
Bruno RemillardHEC Montréal | HEC Montreal · Department of Management Sciences and Quantitative Methods
Bruno Remillard
Doctor of Philosophy
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174
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Introduction
Publications
Publications (174)
We use dynamic programming, finite elements, and parallel computing to design and evaluate two-dimensional financial derivatives. Our dynamic program is flexible, as it divides the evaluation process into two components: one related to the dynamics of the underlying process and the other to the characteristics of the financial derivative. It is eff...
We propose new copula-based models for multivariate time series having continuous or discrete distributions, or a mixture of both. These models include stochastic volatility models and regime-switching models. We also propose statistics for testing independence between the generalized errors of these models, extending previous results of Duchesne,...
We propose a general structural model for valuing risky corporate debt securities within a two-dimensional framework. The state variables in our model include the firm’s asset value, described as a geometric Brownian motion stochastic process, and the short-term interest rate, following a mean-reverting Ornstein–Uhlenbeck stochastic process. Our mo...
In this paper, we consider the pricing problem of European options and spread options for the Hawkes-based model in the limit order book (LOB). We introduce a variant of Hawkes process and consider its limit theorems, namely the exponential multivariate general compound Hawkes process (EMGCHP). We also consider a special case of one-dimensional EMG...
Dedicated to Miklós Csörgő on his 65 th birthday Usually, empirical distribution functions are used to estimate the theoretical distribution function of known functions θ(X) of the observable random variable X. In practice, many researchers are using empirical distribution functions constructed from residuals, which are estimations of a non-observa...
Selecting the number of regimes in Hidden Markov models is an important problem. There are many criteria that are used to select this number, such as Akaike information criterion (AIC), Bayesian information criterion (BIC), integrated completed likelihood (ICL), deviance information criterion (DIC), and Watanabe-Akaike information criterion (WAIC),...
In this article, a copula-based method for mixed regression models is proposed, where the conditional distribution of the response variable, given covariates, is modelled by a parametric family of continuous or discrete distributions, and the effect of a common latent variable pertaining to a cluster is modelled with a factor copula. We show how to...
A convergence theorem for martingales with c\`adl\`ag trajectories (right continuous with left limits everywhere) is obtained in the sense of the weak dual topology on Hilbert space, under conditions that are much weaker than those required for any of the usual Skorohod topologies. Examples are provided to show that these conditions are also very e...
In this article, we define extensions of copula-based dependence measures for data with arbitrary distributions, in the non-serial case, i.e., for independent and identically distributed random vectors, as well as in serial case, i.e., for time series. These dependence measures are covariances with respect to a multilinear copula associated with th...
In this paper, we study the identifiability and the estimation of the parameters of a copula-based multivariate model when the margins are unknown and are arbitrary, meaning that they can be continuous, discrete, or mixtures of continuous and discrete. When at least one margin is not continuous, the range of values determining the copula is not the...
When the limiting compensator of a sequence of martingales is continuous, we obtain a weak convergence theorem for the martingales; the limiting process can be written as a Brownian motion evaluated at the compensator and we find sufficient conditions for both processes to be independent. Examples of applications are provided, notably for occupatio...
In this article, we present a review of important results and statistical applications obtained or generalized by Canadian pioneers and their collaborators, for empirical processes of independent and identically distributed observations, pseudo-observations, and time series. In particular, we consider weak convergence and strong approximations resu...
In this paper, we consider pricing of European options and spread options for Hawkes-based model for the limit order book. We introduce multivariate Hawkes process and the multivariable general compound Hawkes process. Exponential multivariate general compound Hawkes processes and limit theorems for them, namely, LLN and FCLT, are considered then....
This paper aims to incorporate a high order stochastic perturbation into a SIQR epidemic model with transient prophylaxis and lasting prophylaxis. The existence and uniqueness of the global positive solution is proven and a stochastic condition in order to study the extinction of an infectious disease is established. The existence of a stationary d...
Type Package Title Tests of Randomness and Tests of Independence Version 0.1.5 Description Functions for testing randomness for a univariate time series with arbitrary distribution (discrete, continuous, mixture of both types) and for testing independence between random variables with arbitrary distributions. The test statistics are based on the mu...
