014) is remarkably consistent with financial time series of past volatility data as well as with the observed implied volatility surface. Two tractable implementations are derived from the RFSV with the rBergomi model of Bayer et al. (Quant Fina...

Thanks to its outstanding performances, boosting has rapidly gained wide acceptance among actuaries. To speed up calculations, boosting is often applied to gradients of the loss function, not to responses (hence the name gradient boosting). When the model is trained by minimizing Poisson deviance, this amounts to apply the least-squares principle t...

This article investigates the valuation of annuity guarantees under a regime-switching model when the dynamics of the underlying stock price follow a self-exciting switching jump-diffusion process. In this framework, we add a jump component to a regime-switching geometric Brownian for large shocks on the stock price. The intensity of shock arrivals...

Subdiffusions appear as good candidates for modeling illiquidity in financial markets. Existing subdiffusive models of asset prices are indeed able to capture the motionless periods in the quotes of thinly-traded assets. However, they fail at reproducing the jumps and the time-varying volatility observed in the price of these assets. The aim of thi...

This article proposes an interest rate model ruled by mean reverting Lévy processes with a sub-exponential memory of their sample path. This feature is achieved by considering an Ornstein–Uhlenbeck process in which the exponential decaying kernel is replaced by a Mittag–Leffler function. Based on a representation in term of an infinite dimensional...

This article establishes the moment generating function (mgf) of self-excited claim processes with memory functions that admit a Fourier's transform representation. In this case, the claim and intensity processes may be reformulated as an infinite dimensional Markov processes in the complex plane. Approaching these processes by discretization and n...

A time-consistent evaluation is a dynamic pricing method according to which a risk that will be almost surely cheaper than another one at a future date should already be cheaper today. Common actuarial pricing approaches are usually not time-consistent. Pelsser and Ghalehjooghi (2016) derived time-consistent valuation principles from time-inconsist...

We propose a fractional self-exciting model for the risk of corporate default. We study the properties of a time-changed version of an intensity based model. As a time-change, we use the inverse of an α-stable subordinator. Performing such a time-change allows to incorporate two particular features in the survival probability curves implied by the...

Bagging trees and random forests base their predictions on an ensemble of trees. In this chapter, we consider another training procedure based on an ensemble of trees, called boosting trees. However, the way the trees are produced and combined differ between random forests (and so bagging trees) and boosting trees.

In actuarial pricing, the objective is to evaluate the pure premium as accurately as possible. The target is thus the conditional expectation \(\mu (\textit{\textbf{X}})=\text {E}[Y|\textit{\textbf{X}}]\) of the response Y (claim number or claim amount for instance) given the available information \(\textit{\textbf{X}}\).

Actuarial pricing models are generally calibrated so that they minimize the generalization error computed with an appropriate loss function. Model selection is based on the generalization error.

In this chapter, we present the regression trees introduced by Breiman et al. (1984). Regression trees are at the core of this second volume.

Two ensemble methods are considered in this chapter, namely bagging trees and random forests. One issue with regression trees is their high variance. There is a high variability of the prediction over the trees trained from all possible training sets . Bagging trees and random forests aim to reduce the variance without too much altering bias.

The credit crunch of 2007 caused major changes in the market of interbank rates making the existing interest rate theory inconsistent. This article puts forward one way to reconcile practice and theory by modifying the arbitrage-free condition. In this framework, the forward Libor rate is no longer considered as a risk-free rate and the credit and...

Wavelet theory is known to be a powerful tool for compressing and processing time series or images. It consists in projecting a signal on an orthonormal basis of functions that are chosen in order to provide a sparse representation of the data. The first part of this article focuses on smoothing mortality curves by wavelets shrinkage. A chi-square...

We study the pricing of European options when the underlying stock price is illiquid. Due to the lack of trades, the sample path followed by prices alternates between active and motionless periods that are replicable by a fractional jump-diffusion. This process is obtained by changing the time-scale of a jump-diffusion with the inverse of a Lévy su...

Hawkes processes have a self-excitation mechanism used for modeling the clustering of events observed in natural or social phenomena. In the first part of this article, we find the forward differential equations ruling the probability density function and the Laplace’s transform of the intensity of a Hawkes process, with an exponential decaying ker...

