In this page, you can access the abstracts, pdfs and slides of most of my published and unpublished papers. You like maths? I also selected the most representative equation of each of my papers to help you see what is in them. Interested in business and economics? These equations also show the main ideas of the papers. Any comments or suggestions? Send me an email via the contact form of this website!

## Health and Portfolio Choices : a Diffidence Approach

with D. Crainich and L. Eeckhoudt
$E[u_1(w_{0}+\tilde{x},y) \tilde{x}] = 0 \Rightarrow E[u_1(w_{0}+\tilde{x},y+\tilde{\epsilon }) \tilde{x}] \leq 0$

The effect of health status on portfolio decisions has been extensively studied from an empirical viewpoint. In this paper, we propose a theoretical model of individuals’ choice of financial assets under bivariate utility functions depending on wealth and health. Our model relies on the diffidence theorem, which pertains to the class of hyperplane separation theorems. We establish the conditions under which the share of wealth held in risky assets falls as: 1) individuals’ health status deteriorates and; 2) individuals’ health status becomes risky. These conditions are shown to be related to the behaviour of the intensities of correlation aversion and of cross prudence as wealth increases.

European Journal of Operational Research

Vol. 259, N° 1, p. 273-279, 2017

## Inside the Solvency 2 Black Box: Net Asset Values and Solvency Capital Requirements with the least-squares Monte Carlo method

with A. Floryszczak and M. Majri
$\hat{\text{NAV}}_{\phi(p)} = \frac{\sum\limits_{i=-M}^M \hat{\text{NAV}}_{\phi(p)+i}}{2M+1}$

The calculation of Net Asset Values and Solvency Capital Requirements in a Solvency 2 context – and the derivation of sensitivity analyses with respect to the main financial and actuarial risk drivers – is a complex procedure at the level of a real company, where it is illusory to be able to rely on closed-form formulas. The most general approach to perform- ing these computations is that of nested simulations. However, this method is also hardly realistic because of its huge computation resources demand. The least-squares Monte Carlo method has recently been suggested as a way to overcome these difficulties. The present paper confirms that using this method is indeed relevant for Solvency 2 computations at the level of a company.

Insurance: Mathematics and Economics

Vol. 71, p. 15-26, 2016

## Portfolio Optimisation with Jumps: Illustration with a Pension Accumulation Scheme

with F. Menoncin
$\frac{c\left(t\right)^{*}}{\tilde{R}\left(t\right)}=\frac{1}{\int\limits_{t}^{\infty}e^{-\frac{1}{\delta}\int\limits_{t}^{s}\left(\rho+\lambda\left(u\right)-\left(1-\delta\right)r-h\left(\tilde{w}^{*}\right)\right)du}ds}$

In this paper, we address portfolio optimisation when stock prices follow general Lévy processes in the context of a pension accumulation scheme. The optimal portfolio weights are obtained in quasi-closed form and the optimal consumption in closed form. To solve the optimisation problem, we show how to switch back and forth between the stochastic differential and standard exponentials of the Lévy processes. We apply this procedure to both the Variance Gamma process and a Lévy process whose arrival rate of jumps exponentially decreases with size. We show through a numerical example that when jumps, and therefore asymmetry and leptokurtosis, are suitably taken into account, then the
3
NAVφ(p)+i
optimal portfolio share of the risky asset is around half that obtained in the Gaussian framework.

Journal of Banking and Finance

Vol. 60, p. 127-137, 2015

## Imprudent Risk-Lovers are Inconsistent

$\frac{1}{2}U(x)+\frac{1}{2}U(y+k)>\frac{1}{2}U(x+k)+\frac{1}{2}U(y)$

Research characterizes most risk averters as prudent and temperate but devotes little attention to the study of risk lovers. The risk lovers who prefer to combine good with good are prudent and intemperate. This paper shows how the assumption of “combining good with good » can be relaxed, and how similar results can come from the consistency hypothesis for decision makers. Namely, the risk lovers who are consistent are prudent and intemperate with positive derivatives of their utility function for all orders. However, empirically risk lovers do exist who are both imprudent and intemperate. These risk lovers, being imprudent, are inconsistent.

