Project Report on Financial Mathematics

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INSTITUT NATIONAL DE RECHERCHE EN INFORMATIQUE ET EN AUTOMATIQUE

Project-Team Math? Financial Mathematics
Paris - Rocquencourt

Theme : Stochastic Methods and Models

c tivity eport
2009

Table of contents
1. 2. Team . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 Overall Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 2.1. Highlights of the year 2 2.2. Introduction 2 Scienti?c Foundations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 3.1. Simulation of Stochastic Differential Equations 2 3.2. Numerical methods for option pricing and hedging and model calibration 2 3.3. Malliavin calculus and applications in ?nance 3 3.4. Optimal stopping 5 3.5. Stochastic Control and Backward Stochastic Differential equations (BSDEs) 5 Application Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Software . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 5.1.1. Description of Premia 6 5.1.2. Content of Premia 6 5.1.3. Latest features 7 5.1.4. Software organization 9 5.1.5. Consortium Premia 9 New Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 6.1. Discretization of stochastic volatility models 9 6.2. Monte Carlo simulations and stochastic algorithms for option pricing 10 6.3. Optimal stopping and American Options 10 6.4. Risk 11 6.5. Stochastic control of jump diffusions, Stochastic Maximum principles and BSDEs 11 6.6. Pricing and hedging in incomplete markets 12 6.7. Stochastic analysis and Malliavin calculus 12 Contracts and Grants with Industry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 Other Grants and Activities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 8.1. ANR programs 13 8.2. Pôle compétitivité 13 8.3. International cooperations 13 Dissemination . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 9.1. Conference and Seminar organisation 13 9.2. Editorship 14 9.3. Services to the scienti?c community 14 9.4. Teaching 15 9.5. Internship advising 16 9.6. PhD defences 16 9.7. PhD advising 17 9.8. PhD reports 17 9.9. Participation to workshops, conferences and invitations 18 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

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1. Team
Research Scientist Agnès Sulem [ Team leader, DR, INRIA, HdR ] Benjamin Jourdain [ Professor, ENPC, HdR ] Bernard Lapeyre [ Professor ENPC, HdR ] Aurélien Alfonsi [ ENPC ] Francesco Russo [ Professor Paris 13, on leave at INRIA from October 2008, HdR ] Faculty Member Vlad Bally [ Professor, Université Paris-Est Marne la Vallée, HdR ] Damien Lamberton [ Professor, Université Paris-Est Marne la Vallée, HdR ] Antonino Zanette [ Assistant professor, University of Udine - Italy ] External Collaborator Jean-Philippe Chancelier [ ENPC, Cermics ] Céline Labart [ Assistant professor University Paris 6 ] Jérôme Lelong [ Assistant professor ENSTA, from October 2009, ENSIMAG Grenoble ] Ahmed Kebaier [ Assistant professor University Paris 13 ] Technical Staff Amaury de la Vaissière [ until February 2009 ] Ismail Laachir David Pommier PhD Student Lokmane Abbas-Turki [ Pôle France-Innovation, Université Paris-Est Marne la Vallée ] José Infante Acevedo [ ENPC ] Abdelkoddousse Ahida [ Pôle Finance-Innovation, Université Paris-Est Marne la Vallée ] Ayech Bouselmi [ MENRT grant, Université Paris-Est Marne la Vallée ] El Hadj Dia [ MENRT grant, Université Paris-Est Marne la Vallée ] Stefano Di Marco [ MENRT grant, Université Paris-Est Marne la Vallée and Pise ] Maxence Jeunesse [ ENPC ] Mohamed Mikou [ MENRT grant, Université Paris-Est Marne la Vallée ] Andreea Minca [ Fondation Natixis grant, INRIA and Paris 6 ] Sidi Mohamed Ould Aly [ MENRT grant, Université Paris-Est Marne la Vallée ] Victor Rabiet [ Université Paris-Est Marne la Vallée ] Mohamed Sbai [ ENPC ] Post-Doctoral Fellow Teitur Arnarson [ INRIA, January-July ] Alessandra Cretarola [ INRIA, from May 2009, funded by Creditnext, Pôle Finance Innovation ] John Joseph Absalom Hosking [ INRIA, from October 2009 ] Visiting Scientist Vadim Zherder [ Russian Customs Academy Rostov Branch - Department of Informatics, Russia (2 months) ] Oleg Kudryavtsev [ Moscow, Russia (1 month) ] Xiao Wei [ Beijing University, China (2 months) ] Administrative Assistant Martine Verneuille [ AI, INRIA ] Other Cédric Allali [ Master 2 Université Paris-Diderot, Internship, INRIA, 5 months ] Kaouther Hajji [ Ecole Polytechnique de Tunis Internship, INRIA, Since November 2009 ] Pierre-Yves Lagrave [ ENPC internship ] Yisheng Wang [ ENSTA internship, INRIA, 3 months ]

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Activity Report INRIA 2009

2. Overall Objectives
2.1. Highlights of the year
Concerning research, a lot of effort has been put again this year in the study of Lévy processes in ?nancial modelling. The project Credinext, supported by Pôle Finance Innovation has started.

2.2. Introduction
MathFi is a joint INRIA project-team with ENPC (CERMICS) and the University Paris-est Marne la Vallée, located in Rocquencourt and Marne la Vallée. The development of increasingly complex ?nancial products requires the use of advanced stochastic and numerical analysis techniques. The scienti?c skills of the MathFi research team are focused on probabilistic and deterministic numerical methods and their implementation, stochastic analysis, stochastic control. Main applications concern evaluation and hedging of derivative products, dynamic portfolio optimization in incomplete markets, calibration of ?nancial models, risk management. Special attention is paid to models with jumps, stochastic volatility models, asymmetry of information. The MathFi project team develops the software Premia dedicated to pricing and hedging options and calibration of ?nancial models, in collaboration with a consortium of ?nancial institutions. Premia web site: http://www.premia.fr .

3. Scienti?c Foundations
3.1. Simulation of Stochastic Differential Equations
Participants: B. Jourdain, A. Alfonsi, D. Lamberton, M. Sbai. Most ?nancial models are described by SDEs. Except in very special cases, no closed-form solution is available for such equations and one has to approximate the solution via time-discretization schemes in order to compute options prices and hedges by Monte Carlo simulations. Usually this is done by using the standard explicit Euler scheme since schemes with higher order of strong convergence involve multiple stochastic integrals which are dif?cult to simulate. In addition, the weak order of convergence of the explicit Euler scheme can be improved by using Romberg-Richardson’s extrapolations. Nethertheless, some schemes with weak order of convergence two or more have been designed recently. The idea is either to replace the multiple Brownian integrals by discrete random variables which share their moments up to a given order or to integrate Ordinary Differential Equations associated with the vector ?elds giving the coef?cients of the Stochastic Differential Equation up to well-chosen random time-horizons. Another interesting new direction of investigation is the design of exact simulation schemes. Three directions of research have been investigated in the Math? project. First, ?ne properties of the Euler scheme have been studied [72], [6], [74]. Secondly, concerning SDEs for which the Euler scheme is not feasible, A. Alfonsi and V. Lemaire [1] have proposed and analysed new schemes respectively for CoxIngersoll-Ross processes and for equations with locally but not globally Lipschitz continuous coef?cients. Last, the team has contributed to the new directions of research described above. For CIR processes, A. Alfonsi has designed a scheme with weak order two even for large values of the volatility parameter. Adapting exact simulation ideas, B. Jourdain and M. Sbai [7] have proposed an unbiased Monte Carlo estimator for the price of arithmetic average Asian options in the Black-Scholes model.

3.2. Numerical methods for option pricing and hedging and model calibration
Participants: B. Jourdain, A. Alfonsi, D. Lamberton, M. Sbai, V. Bally, B. Lapeyre, A. Sulem, A. Kebaier, C. Labart, J. Lelong, D. Pommier, L. Abbas-Turki, A. Ahida, A. Zanette, E. Dia.

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Ef?cient computations of prices and hedges for derivative products is a major issue for ?nancial institutions (see [76]) . This is done by using either Monte-Carlo methods or partial differential equations techniques. Monte-Carlo simulations are widely used because of their implementation simplicity and because closed formulas are usually not available. Speeding up the algorithms is a constant preoccupation in the developement of MonteCarlo simulations. The team is mainly concerned with adaptive versions which improve the Monte-Carlo estimator by relying only on stochastic simulations. The team has also been active on numerical methods in models with jumps and large dimensional problems. This activity in the MathFi team is strongly related to the development of the Premia software. Model calibration: The modeling of the so called implied volatility smile which indicates that the BlackScholes model with constant volatility does not provide a satisfactory explanation of the prices observed in the market has led to the appearance of a large variety of extensions of this model as the local volatility models (where the stock price volatility is a deterministic function of price level and time), stochastic volatility models, models with jump, and so on. An essential step in using any such approach is the model calibration, that is, the reconstruction of model parameters from the prices of traded options. This is an inverse problem to that of option pricing and as such, typically ill-posed. The calibration problem is yet more complex in the interest rate markets since in this case the empirical data that can be used includes a wider variety of ?nancial products from standard obligations to swaptions (options on swaps). The underlying model may belong to the class of short rate models like Hull-White [73], [66], CIR [69], Vasicek [84] etc. or to the popular class of LIBOR (London Interbank Offered Rates) market models like BGM [67]. The choice of a particular model depends on the ?nancial products available for calibration as well as on the problems in which the result of the calibration will be used. The calibration problem is of particular interest for MathFi project because due to its high numerical complexity, it is one of the domains of mathematical ?nance where ef?cient computational algorithms are most needed.