This book summarizes the state of the art in tree-based methods for insurance: regression trees, random forests and boosting methods. It also exhibits the tools which make it possible to assess the predictive performance of tree-based models. Actuaries need these advanced analytical tools to turn the massive data sets now at their disposal into opp...

We propose a new approach for bivariate financial time series modelling which allows for mutual excitation between shocks. Jumps are triggered by changes of regime of a hidden Markov chain whose matrix of transition probabilities is constructed in order to approximate a bivariate Hawkes process. This model, called the Bivariate Mutually-Excited Swi...

In this chapter, we study a particular type of neural networks that are designed for providing a representation of the input with a reduced dimensionality. These networks contains a hidden layer, called bottleneck, that contains a few nodes compared to the previous layers. The output signals of neurons in the bottleneck carry a summarized informati...

The learning of large neural networks is an ill-posed problem and there is generally a continuum of possible set of admissible weights. In this case, we cannot rely anymore on asymptotic properties of maximum likelihood estimators to approximate confidence intervals. Applying the Bayesian learning paradigm to neural networks or to generalized linea...

In Chap. 1, our empirical analysis was based on neural networks with a single hidden layer. These networks, called shallow, are in theory universal approximators of any continuous function. Deep neural networks use instead a cascade of multiple layers of hidden neurons. Each successive layer uses the output from the previous layer as input. As with...

The main objective of time series analysis is to provide mathematical models that offer a plausible description for a sample of data indexed by time. Time series modelling may be applied in many different fields. In finance, it is used for explaining the evolution of asset returns. In actuarial sciences, it may be used for forecasting the number of...

The most frequent approach to data-driven modeling consists to estimate only a single strong predictive model. A different strategy is to build a bucket, or an ensemble of models for some particular learning task. One can consider building a set of weak or relatively weak models like small neural networks, which can be further combined altogether t...

This chapter introduces the general features of artificial neural networks. After a presentation of the mathematical neural cell, we focus on feed-forward networks. First, we discuss the preprocessing of data and next we present a survey of the different methods for calibrating such networks. Finally, we apply the theory to an insurance data set an...

Gradient boosting machines form a family of powerful machine learning techniques that have been applied with success in a wide range of practical applications. Ensemble techniques rely on simple averaging of models in the ensemble. The family of boosting methods adopts a different strategy to construct ensembles. In boosting algorithms, new models...

Feed-forward neural networks are algorithms with supervised learning. It means that we have to a priori identify the most relevant variables and to know the desired outputs for combinations of these variables. For example, forecasting the frequency of car accidents with a perceptron requires an a priori segmentation of some explanatory variables li...

technical price and commercial premiums

This chapter is devoted to the study of the family of Exponential Dispersion (or ED) distributions that are central to insurance data analytics techniques. The objective functions used to calibrate the regression models described in this book correspond to log-likelihoods taken from this family. This is why a good knowledge of these models is the n...

With GLMs, mean responses are modeled as monotonic functions of linear scores. The assumed linearity of the score is not restrictive for categorical features coded by means of binary variables. However, this assumption becomes questionable for continuous features which may have a nonlinear effect on the score scale. This chapter is devoted to Gener...

Data sets exhibiting a hierarchical or nested structure, or including longitudinal or spatial elements often arise in insurance studies. This generally results in correlation among the responses within the same group, casting doubts about the outputs of analyses assuming mutual independence. Random effects offer a convenient way to model such group...

In this chapter, the modeling of the mean response is supplemented with additional scores linked to other parameters of the distribution, like dispersion, scale, shape or probability mass at the origin, for instance. This allows the actuary to let the available information enter other dimensions of the response, such as volatility or no-claim proba...

Generalized Linear Models are widely known under their famous acronym GLMs. Today, GLMs are recognized as an industry standard for pricing personal lines and small commercial lines of insurance business. This chapter reviews the GLM methodology with a special emphasis to insurance problems. The statistical framework of GLMs allows the actuary to ma...