Working Paper

## Q-Credibility

$Z_q = \frac{n\left[a(nc+h)-b(nb+g)\right]}{(na+v)(nc+h)-(nb+g)^2}$

This article extends credibility theory by making quadratic adjustments that take into account the squared values of past observations. This approach amounts to introducing non-linearities in the framework, or to considering higher order cross moments in the com- putations. We first describe the full parametric approach and, for illustration, we examine the Poisson-gamma and Poisson-single Pareto cases. Then, we look at the non-parametric approach where premiums must be estimated based on data only, without postulating any type of distribution. Finally, we examine the semi-parametric approach where the con- ditional distribution is Poisson but the unconditional distribution is unknown. The goal of this paper is not to claim that q-credibility always brings better results than standard credibility, but it is to provide several building blocks for understanding how credibility changes when quadratic corrections are added.

Forthcoming in Variance

## Credit Risk and Solvency Capital Requirements

with J. Allali and M. Majri
$\psi = \frac{ \sum\limits_{k=1}^K \sum\limits_{j=1}^{M_k} (1-R^{j,k}) \sum\limits_{i=1}^{N^{j,k}} C_i^{j,k} \ P(0,T_i) \ \psi_{k,K+1}(0, T_i) \ p_{k,K+1}(0,T_i) }{ \sum\limits_{k=1}^K \sum\limits_{j=1}^{M_k} (1-R^{j,k}) \sum\limits_{i=1}^{N^{j,k}} C_i^{j,k} \ P(0,T_i) \ p_{k,K+1}(0,T_i) }$

Credit risk permeates the assets of most insurance companies. This article develops a framework for computing credit capital requirements under the constant position paradigm and taking into account recovery rates. Although this framework was originally derived under the Solvency 2 regulation, it also provides concepts that can be useful under other international regula- tions. After a brief survey of the existing technology on rating transitions and default probabilities, the paper provides new results on risk premium adjust- ment factors. Then, three different procedures for reconstructing constant po- sition market-consistent histories of credit portfolios from quoted Merrill Lynch indices are given. The reconstructed historical credit values are modeled via mixed empirical-Generalized Pareto Distribution (GPD) dynamics and a de- tailed parameter estimation is performed. Several validations of the estimation are also provided. Finally, credit Solvency Capital Requirements are computed and an analysis of the results per rating class is given.

European Actuarial Journal

Vol. 8, N°2, p. 487–515, 2018

## Some Further Results on the Tempered Multistable Approach

$L_{II}(t)=\sum\limits_{i=1}^{\infty} C_{\alpha(V_i)}^{1/ \alpha(V_i)} \gamma_i \Gamma_i^{-1/\alpha(V_i)}\mathbf{1}_{(V_i \leq t)}$

Typical approaches incorporating jumps in financial dynamics, such as the Variance Gamma and CGMY models, can be made to depart from the i.i.d. hypothesis by using a stochastic clock. In such a context, the introduction of a dispersion of the clock is equiva- lent to the introduction of a dispersion of the volatility itself. A distinct route that yields
1
comparable features is that of adding a jump component to a stochastic volatility process, or of considering, in discrete time, leptokurtic innovations within a GARCH process. In this article, we take a third route and we provide a study on tempered multistable pro- cesses, which convey both jumps and autocorrelation from their very construction, and on some of their applications in finance. We obtain the multivariate characteristic function of the asymmetrical field-based tempered multistable process and we study the autocorrela- tions that stem from the use of this model. We concentrate on three types of applications in finance: we study the term structures of Value-at-Risk that can be obtained with this model, we perform a calibration on stock index data, and we also conduct a calibration on derivatives prices.

Asia-Pacific Financial Markets

Vol. 25, N°2, p. 87–109, 2018

## An Intensity Model for Credit Risk with Switching Lévy Processes

with D. Hainaut
$\frac{\partial \tilde{A}(t)}{\partial t} + \text{diag}(F(t)) \tilde{A}(t) + Q \tilde{A}(t)=0$

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 observed in credit spreads. Also, due to the presence of jumps, probabilities do not vanish at very short maturities, contrary to models based on Brownian dynamics. Furthermore, as parameters of the Lévy process are modulated by a hidden Markov process, our approach is well suited to model changes of volatility trends in credit spreads, related to modifications of unobservable economic factors.

Quantitative Finance

Vol. 14, N°8, p. 1453-1465, 2014

## Decreasing Downside Risk Aversion and Background Risk

with D. Crainich and L. Eeckhoudt
$\zeta_w(t) \leq \zeta_w'(0) \ t \quad \forall t \; \Rightarrow \; \text{DDRA}$

In this paper, we show that risk vulnerability can be associated with the concept of down- side risk aversion (DRA) and an assumption about its behavior, namely that it is decreas- ing in wealth. Specifically, decreasing downside risk aversion in the Arrow-Pratt and Ross senses are respectively necessary and sufficient for a zero-mean background risk to raise the aversion to other independent risks.

