Thierry Roncalli's Home Page
Welcome to my home page. I made available on it some of my academic works. Please contact me for any comments or questions related to these materials.
Send e-mail: thierry_dot_roncalli_at_lyxor_dot_com
My SSRN page is http://ssrn.com/author=903940
- Head of Investment Products and Strategies, Structured Asset Management, SGAM Alternative Invesments (2005-2009)
- Head of Risk Analytics, Groupe de Recherche Opérationnelle, Crédit Agricole SA (2004-2005)
- Financial Engineer, Groupe de Recherche Opérationnelle, Crédit Lyonnais (1999-2003)
- Research Fellow at Financial
Econometrics Research Centre, City University Business School, London (1998-1999)
- Ph.D. Student at Laboratoire d'Analyse et de Recherche Economique,
University of Bordeaux IV (1993-1998)
- La Gestion des Risques Financiers (deuxième édition)
Auteur
T. Roncalli
Date
Octobre 2009
Résumé
Cette nouvelle édition a été l'occasion de revoir entièrement le texte, de supprimer un certain nombre de développement
qui ne sont plus d'actualité et d'apporter un éclairage nouveau par rapport à la crise actuelle. Elle contient aussi de
nouvelles illustrations et de nouvelles applications afin de mieux préciser certains concepts qui peuvent apparaître complexes.
Plusieurs sujets ont fait l'objet de nouveaux développements comme par exemple :
- Le risque de liquidité
- Les produits exotiques
- Le risque de contrepartie sur opération de marché
- Les produits structurés
- La gestion du risque de marché dans la gestion d'actifs
- La dépendance dans les dérivés de crédit
- La contribution en risque
Enfin, cette deuxième édition est accompagnée d'exercices permettant de vérifier ses connaissances.
- Editions Economica, Collection Finance, 560 pages
- Télécharger les programmes Gauss et les applications numériques du livre
- Télécharger les erratas du livre
- Télécharger la table des matières
- La Gestion des Risques Financiers
Auteur
T. Roncalli
Date
Août 2004
Résumé
La gestion des risques financiers est en pleine évolution sous la
pression de la réglementation prudentielle et du développement des outils
pour mieux les maîtriser. Le Comité de Bâle a publié le Nouvel Accord
sur le ratio international de solvabilité (Bâle II) le 26 juin 2004,
et la Commission Européenne a déjà adopté les différentes propositions
de cet accord. Cet accord a été accueilli favorablement par la profession
bancaire et les établissements financiers ont maintenant deux ans et demi
pour mener à bien cette réforme afin d'en bénéficier pleinement.
Les banques n'ont cependant pas attendu le Nouvel Accord pour
moderniser leur gestion des risques. Depuis dix ans, on assiste en
effet à un développement technique du risk management et les modèles pour
mesurer les risques sont de plus en plus sophistiqués. Le Nouvel Accord
participe d'ailleurs à cette évolution, puisqu'il vise à définir un capital
réglementaire plus proche du capital économique obtenu avec les modèles internes.
Le présent ouvrage s'inscrit dans ces deux lignes directrices : réglementation du
risque et modélisation du risque. Il s'adresse aussi bien à des étudiants de
troisième cycle, qui désirent acquérir une culture financière du risque et
de sa gestion, qu'à des professionnels qui cherchent à mieux comprendre
les fondements de la modélisation mathématique du risque.
- Editions Economica, Collection Gestion, 455 pages
- Télécharger les programmes Gauss et les applications numériques du livre
- TSM (Time Series and Wavelets for Finance)
Author
T. Roncalli
Date
1996
Summary
TSM is a GAUSS library for time series modeling in both time domain and frequency domain.
It is primarily designed for the analysis and estimation of ARMA, VARX processes, state space models, fractional processes and structural models.
To study these models, special tools have been developed like procedures for simulation, spectral analysis, Hankel matrices, etc.
Estimation is based on the Maximum Likelihood principle or Gnereralized Method of Moments and linear restrictions may be easily imposed.
It also contains several filtering methods (Kalman Filter, FLS and GFLS) and several procedures for Time-Frequency analysis of 1-D signal
(wavelet analysis and wavelet packet analysis).
- Source code: 300 Ko, Examples code: 390 Ko, Manual: 230 pages.
- TSM description at Gauss Aptech Systems webpage
- Download TSM examples
- Introduction à la programmation sous Gauss
Auteur
T. Roncalli
Date
1995
- Global Design, 660 pages
- Voir la page correspondante de Ritme Informatique
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- T. Roncalli and J. Teiletche
- Journal of Financial Transformation, 2008
- Download the paper.
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- P. Demey, J-F. Jouanin, T. Roncalli and C. Roget
- Risk, November 2004
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Authors
P. Hereil, P. Mitaine, N. Moussavi and T. Roncalli
Date
June 2010
Abstract
This paper studies the persistence of mutual fund performance.
Academic research often focuses on fund returns, sometimes
adjusted for style and market cap biases. Because fund rating
systems play a central role in the asset management industry, we
consider another approach in this paper. Using a Markov modeling of
these ratings, we illustrate that the persistence of the performance
is relatively poor with respect to the time horizon of investors. We
show that two facts may explain these results. First, the rating
system is not necessarily time-homogeneous. Second, the importance
of style is crucial when comparing the ratings of mutual funds. However, we
show that it is extremely difficult to characterize quantitatively
the style of a mutual fund. We conclude that fund selection is more
art than science, and that quantitative analysis must be combined with
qualitative insight.
Keywords
Mutual funds, rating system, style analysis, Markov chain, active management..
- Download the PDF file
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Authors
T. Roncalli
Date
May 12, 2010
Conference slides
- Download the PDF file
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Authors
P. Demey, S. Maillard and T. Roncalli
Date
March 2010
Abstract
A capitalization-weighted index is the most common way to gain access to broad
equity market performance. These portfolios are generally
concentrated in a few stocks and present some lack of diversification. In order to avoid
this drawback or to simply diversify market exposure,
alternative indexation methods have recently prompted great interest,
both from academic researchers and market practitioners. Fundamental
indexation computes weights with regard to economic measures, while risk-based
indexation focuses on risk and diversification criteria. This paper
describes risk-based indexation methodologies,
highlights potential practical issues when implemented, and
illustrates these issues as it applies to the Euro Stoxx 50 universe.
