
Jean-François RenaudUniversity of Quebec in Montreal | UQAM · Department of Mathematics
Jean-François Renaud
Ph.D. Applied Mathematics (U. de Montréal)
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61
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August 2007 - March 2008
September 2010 - present
July 2008 - August 2010
Publications
Publications (61)
We consider de Finetti's stochastic control problem for a spectrally negative L\'evy process in an Omega model. In such a model, the (controlled) process is allowed to spend time under the critical level but is then subject to a level-dependent intensity of bankruptcy. First, before considering the control problem, we derive some analytical propert...
We consider a classical stochastic control problem in which a diffusion process is controlled by a withdrawal process up to a termination time. The objective is to maximize the expected discounted value of the withdrawals until the first-passage time below level zero. In this work, we are considering absolutely continuous control strategies in a ge...
We consider De Finetti’s control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal in a Brownian model. In order to solve this problem, we need to deal with a nonlinear Ornstein–Uhlenbeck process. Despite the level of generality of the bound...
We consider De Finetti's control problem for absolutely continuous strategies with control rates bounded by a concave function and prove that a generalized mean-reverting strategy is optimal. In order to solve this problem, we need to deal with a nonlinear Ornstein-Uhlenbeck process. Despite the level of generality of the bound imposed on the rate,...
Aiming for more realistic optimal dividend policies, we consider a stochastic control problem with linearly bounded control rates using a performance function given by the expected present value of dividend payments made up to ruin. In a Brownian model, we prove the optimality of a member of a new family of control strategies called delayed linear...
In this paper, we study a stochastic control problem faced by an insurance company allowed to pay out dividends and make capital injections. As in (Løkka and Zervos (2008); Lindensjö and Lindskog (2019)), for a Brownian motion risk process, and in Zhu and Yang (2016), for diffusion processes, we will show that the so-called Løkka–Zervos alternative...
We consider de Finetti’s stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strate...
We consider de Finetti's stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative L\'evy model with exponential Parisian ruin. We show that, under mild assumptions on the L\'evy measure, an optimal st...
We consider de Finetti's stochastic control problem when the (controlled) process is allowed to spend time under the critical level. More precisely, we consider a generalized version of this control problem in a spectrally negative Lévy model with exponential Parisian ruin. We show that, under mild assumptions on the Lévy measure, an optimal strate...
In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a...
In an option contract, the obligation to buy the underlying asset at the maturity date is replaced by the possibility to run away whenever the situation is not profitable. The basic options are call options, long put options, and short put options. This chapter helps readers to understand the options to buy (call) and options to sell (put) an asset...
This chapter provides a thorough treatment of its two‐period binomial tree model so as to better understand the construction of the general binomial tree and the dynamics of the underlying asset over time and focus on the replication procedure over more than one period, which is important for risk management purposes. It also helps to analyze the c...
This chapter lays the foundations of the famous Black‐Scholes‐ Merton market model and its pricing formula. It provides a heuristic approach to this formula by linking as much as possible the derivations to the binomial model of Part I using a limiting argument. The chapter helps the reader to understand the main assumptions of the Black‐Scholes‐Me...
Forward contracts and futures contracts can be used for fixing today the price of a good to be bought in the future. This chapter provides an introduction to forwards and futures. It analyzes equity forwards, also known as forwards on stocks, and looks at stocks not paying dividends; stocks paying discrete and fixed dividends; and stocks paying con...
In statistics and actuarial science, simulation is used to generate artificial random scenarios from a specified model. This chapter aims to apply simulation techniques to compute approximations of the no‐arbitrage price of derivatives when the underlying market model is a BSM model and also to apply variance reduction techniques, such as stratifie...
This chapter familiarizes the reader with exotic/path‐dependent options and event‐triggered derivatives. It looks at the four categories of exotic options: barrier options, Asian options, lookback options, and exchange options. The two main types of barrier options: knock‐in and knock‐out. Lookback options are options whose payoff is based upon the...
A swap usually involves exchanging periodically variable (random) payments, whose value is based on a financial benchmark or an underlying asset (a stock, a bond, an index or some economic or financial quantity), for fixed payments.This chapter provides an introduction to swaps with an emphasis on those used in the insurance industry, namely intere...
The chapter analyzes the pricing of options and other derivatives such as options on dividend‐paying assets, currency options and futures options, but also insurance products such as investment guarantees, equity‐indexed annuities and variable annuities, as well as exotic options (Asian, lookback and barrier options). In most of these cases, we can...