Journal of Mathematical Economics

Vol. 53, 2014

## The Computation of Risk Budgets under the Lévy Process Assumption

with C. Walter
$F(x) = \frac{e^{a x}}{2\pi} \int\limits_{-\infty}^{+\infty} e^{iux} \ \frac{\Phi (-u+ia)}{a+iu} \ du$

This paper revisits the computation of Value-at-Risk and other risk indicators based on the use of Lévy processes. We first provide a new presentation of Variance Gamma Pro- cesses with Drift: we reconstruct them in an original way, starting from the exponential distribution. Then, we derive general Fourier formulas that allows to compute VaR quickly and efficiently, but also other typical indicators like Tail Conditional Expectation (TCE), TailVaR or Expected Shortfall. Based on this formula, we conduct a study of the term structure of VaR, and provide a discussion of the Basle 2 and Solvency II agreements.

Revue Finance

Vol. 35, N°2, 2014

## On Surrender and Default Risks

with H. Nakagawa
$dA_t = A_{t-}\left( r_t dt + \sigma_tdW_t^Q - \frac{dN_t}{I - N_{t-}}\right)$

This article examines the impact that surrender risk can have on the default of insurance 4
companies. The companies that we study issue contacts similar to the ones studied earlier in the literature by Briys and de Varenne (2001), Grosen and Jorgensen (2000), or Bernard, Le Courtois and Quittard-Pinon (2005). They are subject to interest rate and default risk; they offer a guaranteed amount plus a bonus indexed on the performance of the underlying portfolio. In this article, we assume in addition that policyholders have the option to surrender. Surrender risk has been extensively studied in an arbitrated market by Bacinello (2001), using trees for the valuations. We deal with surrender risk in another way, assuming policyholders have sets of information and preferences that differ from the ones of financial market agents. In particular, policyholders are supposed to be only partially rational (at least in the financial sense).

Mathematical Finance

Vol. 23, N°1, p. 143-168, 2013

## On the Bankruptcy Risk of Insurance Companies

with R. Randrianarivony
$R(u,t) = 1 - \mathcal{L}^{-1}_q \otimes \mathcal{L}^{-1}_z \left[\frac{\psi^-_{SL} (q,z) }{qz} \right]$

The fall of AIG is another confirmation that the insurance business is not immune to bankruptcy. Contrary to the actuarial literature which postulates that insurance firms can survive forever, we believe that this is not the case, and that a realistic and business- oriented risk management approach needs to be designed in order to prevent the actual, finite-time, bankruptcy of insurance companies. In this article we model the surplus process of an insurance firm firstly by a stable Lévy process, secondly by a double exponential compound Poisson process. We compute finite-time survival and bankruptcy probabilities under such hypotheses. To achieve this, we make use of the Wiener-Hopf factorization and compute bankruptcy formulas written in terms of inverse Laplace transforms. The Abate and Whitt, and Gaver-Stehfest algorithms are used to obtain numerical estimations.

Revue Finance

Vol. 34, N°1, p. 43-72, 2013

## Concentration des portefeuilles boursiers et asymétrie des distributions de rentabilités d’actifs

with C. Walter
$E[U(V)] = \alpha + \beta E[V] + \gamma E[V^2] + \delta E[V^3]$

This article develops on the link between the asymmetry of asset return distributions and the concentration of portfolios. We start by recalling the rationale behind the theory of diversification, in order to let appear that this theory relies on a reduction of risk viewed only at order 2 and on the related application of the theory of errors, as developed during the XVIIIth century. We also expose the controversy opened by E. Fama in 1965 on this theory of errors, in order to let appear that a change in the type of underlying randomness can lead to the concentration and not the diversification of portfolios. Then, we examine how the inclusion in the optimization program of the asymmetry between gains and losses can lead to a propensity to concentrate.We recall the main aspects of the Mitton and Vorkink (2007) model, and then we propose a new approach in the spirit of this model. We end up with an illustration of the latter framework on American data, letting appear important differences between the performance obtained with a classically diversified port- folio, a portfolio concentrated along existing models, and a portfolio concentrated along the model that we propose.