3.3. Malliavin calculus and applications in ?nance
Participants: V. Bally, A. Kohatsu-Higa, A. Sulem, A. Zanette. The original Stochastic Calculus of Variations, now called the Malliavin calculus, was developed by Paul Malliavin in 1976 [77]. It was originally designed to study the smoothness of the densities of solutions of stochastic differential equations. One of its striking features is that it provides a probabilistic proof of the celebrated Hörmander theorem, which gives a condition for a partial differential operator to be hypoelliptic. This illustrates the power of this calculus. In the following years a lot of probabilists worked on this topic and the theory was developed further either as analysis on the Wiener space or in a white noise setting. Many applications in the ?eld of stochastic calculus followed. Several monographs and lecture notes (for example D. Nualart [79], D. Bell [65] D. Ocone [81], B. Øksendal [85]) give expositions of the subject. See also V. Bally [64] for an introduction to Malliavin calculus. From the beginning of the nineties, applications of the Malliavin calculus in ?nance have appeared : In 1991 Karatzas and Ocone showed how the Malliavin calculus, as further developed by Ocone and others, could be used in the computation of hedging portfolios in complete markets [80]. Since then, the Malliavin calculus has raised increasing interest and subsequently many other applications to ?nance have been found [78], such as minimal variance hedging and Monte Carlo methods for option pricing. More recently, the Malliavin calculus has also become a useful tool for studying insider trading models and some extended market models driven by Lévy processes or fractional Brownian motion.

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Activity Report INRIA 2009

Let us try to give an idea why Malliavin calculus may be a useful instrument for probabilistic numerical methods. We recall that the theory is based on an integration by parts formula of the form E (f (X )) = E (f (X )Q). Here X is a random variable which is supposed to be “smooth” in a certain sense and non-degenerated. A basic example is to take X = ? ? where ? is a standard normally distributed random variable and ? is a strictly positive number. Note that an integration by parts formula may be obtained just by using the usual integration by parts in the presence of the Gaussian density. But we may go further and take X to be an aggregate of Gaussian random variables (think for example of the Euler scheme for a diffusion process) or the limit of such simple functionals. An important feature is that one has a relatively explicit expression for the weight Q which appears in the integration by parts formula, and this expression is given in terms of some Malliavin-derivative operators. Let us now look at one of the main consequenses of the integration by parts formula. If one considers the Dirac function ?x (y ), then ?x (y ) = H (y ? x) where H is the Heaviside function and the above integration by parts formula reads E (?x (X )) = E (H (X ? x)Q), where E (?x (X )) can be interpreted as the density of the random variable X . We thus obtain an integral representation of the density of the law of X . This is the starting point of the approach to the density of the law of a diffusion process: the above integral representation allows us to prove that under appropriate hypothesis the density of X is smooth and also to derive upper and lower bounds for it. Concerning simulation by Monte Carlo methods, suppose that you want to compute M 1 i 1 M E (?x (y )) ? M is a sample of X . As X has a law which is absolutely coni=1 ?x (X ) where X , ..., X tinuous with respect to the Lebesgue measure, this will fail because no X i hits exactly x. But if you are able to simulate the weight Q as well (and this is the case in many applications because of the explicit form mentioned M 1 i i above) then you may try to compute E (?x (X )) = E (H (X ? x)Q) ? M i=1 E (H (X ? x)Q ). This basic remark formula leads to ef?cient methods to compute by a Monte Carlo method some irregular quantities as derivatives of option prices with respect to some parameters (the Greeks) or conditional expectations, which appear in the pricing of American options by the dynamic programming). See the papers by Fournié et al [71] and [70] and the papers by Bally et al., Benhamou, Bermin et al., Bernis et al., Cvitanic et al., Talay and Zheng and Temam in [75]. L. Caramellino, A. Zanette and V. Bally have been concerned with the computation of conditional expectations using Integration by Parts formulas and applications to the numerical computation of the price and the Greeks (sensitivities) of American or Bermudean options. The aim of this research was to extend a paper of Reigner and Lions who treated the problem in dimension one to higher dimension - which represent the real challenge in this ?eld. Signi?cant results have been obtained up to dimension 5 [5] and the coresponding algorithms have been implemented in the Premia software. Moreover, there is an increasing interest in considering jump components in the ?nancial models, especially motivated by calibration reasons. Algorithms based on the integration by parts formulas have been developped in order to compute Greeks for options with discontinuous payoff (e.g. digital options). Several papers and two theses (M. Messaoud and M. Bavouzet defended in 2006) have been published on this topic and the corresponding algorithms have been implemented in Premia. Malliavin Calculus for jump type diffusions and more general for random variables with localy smooth law - represents a promising ?eld of research, also for applications to credit risk problems. More recently the Malliavin calculus has been used in models of insider trading. The "enlargement of ?ltration" technique plays an important role in the modeling of such problems and the Malliavin calculus can be used to obtain general results about when and how such ?ltration enlargement is possible. See the paper by P.Imkeller in [75]). Moreover, in the case when the additional information of the insider is generated by adding the information about the value of one extra random variable, the Malliavin calculus can be used to ?nd explicitly the optimal portfolio of an insider for a utility optimization problem with logarithmic utility. See the paper by J.A. León, R. Navarro and D. Nualart in [75]). A. Kohatsu Higa and A. Sulem have studied a controlled stochastic system whose state is described by a stochastic differential equation with anticipating coef?cients. These SDEs can be interpreted in the sense of forward integrals, which are the natural generalization of the semimartingale integrals, as introduced by Russo and Valois [83]. This methodology has been applied for utility maximization with insiders.

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3.4. Optimal stopping
Participants: A. Alfonsi, B. Jourdain, D. Lamberton. The theory of American option pricing has been an incite for a number of research articles about optimal stopping. Our recent contributions in this ?eld concern optimal stopping for one dimensional diffusions and American options in exponential Lévy models. In the context of general one-dimensional diffusions, we have studied optimal stopping problems with bounded measurable payoff functions. We have obtained results on the continuity of the value function and its characterization as the unique solution of a variational inequality in the sense of distributions, both in ?nite and in?nite horizon problems (collaboration between D. Lamberton and Michail Zervos, London School of Economics). We have explained how to calibrate a continuous and time-homogeneous local volatility function from the prices of perpetual American Call and Put options (A. Alfonsi and B. Jourdain). The use of jump diffusions in ?nancial models goes back to Merton (1976). More recently, there has been a growing interest for more sophisticated models, involving Lévy processes with no diffusion part and in?nite activity (see, in particular, papers by Carr, Geman, Madan and Yor). One of our PhD students (M. Mikou) works on the qualitative properties of American option prices in exponential Lévy models. A number of results on the exercice boundary and on the so called smooth ?t property have been established.

3.5. Stochastic Control and Backward Stochastic Differential equations (BSDEs)
Participants: V. Bally, J.-Ph. Chancelier, M.C. Kammerer-Quenez, A. Sulem. B. Øksendal (Oslo University) and A.Sulem have written a book on Stochastic control of Jump diffusions [11]). The types of control problems covered include classical stochastic control, optimal stopping, impulse control and singular control. Both the dynamic programming method and the maximum principle method are discussed, as well as the relation between them. Corresponding veri?cation theorems involving the HamiltonJacobi Bellman equation and/or (quasi-)variational inequalities are formulated. There are also chapters on the viscosity solution formulation and numerical methods. In the second edition (2007), a chapter on optimal control of stochastic partial differential equations driven by Lévy processes and a section on optimal stopping with delayed information have been added. Applications to portfolio optimization problems and insurance problems have been studied. In the context of risk measures, M.C. Quenez (assistant professor at UPEMLV until 2007, now Prof Paris VII) has shown how some dynamic measures of risk can be induced by Backward Stochastic Differential Equations and A. Sulem and B. Øksendal in [35] have studied risk-indifference pricing in incomplete markets with jumps using stochastic control theory and PDE methods.

4. Application Domains
4.1. Application domains
• • • • • • Option pricing and hedging Calibration of ?nancial models Portfolio optimization Risk management Insurance-reinsurance optimization policy Insider modeling, asymmetry of information

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Activity Report INRIA 2009

5. Software
5.1. Development of the software PREMIA for ?nancial option computations
Participants: A. Alfonsi, V. Bally, J-Ph. Chancelier, A. De la Vaissière, B. Jourdain, A. Kebaier, O Kudryavtsev, I. Laachir, C. Labart, A. Kolotaev, B. Lapeyre, J. Lelong, D. Pommier, A. Sulem, X. Wei, A. Zanette, V. Zherder.

Figure 1.