This chapter discusses a statistical modeling strategy based on extreme value theory to describe the behavior of data far in the tails of the distributions, with a particular emphasis on large claims in property and casualty insurance and mortality at oldest ages in life insurance. Large claims generally affect liability coverages and require a sep...

This chapter recalls the basics of the estimation method consisting in maximizing the likelihood associated to the observations. The resulting estimators enjoy convenient theoretical properties, being optimal in a wide variety of situations. The maximum likelihood principle will be used throughout the next chapters to fit the supervised learning mo...

With GLMs, scores are linear functions of the regression parameters. GAMs allow the actuary to include in the score nonlinear effects of the features, to be learned from the data. GAMs can be fitted with the help of local versions of GLMs or by decomposing the nonlinear effects of the features in an appropriate spline basis so that the working scor...

This study proposes a new Markov switching process with clustering effects. In this approach, a hidden Markov chain with a finite number of states modulates the parameters of a self-excited jump process combined to a geometric Brownian motion. Each regime corresponds to a particular economic cycle determining the expected return, the diffusion coef...

This article proposes a microstructure model for stock prices in which parameters are modulated by a Markov chain determining the market behaviour. In this approach, called the switching microstructure model (SMM), the stock price is the result of the balance between the supply and the demand for shares. The arrivals of bid and ask orders are repre...

This book summarizes the state of the art in generalized linear models (GLMs) and their various extensions: GAMs, mixed models and credibility, and some nonlinear variants (GNMs). In order to deal with tail events, analytical tools from Extreme Value Theory are presented. Going beyond mean modeling, it considers volatility modeling (double GLMs) an...

Artificial intelligence and neural networks offer a powerful alternative to statistical methods for analyzing data. This book reviews some of the most recent developments in neural networks, with a focus on applications in actuarial sciences and finance. The third volume of the trilogy simultaneously introduces the relevant tools for developing and...

This article explores the capacity of self-organizing maps (SOMs) for analysing non-life insurance data. Contrary to feed forward neural networks, also called perceptron, a SOM does not need any a priori information on the relevancy of variables. During the learning procedure, the SOM algorithm selects the most relevant combination of explanatory v...

This paper analyzes the efficiency of hedging strategies for stock options, in presence of jump clustering. In the proposed model, the asset is ruled by a jump-diffusion process wherein the arrival of jumps is correlated to the amplitude of past shocks. This feature adds feedback effects and time heterogeneity to the initial jump diffusion. After a...

This article studies hedging strategies of crop harvest incomes with futures and options on indexes of cumulated average temperatures (CAT). To account for the time and space dependence, temperatures and crop yields are modeled by three dimensions Gaussian fields. In this framework, we study the features and dynamics of CAT futures and CAT basket o...

A way to model the clustering of jumps in asset prices consists in combining a diffusion process with a jump Hawkes process in the dynamics of the asset prices. This article proposes a new alternative model based on regime switching processes, referred to as a self-exciting switching jump diffusion (SESJD) model. In this model, jumps in the asset p...

Most of the models leading to an analytical expression for option prices are based on the assumption that underlying asset returns evolve according to a Brownian motion with drift. For some asset classes like commodities, a Brownian model does not fit empirical covariance and autocorrelation structures. This failure to replicate the covariance intr...

This paper considers the capital structure of a bank in a continuous-time regime-switching economy. The modeling framework takes into account various categories of instruments, including equity, contingent convertible debts, straight debts, deposits and deposits insurance. Whereas previous researches concentrate on the determination of the capital...

This article proposes a neural-network approach to predict and simulate human mortality rates. This semi-parametric model is capable to detect and duplicate non-linearities observed in the evolution of log-forces of mortality. The method proceeds in two steps. During the first stage, a neural-network-based generalization of the principal component...

This paper proposes two jump diffusion models with and without mean reversion,for stocks or commodities, capable to fit highly leptokurtic processes. The jump component is acontinuous mixture of independent point processes with Laplace jumps. As in financial markets,jumps are caused by the arrival of information and sparse information has usually m...

This study analyses the impact of contagion between financial and non-life insurance markets on the asset-liability management policy of an insurance company. The indirect dependence between these markets is modeled by assuming that the assets return and non-life insurance claims are led respectively by time-changed Brownian and jump processes, for...