Journal de la Société Française de Statistiques

Vol. 153, N°2, p. 1-20, 2012

## Asset Risk Management of Participating Contracts

with C. Bernard
$C_{k+1} = C_k \ (1+m \ R_{k+1} + (1-m) \ r)$

In this paper we study the asset-liability management of an insurance company selling “participating contracts”. Participating contracts are typical insurance policies sold world- wide. The payoff of a participating policy is linked to the portfolio or the surplus of the insurance company. We examine the impact of the choice of assets’ investment strategy on the company value, its solvency, and how well the company may meet the commitments associated with its liabilities. Four strategies are investigated and compared: a simple buy-and-hold strategy, a dynamic CPPI (Constant Proportion Portfolio Insurance), an investment in Equity Default Swaps (EDS), and a protection by ways of forward-starting puts. For example it is shown that an active protection strategy by CPPI may significantly reduce the company’s default risk but is very costly to policyholders. Our study illustrates how to compare asset management strategies and how to choose the parameters of a suit- able allocation such that the policyholders’ market value is preserved and the default risk is reduced.

Asia-Pacific Journal of Risk and Insurance

Vol. 6, N°2, p. -, 2012

## Performance Regularity: a New Class of Executive Compensation Packages

with C. Bernard
$G = \inf\left\{ t > 0 \ | \ \max\limits_{u \in [t-D_L,t]} S_u \le L \right\}$

The ability of standard executive stock options to incite managers to adequately select the assets of their firm has been extensively questioned by academics and practitioners. However, very few alternatives exist or have been proposed to better control the investment strategies of top managers. The present article studies the evaluation and sensitivity of a new class of executive stock options well designed for the control of managers. Such packages are aimed at giving incentives to CEOs to maintain a regular performance over time and a stable volatility level. The importance and implications of the choice of the different parameters as well as their robustness with respect to standard financial criteria are examined. In brief, this article studies in a utility-based framework a new type of executive stock options that can be useful to protect and enhance the economic performance of corporations.

Asia-Pacific Financial Markets

Vol. 19, N°4, p. 353-370, 2012

## Protection of a Company Issuing a Certain Class of Participating Policies in a Complete Market Framework

with C. Bernard and F. Quittard-Pinon
$\phi \ \zeta = \frac{\hat{S}_0}{E_Q\left( \frac{A_0}{2} \, e^{-r \tau} \, \mathbb{1}_{\tau

In this article, we examine to what extent policyholders buying a certain class of partici- pating contracts (in which they are entitled to receive dividends from the insurer) can be described as standard bondholders. Our analysis extends the ideas of Bühlmann , and sequences the fundamental advances of Merton , Longstaff and Schwartz , and Briys and de Varenne [1994, 2001]. In particular, we develop a setup where these
6
participating policies are comparable to hybrid bonds but not to standard risky bonds (as done in most papers dealing with the pricing of participating contracts). In this mixed framework, policyholders are only partly protected against default consequences. Continu- ous and discrete protections are also studied in an early default Black and Cox  type setting. A comparative analysis of the impact of various protection schemes on ruin prob- abilities and severities of a Life Insurance company which only sells this class of contracts concludes this work.

North American Actuarial Journal

Vol. 14, N°1, p. 131-149, 2010

## Fair Valuation of Participating Life Insurance Contracts with Jump Risk

with F. Quittard-Pinon
$\tau > T \; \Leftrightarrow \; \min\limits_{t\in [0,T]} X_t^* > b$

The purpose of this article is to value participating life insurance contracts when the linked portfolio is modeled by a jump-diffusion. More precisely, this process has a Brownian com- ponent and a compound Poisson one, where the jump size is driven by a double exponential distribution. Specifically here, the bankruptcy risk of the insurance company is considered. Thus, market and credit risks are taken into account. A quasi-closed-form formula is ob- tained in fair value for the price of the considered life insurance contract. This allows us to investigate the impact of strategic parameters as well as structural ones, as is shown in the numerical section of this paper. In particular, we study the impact on the contract of the volatility, jump intensity, jump asymmetry, company leverage, guaranteed rate, participa- tion rate and level of the default barrier, and comment on how they are likely to increase the probability of early default of the issuer.