Keywords
Risk-based indexation, fundamental indexation, market capitalization, equity indexes, diversification, portfolio optimization, robust estimation,
minimum-variance indexation, equally-weighted indexation, erc indexation, mdp/msr indexation.
- Download the PDF file
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Authors
T. Roncalli and G. Weisang
Date
September 2009
Abstract
Of the three main challenges of hedge fund replication, only replication of the well known
nonlinearities of their returns remains undisputed. Recent advances in hedge fund replication
using factor models have shown that the use of Bayesian filters helps greatly
in capturing the dynamic allocation of assets of hedge fund managers, particularly in the
case of aggregates of hedge funds. Furthermore, from a practitioner’s perspective,
access to the alpha of the funds can be provided on top of capturing the dynamic exposures
by adopting a core/satellite approach to building the replication portfolio. In this
working paper, we explore tentatively the solutions that Bayesian filters could provide to
the replication of hedge fund nonlinearities. Although, not entirely successful, our results
show promises and open new grounds for the field.
Keywords
Particle filters, hedge funds, non-linearity.
- Download the PDF file
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Authors
P. Clauss, T. Roncalli and G. Weisang
Date
November 12, 2009
Conference slides
- Northeastern Section of the Mathematical Association of America, Spring 2009 Meeting, Fairfield University, May 29-30, 2009
- The 23rd New England Statistics Symposium, University of Connecticut, April 25, 2009
- Conseil Scientifique de l'AMF, Paris, November 12, 2009
- Download the PDF file
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Authors
P. Clauss, T. Roncalli and G. Weisang
Date
March 2, 2009
Abstract
In December 2008, as the financial and economic crisis continued on
its devastating course, a new scandal bursts. After the 1998's
failure of Long-Term Capital Management, Madoff's fraud brings once
again the discredit on the hedge funds industry. This one is however
of a different kind. Indeed, Madoff's firm is not a standard hedge
fund but a developed Ponzi scheme. By explaining Madoff's system and
exploring the reasons to its collapse, this paper draws risk
management lessons from this fraud, especially for operational risk
management. Risk management rules as applied nowadays partially
failed to prevent Madoff's scandal. This paper presents the issues
for risk capital requirements raised by Madoff collapse.
Implications for due diligence processes, including the use of
quantitative replication to assess hedge fund performance's
credibility, are also considered. Finally, consideration is given to
the regulatory and standardizing approaches of the hedge fund
industry as an answer to frauds of Madoff's kind.
Keywords
Madoff fraud, Ponzi scheme, operational risk, due diligence, supervision, hedge funds, bull spread strategy, split strike conversion.
- Download the PDF file
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Authors
T. Roncalli and G. Weisang
Date
February 11, 2009
Conference slides
- Brown Bag Seminar Series, IE Business School, Madrid, November 18, 2008
- Petits Déjeuners de la Finance, Palais Brogniart, Paris, February 11, 2009
- Laboratoire J. A. Dieudonné, Université de Nice Sophia-Antipolis, April 29, 2009
- Download the PDF file
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Authors
T. Roncalli and G. Weisang
Date
December 24, 2008
Abstract
As hedge fund replication based on factor models has encountered
growing interest among professionals and academics, and despite the
launch of numerous products (indexes and mutual funds) in the past
year, it faced many critics. In this paper, we consider three of the
main critiques, namely the lack of reactivity of hedge fund
replication and its deficiency in capturing tactical allocations;
its failure to apprehend non-linear positions of the underlying
hedge fund industry and higher moments of hedge fund returns; and,
finally, the lack of access to the alpha of hedge funds. To address
these problems, we consider hedge fund replication as a general
tracking problem which may be solved by means of Bayesian filters.
Using the linear Gaussian model as a basis for discussion, we
provide the reader with an intuition for the inner tenets of the
Kalman filter and illustrate the results' sensitivity to the
algorithm specification choices. This part of the paper includes
considerations on the type of strategies which can be replicated, as
well as the problem of selecting factors. We then apply more
advanced Bayesian filters' algorithms, known as particle filters, to
capture the non-normality and non-linearities documented on hedge
fund returns. Finally, we address the problem of accessing the pure
alpha by proposing a core/satellite approach of alternative
investments between high-liquid alternative beta and less liquid
investments.
Keywords
Tracking problem, hedge fund replication,
alternative beta, global tactical asset allocation, Bayes filter, Kalman filter, particle filter, numerical algorithms (SIS, GPP, SIR and RPF),
skewness, kurtosis, non-linear exposure, alpha.
- Download the PDF file
- Download the GAUSS library of Particle Filters
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Authors
S. Maillard, T. Roncalli and J. Teiletche
Date
September 2008
Abstract
Slides (not yet presented)
Keywords
risk contributions, minimum-variance, 1/n portfolio,
diversification, equity market neutral hedge funds.
- Download the PDF file
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Authors
S. Maillard, T. Roncalli and J. Teiletche
Date
July 1, 2008
Abstract
Minimum variance and equally-weighted portfolios have recently
prompted great interest both from academic researchers and market
practitioners, as their construction does not rely on expected
average returns and is therefore assumed to be robust. In this
paper, we consider a related approach, where the risk contribution
from each portfolio components is made equal, which maximizes
diversification of risk (at least on an ex-ante basis). Roughly
speaking, the resulting portfolio is similar to a minimum variance
portfolio subject to a diversification constraint on the weights of
its components. We derive the theoretical properties of such a
portfolio and show that its volatility is located between those of
minimum variance and equally-weighted portfolios. Empirical
applications confirm that ranking. All in all, equally-weighted risk
contributions portfolios appear to be an attractive alternative to
minimum variance and equally-weighted portfolios and might be
considered a good trade-off between those two approaches in terms of
absolute level of risk, risk budgeting and diversification.
Keywords
Asset allocation, risk contributions, minimum-variance, portfolio construction, risk budgeting, portfolio diversification.
- Download the PDF file
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Authors
A-S. Duret, P. Hereil, P. Mitaine, N. Moussavi and T. Roncalli
Date
April 16, 2008
Abstract
This paper studies the performance predictability of external fund rating systems.
Most investors use 5 stars rated funds to build their portfolios. The underlying idea is
that funds which were the best during the last three years will be better performers than the
other funds in the future. It implies that the 5 stars rating is a good persistence measure
of the performance. Using a Markov modelling and the seminal empirical work of Garnier and Pujol (2007),
we show that ratings persistence is poor. It means that fund selection or a fund picking process
may not be reduced to choose funds in a 5 stars rated universe.