In this article we show that under weak assumptions, the change-point tests designed for independent random vectors can also be used with pseudo-observations for testing change-point in the joint distribution of non-observable random vectors, the associated copula, or the margins, without modifying the limiting distributions. In particular, change-...
In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. The law of large numbers (LLN) and two functional central lim...
In this paper, we focus on a new generalization of multivariate general compound Hawkes process (MGCHP), which we referred to as the multivariate general compound point process (MGCPP). Namely, we applied a multivariate point process to model the order flow instead of the Hawkes process. Law of large numbers (LLN) and two functional central limit t...
Lévy processes provide a solution to overcome the shortcomings of the lognormal hypothesis. A growing literature proposes the use of pure‐jump Lévy processes, such as the variance‐gamma (VG) model. In this setting, explicit solutions for derivative prices are unavailable, for instance, for the valuation of American options. We propose a dynamic pro...
Annual inflow forecasts are often based on historical time series, where every year is considered equally likely to reoccur. This process ignores the persistence of dry/wet conditions often observed in time series, behavior that is of utmost importance for hydroelectric energy producers. However, the modeling of persistence properties is challengin...
We consider several time series and for each of them, we fit an appropriate dynamic parametric model. This produces serially independent error terms for each time series. The dependence between these error terms is then modeled by a regime-switching copula. The EM algorithm is used for estimating the parameters and a sequential goodness-of-fit proc...
We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.
In this paper we build a discrete time model for the structure of the limit order book, so that the price per share depends on the size of the transaction. We deduce the value of a portfolio when the investor trades using market orders and a bank account with different interest rates for lending and borrowing. We also deduce conditions to rule out...
For nearest neighbor univariate random walks in a periodic environment, where the probability of moving depends on a periodic function, we show how to estimate the period and the function. For random walks in non-periodic environments, we find that the asymptotic limit of the estimator is constant in the ballistic case, when the random walk is tran...
In this paper, we consider non-stationary random vectors, where the marginal distributions and the associated copula may be time-dependent. We propose estimators for the unknown parameters and we establish the limiting distribution of the estimators of the copula and the conditional copula, together with a parametric bootstrap method for constructi...
In this paper, we propose an intuitive way to couple several dynamic time series models even when there are no innovations. This extends previous work for modeling dependence between innovations of stochastic volatility models. We consider time-dependent and time-independent copula models and we study the asymptotic behavior of some empirical proce...
Statistics are proposed for testing the hypothesis that arbitrary random variables are mutually independent. The tests are consistent and well behaved for any marginal distributions; they can be used, for example, for contingency tables which are sparse or whose dimension depends on the sample size, as well as for mixed data. No regularity conditio...
In this paper, we find necessary and sufficient conditions so that copula-based conditional distributions of a response variable with respect to covariates, are ordered with respect to the simple stochastic order introduced by Lehmann. These conditions do not depend on the marginal distributions of the random variables. As a result, we have conditi...
Parrondo’s paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec’s theorem. The regi...
In this paper we introduce two new Hawkes processes, namely, compound and regime-switching compound Hawkes processes, to model the price processes in limit order books. We prove Law of Large Numbers and Functional Central Limit Theorems (FCLT) for both processes. The two FCLTs are applied to limit order books where we use these asymptotic methods t...
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. This class of Riccati diffusion is quite general, and arises, for example, in data assimilation applications, and more particularly in ensemble (Kalman-type) filtering theory. These one-dimensional stochastic differ...
This article is concerned with the fluctuation analysis and the stability properties of a class of one-dimensional Riccati diffusions. These one-dimensional stochastic differential equations exhibit a quadratic drift function and a non-Lipschitz continuous diffusion function. We present a novel approach, combining tangent process techniques, Feynma...
Parrondo's paradox is extended to regime switching random walks in random environments. The paradoxical behavior of the resulting random walk is explained by the effect of the random environment. Full characterization of the asymptotic behavior is achieved in terms of the dimensions of some random subspaces occurring in Oseledec's theorem. The regi...
In this paper we solve the discrete time mean-variance hedging problem when asset returns follow a multivariate autoregressive hidden Markov model. Time dependent volatility and serial dependence are well established properties of financial time series and our model covers both. To illustrate the relevance of our proposed methodology, we first comp...