5.1.1. Description of Premia
PREMIA is a platform dedicated to the development of algorithms and scienti?c documentation for option pricing, hedging and model calibration (http://www.premia.fr). This project keeps track of the most recent advances in the ?eld of computational ?nance in a well-documented way. It focuses on the implementation of numerical analysis techniques for both probabilistic and deterministic numerical methods. An important feature of the platform Premia is the detailed documentation which provides extended references in option pricing. Premia is thus a powerful tool to assist Research & Development professional teams in their day-to-day duty. It is also a useful support for academics who wish to perform tests on new algorithms or pricing methods without starting from scratch. Besides being a single entry point for accessible overviews and basic implementations of various numerical methods, the aim of the Premia project is: 1. to be a powerful testing platform for comparing different numerical methods between each other; 2. to build a link between professional ?nancial teams and academic researchers; 3. to provide a useful teaching support for Master and PhD students in mathematical ?nance. The development of Premia started in 1999 and 11 are released up to now and registered at the APP agency.

5.1.2. Content of Premia
Premia contains various numerical algorithms (Finite-differences, trees and Monte-Carlo) for pricing vanilla and exotic options on equities, interest rate, credit and energy derivatives. 1. Equity derivatives: The following models are considered: Black-Scholes model (up to dimension 10), stochastic volatility models (Hull-White, Heston, Fouque-Papanicolaou-Sircar), models with jumps (Merton, Kou, Tempered stable processes, Variance gamma, Normal inverse Gaussian), Bates model.

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For high dimensional American options, Premia provides the most recent Monte-Carlo algorithms: Longstaff-Schwartz, Barraquand-Martineau, Tsitsklis-Van Roy, Broadie-Glassermann, quantization methods Malliavin calculus based methods. Dynamic Hedging for Black-Scholes and jump models is available. Calibration algorithms for some models with jumps, local volatility and stochastic volatility are implemented. 2. Interest rate derivatives The following models are considered: HJM and Libor Market Models (LMM): af?ne models, Hull-White, CIR++, Black-Karasinsky, Squared-Gaussian, Li-Ritchken-Sankarasubramanian, Bhar-Chiarella, Jump diffusion LMM, Markov functional LMM, LMM with stochastic volatility. Premia provides a calibration toolbox for Libor Market model using a database of swaptions and caps implied volatilities. 3. Credit derivatives: CDS, CDO Reduced form models and copula models are considered. Premia provides a toolbox for pricing CDOs using the most recent algorithms (Hull-White, LaurentGregory, El Karoui-Jiao, Yang-Zhang, Schönbucher) 4. Hybrid products: PDE solver for pricing derivatives on hybrid products like options on in?ation and interest or change rates is implemented. 5. Energy derivatives: swing options Mean reverting and jump models ar considered. Premia provides a toolbox for pricing swing options using ?nite differences, Monte-Carlo Malliavinbased approach and quantization algorithms.

5.1.3. Latest features
Premia 11 has been delivered to the consortium members in February 2009. This year we have developped and implemented new algorithms for pricing equity derivatives in stochastic volatility models and models with jumps, further developped routines for pricing American options in the LIBOR interest rate market and for credit derivatives with dynamic models. J. Lelong has performed modi?cation of the testing procedure to run on parallel architectures. New algorithms for the release 12 of Premia to be delivered in March 2010 to the Consortium: • Interest Rate Derivatives – – – • A stochastic volatility Libor model and its robust calibration Working paper Belomestny Mathew Schoenmakers (2007) True upper bounds for Bermudean products via Non-Nested Monte Carlo. D. Belostomeny, C. Bender, J. Schoenmakers. Mathematical Finance Volume 191 January 2009 Pricing and calibration in HW2D model.

Credit Risk Derivatives(A. Kebaier, C. Labart, J. Lelong) – – Default Contagion in Large Homogeneous Portfolios Alexander Herbertsson, No 272, Working Papers in Economics from Göteborg University, Department of Economics Advanced credit portfolio modeling and CDO pricing. Eberlein R.Frey E. A. von Hammerstein in Mathematics: Key Technology for the Future, W. Jager, and H.-J. Krebs, (Eds.), Springer (2008), p.p. 253-280

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Activity Report INRIA 2009

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Hedging default risks of CDOs in Markovian contagion models J.-P. Laurent, A. Cousin, J-D. Fermanian Dynamic hedging of synthetic CDO-tranches with spread-and contagion risk Frey, R. and Backhaus, J., preprint, department of mathematics, Universitat Leipzig Pricing Credit from the top down with af?ne point processes, Errais, Giesecke, Goldberg (2006). A.Alfonsi J.Lelong A Closed-form extension to Black-Cox formula.

Insurance – A bivariate model for evaluating fair premiums of equity-linked policies with maturity guarantee and surrender option. Costabile, M., Gaudenzi, M., Massabo, I., Zanette, A. 2009 Insurance: Mathematics and Economics 45 (2), pp. 286-295 Equity – A. Achdou, T. Arnarson and D. Pommier: Calibration of American options in Levy models. Finite Difference. Option Pricing Using Fourier Transforms: A Numerically Ef?cient Simpli?cation. M. Attari – D. Pommier: Sparse grid pricing of basket options. – T. Lelièvre and D. Pommier: Low rank approximation method for basket options. – E. Dia and D. Lamberton: Montecarlo methods for in?nite activity Levy models for Lookback options. – Pricing Variance Swap: Pricing options on realized variance in the Heston model with jumps in returns and volatility Artur Sepp: Journal of Computational Finance, Volume 11 / Number 4, Fall 2008 – O. Kudryavtsev: Pricing in Regime Switching models. – A Multinomial Approximation for American Option Price In Levy Process Models(MallerSolomon-Szymaier Mathematical Finance October 2006 - Vol. 16 Issue 4 Page 589-694) – A generalization of the Hull and White formula with applications to option pricing approximation E. Alos: Finance and Stochastics 10 (3), 2006, p. 353-365 – Polynomial Processes and their applications to mathematical Finance.C.Cuchiero, M. Keller-Ressel, J.Teichmann: preprint arXiv/0812.4740 – Adaptive control variates for pricing multi-dimensional American options Samuel M. T. Ehrlichman and Shane G. Henderson Journal of Computational Finance, Volume 11 / Number 1, Fall 2007 – A Tree-based Method to price American Options in the Heston Model, Vellekoop, M.H. and Nieuwenhuis, J.W. to appear Journal of Computational Finance – Pricing American Options under Stochastic Volatility and Stochastic Interest Rates Alexey N. Medvedev O. Scaillet – A Stochastic Volatility Alternative to SABRL.C.G. Rogers L.A.M. Veraart J. Appl. Probab. Volume 45, Number 4 (2008), 1071-1085. – Multi-level Monte Carlo path simulation. M.B. Giles, Operations Research, 56(3):607617, 2008. – A. Alfonsi, A. Ahdida: High order discretization of Wishart process. – Time dependent Heston model E. Benhamou, E. Gobet and M. Miri, Time Dependent Heston Model(March 24, 2009). – E. Voltchkova: Localization of the Black-Scholes equation using transparent boundary conditions. – M. Gaudenzi, M.A. Lepellere, A. Zanette: The Singular Points method for Pricing American Path-Dependent Options.to appear Journal of Computational Finance

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5.1.4. Software organization
The software Premia provides a collection of C/C++ routines and scienti?c documentation in PDF and HTML. More precisely, Premia is composed of : • • • a library designed to describe derivative products, models, pricing methods and which provides basic input/output functionalities. a collection of pricing routines: in this way, the routines of Premia can easily be plugged into other ?nancial softwares. a scienti?c documentation system. It is created from hyperlinked PDF ?les which describe the pricing routines and the general numerical methods involved like Monte Carlo methods, lattice methods, etc.

Premia is available for Windows and Linux operating systems. It provides Excel and Scilab/Nsp interfaces. Reports in PDF can be automatically generated at the end of each computation session. The size of Premia is around 500 Mb, with 12 Mb of source code in C. J. Lelong has continued to improve and maintain the Nsp interface for Premia. The manual, which he started to write a year ago, now documents almost all functionalities provided by the interface. J. Lelong and D. Pommier are developing a numerical library for Premia to give the contributors a uni?ed scienti?c library (PNL). The version 1.0 (November 2009) (around 27000 lignes of code C) will be registered at the APP agency. Here are the major contributions of J. Lelong in the development of the PNL: • • • Interface for the Amos library which provides approximations for the complex Bessel functions. Complete reorganisation of the source code. Linear algebra : matrix exponential, complex matrices, ef?cient matrix vector operations (inspired from Blas, D. Pommier also worked on that point), resolution of linear systems, interfaces for some Lapack functions (LU decomposition, matrix logarithm, computation of eigenvalues and eigenvectors). Root ?nding functions. Interface for the Fast Fourier Transform library FFTPack. Laplace inversion functions. Sorting functions. Numerical integration methods for one and two variate functions.

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5.1.5. Consortium Premia
Premia is developed in interaction with a consortium of ?nancial institutions or departments presently composed of: CALYON, Société Générale, Natixis, Bank Austria and RZB (Raiffeisen Zentralbank Österreich AG). The participants of the consortium contribute to ?nance the development of Premia and help to determine the directions in which the project evolves. They have access to the complete software with the source and the documentation. Every year, a new release is delivered to the Consortium members. Moreover, a restricted version of Premia is available on Premia web site http://www.premia.fr and can be downloaded with a special license for academic and evaluation purposes.