Keywords
fund ratings, performance predictability,
markov generator, transition matrix, hurst exponent, fund picking, statistical persistence.
- Download the PDF file
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Authors
T. Roncalli and J. Teiletche
Date
April 1, 2007
Abstract
Hedge fund replication based on factor models is encountering growing interest.
In this paper, we investigate the implications of substituting standard rolling windows
regressions, which appear ad-hoc, with more efficient methodologies like the Kalman Filter.
We show that the copycats constructed this way offer risk-return profiles which share several
characteristics with the ones posted by hedge funds indices: Sharpe ratios above buy-and-hold
strategies on standard assets, moderate correlation with standard assets and limited drawdowns
during equity downward trends. An interesting result is that the shortfall risk seems less important
than with hedge fund indices and regressions based-trackers. We finally propose new breakdowns of
hedge fund performance into alpha, traditional beta and alternative beta.
Keywords
Hedge funds, factor models, beta, alpha, replication, Kalman filter.
- Download the PDF file
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Auteur
T. Roncalli
Date
3 Juin 2005
Résumé
Dans cette note, on propose une approche économique pour expliquer et donc prendre en
compte le skew de corrélation des CDO. On présente des premiers résultats basés sur l'indice iTraxx.
Mots-clés
CDO, skew correlation, copula.
- Télécharger la fichier PDF
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Authors
P. Demey and T. Roncalli
Date
September 29, 2004
Abstract
Slides of the seminar
"Petit Dejeuner de la Finance" organized by Frontiers in Finance.
Keywords
Default probability, maximum likelihood, spread jumps,
- Download the PDF file (400 KB)
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Authors
P. Demey, J-F. Jouanin, C. Roget and T. Roncalli
Date
November 2004
Abstract
Estimating asset correlations is difficult in practice since there is little available data and
many parameters have to be found. Paul Demey, Jean-Frédéric Jouanin, Céline Roget and
Thierry Roncalli present a tractable version of the multi-factor Merton model in which firms
are sorted into homogeneous risk classes. They derive a simplified maximum likelihood
approach that provides estimates in a reasonable computational time. As an application of
this methodology, industrial sector correlations are estimated from S&P’s data.
Keywords
default correlations, factor models.
- Download the PDF file (167 KB)
- Download the correction note (93 KB)
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Authors
A. Frachot, T. Roncalli and E. Salomon
Date
January 23, 2004
Abstract
This paper demonstrates that aggregate losses are necessarily low as long as we remain under the
standard assumptions of LDA models. Moreover empirical findings show that the correlation between two
aggregate losses is typically below 5%, which opens a wide scope for large diversification effects, much larger
than those the Basel Committee seems to have in mind. In other words, summing up capital charges is in
substantial contradiction with the type of correlation consistent with the standard LDA model.
Keywords
Operational risk, LDA model,
severity correlation, frequency correlation, aggregate loss correlation.
- Download the PDF file (1.1 MB)
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Authors
J-F. Jouanin, G. Riboulet and T. Roncalli
Date
July 15, 2003
Abstract
Copula functions have been introduced recently in finance. They are a general tool to construct
multivariate distributions and to investigate dependence structure between random variables. In this
paper, we show that copula functions may be extensively used to solve many financial problems. As
examples we show how to monitor the market risk of basket products, to measure the credit risk
of a large pool of loans and to compute capital requirements for operational risk.
Keywords
copula, risk management, market risk, credit risk, operational risk.
- Download the PDF file (1.1 MB)
- Download the zipped PS file (1.0 MB)
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Authors
A. Frachot, O. Moudoulaud and T. Roncalli
Date
May 07, 2003
Abstract
This paper follows the different steps necessary for implementing a LDA in practice:
- Step 1: Severity Estimation
- Step 2: Frequency Estimation
- Step 3: Capital Charge Computations
- Step 4: Confidence Interval
- Step 5: Self Assesment and Scenario Analysis
For each of these steps, we try to give illustrative examples and we gather all demanding mathematics
into subsections named Technical Appendix. We hope it will allow for a more reader-friendly paper.
Keywords
Operational risk, estimation,
confidence interval, self assesment and scenario analysis.
- Download the PDF file (300 KB)
- A critical approach to the copula model for credit derivatives
Author
J-F. Jouanin, G. Riboulet and T. Roncalli
Date
February 7, 2003
Abstract
Slides of the conference "Risque de Crédit", Evry.
Keywords
Credit derivatives, pricing, hedging.
- Download the PDF file (500 KB)
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Author
T. Roncalli
Date
February 6, 2003
Abstract
Slides of the conference "Séminaire CREST / LFA", Paris.
Keywords
Basle II, credit risk measurement,
credit portfolio management, time-inconsistency problems.
- Download the PDF file (1.0 MB)
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Authors
N. Baud, A. Frachot and T. Roncalli
Date
December 01, 2002
Abstract
Intense reflections are being conducted at the moment regarding the way to pool heteregenous data
coming from both banks’ internal systems and industry-pooled databases. We propose here a sound
methodology. As it relies on maximum likelihood principle, it is thus statistically rigorous and should
be accepted by supervisors. We believe that it solves the most part of data heterogeneity and scaling
issues.
Keywords
Operational risk, capital charge,
threshold, conditional distribution, maximum likelihood.
- Download the PDF file (160 KB)
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Authors
N. Baud, A. Frachot and T. Roncalli
Date
June 01, 2002
Abstract
It is widely recognized that calibration on internal data may not
suffice for computing an accurate capital charge against
operational risk. However, pooling external and internal data lead
to unacceptable capital charges as external data are generally
skewed toward large losses. In a previous paper, we have
developped a statistical methodology to ensure that merging both
internal and external data leads to unbiased estimates of the loss
distribution. This paper shows that this methodology is applicable
in real-life risk management and that it permits to pool internal
and external data together in an appropriate way. The paper is
organized as follows.\ We first discuss how external databases are
designed and how their design may result in statistical flaws.
Then we develop a model for the data generating process which
underlies external data.\ In this model, the bias comes simply
from the fact that external data are truncated above a specific
threshold while this threshold may be either constant but known,
or constant but unknown, or finally stochastic. We describe the
rationale behind these three cases and we provide for each of them
a methodology to circumvent the related bias. In each case,
numerical simulations and practical evidences are given.