This chapter analyzes various risk management practices, mostly hedging strategies used for interest rate risk and equity risk management. It helps readers to apply cash‐flow matching or replication to manage interest rate risk and equity risk, and also to apply duration‐(convexity) matching to assets and liabilities sensitive to interest rate risk...
Financial market and financial securities have an impact on actuarial liabilities. This chapter provides an introduction to the financial market and financial securities, especially stocks, bonds and derivatives, from the point of view of an actuary. Bonds are of two types: coupon bonds and zero‐coupon bonds. The chapter provides an understanding o...
When the time step in a binomial tree becomes increasingly small, the geometric random walk followed by the stock price gradually approaches that of a geometric Brownian motion. Although it provides a lot of the intuition and explains where many results come from, the approach lacks some rigor. This chapter intends to fill these gaps by providing a...
The one‐period and two‐period binomial trees had the advantage of introducing important concepts and procedures, such as replication, portfolio dynamics and risk‐neutral formulas, in fairly simple setups. This chapter seeks to generalize these ideas to a model with more than two time steps. It introduces the general multi‐period binomial tree model...
Usually, in most textbooks and research papers, the evolution of the stock price in the Black‐Scholes‐Merton (BSM) model is given by a so‐called stochastic differential equation. To better understand these concepts, this chapter provides a heuristic introduction to stochastic calculus. Stochastic calculus arises naturally in continuous‐time actuari...
This chapter describes the basic assets available in a one‐period binomial model and identifies the assumptions on which a one‐period binomial model is based. In a one‐period model there are two time points: the beginning of the period is time 0 and the end of the period is time 1. In the one‐period binomial tree model, all derivatives can be repli...
Actuaries are professionals dealing with assets and liabilities of insurance companies and pension plans. They play an active role in insurance and financial markets, and have to manage several types of risks whether they occur over the short or the long term, and whether they are systematic or diversifiable. This chapter puts into context the role...
Market incompleteness is everywhere in actuarial science, as typical insurable risks embedded in life insurance or homeowner's insurance cannot be (exactly) replicated using tradable assets. This chapter introduces the concept of market incompleteness and presents a few one‐period trinomial tree models with the main objective of understanding the m...
Equity‐linked insurance or annuity (ELIA) competes with mutual funds with the important distinction that ELIAs include various guarantees at the maturity of the contract and/or on the death of the policyholder. This chapter introduces the reader to a large class of insurance products known as ELIAs and to link their cash flow structure to financial...
The first application of Brownian motion in finance can be traced back to Louis Bachelier in 1900 in his doctoral thesis titled Theorie de la speculation. This chapter aims at providing the necessary background on Brownian motion to understand the Black‐Scholes‐Merton model and how to price and manage (hedge) options in that model. It commences wit...
The main objective of this chapter is to extend the binomial tree to more realistic situations. The specific objectives are to determine replicating portfolios; compute no‐arbitrage prices; and derive risk‐neutral formulas, for American put options, options on stocks paying continuous dividends; currency options. and futures options. American optio...
With the growth of derivatives markets in the 1990s, the term financial engineering became increasingly popular to designate the study of the structure and design of financial derivatives. This chapter focuses on the mathematical structure of payoffs with the goal of designing and replicating simple financial products, and also obtaining parity rel...
In this short paper, we study a VaR-type risk measure introduced by Guérin and Renaud and which is based on cumulative Parisian ruin. We derive some properties of this risk measure and we compare it to the risk measures of Trufin et al. and Loisel and Trufin.
In this paper, we unify two popular approaches for the denition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new denition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the rst time an excursion in the red zone lasts longer than an implementation delay with a deter...
In this paper, we unify two popular approaches for the definition of actuarial ruin with implementation delays, also known as Parisian ruin. Our new definition of ruin includes both deterministic delays and exponentially distributed delays: ruin is declared the first time an excursion in the red zone lasts longer than an implementation delay with a...
In this paper, we investigate Parisian ruin for a L\'evy surplus process with an adaptive premium rate, namely a refracted L\'evy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probabilit...
We introduce the concept of cumulative Parisian ruin, which is based on the
time spent in the red by the underlying surplus process. Our main result is an
explicit representation for the distribution of the occupation time, over a
finite-time horizon, for a compound Poisson process with drift and exponential
claims. The Brownian ruin model is also...
Inspired by the works of Landriault
et al.
(2011), (2014), we study the Gerber–Shiu distribution at Parisian ruin with exponential implementation delays for a spectrally negative Lévy insurance risk process. To be more specific, we study the so-called Gerber–Shiu distribution for a ruin model where at each time the surplus process goes negative, an...