6. New Results
6.1. Discretization of stochastic volatility models
Participants: M. Sbai, B. Jourdain, A. Alfonsi, A. Ahida.

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Activity Report INRIA 2009

In usual stochastic volatility models, the process driving the volatility of the asset price evolves according to an autonomous one-dimensional stochastic differential equation. B. Jourdain and M. Sbai assume that the coef?cients of this equation are smooth. Using Itô’s formula, they get rid, in the asset price dynamics, of the stochastic integral with respect to the Brownian motion driving this SDE. Taking advantage of this structure, they propose • • a scheme, based on the Milstein discretization of this SDE, with order one of weak trajectorial convergence for the asset price, a scheme, based on the Ninomiya-Victoir discretization of this SDE, with order two of weak convergence for the asset price [61].

They also propose a speci?c scheme with improved convergence properties when the volatility of the asset price is driven by an Orsteim-Uhlenbeck process. A. Alfonsi and A. Ahida are working on the simulation of Wishart processes. Wishart matrices process are increasingly used in ?nance to model a multiname volatility. A. Alfonsi and A. Ahida have obtained an exact simulation procedure, and a second-order scheme for general af?ne diffusions on symmetric positive matrices.

6.2. Monte Carlo simulations and stochastic algorithms for option pricing
Participants: E.H.A. Dia, B. Jourdain, C. Labart, D. Lamberton, J. Lelong. D. Lamberton and El Hadj Aly DIA develop Monte-Carlo methods for exotic options in models with jumps. Some results concerning lookback and barrier options in jump-diffusion models have been obtained, in particular, estimates for expectations involving discrete vs continuous maxima of the sample paths of a Lévy process. They also have results on the effect of truncating small jumps and replacing them by Brownian motion. They also study the valuation of lookback and digital barrier options in exponential Lévy models without positive jump. D. Lamberton and S. M. Ould Aly study exotic options and stochastic volatility models. They have obtained results on the effective computation of option prices in a stochastic volatility model, in the context of variance swap modelling.

6.3. Optimal stopping and American Options
Participants: D. Lamberton, M. Mikou, B. Jourdain, M. Vellekoop, D. Pommier, T. Arnarson. - Optimal stopping of one-dimensional diffusions: In continuation of joint work with Mihail Zervos (London School of Economics), D. Lamberton has been working on an example of optimal stopping with irregular payoff (the staircase option). - American options in exponential Lévy models: D. Lamberton and M. Mikou [12] have obtained results on the regularity of the American put price in the case of general exponential Lévy models. They also derive the asymptotic behavior of the early exercise boundary near maturity in non-classical cases. - American Put option with discrete dividends : B. Jourdain and M. Vellekoop (Twente University) are interested in the regularity of the optimal exercise boundary for the American Put option when the underlying asset pays a discrete dividend at a known time during the lifetime of the option. The ex-dividend asset price process is assumed to follow Black-Scholes dynamics and the dividend amount is a deterministic function of the ex-dividend asset price just before the dividend date. The solution to the associated optimal stopping problem can be characterised in terms of an optimal exercise boundary which, in contrast to the case when there are no dividends, is no longer monotone. They prove monotonicity, continuity and a high contact principle at times in a left-hand neighbourhood of the dividend date for different dividend payment functions. - Calibration of American options in Lévy models. D. Pommier, T. Arnarson and Y. Achdou (Paris 7 University) are investigating a numerical procedure for the calibration of American option in Lévy models, based on optimality conditions on the parameters for minimization of a least square problem involving the observed option prices.

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6.4. Risk
• Credit Risk: A. Alfonsi J. Lelong have studied a very simple extension of the Black-Cox model for a single default. Namely, they have considered the case where the default intensity can take different levels according to the ?rm value. In this model, they know the Laplace transform of the default time, and are able to calibrate the model rather quickly to the CDS prices [55]. Liquidity risk: A. Alfonsi, A. Schied and A. Slynko have continued to work on a simple limit order book model [57]. First they have obtained the optimal strategy for buying a large amount of shared with a given number of trades, when the market resilience is exponential. This problem is closely related to the absence of Price Manipulation Strategies. Then, the case of more general resilence kernels has been considered. The model is no longer Markovian, and a new kind of manipulation strategies have been identi?ed.

•

6.5. Stochastic control of jump diffusions, Stochastic Maximum principles and BSDEs
Participants: A. Cretarola, J.J.A Hosking, F. Russo, A. Sulem. Robust control. A. Sulem and B. Øksendal study robust (worst case scenario) optimal stochastic control of jump diffusions and equivalent martingale measures. Singular stochastic control. A. Sulem in a recent joint work with B. Øksendal study general singular control problems of Lévy processes, in which the controller has only partial information and the system is not necessarily Markovian. Two different approaches are considered: (i) by using Malliavin calculus, leading to generalized variational inequalities for partial information singular control (of possibly non-Markovian systems) (ii) by introducing a singular control version of the Hamiltonian and using backward stochastic differential equations (BSDEs) to obtain a partial information maximum principle for such problems. They show that the two methods are related, and ?nd a connection between them. They then study the relation between the generalized variational inequalities found in (i) and general re?ected backward stochastic differential equations (RBSDEs) for Lévy processes. These are again shown to be equivalent to general optimal stopping problems for such processes. Combining this, a connection between singular control and optimal stopping is obtained. Optimal control of forward-backward stochastic differential equations with jumps. In [34], A. Sulem and B. Øksendal present various versions of the maximum principle for optimal control of forward-backward SDEs (FBSDEs) with jumps. This study is motivated by risk minimization via g-expectations. They ?rst prove a general suf?cient maximum principle for optimal control with partial information of a stochastic system consisting of a forward and a backward SDE driven by Lévy processes. They then present a Malliavin calculus approach which allows them to handle non-Markovian systems. Finally they give examples of applications in ?nance, namely they study the risk minimizing portfolio problem and a utility optimization problem under risk constraint. Risk-Minimizing Stopping and Backward Stochastic Differential Equations Following [34], A. Cretarola and A. Sulem consider a coupled system of FBSDEs with jumps and study an optimal stopping problem in which the decision maker uses a dynamic convex risk measure to evaluate the risk of the ?nancial standing. Backward stochastic differential equations under partial information A. Cretarola and Francesco Russo are studying the existence and the uniqueness of a solution to BSDEs of the form
T T

Yt = ? +
t

f (s, Ys , Zs )d M

s?

Zs dMs ? (OT ? Ot ) ,
t

0?t?T

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Activity Report INRIA 2009

driven by the general martingale M and investigate connections to dynamic risk measures and applications to hedging of derivatives or insider trading. As a ?rst step, the linear case has been studied. Then, they have considered the nonlinear case, by assuming the continuity of the martingale M and they have shown existence and uniqueness results by using similar arguments as in [82]. Now they are focusing on the jump case. They have already proved existence by using some results that can be found in [68]. Maximum principles for stochastic differential games. John Joseph Absalon is starting to look with Agnès Sulem at maximum principles for games with spike perturbation method.

6.6. Pricing and hedging in incomplete markets
Participants: F. Russo, A. Sulem. Risk indifference pricing in jump diffusion markets. In [35], A. Sulem and B. Øksendal study the risk indifference pricing principle in incomplete markets: The (seller’s) risk indifference price pseller risk is the initial payment that makes the risk involved for the seller of a contract equal to the risk involved if the contract is not sold, with no initial payment. We use stochastic control theory and PDE methods to ?nd a formula for pseller risk and similarly for pbuyer . In particular, we prove that risk
seller plow ? pbuyer risk ? prisk ? pup ,

where plow and pup are the lower and upper hedging prices, respectively. Variance Optimal Hedging. In [53], F. Russo, with S. Goutte and N. Oudjane address pricing and hedging issues in incomplete markets and consider applications to the electricity markets. The case when the asset price is a process with independent increments or an exponential of those processes is studied. An anticipative stochastic calculus approach to pricing in markets driven by Lévy processes. In [54], A. Sulem and B. Øksendal use the Itô-Ventzell formula for forward integrals and Malliavin calculus to study the stochastic control problem associated to utility indifference pricing in a market driven by Lévy processes. This approach allows them to consider general possibly non-Markovian systems, general utility functions and possibly partial information based portfolios. In the case of exponential utility function U? = ? exp (??x) ; ? > 0, asymptotics properties for vanishing ? are obtained. In the special case of full information based portfolios and no jumps a recursive formula for the optimal portfolio in a non-Markovian setting is given.

6.7. Stochastic analysis and Malliavin calculus
Participants: V. Bally, F. Russo, A. Sulem. The classical Malliavin Calculus for jump type processes as it was developped in the book of Bichteler Gravereu and Jacod 1985 concerns Poisson point measure in which the jump amplitudes represent independent random variables. V. Bally considers equations driven by Poisson random measures in which the law of the jumps depends on the position of the solution of the stochastic equation. This corresponds to physical models as the Bolzmann equation for example. The two preprints of V. Bally in colaboration with E. Clément and N. Fournier [59], [60] concern this topic. This is also a continuation of the work done with M.P. Bavouzet and M. Marouen [42] concerning the sensitivity computations in jump type models in ?nance. V. Bally is studying the problem of lower bounds for the density of a functionalthe since several years. The last two contributions in this area are papers in collaboration with A. Meda and B. Fernandez from the University of Mexico (tubes estimates) [18] and another one in collaboration with A. Kohatsu-Higa [19]. Following an idea of Malliavin and Thalmaier, V. Bally and L. Caramellino (Tor Vergata University) use the Riesz transform in order to give regularity criterions for the law of a random variable under week hypothesis [58].