Keywords
Operational risk, internal data,
external data, consortium data, threshold.
- Download the PDF file (700 KB)
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Authors
N. Baud, A. Frachot and T. Roncalli
Date
May 22, 2002
Abstract
Slides of the conference
"Seminarios de Matemática Financiera",
Instituto MEFF -
Risklab, Madrid.
Keywords
Operational risk, LDA, internal data,
external data, implied threshold.
- Download the PDF file (1.4 MB)
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Authors
J-F. Jouanin, G. Riboulet and T. Roncalli
Date
January 31, 2002
Abstract
Non-technical version of the paper
"Modelling dependence for credit derivatives with copulas".
Keywords
Copulas, intensity models, Moody's diversity score.
- Download the PDF file (280 KB)
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Authors
A. Frachot and T. Roncalli
Date
Januray 29, 2002
Abstract
The Loss Distribution Approach has many appealing features since it is expected to be much more
risk-sensitive than any other methods taken into consideration by the last proposals by the Basel
Committee. Thus this approach is expected to provide significantly lower capital charges for banks
whose track record is particularly good relatively to their exposures and compared with industry-wide
benchmarks.
Unfortunately LDA when calibrated only on internal data is far from being satisfactory from a regu-
latory perspective as it could likely underestimate the necessary capital charge. This happens for two
reasons. First if a bank has experienced a lower-than-average number of events, it will benefit from
a lower-than-average capital charge even though its good track record happened by chance and does
not result from better-than-average risk management practices. As a consequence, LDA is acceptable
as long as internal frequency data are tempered by industry-wide references. As such, it immediately
raises the issue of how to cope with both internal frequency data and external benchmarks. This
paper proposes a solution based on credibility theory which is widely used in the insurance industry
to tackle analogous problems. As a result, we show how to make the statistical adjustment to temper
the information conveyed by internal frequency data with the use of external references.
Similarly if the calibration of severity parameters ignores external data, then the severity distribution
will likely be biased towards low-severity losses since internal losses are typically lower than those
recorded in industry-wide databases. Again from a regulatory perspective LDA cannot be accepted
unless both internal and external data are merged and the merged database is used in the calibration
process. Here again it raises the issue regarding the best way to merge these data. Obviously it cannot
be done without any care since if internal databases are directly fuelled with external data, severity
distributions will be strongly biased towards high-severity losses. This paper proposes also a statistical
adjustment to make internal and external databases comparable with one another in order to permit
a safe and unbiased merging.
Keywords
Operational risk, LDA, internal data, external data, credibility theory.
- Download the PDF file (200 KB)
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Author
T. Roncalli
Date
October 26, 2001
Abstract
Slides of the conference
"Séminaire de Mathématiques et Finance Louis Bachelier", Institut Henri Poincaré.
Keywords
Copulas, credit derivatives, multi-asset options.
- Download the PDF file (1.7 MB)
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Authors
S. Coutant, V. Durrleman, G. Rapuch and T. Roncalli
Date
September 5, 2001
Abstract
In this paper, we use copulas to define multivariate risk-neutral
distributions. We can then derive general pricing formulas for multi-asset
options and best possible bounds with given volatility smiles. Finally, we
then apply the copula framework to define `forward-looking' indicators of
the dependence function between asset returns.
Keywords
Copulas, risk-neutral distribution, change of numéraire, option pricing, implied multivariate RND.
- Download the PDF file (2.2 MB)
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Authors
J-F. Jouanin, G. Rapuch, G. Riboulet and T. Roncalli
Date
August 25, 2001
Abstract
In this paper, we address the problem of incorporating default dependency in
intensity-based credit risk models. Following the works of Li [2000], Giesecke [2001] and
Schonbucher and Schubert [2001], we use copulas to model the joint distribution of the
default times. Two approaches are considered. The first one consists in
modelling the joint survival function directly with survival copulas of
default times, whereas in the second approach, copulas are used to correlate
the threshold exponential random variables. We compare these two approaches
and give some results about their relationships. Then we try some
simulations of simple products, such as first-to-defaults. Finally, we
discuss the calibration issue according to Moody's diversity score.
Keywords
Copulas, intensity models, Cox processes, Bessel processes, Moody's diversity score.
- Download the PDF file (3.0 MB)
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Authors
G. Rapuch and T. Roncalli
Date
July 16, 2001
Abstract
In this short note, we consider some problems of two-asset options pricing.
In particular, we investigate the relationship between options prices and
the `correlation' parameter in the Black-Scholes model. Then, we consider
the general case in the framework of the copula construction of risk-neutral
distributions. This extension involves results on the supermodular order
applied to the Feynman-Kac representation. We show that it could be viewed
as a generalization of a maximum principle for parabolic PDE.
Keywords
Copulas, two-asset options (Spread, Basket, Min, Max, BestOf, WorstOf),
supermodular order, concordance order, Fréchet bounds, Feynman-Kac representation, maximum principle, parabolic PDE.
- Download the PDF file (2.0 MB)
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Author
T. Roncalli
Date
July 8, 2001
Abstract
Slides of the conference
"Statistics 2001", Concordia University, Montréal, Canada.
Keywords
Copulas, gaussian assumption,
operational risk, risk-neutral copula, Heston model.
- Download the PDF file (1.4 MB)
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Authors
P. Georges, A-G. Lamy, E. Nicolas, G. Quibel and T. Roncalli
Date
May 28, 2001
Abstract
In this paper, we review the use of copulas for multivariate survival modelling.
In particular, we study properties of survival copulas and discuss the dependence measures associated to this construction.
Then, we consider the problem of competing risks. We derive the distribution of the failure time and order statistics.
After having presented statistical inference, we finally provide financial applications which concern the life time
value (attrition models), the link between default, prepayment and credit life, the measure of risk for a credit
portfolio and the pricing of credit derivatives.
Keywords
Survival copula, frailty model, ageing concepts, competing risks,
failure time, order statistics, prepayment, credit risk measure, default mode, correlated defaults, risk-bucket capital
charge, default digital put, credit default swap, first-to-default.
- Download the PDF file (3.6 MB)
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Author
T. Roncalli
Date
April 23, 2001
Abstract
Slides of the seminar
"Stochastic Models in Finance", Ecole Polytechnique, Paris, 23/04/2001.
Keywords
Copulas, quantile regression, markov copulas,
credit risk, uniform convergence, operations on distribution functions.