In this paper we solve the discrete time mean-variance hedging problem when asset returns follow a multivariate autoregressive hidden Markov model. Time dependent volatility and serial dependence are well established properties of financial time series and our model covers both. To illustrate the relevance of our proposed methodology, we first comp...
We propose a simple stochastic model for the dynamics of a limit order book, extending the recent work of Cont and de Larrard (2013), where the price dynamics are endogenous, resulting from market transactions. We also show that the conditional diffusion limit of the price process is the so-called Brownian meander.
The empirical checkerboard copula is a multilinear extension of the empirical copula, which plays a key role for inference in copula models. Weak convergence of the corresponding empirical process based on a random sample from the underlying multivariate distribution is established here under broad conditions which allow for arbitrary univariate ma...
Recently, two different copula-based approaches have been proposed to estimate the conditional quantile function of a variable with respect to a vector of covariates : the first estimator is related to quantile regression weighted by the conditional copula density, while the second estimator is based on the inverse of the conditional distribution f...
In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown...
In this paper, we study the asymptotic behavior of the sequential empirical process and the sequential empirical copula process, both constructed from residuals of multivariate stochastic volatility models. Applications for the detection of structural changes and specification tests of the distribution of innovations are discussed. It is also shown...
In this paper, we first present a review of statistical tools that can be used in asset management either to track financial indexes or to create synthetic ones. More precisely, we look at two important replication methods: the strong replication, where a portfolio of very liquid assets is created and the goal is to track an actual index with the p...
In this paper, one studies the asymptotic behavior of empirical processes based on consecutive residuals of univariate conditional mean and variance models. These processes are then used to develop tests of serial independence of the innovations. Even if the limiting distributions of the empirical processes depend on unknown parameters, it is shown...
Strong solutions of p-dimensional stochastic differential equations that can be represented locally in explicit simulation form are considered. The following three-way equivalence is established: 1) There exists such a representation from all starting points, 2) the representation pair satisfies a set differential equations, and 3) the stochastic d...
Strong solutions of p-dimensional stochastic differential equations that can be represented locally in explicit simulation form are considered. The following three-way equivalence is established: 1) There exists such a representation from all starting points, 2) the representation pair satisfies a set differential equations, and 3) the stochastic d...
We propose dynamic programming coupled with finite elements for valuing American-style options under Gaussian and double exponential jumps à la Merton [J. Financ. Econ., 1976, 3, 125–144] and Kou [Manage. Sci., 2002, 48, 1086–1101], and we provide a proof of uniform convergence. Our numerical experiments confirm this convergence result and show the...
In this paper we present a forecasting method for time series using copula-based models for multivariate time series. We study how the performance of the predictions evolves when changing the strength of the different possible dependencies, as well as the structure of the dependence. We also look at the impact of the marginal distributions. The imp...
We propose an innovative approach for dynamic portfolio insurance that overcomes many of the limitations of the earlier techniques. We transform the Payoff Distribution Model, originally introduced by Dybvig [J. Business, 1988, 61(3), 369–393] as a performance measure, into a fund management tool. This approach allows us to generate funds with pre-...
We propose an equivalent martingale measure for the regime-switching geometric Brownian motion, together with a hedging formula. It is shown that this choice is optimal in the sense that it minimizes the quadratic mean between the payoff at maturity and the value of the hedging portfolio, under the objective measure. The solution is easy to impleme...
This article is concerned with robust conditional variance and value-at-risk (VaR) estimation. Losses due to idiosyncratic
events can have a disproportionate impact on traditional VaR estimates, upwardly biasing these estimates, increasing capital
requirements, and unnecessarily reducing the available capital and profitability of financial institut...
Continuation refers to the operation by which the cumulative distribution
function of a discontinuous random vector is made continuous through
multilinear interpolation. The copula that results from the application of this
technique to the classical empirical copula is either called the multilinear or
the checkerboard copula. As shown by Genest and...
In this paper, one studies the asymptotic behavior of multivariate empirical processes based on consecutive residuals of univariate stochastic volatility models. These processes are used to develop tests of randomness about the innovations. Even if the limiting distributions of the empirical processes depend on unknown parameters, it is shown that...