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F. Russo studies modelling of vortex ?laments via stochastic integrals via regularisation and the probabilistic representation of a non-linear PDE of porous medium type and discontinuous coef?cient. Both the nondegenerate case and the degenerate case are considered.

7. Contracts and Grants with Industry
7.1. Consortium Premia
The consortium Premia is centered on the development of the pricer software Premia. It is presently composed of the following ?nancial institutions: CALYON, Société Générale, Natixis, Bank Austria, Raiffeisen Zentralbank Österreich AG.

8. Other Grants and Activities
8.1. ANR programs
• ANR program GCPMF "Grid Computation for Financial Mathematics" February 2006-June 2009. Partners : Calyon, Centrale, EDF, ENPC, INRIA, Ixis, Paris 6, Pricing Partner, Summit, Supelec. Global coordinator: B. Lapeyre. • ANR program BigMC (Monte Carlo Methods in large dimension). Partners ENST, ENPC, University Paris-Dauphine. ENPC coordinator: B. Jourdain

8.2. Pôle compétitivité
The project “Credinext” on credit risk derivatives has started in the “Pôle Finance Innovation”. Partners: Euronext Paris, Lunalogic, Pricing Partners, CMAP (Ecole Polytechnique), CERMICS/ENPC, Université Paris-Est Marne la Vallée (Laboratoire de Mathématiques Appliqués), INRIA (projet Math?).

8.3. International cooperations
• • Part of the European network "Advanced Mathematical Methods for Finance" (AMaMef). This network is supported by the European Science Foundation (ESF). Collaborations with the Universities of Oslo, Bath, Chicago, Mexico, Osaka, Rome II and III, Tokyo Institute of Technology

9. Dissemination
9.1. Conference and Seminar organisation
• • • A. Alfonsi - Co-organizer of the working group seminar of MATHFI “Stochastic methods and ?nance”. A. Alfonsi, J.-F. Delmas, B. Jourdain, B. Lapeyre: - 3rd conference on numerical methods in ?nance, Ecole des Ponts ParisTech, 15-17 april 2009. B. Lapeyre - Member of the program committee of the "Parallel and Distributed Computing in Finance" May 25-29, 2009, Roma (Italy).

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Activity Report INRIA 2009

•

F. Russo - Coorganisation (with E. Valkeila, J.M. Corcuera and F. Biagini) of an AMAMEF workshop on “No semimartingale models in ?nance”, May 2009, Helsinki, Finland. - Member of the Scientidic Committee of the Conference Dé?s actuels de la ?nance, University Paris 13, November 2009.

•

A. Sulem - Member of the Scienti?c Committee of the Optimal stopping with applications symposium, Turku Finland 15-19 June 2009. http://web.abo.?/fak/mnf/mate/gradschool/optimalstopping2009/

9.2. Editorship
• D. Lamberton - Associate Editor of Mathematical Finance - Co-Editor of ESAIM P&S. • A. Sulem - Associate editor of SIAM Journal on Financial mathematics SIFIN - Associate editor of Journal of Applied Mathematics and Stochastic Analysis JAMSA

9.3. Services to the scienti?c community
• • B. Jourdain - Deputy head of the doctoral school ICMS, University Paris-Est C. Labart - Reviewer for Mathematical Reviews and Referee for Stochastic Processes and Their Applications, Mathematical Finance and Mathematics of computation. • D. Lamberton - in charge of the master program “Mathématiques et Aplications" (Universities of Marne-la-Vallée, Créteil and Evry, and Ecole Nationale des Ponts et Chaussées). - Director of the Mathematical department (UFR de mathématiques), Université Paris-Est Marne-laVallée. - Member of the Steering Committee of the ESF European Network "Amamef" (http://www.iac. rm.cnr.it/amamef/); in charge of the GDR "Méthodes Mathématiques pour la ?nance", which is the national CNRS group related to the network(until the end of 2009). • • B. Lapeyre - President of the Doctoral Department at Ecole des Ponts F. Russo - Member of the Professors council of the graduate program of LUISS University (Rome). Title: Mathematical methods for ?nance, economics and insurance. - Advisor (Commiassario) for teaching of mathematics at the “Liceo cantonale Lugano” (Switzerland). • A. Sulem - Member of the evaluation committee of the University Paris-Dauphine. - Member of the doctoral board of the the University Paris-Dauphine.

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9.4. Teaching
• A. Alfonsi - Probabilités et Statistiques”, ?rst year course at the Ecole des Ponts. - Modéliser, Programmer et Simuler”, second year course at the Ecole des Ponts. • - “Calibration, Volatilité Locale et Stochastique”, third-year course at ENSTA (Master with Paris I). V. Bally - Malliavin Calculus and applications in ?nance (30h) Master II, UMLV-ENPC, Finance. - Interest rates. (20h) Master II, UMLV-ENPC, Finance. - Risk analysis. Master II, IMIS UMLV. • - Probability Master I, UMLV. DIA El Hadj Aly - Tutor of a Master Thesis in Applied Mathematics preparing to a PhD’s program, August-September 2009. B. Jourdain - course "Probability theory and statistics", ?rst year ENPC - course "Introduction to probability theory", 1st year, Ecole Polytechnique - course "Stochastic numerical methods", 3rd year, Ecole Polytechnique • - projects in ?nance and numerical methods, 3rd year, Ecole Polytechnique B. Jourdain, B. Lapeyre - course "Monte-Carlo methods in ?nance", 3rd year ENPC and Master Recherche Mathématiques et Application, university of Marne-la-Vallée J.-F. Delmas, B. Jourdain - course "Jump processes with applications to energy markets", 3rd year ENPC and Master Recherche Mathématiques et Application, university of Marne-la-Vallée C. Labart - Lectures on “Discrete time models for ?nance” at ENSTA (2nd year course), 18 hours and at University Pierre et Marie Curie (Master “Probabilités et Finance”) 12 hours. - Lectures on “Stochastic Calculus : applications of Itô’s formula and stochastic differential equations” at University Pierre et Marie Curie (Master IFMA) 12 hours. - Lectures on “Random Models: Markov chains and Markov Processes”, at University Pierre et Marie Curie (Master IFMA), 28 hours. - Practicals on “Probability”, 12 hours, Polytech’Paris. - Practicals on “Numerical methods for differential equations”, 48 hours at University Pierre et Marie Curie (Licence 3). D. Lamberton - Third year of Licence de mathématiques (differential calculus, differential equations), Université Paris-Est Marne-la-Vallée. -Préparation à l’agrégation interne de mathématiques, Université Paris-Est Marne-la-Vallée. - Master course “Calcul stochastique et applications en ?nance", Université Paris-Est Marne-laVallée. B. Lapeyre - Ecole des Ponts, 2nd year, "Introduction to mathematical methods for ?nance", 2009

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•

•

•

•

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Activity Report INRIA 2009

•

J. Lelong - Lectures on “Parallel programming in ?nancial mathematics” at Ensimag (third year course) - Lectures on “Numerical Methods in Finance” at Ecole Nationale Supérieure de Techniques Avancées (second year course) - Supervision of ?nancial engineering projects at Ecole Nationale Supérieure de Techniques Avancées (second year course) - Lectures on “Discrete time martingales” at Ecole Nationale Supérieure de Techniques Avancées (second year course) - Lectures on “Monte Carlo methods for American option pricing” at Collège de l’Ecole Polytechnique.

•

A. Sulem - Course on numerical methods in ?nance, Master II MASEF University Paris-Dauphine (21 hours)

9.5. Internship advising
• B. Jourdain: - Pierre-Yves Lagrave : Implementation into Premia of approximation formulas for options in stochastic volatility models given by Elisa Alos in "A generalisation of the Hull and White formula with applications to option pricing approximation", Finance and Stochastics, 10(3) 353–365, 2006 • A. Kebaier: - Kaouther Hajji, Ecole Polytechnique de Tunis (November 2009 to April 2010) months internship) on Interacting particle systems for the computation of rare credit portfolio losses • C. Labart: - Cédric Allali, student of Master 2 “Statistique et Modèles aléatoires en ?nance”, University ParisDiderot for 5 months on the hedging of CDOs based on the following papers – – – – Hedging Default Risks of CDOs in Markovian Contagion Models. by J.P. Laurent A.Cousin J.D.Fermanian Dynamic hedging of synthetic CDO-tranches with spread risk and default contagion, by R. Frey and J. Backhaus, 2007 Pricing and Hedging of Portfolio Credit Derivatives with Interacting Default intensities, by R. Frey and J. Backhaus, 2007 Portfolio Credit Risk Models with Interacting Default Intensities: a Markovian Approach, by R. Frey and J. Backhaus, 2004

leading to two algorithms implemented in PREMIA. - Yisheng Wang, student of ENSTA (2nd year) for 3 months on the pricing of parisian options in local volatility models. PDE and Monte Carlo approaches have been implemented in Premia.