- Download the PDF file (3.2 MB)
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Authors
A. Frachot, P. Georges and T. Roncalli
Date
March 30, 2001
Abstract
In this paper, we explore the Loss Distribution Approach (LDA)
for computing the capital charge of a bank for operational risk where LDA refers to statistical/actuarial
methods for modelling the loss distribution. In this framework, the capital charge is calculated using a
Value-at-Risk measure. In the first part of the paper, we give a detailed description of the LDA
implementation and we explain how it could be used for economic capital allocation. In particular, we show
- how to compute the aggregate loss distribution by compounding the loss severity distribution and the
loss frequency distribution,
- how to compute the total Capital-at-Risk using copulas,
- how to control the upper tail of the loss severity distribution with order statistics.
In the second part of the paper, we compare LDA with the Internal Measurement Approach (IMA) proposed by the Basel
Committee on Banking Supervision to calculate regulatory capital for operational risk. LDA and IMA
are bottom-up internal measurement models which are apparently different. Nevertheless, we could map LDA
into IMA and give then some justifications about the choice done by regulators to define IMA. Finally, we
provide alternative ways of mapping both methods together.
Keywords
Operational risk, aggregated loss,
compound distribution, loss severity, loss frequency, Panjer algorithm, Capital-at-Risk, economic capital allocation,
order statistics, LDA, IMA, RPI, copulas.
- Download the PDF file (5.5 MB)
- Download the PDF zipped file (1.3 MB)
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Authors
V. Durrleman, A.Nikeghbali and T. Roncalli
Date
March 17, 2001
Abstract
Slides for the
International Finance Conference, Hammam-Sousse, Tunisia, 03/17/2001.
Keywords
Copulas, risky
dependence function, singular copulas, extreme points, quantile aggregation,
spread option.
- Download the PDF file (1.5 MB)
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Author
T. Roncalli
Date
January 26, 2001
Abstract
Slides of the seminar
"Statistical Methods in Integrated Risk Management" organized by Frontiers in Finance.
Keywords
Copulas, 2D option pricing,
markov processes, credit risk, CreditMetrics, CreditRisk+, first-to-default.
- Download the PDF file (3.4 MB)
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Authors
J. Bodeau, G. Riboulet and T. Roncalli
Date
December 15, 2000
Abstract
In this paper,
we consider non-uniform grids to solve PDE. We derive the theta-scheme
algorithm based on finite difference methods and show its consistency.
We then apply it to different option pricing problems.
Keywords
Theta-scheme, non-uniform
grids, temporal grids, cubic spline interpolation, european option,
american option, barrier option.
- Download the PDF file (830 KB)
- Download the corresponding GAUSS library
-
Author
T. Roncalli
Date
November 16, 2000
Abstract
Slides of the seminar
"Financial Applications of Copulas".
Keywords
Copulas, financial
applications, risk management, statistical modelling, probabilistic
metric spaces, markov operators, quasi-copulas.
- Download the PDF file (2.8 MB)
-
Authors
V. Durrleman, A.Nikeghbali and T. Roncalli
Date
November 23, 2000
Abstract
In this paper,
we consider the open question on Spearman's rho and Kendall's
tau of Nelsen [1991]. Using a technical hypothesis, we can answer
in the positive. One question remains open: how can we understand
the technical hypothesis? Because this hypothesis is not right
in general, we could find some pathological cases which contradict
Nelsen's conjecture.
Keywords
Spearman's rho, Kendall's tau, cubic copula.
- Download the PDF file (1.0 MB)
-
Authors
E. Bouyé, V. Durrleman, A. Nikeghbali G. Riboulet and T. Roncalli
Date
March 23, 2001 (First version: November 10, 2000)
Abstract
In this paper,
we show that copulas are a very powerful tool for risk management
since it fulfills one of its main goals: the modelling of dependence
between the individual risks. That is why this approach is an
open field for risk.
Keywords
Copulas, market risk,
credit risk, operational risk.
- Download the PDF file (1.3 MB)
-
Authors
A. Costinot, T.Roncalli and J. Teïletche
Date
October 24, 2000
Abstract
We consider the
problem of modelling the dependence between financial markets.
In financial economics, the classical tool is the Pearson (or
linear correlation) coefficient to compare the dependence structure. We show
that this coefficient does not give a precise information on the
dependence structure. Instead, we propose a conceptual framework
based on copulas. Two applications are proposed. The first one
concerns the study of extreme dependence between international
equity markets. The second one concerns the analysis of the East
Asian crisis.
Keywords
Linear correlation,
extreme value theory, quantile regression, concordance order,
Deheuvels copula, contagion, Asian crisis.
- Download the PDF file (3.7 MB)
-
Auteurs
A. Costinot, G. Riboulet et T. Roncalli
Date
September 15, 2000
Résumé
Les banques ont aujourd'hui
la possibilité de mettre en place un modèle interne de risque de marché. L'une des composantes
indispensables de ce modèle est la création d'un programme de stress testing.
Cet article présente un outil potentiel pour la construction d'un tel programme :
la théorie des valeurs extrêmes. Après avoir rappelé la réglementation propre au stress
testing et les principaux résultats de cette théorie, nous montrons comment les utiliser
pour construire des scénarios unidimensionnels, multidimensionnels et enfin pour quantifier
des scénarios de crise élaborés à partir de méthodologies différentes. Aux considérations méthodologiques
sont adjoints les résultats des simulations que nous avons réalisées sur différentes séries financières.
Mots-clés
Copules, fonction de dépendance de queue stable,
théorie des valeurs extrêmes, stress testing.
- Télécharger le fichier PDF (430 KB)
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Authors
V. Durrleman, A. Nikeghbali and T. Roncalli
Date
September 10, 2000
Abstract
In this paper,
we consider the problem of bounds for distribution convolutions
and we present some applications to risk management. We show that
the upper Fréchet bound is not always the more risky dependence
structure. It is in contradiction with the belief in finance that
maximal risk corresponds to the case where the random variables
are comonotonic.
Keywords
Triangle functions,
dependency bounds, infimal, supremal and sigma-convolutions, Makarov
inequalities, Value-at-Risk, "square root" rule, Dall'aglio problem,
Kantorovich distance.