Geneva Risk and Insurance Review

Vol. 33, N°2, p. 109-136, 2008

## Pricing Derivatives with Barriers in a Stochastic Interest Rate Environment

with C. Bernard and F. Quittard-Pinon
$\mathcal{B} \approx\ \sum\limits_{j=0}^{n_T}\sum\limits_{i=0}^{n_r} \,\mathcal{N}\left(\frac{\ln(K)-\hat \mu_{t_j,T}}{\sqrt{\hat \Sigma^2_{t_j,T}}}\right)q^u(i,j)$

This paper develops a general valuation approach to price barrier options when the term structure of interest rates is stochastic. These products’ barriers may be constant or stochastic, in particular we examine the case of discounted barriers (at the instantaneous interest rate). So, in practice, we extend Rubinstein and Reiner (1991), who give closed- form formulas for pricing barrier options in a Black and Scholes context, to the case of a Vasicek modeling of interest rates. We are therefore in the situation of pricing barrier options semi-explicitly or explicitly (depending on the shape of the barrier) with stochastic Vasicek interest rates. The model is illustrated with a specific contract, an up and out call with rebate, hence a typical barrier option. This example is merely here to show how any standard barrier option can be priced and its Greeks be obtained in such a context. The validity of the approximation is analyzed and the sensitivity to the barrier level and to discretization schemes are also derived.

Journal of Economic Dynamics and Control

Vol. 32, N°9, p. 2903-2938, 2008

## The Optimal Capital Structure of the Firm with Stable Lévy Assets Returns

with F. Quittard-Pinon
$\mathcal{V} = V_0 + \frac{\tau C}{r} E\left( 1 - e^{-r H_x} \right) - \hat{\alpha} V_0 E \left(e^{\gamma X_{H_x}-r H_x} \right)$

This article builds a new structural default model under the assumption that a firm’s assets return follows a dynamics displaying jumps of both signs. In essence, we expand the work of Hilberink and Rogers (itself an extension of the Leland and Toft framework), which deals only with negative jumps. In contrast, we make use of stable Lévy processes, and we compute the values of the firm, debt and equity under this assumption. Theoretical credit spreads can also be obtained in our framework. They prove to be consistent with the empirical credit spreads observed in financial markets.

Decisions in Economics and Finance

Vol. 31, N°1, p. 51-72, 2008

## Risk-Neutral and Actual Default Probabilities with an Endogenous Bankruptcy Jump- Diffusion Model

with F. Quittard-Pinon
$\hat{M}_t = \sum\limits_{k=1}^{\hat{N}_t} (\hat{Z}_k-1) - \hat{\lambda} \ \hat{\xi} \ t$

This paper focuses on historical and risk-neutral default probabilities in a structural model, when the firm assets dynamics are modeled by a double exponential jump diffusion process. Relying on the Leland [1994a, 1994b] or Leland and Toft  endogenous structural approaches, as formalized by Hilberink and Rogers , this article gives a coherent construction of historical default probabilities. The risk-neutral world where evolve the firm assets, modeled by a geometric Kou process, is constructed based on the Esscher measure, yielding useful and new analytical relations between historical and risk-neutral probabilities. We do a complete numerical analysis of the predictions of our framework, and compare these predictions with actual data. In particular, this new framework displays a greater predictive power than current Gaussian endogenous structural models.

Asia-Pacific Financial Markets

Vol. 13, p. 11-39, 2006

## Development and Pricing of a New Participating Contract

with C. Bernard et F. Quittard-Pinon
$E_1=Q_T\left(\inf\limits_{u\in[0,T[}\left(\frac{A_u}{P(u,T)}\right)< \lambda_1\,l_T^g\right)$

This article designs and prices a new type of participating life insurance contract. Partic- ipating contracts are popular in the United States and European countries; they present many different covenants and depend on national regulations. In the present article, we design a new type of participating contract very similar to the one considered in other studies, but with the guaranteed rate matching the return of a government bond. We prove that this new type of contract can be valued in closed form when interest rates are stochastic and when the company can default.

North American Actuarial Journal

Vol. 10, N°4, p. 179-195, 2006

## Le Point Sur… Les Options Parisiennes et leurs Applications

with C. Bernard
$\hat{h_1}(\lambda,y) = e^{(2b-y)\sqrt{2 \lambda}} \ \frac{\Psi(-\sqrt{2\lambda D})}{\sqrt{2 \lambda} \ \Psi(\sqrt{2\lambda D})}$

This survey paper is dedicated to some options that are not extremely well-known, but that bring along very promising applications: Parisian options. We start by describing these products, then we detail their use in the contexts of bank deposit insurance, the theory of real options, and finally the structural theory of default. The reader will consider these examples to be illustrative, though not exhaustive.