This work contributes to the literature over time-changed processes in two directions. Firstly, this is the first thorough study of theoretical properties of Lévy processes, subordinated by a self-excited random clock. The process observed on this new time scale, called clustered Lévy process, presents interesting features for financial modeling li...

This study analyses the impact of volatility clustering in stock markets on the evaluation and risk management of equity indexed annuities (EIA). To introduce clustering in equity returns, the reference index is modelled by a diffusion combined with a bivariate self-excited jump process. We infer a semi-closed form or parametric expression of the m...

This paper proposes a continuous time model for interest rates, based on a bivariate mutually exciting point process. The two components of this process represent the global supply and demand for fixed income instruments. In this framework, closed form expressions are obtained for the first moments of the short term rate and for bonds, under an equ...

We propose a model for short-term rates driven by a self-exciting jump process to reproduce the clustering of shocks on the Euro overnight index average (EONIA). The key element of the model is the feedback effect between the absolute value of jumps and the intensity of their arrival process. In this setting, we obtain a closed-form solution for th...

In this study, we propose a modelling framework for evaluating companies financed by random liabilities, such as insurance companies or commercial banks. In this approach, earnings and costs are driven by double exponential jump diffusion processes and bankruptcy is declared when the income falls below a default threshold, which is proportional to...

In defined benefit pension plans, allowances are independent from the financial performance of the fund. And the sponsoring firm pays regularly contributions to limit deviations of fund assets from the mathematical reserve, necessary for covering the promised liabilities. This research paper proposes a method to optimize the timing and size of cont...

This paper studies a switching regime version of Merton's structural model for the pricing of default risk. The default event depends on the total value of the firm's asset modeled by a switching Lévy process. The novelty of this approach is to consider that firm's asset jumps synchronously with a change in the regime. After a discussion of dynamic...

Optimal timing for annuitization is developped along three approaches. Firstly, the mutual fund in which the individual invests before annuitization is modeled by a jump diffusion process. Secondly, instead of maximizing an economic utility, the stopping time is used to maximize the market value of future cash-flows. Thirdly, a solution is proposed...

This paper proposes a statistical model for insurance claims arising from climatic events, such as tornadoes in the USA, that exhibit a large variability both in frequency and intensity. To represent this variability and seasonality, the claims process modelled by a Poisson process of intensity equal to the product of a periodic function, and a mul...

This paper presents a switching regime version of the Merton's structural model for the pricing of default risk. The default event depends on the total value of the firm's asset modeled by a Markov modulated Lévy process. The novelty of our approach is to consider that firm's asset jumps synchronously with a change in the regime. After a discussion...

We develop a switching regime version of the intensity model for credit risk pricing. The default event is specified by a Poisson process whose intensity is modeled by a switching Lévy process. This model presents several interesting features. Firstly, as Lévy processes encompass numerous jump processes, our model can duplicate sudden jumps observe...

This work intend to shed some light on a new use of Phase-type distributions in credit risk, taking into account different flows of information without huge numerical calculations. We consider credit migration models with partial information and study the influence of a deficit of information on prices of credit linked securities. The transitions t...

In defined benefit pension plans, allowances are independent from the financial performance of the fund. And the sponsoring firm pays regularly contributions to limit deviations of fund assets from the mathematical reserve, necessary for covering the promised liabilities. This research paper proposes a method to optimize the timing and size of cont...

This paper studies the valuation of credit risk for firms that own several subsidiaries or business lines. We provide simple analytical approximating expressions for probabilities of default, and for equity-debt market values, both in the case when the information is available in continuous time as well as in the case that it is not instantaneously...

This paper develops a new version of the Hull-White's model of interest rates, in which the volatility of the short term rate is driven by a Markov switching multifractal model. The interest rate dynamics is still mean reverting but the constant volatility of the Brownian motion is replaced by a multifractal process so as to capture persistent vola...

a b s t r a c t This paper proposes a multidimensional Lee-Carter model, in which the time dependent components are ruled by switching regime processes. The main feature of this model is its ability to replicate the changes of regimes observed in the mortality evolution. Changes of measure, preserving the dynamics of the mortality process under a p...