- Download the PDF file (570 KB)
-
Authors
V. Durrleman, A.Nikeghbali and T. Roncalli
Date
August 21, 2000
Abstract
In this paper,
we study the approximation procedures introduced by Li, Mikusinski,
Sherwood and Taylor [1997]. We show that there exists a bijection
between the set of the discretized copulas and the set of the
doubly stochastic matrices. For the Bernstein and checkerboard
approximations, we then provide analytical formulas for the Kendall's
tau and Spearman's rho concordance measures. Moreover, we demonstrate
that these approximations do not exhibit tail dependences. Finally,
we consider the general case of approximations induced by partitions
of unity. Moreover, we show that the set of copulas induced by
partition of unity is a Markov sub-algebra with respect to the
*-product of Darsow, Nguyen and Olsen [1992].
Keywords
Doubly stochastic
matrices, Bernstein polynomials approximation, checkerboard copula,
partitions of unity, Markov algebras, product of copulas.
- Download the PDF file (610 KB)
-
Authors
V. Durrleman, A.Nikeghbali and T. Roncalli
Date
August 25, 2000
Abstract
In this paper,
we give a few methods for the choice of copulas in financial modelling.
Keywords
Maximum likelihood
method, inference for margins, CML method, point estimator, non
parametric estimation, Deheuvels copula, copula approximation,
discrete $L^{p}$ norm.
- Download the PDF file (1.2 MB)
-
Authors
V. Durrleman, A.Nikeghbali and T. Roncalli
Date
July 31, 2000
Abstract
We study how copulas
properties are modified after some suitable transformations. In
particular, we show that using appropriate transformations permits
to fit the dependence structure in a better way.
Keywords
$\gamma$-transformation,
Kendall's tau, Spearman's rho, upper tail dependence.
- Download the PDF file (2.1 MB)
-
Authors
V. Durrleman, A.Kurpiel, G. Riboulet and T. Roncalli
Date
June 14, 2000
Abstract
In this paper,
we consider 2D option pricing. Most of the problems come from
the fact that only few closed-form formulas are available. Numerical
algorithms are also necessary to compute option prices. This paper
examines some topics on this subject.
Keywords
Numerical integration
methods, Gauss quadratures, Monte Carlo, Quasi Monte Carlo, Sobol
sequences, Faure sequences, two-dimensional PDE, Hopscotch, LOD,
ADI, MOL, Stochastic volatility model, Malliavin calculus.
- Paper presented at the 17th International Conference in Finance
organized by the French Finance Association, Paris (June 28, 2000).
- Download the PDF file (3.2 MB)
-
Authors
E. Bouyé, V. Durrleman, A. Nikeghbali, G. Riboulet and T. Roncalli
Date
Mars 7, 2000
Abstract
Copulas are a general
tool to construct multivariate distributions and to investigate
dependence structure between random variables. However, the concept
of copula is not popular in Finance. In this paper, we show that
copulas can be extensively used to solve many financial problems.
Keywords
Multivariate distribution,
dependence structure, concordance measures, scoring, Markov processes,
risk management, extreme value theory, stress testing, operational
risk, market risk, credit risk.
- Paper presented at the 17th International Conference in Finance
organized by the French Finance Association, Paris (June 27, 2000)
and at First World Congress of the Bachelier Finance Society (June 29, 2000).
- Download the PDF file (3.9 MB)
-
Auteurs
N. Baud, P. Demey, D. Jacomy, G. Riboulet et T. Roncalli
Date
1er Mars 2000
Résumé
Comme
son nom l'indique, le Plan Epargne Logement est un produit d'épargne
qui permet d'acquérir des droits à prêts pour
financer un éventuel achat immobilier. Pour que les établissements
financiers et les particuliers y trouvent un intérêt
commun, le législateur a mis en place un système
de prime pendant la phase d'épargne. Celui-ci est perçu
comme un système incitatif pour le particulier et doit
permettre d'assurer la rentabilité du produit pour la banque.
Une note rédigée par le Trésor en 1996 conclut
à la rentabilité du PEL pour les banques. L'argument
repose sur le fait que les pertes (éventuelles) supportées
par la banque pendant la phase d'emprunt sont largement compensées
par les revenus de la phase d'épargne. En réponse
à cette note l'AFB s'est attachée à montrer
le contraire en incluant les coûts liés aux risques
de taux (Note de l'AFB du 16/12/1996).
Il n'est donc pas du tout certain que le système mis en
place soit rentable pour l’établissement financier. D'autant
plus que le Plan Epargne Logement est un produit financier relativement
complexe et que celui-ci contient différentes options cachées.
Le calcul de sa rentabilité est donc beaucoup plus difficile
que ceux présentés par le Trésor ou l'AFB.
C'est pourquoi le GRO a tenté de modéliser les options
cachées du PEL, de les valoriser et de calculer la rentabilité
finale de ce produit.
Mots-clés
Plan d’épargne
logement, option cachée de conversion, option américaine,
problème de contrôle optimal.
- Télécharger le fichier
PDF (1.2 MB)
-
Authors
N. Baud, A. Frachot, P. Igigabel, P. Martineu and T. Roncalli
Date
December 1, 1999
Abstract
Capital allocation
within a bank is getting more important as the regulatory requirements
are moving towards economic-based measures of risk. Banks are
urged to build sound internal measures of credit and market risks
for all their activities. Internal models for credit, market and
operational risks are fundamental for bank capital allocation
in a bottom-up approach. But this approach has to be completed
by a top-down approach in order to give to bank managers
a more comprehensive (but less detailed) vision of the allocation
efficiency.
From a top-down viewpoint, we are considering the different
business lines of a bank as assets. Then the capital has to be
allocated in order to balance a portfolio in an optimal way. In
this respect, a bank has to evaluate not only the expected return
and the risk of every business line, but also the correlation
matrix of these business lines returns. If a bank usually has
a good knowledge of its expected returns and risks, the problem
is more complex in the case of the correlation matrix: to cope
with the lack of internal data and information, we develop an
approach based on a Market Factor Model and estimate an implied
correlation matrix using the returns of a panel of banks.
The allocation problem is not exactly the problem a bank is confronted
to. It more precisely deals with capital reallocation. Moving
from an allocation to a new one generates costs that have to be
taken into account to ensure that the new allocation is better
than the former one. That is why reallocation signals are more
interesting: they do not point out the optimal allocation but
they allow the implementation of a dynamic policy that leads to
an optimal situation.
Keywords
Capital allocation,
top-down, bottom-up, factor model, optimisation problem, Lagrange
multipliers.