We study existence, uniqueness and mass conservation of signed measure valued
solutions of a class of stochastic evolution equations with respect to the
Wiener sheet, including as particular cases the stochastic versions of the
regularized two-dimensional Navier-Stokes equations in vorticity form
introduced by Kotelenez.
Tie-corrected versions of Spearman’s rho are often used to measure the dependence in a pair of non-continuous random variables. Multivariate extensions of this coefficient, and estimators thereof, have recently been proposed by Quessy (2009a) [23] and Mesfioui and Quessy (2010) [19]. Asymptotically equivalent but numerically much simpler estimators...
While many financial engineering books are available, the statistical aspects behind the implementation of stochastic models used in the field are often overlooked or restricted to a few well-known cases. Statistical Methods for Financial Engineering guides current and future practitioners on implementing the most useful stochastic models used in f...
In this paper, we present a forecasting method for time series using copula-based models for multivariate time series. We study how the performance of the predictions evolve when changing the strength of the different possible dependencies and compare it with the univariate version of the forecasting method introduced by Sokolinskiy & Van Dijk. Mor...
Building on the work of Schweizer (1995) and Cern and Kallseny (2007), we
present discrete time formulas minimizing the mean square hedging error for
multidimensional assets. In particular, we give explicit formulas when a
regime-switching random walk or a GARCH-type process is utilized to model the
returns. Monte Carlo simulations are used to comp...
Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more than two time series, checking pair...
In this paper, using simulations, we compare specification procedures for testing the null hypothesis of a Gaussian distribution for the innovations of GARCH models. More precisely, Cramer-von Mises and Kolmogorov-Smirnov type statistics are computed for empirical processes based on the standardized residuals and their squares. For computing P-valu...
Trying to perform non-parametric change point tests for multivariate data using empirical processes is much more difficult that in the univariate case, since the limiting distribution depends on the unknown joint distribution function or its associated copula. In order to solve this problem, we extend the multiplier central limit theorem to empiric...
It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not pre...
Extending the multiplier central limit theorem and resampling bootstrap to statistics and empirical processes of pseudo-observations, it is shown how to build asymptotically independent copies of statistics and empirical processes to perform statistical tests. Application to parametric and semi-parametric specification tests for the innovations of...
In this paper, we extend copula-based univariate time series models studied in Chen & Fan (2006) to multivariate time series. Doing so, we tackle at the same time serial dependence as well as interdependence between several time series. The proposed methodology is totally different from the usual time-varying copula modeling of time series in which...
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...
It is shown that parametric bootstrap can be used for computing P-values of goodness-of-fit tests of multivariate time series parametric models. These models include Markovian models, GARCH models with non-Gaussian innovations, regime-switching models, as well as semi parametric models involving copulas of multivariate time series. The methodology...
Test statistics for checking the independence between the innovations of several time series are developed. The time series models considered allow for general specifications for the conditional mean and variance functions that could depend on common explanatory variables. In testing for independence between more that two time series, checking pair...
The asymptotic behaviour of the empirical copula constructed from residuals of stochastic volatility models is studied. It is shown that if the stochastic volatility matrix is diagonal, then the empirical copula process behaves like if the parameters were known, a remarkable property. However, that is not true if the stochastic volatility is genuin...
In this article we study the price of an American style option based on hedging the underlying assets at discrete time. Like its European style analog, the value of the option is not given in general by an expectation with respect to an equivalent martingale measure. We provide the optimal solution that minimizes the hedging error variance. When th...
It can be shown that when the payoff function is convex and decreasing (respectively increasing) with respect to the underlying (multidimensional) assets, then the same is true for the value of the associated American option, provided some conditions are satisfied. In such a case, all Monte Carlo methods proposed so far in the literature do not pre...
Discussion on "Brownian distance covariance" by G\'abor J. Sz\'ekely and Maria L. Rizzo [arXiv:1010.0297] Comment: Published in at http://dx.doi.org/10.1214/09-AOAS312F the Annals of Applied Statistics (http://www.imstat.org/aoas/) by the Institute of Mathematical Statistics (http://www.imstat.org)
In view of applications to diagnostic tests of ARMA models, the asymptotic behavior of multivariate empirical and copula processes based on residuals of ARMA models is investigated. Multivariate empirical processes based on squared residuals and other functions of the residuals are also investigated. It is shown how these processes can be used to d...