9.6. PhD defences
• Mohamed Mikou: Options américaines dans le modèle exponentiel de Lévy, Defended December 2th at Université Paris-Est Marne la Vallée, Adviser: D. Lamberton • Mohamed Sbai: Dependence modelling and simulation of stochastic processes in ?nance, Defended November 25th at Ecole des Ponts, Adviser: B. Jourdain.

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9.7. PhD advising
• • A. Alfonsi and T. Lelièvre - José Infante Acevedo (1st year). Half of this thesis will be denoted to liquidity risk. V. Bally - Stephano di Marco, 3rd year, student in Scola Normale Superiore di Pisa, in collaboration with M. Pratelli from the University of Pisa. • - Victor Rabiet, 1st year. Grant of ENS Cachant. B. Jourdain - Mohamed Sbai, "Dependence modelling and simulation of stochastic processes in ?nance", Mohamed Sbai, defended November 25th 2009, ENPC - Maxence Jeunesse, 1st year • "Study of some numerical methods in ?nance", D. Lamberton - Mohamed Mikou. American options in models with jumps. ATER at Université Paris-Est Marnela-Vallée. Thesis defense on December 2nd, 2009 - El Hadj Aly Dia (4th year). Monte-Carlo methods for exotic options in models with jumps. Ingénieur-expert at INRIA. - Sidi Mohamed Ould Aly (3rd year). Exotic options and stochastic volatility models. Allocataire de recherche, Université Paris-Est, doctorant-conseil at Natixis. - Lokmane Abbasturkki (1st year, started in March 2009). Modelling of correlation in high dimensions. This thesis is funded by Credinext. - Ayech Bouselmi (1st year, started in October 2009). Allocataire de recherche, Université Paris-Est. Lévy processes and multi-dimensional models in ?nance. B. Lapeyre and A. Alfonsi - Abdelkoddousse Ahdida (2nd year, Funding "Pôle Finance Innovation" - Créditnext Project) Current work on high order schemes and exact simulation of Wishart processes. Further work on the application of these processes to ?nance. F. Russo - Nadia Belaribi. Probabilistic representation of PDE’s of porous media type equations: probabilistic and deterministic numerical simulations. - Cristina Di Girolami (in collaboration with Luiss University). Calculus via regularisation in in?nite dimension and applications to mathematical ?nance. - Stéphane Goutte (in collaboration with Luiss University). Mean-variance hedgine in incomplet markets and applications to the electricity markets. - Ida Kruk. Malliavin calculus for general Gaussian processes. - Stefano Mega (in collaboration with Luiss University). Pricing of services related to telecommunications network. A. Sulem and R. Cont (University Paris 6) - Andrea Minca (1st year), funded by the Natixis Foundation for Quantitative Research. "Contagion in Financial Markets"

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9.8. PhD reports

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Activity Report INRIA 2009

•

B. Jourdain - Referee for the PhD thesis of Romain Deguest, "Model uncertainty in ?nance : risk measure and model calibration", Ecole Polytechnique, 12 november 2009

•

F. Russo - Referee of the PhD or Habilitation theses of – – – Eulalia Nualart (I presented her habilitation). Mohammed SBAI, Ecole Des Ponts, Paristech. Makrem Sghairi, Paris 5.

•

A. Sulem - Referee for the PhD thesis of D. Andersson: Contributions to the Stochastic maximum principle, KTH, Stockholm, Sweden, October 2009.

9.9. Participation to workshops, conferences and invitations
• A. Alfonsi Conferences: - “A closed-form extension to the Black-Cox model”, Nice 30th September, Recent Advancements in the Theory and Practice of Credit Derivatives. Talks: - “High order discretization schemes for the CIR process: application to Heston and Af?ne models” (in January at Paris XIII, in February at Oxford, and in May at Le Mans). - “Optimal execution and price manipulations in limit order book models”, November, Ecole Polytechnique. Invitations: - Oxford University, by Jan Obloj (18-02 to 20-02) • V. Bally -16-25 April and 23 September - 10 October: visits to the University Tor Vergata for scienti?c collaboration with Lucia Caramellino. - 25-26 June: talk on "Lower Bounds for the Densities of Asian Type Stochastic Differential Equations." in the workshop organized by Denis Talay in INRIA Sophia Antipolis. - 13-17 July: invited to the 7th International ISSAC Congress at Imperial College London. Talk on "Integration by parts formulas and applications to equations with jumps.". - 27-31 July: Invited to the SPA Congress in Berlin. Talk on "Integration by parts formulas and applications to equations with jumps.". • El Hadj Aly Dia Conferences: - Fourth General Conference on Advances Mathematical Methods in Finance, University of Oslo, Alesund (Norway), 4-10 May 2009. - Third conference on Numerical Methods in Finance, Ecole Nationale des Ponts et Chaussées, Paris, 15-17 April 2009. Workshops: - Journée des Doctorants du Séminaire Bachelier, Institut Henri-Poincaré, Paris, 06 Novemeber 2009. - Second SMAI Summer School in Financial Mathematics, Ecole Polytechnique, Paris, 2429 August 2009. - Premia 11 meeting, Institut Louis Bachelier, Paris, 11 February 2009.

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•

B. Jourdain - BigMC seminar, 29 january, Exact simulation of stochastic differential equations, according to Beskos, Papaspiliopoulos and Roberts. - workshop computational Finance, RIMS Kyoto university, 10-12 August, High order discretization schemes for stochastic volatility models. - seminar of the chair "?nancial risks", 21 september, High order discretization schemes for stochastic volatility models. - petit déjeuner de la ?nance, 23 september, Coupling index and stocks. - BigMC seminar, 24 september, Robust adaptive importance sampling for Normal random vectors. - workshop on Optimal stopping and singular stochastic control problems in ?nance, National University of Singapore, 9-15 december : Exercise boundary of the American put option in the black-Scholes model with discrete dividends.

•

A. Kebaier - congress on Non-Semimartingale Techniques in Mathematical Finance May 26-28, 2009, Helsinki University of Technology

• •

C. Labart - Seminar at Univesity Paris 13, January 2009. D. Lamberton Invitations: - 8th Winter School on Mathematical Finance in the Netherlands, Lunteren, January 2009. Special invited lecture "Some option pricing problems in exponential Lévy models". - AMaMeF 4th General Conference, Aalesund, Norway, May 2009, . "Discrete vs continuous supremum of Lévy processes". - Summer School in Financial Mathematics, Ljubljana, September 2009. A short course on optimal stopping and American options. - Finance and Stochastics Seminar, Imperial College, London October 2009. "On the approximation of the supremum of a Lévy process"

• •

B. Lapeyre J. Lelong - Workshop on stochastic algorithms, Dijon (University of Bourgogne), November 2009, invited speaker. - Seminar of Probability, LJK University of Grenoble, September 2009. - IEEE International Symposium on Parallel and Distributed Processing, Roma, May 2009. - Third Conference on Numerical Methods in Finance, Marne-la-Vallée (Ecole de Ponts), April 2009. - Seminar of Université du Maine, May 2009. - SMAI 2009 conference, La Colle sour Loup, May 2009.

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A. Minca Attended Schools: - 16-20 March 2009, CIRM, Luminy, Journées ALEA 2009 (Thematic School CNRS): Combinatorics, Random Graphs

•

6-17 July 2009, Ithaca, NY, 5th Cornell Probability Summer School: Mixing time of Markov chains, matching and point processes

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Activity Report INRIA 2009

Conferences: - 18 September, London, Standard and Poor’s Credit Risk Summit, invited talk on "Credit Default Swaps and Systemic Risk" - 16-19 December, Sydney, Quantitative Methods in Finance, contributed talk on "Contagious Defaults and Systemic Risk in Financial Networks" Seminars: - 16 October, Groupe de travail Methodes Stochastiques en Finance, ENPC-INRIA-UPEMLV, talk on joint article with Rama Cont, "Recovering portfolio default intensities implied by CDO quotes" - 29 October, Groupe de travail Probabilités et Finance, LPMA, Paris 6, "Recovering portfolio default intensities implied by CDO quotes" • D. Pommier - 3rd Workshop on High-Dimensional Approximation, Sydney (Australia), February 2009, talk on High Dimensional PDE’s methods applied to option pricing • F. Russo Presented Conferences and Seminars: - Sydney (Australie), february 2009. University Seminar. - Koblenz (Allemagne), april 2009. Presentation to Debeka Versicherung (Insurance company) - Marne-la-Vallée, may 2009. - Helsinki (Fin), May 2009. No-semimartingale models in mathematical ?nance (Workshop). - Tianjin (China), June 2009. Workshop on Random dynamical systems and related topics. - Manchester (UK), August 2009. Conference on SDEs, SPDEs and related topics. - Purdue (USA), September 2009. Workshop Stochastic Analysis at Purdue 09. - Princeton (USA), October 2009. Stochastic analysis seminar, Bendheim center for ?nance. - New York (USA), October 2009 Stochastic analysis and mathematical physics seminar. - Oxford (UK), November 2009. Stochastic analysis seminar of the Man Institute. - Seoul (Corea), December 2009. Invitation at the Department of Mathematical Sciences, Seoul National University Presentation of a conference. Longer invitation stays abroad - Sydney (Australia), february 2009, 2 weeks. - Bielefeld (Allemagne), jul, aug. 2009, 6 weeks. - Purdue (USA), oct; 2009, 1 week. - Cambridge (UK), Newton Institute, march-april 2010, 6 weeks. • A. Sulem - 29 Avril: seminar “Analysis on the Wiener space”, IHP, Paris talk on : Singular stochastic control with partial information of jump diffusions • A. Zanette - Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model, Third Conference on Numerical Methods in Finance ENPC Paris 2009, - International Congress on Insurance: Mathematics and Economics (IME2009) Istanbul 2009, - AMASES 2009 Parma