- Paper presented at "Les petits déjeuners de la Finance",
Paris (January 27, 2000).
- Download the PDF file (460 KB)
-
Author
T. Roncalli
Date
January 13, 1999
Abstract
In this paper, we consider the use of interest rate contingent
claims as indicators for the monetary policy. We analyze two approches: one based on the term
structure of zero bonds and another based on interest-rate option derivatives. We show how traditional tools
based on the Black framework could be biased to build indicators for monetary policy.
In fact, the second approach could not be viewed as an alternative approach, but as a complementary
approach of the term structure approach.
Keywords
Yield curve, Hull-White trinomial model, monetary policy.
- Download the PDF file (570 KB)
-
Authors
A. Kurpiel and T. Roncalli
Date
December 8, 1998
Abstract
The purpose of this paper is to analyse different
implications of the stochastic behavior of asset prices volatilities for option hedging purposes.
We present a simple stochastic volatility model for option pricing and illustrate its consistency
with financial stylized facts. Then, assuming a stochastic volatility environment, we study the
accuracy of Black and Scholes implied volatility-based hedging. More precisely, we analyse the
hedging ratios biases and investigate different hedging schemes in a dynamic setting.
Keywords
option hedging, stochastic volatility, Heston model, delta, gamma, vega.
- Download the PDF file (530 KB)
-
Authors
A. Kurpiel and T. Roncalli
Date
November 17, 1998
Abstract
In this paper, we consider Hopscotch methods for solving two-state
financial models. We first derive a solution algorithm for two-dimensional partial differential equations with
mixed boundary conditions. We then consider a number of financial applications including stochastic volatility
option pricing, term structure modelling with two states and elliptic irreversible investment problems.
Keywords
Two-dimensional PDE, Hopscotch method, parabolic financial models,
elliptic problems.
- Download the PDF file (670 KB)
- Download the corresponding GAUSS library
-
Auteur
T. Roncalli
Date
23 mars 1998
Résumé
Thèse de l'Université de Montesqieu-Bordeaux IV.
Mots-clés
Structure par terme, taux zéro, taux forward, méthode de Nelson-Siegel,
modèles factoriels, processus de diffusion, modèle de Black-Derman-Toy, modèle de Hull-White.
- Télécharger le fichier PDF (5.9 MB)
- Télécharger la bibliothèque Gauss
-
-
-
-
-
All the Gauss code provided in this web page may be used under the MIT License:
Copyright (c) 2009 Thierry Roncalli
Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
copies of the Software, and to permit persons to whom the Software is
furnished to do so, subject to the following conditions:
The above copyright notice and this permission notice shall be included in
all copies or substantial portions of the Software.
THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
- Gauss Ressources
-
- 2009
- GDR est la bibliothèque Gauss qui accompagne le livre La Gestion des Risques Financiers.
- Cette bibliothèque contient plus de 250 programmes sur les thèmes suivants :
Basel II and Cooke Ratio (Market risk / Credit Risk / Operational Risk), Multiple Risks Management (Copula functions / Risks Agregation),
Credit Risk Management (Credit Derivatives / PD / LGD / Default Correlations / Credit Portfolio Management / RAROC).
- Télécharger la bibliothèque Gauss
-
- May 2009 (work in progress)
- QAM is the Gauss library which has been developped for the lecture notes on Quantitative Asset Management.
- This library contains procedures:
- for computing backtest (monthly rebalancing, currency hedging, strategy leveraging, fees managing, performance reporting, etc.).
- for portfolio allocation (Black-Litterman, Markowitz, ERC, MDP, risk bubdgeting, index sampling, 130/30, MSR, Sharpe style analysis, etc.).
- for computing numerical algorithms (simplex set, Markov generator, quadrature rules, Fokker-Planck equation, etc.).
- for derivatives pricing (dynamic delta hedging, Hedge fund replication, etc.).
- for statistical methods (Artificial neural networks, copula, CLS, FLS, GMM, Huber, LAD, Logit, MARS, ML, NLS, PCA, Probit, Quantile regression, QP, Robust, Non-parametric Kernel regression, RBS, Tobit, factor models, etc.).
- for time series analysis (arch, garch, vecm, spectral analysis, wavelets, etc.)
- for strategy backtesting (covered call, bull-spread, carry trade, variance swaps, vix, long/short equity, absolute return strategy, trend-following strategy, etc.);
- for stock screening (gini optimization, scoring methods, boosting, bagging method, etc.)
- for risk management (stop loss strategy, take profit strategy, concentration, etc.)
- Download the manual of the Gauss library
- Download the Gauss library
- Download the lecture notes (only available in French)
-
- December 2008
- PF is a Gauss library written with Guillaume Weisang for computing particle filters using the numerical algorithms described in
S. Arulampalam, S. Maskell, N.J. Gordon and T. Clapp [2002], A tutorial on particle filters for online nonlinear/non-Gaussian
Bayesian tracking, IEEE Transaction on Signal Processing, 50:2, 174-188.
- This library contains procedures for computing
- Generic Particle Filter (GPF) algorithm
- Regularized Particle Filter (RPF) algorithm
- Sampling Importance Resampling (SIR) algorithm
- Sampling Importance Sampling (SIS) algorithm
- Particle smoother (PS) algorithm
- Download the manual of the Gauss library
- Download the Gauss library
-
- 2008 (the first version of this library was written in 1998 when I worked in FERC. I have added some new procedures based on SABR and Durrleman models.)
- This library contains procedures:
- for the following models: Black and Scholes [1973], Merton [1976], Cox, Ross and Rubinstein [1979],
Barone-Adesi and Whaley [1987], Bates [1991], Rubinstein et Reiner [1991], Heston [1993], Dupire [1993],
Chang, Chang and Lim [1998), Hagan et al. [2002] (SABR model).
- for the following payoff functions: European, American, Barrier (DIC, DIP, DOC, DOP, KIC, KIP, KOC, KOP, UIC, UIP, UOC, UOP), Binary,
Asian (Fixed and Floating stike), Lookback, Spread, Corridor, Partial Barrier, Window Barrier, Bermudean.
- for computing greeks and implied volatility.
- for estimating local volatility.
- for estimating density (Breeden and Litzengerber [1978], Durrleman [2004]).
- for solving backward PDE, forward PDE and variational inequalities.