Banque et Marchés

N° 82, p. 81-90, 2006

## A New Procedure for Pricing Parisian Options

with C. Bernard and F. Quittard-Pinon
$\lambda \rightarrow \sum\limits_{i=1}^{N} \alpha_i \frac{1}{\lambda^{i/j}}$

Parisian options extend barrier options in that their covenant depends on the time spent by the underlying beyond a given threshold. Due to their very nature, they are hard to price and hedge, though some quite involved material has been made available in that direction. Valuation of Parisian Options can be done by using four different main methods: Monte- Carlo simulations, lattices, partial differential equations, inverse Laplace transform. In this article, we develop a new inverse Laplace transform method, quick and well-suited to the problem under study. This method could also be used to treat other financial problems where inversion of a Laplace transform is required.

Journal of Derivatives

Vol. 12, N°4, p. 45-53, 2005

## Market Value of Life Insurance Contracts under Stochastic Interest Rates and Default Risk

with C. Bernard and F. Quittard-Pinon
$q(i,j)=\Phi(\,r_i,\,t_j\,)-\sum\limits_{v=0}^{j-1}\sum\limits_{u=0}^{n_r}q(\,u,\,v\,)\ \Psi(\,r_i,\,t_j\,|\,r_u,\,t_v\,)$

The purpose of this article is to value some life insurance contracts in a stochastic interest rate environment taking into account the default risk of the underlying insurance company. The participating life insurance contracts considered here can be expressed as portfolios of barrier options as shown by Grosen and Jørgensen . In order to price these options, the Longstaff and Schwartz  methodology is used with the Collin-Dufresne and Goldstein  correction.

Insurance: Mathematics and Economics

Vol. 36, N°3, p. 499-516, 2005

## A Study of Mutual Insurance for Bank Deposits

with C. Bernard and F. Quittard- Pinon
$O_G = \theta E_Q \left[ e^{-rT} (D_T - V_T -K)^+ \right]$

This article displays a study on the mutual insurance of bank deposits. A system where deposits are first insured by a consortium then by the Government is envisaged. We wish to compute the fair premia due to both the consortium and the Government. Various types of covenants aiming at making banks reduce their risks are detailed. These provisions can be, as is the case in Chapter 11, of a Parisian type. This means that surveillance is based on the path followed by the assets or the leverage. We compare these various types of covenants and conclude on the proposal for new regulatory provisions.

Geneva Risk and Insurance Review

Vol. 30, N°2, p.129-146, 2005

## Changes of Probability Measures in Finance and Insurance: A Synthesis

with F. Quittard-Pinon
$\left(\frac{dQ_{2}}{dQ_{1}}\right)_{t}=\frac{S(t)^\theta }{E(S(t)^\theta )}$

Changes of probability measures now pave a very efficient way towards the study of many problems in Finance and Insurance. To quote only a few, this approach has proven to be useful in portfolio selection, option pricing and premium valuation in insurance. Many techniques have been developed more or less independently: changes of numéraire, pricing kernels (equivalents of probability measure changes) , Esscher transforms. The aim of this paper is to give a survey on these approaches and illustrate them with the Black and Scholes model taken as a benchmark; and then to see to what extent these changes of measure can be adapted to a more general context than the standard diffusion processes extensively assumed in the financial literature for modelling price processes. More specifically, we explore these techniques when price movements are modelled as Lévy processes. In sections two, three, four, five of this article we concentrate on diffusions – our aim being to exhibit the fundamental ideas behind the commonly used tools. From section six, we move on to the case of jump-diffusions, and in particular to the case of assets following geometric Lévy motions.

Revue Finance

Vol. 25, Special Issue, p. 95-120, 2004

## Modelling Stock Returns with Lévy Processes

$\nu(x) = \left\lbrace \begin{array}{l} C\frac{e^{-G|x|}}{{|x|^{1+Y}}}(\alpha - |x|^\beta)^2 \qquad \forall x < 0\\ C\frac{e^{-M|x|}}{{|x|^{1+Y}}}(\alpha - |x |^\beta)^2 \qquad \forall x > 0 \end{array}\right.$

Several approaches to model stock returns with Lévy Processes have been developed in the past years. Firstly, this article will review existing approaches and compare the latest ones through an analysis of the Lévy density. Secondly, this article will provide a simple but general parameterization for the Lévy density which yields a class of Lévy processes that can be used in a financial context. These processes – titled α-β Lévy motions – will allow for excessive arrival rates of average size jumps, in correspondence to humped return distributions at short time scales.

Banque et Marchés

N° 66, p. 36-46, 2003