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10. Bibliography
Major publications by the team in recent years
[1] A. A LFONSI . On the discretization schemes for the CIR (and Bessel squared) processes, in "Monte Carlo Methods and Applications", vol. 11, no 4, 2005, p. 355–384. [2] B. A ROUNA . Adaptative Monte Carlo Method, A Variance Reduction technique, in "Monte Carlo Methods and Applications", vol. 10, no 1, 2004. [3] V. BALLY. An elementary introduction to Malliavin calculus, no 4718, Inria, Rocquencourt, February 2003, http://hal.inria.fr/inria-00071868, Research Report. [4] V. BALLY, M. BAVOUZET, M. M ESSAOUD . Computations of Greeks using Malliavin Calculus in jump type market models, in "Annals of Applied Probability", vol. 17, 2007, p. 33-66. [5] V. BALLY, L. C ARAMELLINO , A. Z ANETTE . Pricing American options by a Monte Carlo method using a Malliavin calculus approach, in "Monte Carlo methods and applications", vol. 11, no 2, 2005, p. 97–133. [6] E. C LÉMENT, D. L AMBERTON , A. KOHATSU -H IGA . A duality approach for the weak approximation of stochastic differential equations, in "Annals of Applied Probability", vol. 16, no 3, August 2006, p. 11241154. [7] B. J OURDAIN , M. S BAI . Exact retrospective Monte Carlo computation of arithmetic average Asian options, in "Monte Carlo methods and Applications", vol. 13, no 2, 2007, p. 135–171. [8] A. KOHATSU -H IGA , A. S ULEM . Utility maximization in an insider in?uenced market, in "Mathematical Finance", vol. 16, no 1, 2006, p. 153–179. [9] D. L AMBERTON , M. M IKOU . The critical exercise price for the American put in an exponential Lévy model, in "Finance & Stochastics", vol. 12, 2008, p. 561-581. [10] M. N’ ZI , Y. O UKNINE , A. S ULEM . Regularity and representation of viscosity solutions of Partial differential equations via backward stochastic differential equations, in "Stochastic processes and their applications", vol. 116, no 9, 2006, p. 1319–1339. [11] B. Ø KSENDAL , A. S ULEM . Applied Stochastic Control of Jump Diffusions, Universitext, Second Edition, Springer, Berlin, Heidelberg, New York, 257 pages 2007.

Year Publications
Doctoral Dissertations and Habilitation Theses
[12] M. M IKOU . Options américaines dans le modèle exponentiel de Lévy, Université Paris Est Marne la Vallée, 2009, Ph. D. Thesis. [13] M. S BAI . Modélisation de la dépendance et simulation de processus en ?nance, Ecole des Ponts, 2009, Ph. D. Thesis.

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Articles in International Peer-Reviewed Journal
[14] Y. ACHDOU , F. H ECHT, D. P OMMIER . A posteriori error estimates for parabolic variational inequalities, in "Journal of Scienti?c Computing", 2009, accepted for publication. [15] A. A LFONSI . High-order discretization scheme for the CIR process: application to the Heston model, in "Mathematics of Computation", vol. 79, no 269, 2010, p. 209-237. [16] A. A LFONSI , A. F RUTH , A. S CHIED . Optimal execution strategies in limit order books with general shape functions, in "Quantitative Finance", vol. 9, 2009, DOI:10.1080/14697680802595700. [17] A. A LFONSI , B. J OURDAIN . Exact volatility calibration based on a Dupire-type Call-Put duality for perpetual American options, in "Nonlinear Differential Equations and Applications", vol. 16, no 4, 2009, p. 523-554. [18] V. BALLY, B. F ERNANDEZ , A. M EDA . Estimates of the Probability Itô Processes remain around a curve and applications in ?nance, in "Stoch. Proc. Appl.", 2009, to appear. [19] V. BALLY, A. KOHATSU -H IGA . Lower Bounds for the Densities of Asian Type Stochastic Differential Equations, in "J. Functional Anal.", 2009, to appear. [20] M. C OSTABILE , M. G AUDENZI , I. M ASSABÒ , A. Z ANETTE . Evaluating fair premiums of equity-linked policies with surrender option in a bivariate model, in "Insurance: Mathematics and Economics", vol. 45, no 2, 2009, p. 286-295. [21] P. E TORÉ , G. F ORT, B. J OURDAIN , E. M OULINES . On Adaptive Strati?cation, in "Annals of operations research", 2009, accepted for publication, preprint hal-00319157. [22] P. E TORÉ , B. J OURDAIN . Adaptive optimal allocation in strati?ed sampling methods, in "Methodology and Computing in Applied Probability", 2009, accepted for publication. [23] F. F LANDOLI , M. G UBINELLI , F. RUSSO . On the regularity of stochastic currents, fractional Brownian motion and applications to a turbulence model, in "Annales de l’Institut Henri Poincaré", vol. 45, no 2, 2009, p. 545–576. [24] J. D. F ONSECA , M. M ESSAOUD . Libor Market Model in Premia: Bermudan pricer, Stochastic Volatility and Malliavin calculus, in "Bankers, Markets, Investors", vol. Special report: Numerical Methods implemented in the Premia Software, no 99, March-April 2009, p. 44–57. [25] M. G AUDENZI , M. L EPELLERE , A. Z ANETTE . The Singular Points method for Pricing American PathDependent Options, in "Journal of Computational Finance", 2009, to appear. [26] M. G AUDENZI , A. Z ANETTE . Pricing American Barrier options with discrete dividend by binomial trees, in "Decisions in Economics and Finance", vol. 32, no 2, 2009, p. 129-148. [27] M. G AUDENZI , A. Z ANETTE . Pricing Cliquet options by tree methods, in "Computational Management Science", 2009, p. 1-11.

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[28] B. J OURDAIN , J. L ELONG . Robust Adaptive Importance Sampling for Normal Random Vectors, in "Annals of applied probability", vol. 19, no 5, 2009, p. 1687-1718, http://arxiv.org/pdf/0811.1496v1+. [29] C. L ABART, E. G OBET. Solving BSDE with adaptive control variates, in "SINUM", 2009, accepted for publication. [30] C. L ABART, J. L ELONG . Pricing double barrier Parisian Options using Laplace transforms, in "International Journal of Theoretical and Applied Finance", vol. 12, no 1, 2009, p. 19–44, preprint hal-00220470. [31] C. L ABART, J. L ELONG . Pricing Parisian Options using Laplace transforms, in "Bankers, Markets, Investors", vol. Special report: Numerical Methods implemented in the Premia Software, no 99, March-April 2009, p. 29–43. [32] D. L AMBERTON . Optimal stopping with irregular reward functions, in "Stochastic Processes and their Applications", vol. 119, 2009, p. 3253-3284. [33] N. P RIVAULT, X. W EI . Calibration of the LIBOR market model - implementation in Premia, in "Bankers, Markets, Investors", vol. Special report: Numerical Methods implemented in the Premia Software, no 99, March-April 2009, p. 20–29. [34] B. Ø KSENDAL , A. S ULEM . Maximum principles for optimal control of forward-backward stochastic differential equations with jumps, in "SIAM J. Control Optimization", vol. 48, no 5, 2009, p. 2845–2976. [35] B. Ø KSENDAL , A. S ULEM . Risk indifference pricing in jump diffusion markets, in "Mathematical Finance", vol. 19, no 4, 2009, p. 619–637.

Articles in National Peer-Reviewed Journal
[36] A. S ULEM , A. Z ANETTE . Premia: A Numerical Platform for Pricing Financial Derivatives, in "Ercim News", vol. 78, July 2009.

Articles in Non Peer-Reviewed Journal
[37] A. S ULEM . Mathématiques ?nancières; des modèles de plus en plus complexes, in "La Recherche", Avril 2009.

International Peer-Reviewed Conference/Proceedings
[38] J. C HANCELIER , B. L APEYRE , J. L ELONG . Using Premia and Nsp for Contructing a Risk Management Benchmark for Testing Parallel Architecture, in "Proceedings of IEEE International Symposium on Parallel & Distributed Processing, Rome", 2009, http://doi.ieeecomputersociety.org/10.1109/IPDPS.2009.5161144.

Scienti?c Books (or Scienti?c Book chapters)
[39] Numerical Methods implemented in the Premia Software, 2009, Bankers, Markets, Investors, Introduction by A. Sulem and A. Zanette. [40] A. A LFONSI . An introduction to the multiname modelling in credit risk, T. B IELECKI , D. B RIGO , F. PATRAS (editors), Bloomberg Press, 2009, to appear.