- for simulating SDE (GBM, OU, CIR, jump-diffusion processes, mGBM, SDE, mSDE and local volatility processes).
- for performing Monte-Carlo (LCG, Park-Miller, L'Ecuyer) and Quasi-Monte-Carlo (Sobol, Faure) methods using reduction variance techniques (antithetic variables).
- for computing option prices using numerical integrations (Gauss-Legendre, Gauss-Laguerre, Gauss-Hermite and Gauss-Jacobi quadrature methods, Simpson approximation).
- for estimating multi-assets option prices and Asset/Time-sorted options using Monte-Carlo simulations
(BestOf, WorstOf, Bermudean, Rainbow, Average, Himalaya, etc.)
- Download the manual of the Gauss library
- Download the Gauss library
-
- November 2004
- SKEW is a Gauss library for computing pdf, cdf and inverse of the cdf and simulating random numbers for the
SN, ST, MSN and MST distribution functions described in Azzalini, A. and Capitanio A.[2003], Distributions generated by
perturbation of symmetry with emphasis on a multivariate skew t distribution, JRSS B, 65, 367-389.
- Download the manual of the Gauss library
- Download the Gauss library
- Link to the Skew Normal webpage of Adelchi Azzalini
-
- 2004
- GDR est la bibliothèque Gauss qui accompagne le livre La Gestion des Risques Financiers.
- Cette bibliothèque contient plus de 250 programmes sur les thèmes suivants :
Basel II and Cooke Ratio (Market risk / Credit Risk / Operational Risk), Multiple Risks Management (Copula functions / Risks Agregation),
Credit Risk Management (Credit Derivatives / PD / LGD / Default Correlations / Credit Portfolio Management / RAROC).
- Télécharger la bibliothèque Gauss
-
-
- June 2001
- MVT is a Gauss library for computing multivariate t cdf. It is based on the Fortran packages mvt.f and mvtdstpack.f written by Alan Genz.
The Fortran source code and the corresponding articles are available from his web page
http://www.sci.wsu.edu/math/faculty/genz/homepage.
- Download the manual of the Gauss library
- Download the Gauss library
-
-
- August 2000
- This library is a Gauss implementation of the algorithms described in Portnoy and Koenker [1997] and Yu and Jones [1998].
- Download the manual of the Gauss library
- Download the Gauss library
-
- November 1998
- PDE2D is a Gauss library for solving Parabolic and Elliptic Partial Differential Equations (PDE) in a 2 space dimensions.
It includes Hopscotch and theta-schemes algorithms with finite difference methods.
- The working paper "Hopscotch methods for two-state financial models" describes the Hopscotch methods (problem definition,
algorithm, mixed boundary, computational considerations, band and sparse algorithms). It shows also how to apply them
to two-state financial models (2D Feynman-Kac problems, European stochastic volatility pricing, variational
inequalities problems, American stochastic volatility pricing, Term structure modelling with two states, financial
elliptic problems, Perpetual options, McDonald-Siegel-Dixit-Pyndick (irreversible investment) model, equity-based credit
risk model).
- PDE2D is a Gauss implementation of the paper Hopscotch methods for two state financial models.
- Download the manual of the Gauss library
- Download the Gauss library
-
- 1999
- A Gauss library for Artificial Neural Network (ANN).
- Contains procedures for estimating (Steepest Descent, Momentum, Adaptative Learning Rates, Adaptative Learning Rates with Momentum,
BFGS, Scaled BFGS, Self-Scaling DFP, Newton-Raphson; Polak-Ribiere Conjugate Gradient), analyzing
(graphic representation, tree pruning and omission cost) and forecasting ANN .
- Download the Gauss library
-
- 1998
- A Gauss library to manipulate graphs
- Download the Gauss library
-
-
- 1997
- This is an implementation of the envelopment problems of Seiford, L.M. and R.M. Thrall [1990], Recent Developments in DEA,
The mathematical Programming Approach to Frontier Analysis,Journal of Econometrics, 46, 7-38
- The procedure Linear_Programming solves the linear programming problem whereas the proceduress Input_Oriented and
Output_Oriented solve the input-oriented and output-oriented envelopment problems.
- Download the Gauss library
-
- 1997
- This is an implementation of the fitting procedures of Friedman, J.H. [1991], Multivariate Adaptive Regression Splines,
Annals of Statistics, 19, 1-141
- Contains procedures for polynomial spline regression, one and two-dimensional projection index optimization and
projection pursuit regression. The precedure for multivaraite adaptative regression spline is not yet available because it was based
on a dll of the fortran code developped by Jerome Friedman which is now not available in the public domain.
- Download the Gauss library
-
-
- 1997
- Procedures for computing functions of matrices (matrix cosine, matrix exponential, general matrix function and matrix sine)
and tridiagonal matrices (dense matrix to tridiagonal matrix form, inverse of a tridiagonal matrix, tridiagonal matrix form to dense matrix
and solve Xy=d with tridiagonal X matrix)
- Download the Gauss library
- Gauss Courses
-
- AGF Asset Management (1 et 2 juillet 2003, Paris)
- CDC IXIS (15, 16 et 17 avril 2002, Paris)
- INSEE (18, 19, 20 mars 2002 et 8 novembre 2002, Paris)
- Banque Worms (19 et 20 juillet 2001, Paris)
- INRA (27 et 28 septembre 1999, Rennes)
- Télécharger les notes
- Télécharger les programmes Gauss
-
-
-
- CCF, Direction de la Recherche et de l'Innovation (11, 25 septembre, 7 et 23 octobre 1997)
- Télécharger les notes du niveau II
- Télécharger les notes du niveau I
- Télécharger les programmes Gauss
-
- Société Générale (5 novembre 1998)
-
- Caisse Nationale du Crédit Agricole (9 mars 1995)
- Gauss Seminars
-
- Séminaire Gauss organisé par Ritme Informatique, Paris (1 Octobre 2002).
- Télécharger le fichier PDF
-
- Séminaire Gauss organisé par Ritme Informatique, Paris (20 Avril 2000).
- Télécharger le fichier PDF
-
- Seminar organized by FERC and Timberlake Consultants, London (December 9 1998).
- Download the PDF notes
- Download the Gauss code
-
- Séminaire Gauss organisé par Ritme Informatique, Paris (27 Avril 1998).
- Télécharger le fichier PDF
-
- Séminaire Gauss organisé par Ritme Informatique, Paris (17 Mai 1995).