In this short paper, we investigate a definition of Parisian ruin introduced in [3], namely Parisian ruin with an ultimate bankruptcy level. We improve the results originally obtained and, moreover, we compute more general Parisian fluctuation identities.
In this short paper, in order to price occupation-time options, such as (double-barrier) step options and quantile options, we derive various joint distributions of a mixed-exponential jump-diffusion process and its occupation times of intervals.
In this paper, we investigate Parisian ruin for a L\'evy surplus process with an adaptive premium rate, namely a refracted L\'evy process. More general Parisian boundary-crossing problems with a deterministic implementation delay are also considered. Our main contribution is a generalization of the result in Loeffen et al. (2013) for the probabilit...
For a spectrally negative Lévy process $X$, we study the following
distribution: $$ \mathbb{E}_x \left[ \mathrm{e}^{- q \int_0^t
\mathbf{1}_{(a,b)} (X_s) \mathrm{d}s } ; X_t \in \mathrm{d}y \right], $$ where
$-\infty \leq a < b < \infty$, and where $q,t>0$ and $x \in \mathbb{R}$. More
precisely, we identify the Laplace transform with respect to $t$...
In this short paper, in order to price occupation-time options, such as (double-barrier) step options and quantile options, we derive various joint distributions of a mixed-exponential jump-diffusion process and its occupation times of intervals.
In this paper we introduce an insurance ruin model with an adaptive premium rate, henceforth referred to as restructuring/refraction, in which classical ruin and bankruptcy are distinguished. In this model the premium rate is increased as soon as the wealth process falls into the
red zone
and is brought back to its regular level when the wealth pro...
Inspired by Parisian barrier options in finance (see e.g. Chesney et al. (1997)), a new definition of the event "ruin" for an insurance risk model is considered. As in Dassios and Wu (2009), the surplus process is allowed to spend time under a pre-specified default level before ruin is recognized. In this paper, we capitalize on the idea of Erlangi...
In this paper, we identify Laplace transforms of occupation times of
intervals until first passage times for spectrally negative Lévy processes.
New analytical identities for scale functions are derived and therefore the
results are explicitly stated in terms of the scale functions of the process.
Applications to option pricing and insurance risk m...
In this note, using Malliavin calculus for Lévy processes, we compute an explicit martingale representation for the maximum of a square-integrable Lévy process.
A discretization scheme for nonnegative diffusion processes is proposed and the convergence of the corresponding sequence of approximate processes is proved using the martingale problem framework. Motivations for this scheme come typically from finance, especially for path-dependent option pricing. The scheme is simple: one only needs to find a non...
In this paper, we compute the Laplace transform of occupation times (of the negative half-line) of spectrally negative Lévy processes. Our results are extensions of known results for standard Brownian motion and jump-diffusion processes. The results are expressed in terms of the so-called scale functions of the spectrally negative Lévy process and...
In a Lévy insurance risk model, under the assumption that the tail of the Lévy measure is log-convex, we show that either a horizontal barrier strategy or the take-the-money-and-run strategy maximizes, among all admissible strategies, the dividend payments subject to an affine penalty function at ruin. As a key step for the proof, we prove that, un...
We study the distribution of tax payments in the model of Kyprianou and Zhou [Kyprianou, A.E., Zhou, X., 2009. General tax structures and the Lévy insurance risk model. J. Appl. Probab. (in press)], that is a Lévy insurance risk model with a surplus-dependent tax rate. More precisely, after a short discussion on the so-called tax identity, we deriv...
Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Levy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected d...
Using fluctuation theory, we solve the two-sided exit problem and identify the ruin probability for a general spectrally negative Lévy risk process with tax payments of a loss-carry-forward type. We study arbitrary moments of the discounted total amount of tax payments and determine the surplus level to start taxation which maximises the expected d...
In this paper, we construct a Malliavin derivative for functionals of square-integrable Lévy processes and derive a Clark-Ocone formula. The Malliavin derivative is defined via chaos expansions involving stochastic integrals with respect to Brownian motion and Poisson random measure. As an illustration, we compute the explicit martingale representa...
Using Clark-Ocone formula, explicit martingale repre-sentations for path-dependent Brownian functionals are computed. As direct consequences, explicit martingale representations of the extrema of geometric Brownian motion and explicit hedging port-folios of path-dependent options are obtained.
In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.
In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividends paid until ruin in a Lévy insurance risk model with a dividend barrier.
In this short paper, we show how fluctuation identities for Lévy processes with no positive jumps yield the distribution of the present value of dividend payments until ruin in a Lévy insurance risk model with a dividend barrier.