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[41] A. A LFONSI . Cox-Ingersoll-Ross (CIR) model, in "Encyclopedia of Quantitative Finance", R. C ONT (editor), John Wiley and Sons, 2009, to appear. [42] V. BALLY, M. BAVOUZET, M. M ESSAOUD . Malliavin calculus for pure jump processes and applications to ?nance, in "Mathematical Modelling and Numerical Methods in Finance", A. B ENSOUSSAN , Q. Z HANG , P. C IARLET (editors), Handbook of Numerical Analysis, vol. 15, Elsevier, 2009. [43] G. D I N UNNO , A. KOHATSU -H IGA , T. M EYER -B RANDIS , B. Ø KSENDAL , F. P ROSKE , A. S ULEM . Anticipative stochastic control for Lévy processes with application to insider trading, in "Mathematical Modelling and Numerical Methods in Finance", A. B ENSOUSSAN , Q. Z HANG , P. C IARLET (editors), Handbook of Numerical Analysis, vol. 15, Elsevier, 2009. [44] B. J OURDAIN . Probabilités et statistique, Ellipses, 2009. [45] B. J OURDAIN . Adaptive variance reduction techniques in ?nance, in "Advanced Financial Modelling", H. A LBRECHER , W. RUNGGALDIER , W. S CHACHERMAYER (editors), Radon Series on Computational and Applied Mathematics, vol. 8, Gruyter, 2009, to appear. [46] C. L ABART. Parisian options, in "Encyclopedia of Quantitative Finance", R. C ONT (editor), John Wiley and Sons, 2009, (3 pages), to appear. [47] B. L APEYRE . Variance reduction methods for ?nancial models, in "Encyclopedia of Quantitative Finance", R. C ONT (editor), John Wiley and Sons, 2009, to appear. [48] J. L ELONG , A. Z ANETTE . Tree methods in Finance, in "Encyclopedia of Quantitative Finance", R. C ONT (editor), John Wiley and Sons, 2009, to appear. [49] M. Q UENEZ . Backward equations and applications, in "Encyclopedia of Quantitative Finance", R. C ONT (editor), John Wiley and Sons, 2009, to appear.

Books or Proceedings Editing
[50] R. DALANG , M. D OZZI , F. RUSSO (editors). Seminar on stochastic analysis, random ?elds and applications, vol. Proceedings, 2009, in preparation.

Research Reports
[51] V. BARBU , M. R ÖCKNER , F. RUSSO . Probabilistic representation for solutions of an irregular porous media type equation: the degenerate case, INRIA, 2009, http://hal.inria.fr/inria-00410248/fr/, Preprint. [52] R. C OVIELLO , F. RUSSO . Modeling ?nancial assets without semimartingales, no 2006-06-219, Preprint BiBoS, Bielefeld, June 2009, http://arxiv.org/abs/math.PR/0606642, Technical report. [53] S. G OUTTE , N. O UDJANE , F. RUSSO . Variance Optimal Hedging for continuous time processes with independent increments and applications, INRIA, 2009, http://hal.inria.fr/inria-00437984, 67 pages, Preprint. [54] B. Ø KSENDAL , A. S ULEM . An anticipative stochastic calculus approach to pricing in markets driven by Lévy processes, no 7127, INRIA, November 2009, http://hal.archives-ouvertes.fr/inria-00439350/fr/, Technical report.

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Other Publications
[55] A. A LFONSI , J. L ELONG . A closed-form extension to the Black-Cox model, September 2009, http://hal. archives-ouvertes.fr/docs/00/41/42/80/PDF/BC_extension_hal.pdf, hal-00414280. [56] A. A LFONSI , A. S CHIED . Optimal execution and absence of price manipulations in limit order book models, July 2009, hal-00397652. [57] A. A LFONSI , A. S CHIED , A. S LYNKO . Order Book Resilience, Price Manipulation, and the Positive Portfolio Problem, October 2009, http://ssrn.com/abstract=1498514. [58] V. BALLY, L. C ARAMELLINO . Regularity of probability laws using the Riesz transform and Sobolev spaces techiques, 2009, arXiv:0911.2631. [59] V. BALLY, E. C LÉMENT. Integration by parts formulas and applications to equations with jumps, 2009, arXiv:0911.3017. [60] V. BALLY, N. F OURNIER . Regularization properties of the 2D homogeneous Boltzmann equation without cutoff, 2009, arXiv:0911.2614. [61] B. J OURDAIN , M. S BAI . High order discretization schemes for stochastic volatility models, 2009, Preprint Hal 00409861. [62] B. J OURDAIN , M. V ELLEKOOP. Regularity of the Exercise Boundary for American Put Options on Assets with Discrete Dividends, 2009, preprint Hal 00436327. [63] D. L AMBERTON , M. Z ERVOS . On the problem of optimally stopping a one-dimensional Ito diffusion, 2009, submitted for publication.

References in notes
[64] V. BALLY. An elementary introduction to Malliavin calculus, no 4718, Inria, Rocquencourt, February 2003, http://hal.inria.fr/inria-00071868, Research Report. [65] D. B ELL . The Malliavin Calculus, Pitman Monographs and Surveys in Pure and Applied Math., no 34, Longman and Wiley, 1987. [66] F. B LACK , E. D ERMAN , W. T OY. A one factor model of interest rates and its application to treasury bond options, in "Financial Analysts Journal", January-February 1990. [67] A. B RACE , D. G ATAREK , M. M USIELA . The Market Model of Interest Rate Dynamics, in "Mathematical Finance", vol. 7, 1997, p. 127-156. [68] P. B RIAND , B. D ELYON , J. M EMIN . On the robustness of backward stochastic differential equations, in "Stochastic Processes and their Applications", vol. 97, no 2, 2002, p. 229-253. [69] J. C. C OX , J. E. I NGERSOLL , S. A. ROSS . A Theory of the Term Structure of Interest Rate, in "Econometrica", vol. 53, 1985, p. 363-384.

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[70] E. F OURNIÉ , J.-M. L ASRY, J. L EBUCHOUX , P.-L. L IONS . Applications of Malliavin calculus to Monte Carlo methods in Finance, II, in "Finance & Stochastics", vol. 2, no 5, 2001, p. 201-236. [71] E. F OURNIÉ , J.-M. L ASRY, J. L EBUCHOUX , P.-L. L IONS , N. T OUZI . An application of Malliavin calculus to Monte Carlo methods in Finance, in "Finance & Stochastics", vol. 4, no 3, 1999, p. 391-412. [72] J. G UYON . Euler scheme and tempered distributions, in "Stochastic Processes and their Applications", vol. 116, no 6, 2006, p. 877–904. [73] J. H ULL , A. W HITE . Numerical Procedures for Implementing Term Structure Models I:Single Factor Models, in "Journal of Derivatives", vol. 2, 1994, p. 7-16. [74] A. K EBAIER . Statistical Romberg extrapolation: a new variance reduction method and applications to option pricing, in "The Annals of Applied Probability", vol. 15, no 4, 2005, p. 2681–2705. [75] D. L AMBERTON , B. L APEYRE , A. S ULEM . Application of Malliavin Calculus to Finance, in "special issue of Mathematical Finance", January 2003. [76] B. L APEYRE , A. S ULEM , D. TALAY. Simulation of Financial Models: Mathematical Foundations and Applications., Cambridge University Press, 2009, to appear. [77] P. M ALLIAVIN . Stochastic calculus of variations and hypoelliptic operators, in "Proc. Inter. Symp. on Stoch. Diff. Equations, Kyoto", Wiley 1978, 1976, p. 195-263. [78] P. M ALLIAVIN , A. T HALMAIER . Stochastic Calculus of variations in Mathematical Finance, Springer Finance, Springer, 2006. [79] D. N UALART. The Malliavin Calculus and Related Topics, Springer–Verlag, 1995. [80] D. O CONE , I. K ARATZAS . A generalized representation formula with application to optimal portfolios, in "Stochastics and Stochastic Reports", vol. 34, 1991, p. 187-220. [81] D. O CONE . A guide to the stochastic calculus of variations, in "Stochastic Analysis and Related Topics", H. KOERZLIOGLU , S. Ü STÜNEL (editors), Lecture Notes in Math.1316, 1987, p. 1-79. [82] E. PARDOUX . Backward stochastic differential equations and viscosity solutions of systems of semilinear parabolic and elliptic PDEs of second order, in "Stochastic analysis and related topics VI, Boston, MA.", Progr. Probab. 42, Birkhäuser, 1998, p. 79-127. [83] F. RUSSO , P. VALLOIS . Stochastic calculus with respect to continuous ?nite quadratic variation processes, in "Stochastics and Stochastics Reports", vol. 70, 2000, p. 1–40. [84] O. VASICEK . An Equilibrium Characterisation of Term Strucuture, in "Journal of Financial Economics", vol. 5, 1977, p. 177-188. [85] B. Ø KSENDAL . An Introduction to Malliavin Calculus with Applications to Economics, in "Lecture Notes from a course given 1996 at the Norwegian School of Economics and Business Administration (NHH)", September 1996, NHH Preprint Series.



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