MathRisk is a joint Inria project-team with ENPC (CERMICS Laboratory) and the University Paris Est Marne-la-Vallée (UPEMLV, LAMA Laboratory), located in Rocquencourt and Marne-la-Vallée.

http://

The starting point of the development of modern finance theory is traditionally associated to the publication of the famous paper of Black and Scholes in 1973 . Since then, in spite of sporadic crises, generally well overcome, financial markets have grown in a exponential manner. More and more complex exotic derivative products have appeared, on equities first, then on interest rates, and more recently on credit markets. The period between the end of the eighties and the crisis of 2008 can be qualified as the “golden age of financial mathematics”: finance became a quantitative industry, and financial mathematics programs flourished in top universities, involving seminal interplays between the worlds of finance and applied mathematics. During its 12 years existence, the Mathfi project team has extensively contributed to the development of modeling and computational methods for the pricing and hedging of increasingly complex financial products.

Since the crisis of 2008, there has been a critical reorientation of research priorities in quantitative finance with emphasis on risk. In 2008, the “subprime” crisis has questioned the very existence of some derivative products such as CDS (credit default swaps) or CDOs (collateralized debt obligations), which were accused to be responsible for the crisis. The nature of this crisis is profoundly different from the previous ones. It has negatively impacted the activity on the exotic products in general, - even on equity derivative markets-, and the interest in the modeling issues for these products. The perfect replication paradigm, at the origin of the success of the Black and Scholes model became unsound, in particular through the effects of the lack of liquidity. The interest of quantitative finance analysts and mathematicians shifted then to more realistic models taking into account the multidimensional feature and the incompleteness of the markets, but as such getting away from the “lost paradi(gm)” of perfect replication. These models are much more demanding numerically, and require the development of hedging risk measures, and decision procedures taking into account the illiquidity and various defaults.

Moreover, this crisis, and in particular the Lehman Brothers bankruptcy and its consequences, has underlined a systemic risk due to the strong interdependencies of financial institutions. The failure of one of them can cause a cascade of failures, thus affecting the global stability of the system. Better understanding of these interlinkage phenomena becomes crucial.

At the same time, independently from the subprime crisis, another phenomenon has appeared: deregulation in the organization of stock markets themselves. This has been encouraged by the Markets in Financial Instruments Directive (MIFID) which is effective since November, 1st 2007. This, together with the progress of the networks, and the fact that all the computers have now a high computation power, have induced arbitrage opportunities on the markets, by very short term trading, often performed by automatic trading. Using these high frequency trading possibilities, some speculating operators benefit from the large volatility of the markets. For example, the flash crash of May, 6 2010 has exhibited some perverse effects of these automatic speculating trading strategies. These phenomena are not well understood and the theme of high frequency trading needs to be explored.

To summarize, financial mathematics is facing the following new evolutions:

the complete market modeling has become unsatisfactory to provide a realistic picture of the market and is replaced by incomplete and multidimensional models which lead to new modeling and numerical challenges.

quantitative measures of risk coming from the markets, the hedging procedures, and the lack of liquidity are crucial for banks,

uncontrolled systemic risks may cause planetary economic disasters, and require better understanding,

deregulation of stock markets and its consequences lead to study high frequency trading.

The project team MathRisk is designed to address these new issues, in particular dependence modeling, systemic risk, market microstructure modeling and risk measures. The research in modeling and numerical analysis remain active in this new context, motivated by new issues.

The MathRisk project team develops the software Premia dedicated
to pricing and hedging options and calibration of financial models, in collaboration with a consortium of financial institutions.
https://

The MathRisk project is part of the
Université Paris-Est **“Labex” BÉZOUT**.

The volatility is a key concept in modern mathematical finance, and
an indicator of the market stability.
Risk management and associated instruments depend strongly on the
volatility, and volatility modeling has thus become a crucial issue in the finance industry. Of particular importance is the assets *dependence* modeling.
The calibration of models for a single asset can now be well managed by banks but modeling of dependence is the bottleneck to efficiently aggregate such models.
A typical issue is how to go from the individual evolution of
each stock belonging to an index to the joint modeling of these stocks. In
this perspective, we want to model stochastic volatility in a *multidimensional* framework. To handle these questions mathematically, we have to deal with stochastic
differential equations that are defined on matrices in order to model either
the instantaneous covariance or the instantaneous correlation between the
assets. From a numerical point of view, such models are very demanding since
the main indexes include generally more than thirty assets. It is therefore
necessary to develop efficient numerical methods for pricing options and
calibrating such models to market data.
As a first application, modeling the dependence between assets allows us to
better handle derivatives products on a basket. It would give also a way to price
and hedge consistenly single-asset and basket products. Besides, it can be a way to
capture how the market estimates the dependence between assets. This could give
some insights on how the market anticipates the systemic risk.

The financial crisis has caused an increased interest in mathematical finance studies which take into account the market incompleteness issue and the liquidity risk. Loosely speaking, liquidity risk is the risk that comes from the difficulty of selling (or buying) an asset. At the extreme, this may be the impossibility to sell an asset, which occurred for “junk assets” during the subprime crisis. Hopefully, it is in general possible to sell assets, but this may have some cost. Let us be more precise. Usually, assets are quoted on a market with a Limit Order Book (LOB) that registers all the waiting limit buy and sell orders for this asset. The bid (resp. ask) price is the most expensive (resp. cheapest) waiting buy or sell order. If a trader wants to sell a single asset, he will sell it at the bid price. Instead, if he wants to sell a large quantity of assets, he will have to sell them at a lower price in order to match further waiting buy orders. This creates an extra cost, and raises important issues. From a short-term perspective (from few minutes to some days), this may be interesting to split the selling order and to focus on finding optimal selling strategies. This requires to model the market microstructure, i.e. how the market reacts in a short time-scale to execution orders. From a long-term perspective (typically, one month or more), one has to understand how this cost modifies portfolio managing strategies (especially delta-hedging or optimal investment strategies). At this time-scale, there is no need to model precisely the market microstructure, but one has to specify how the liquidity costs aggregate.

On a long-term perspective, illiquidity can be approached via various ways: transactions costs , , , , , , , delay in the execution of the trading orders , , , trading constraints or restriction on the observation times (see e.g. and references herein). As far as derivative products are concerned, one has to understand how delta-hedging strategies have to be modified. This has been considered for example by Cetin, Jarrow and Protter . We plan to contribute on these various aspects of liquidity risk modeling and associated stochastic optimization problems. Let us mention here that the price impact generated by the trades of the investor is often neglected with a long-term perspective. This seems acceptable since the investor has time enough to trade slowly in order to eliminate its market impact. Instead, when the investor wants to make significant trades on a very short time horizon, it is crucial to take into account and to model how prices are modified by these trades. This question is addressed in the next paragraph on market microstructure.

The European directive MIFID has increased the competition between markets (NYSE-Euronext, Nasdaq, LSE and new competitors). As a consequence, the cost of posting buy or sell orders on markets has decreased, which has stimulated the growth of market makers. Market makers are posting simultaneously bid and ask orders on a same stock, and their profit comes from the bid-ask spread. Basically, their strategy is a “round-trip” (i.e. their position is unchanged between the beginning and the end of the day) that has generated a positive cash flow.

These new rules have also greatly stimulated research on
market microstructure modeling. From a
practitioner point of view, the main issue is to solve the so-called
“optimal execution problem”: given a deadline

Solving the optimal execution problem is not only an interesting mathematical challenge. It is also a mean to better understand market viability, high frequency arbitrage strategies and consequences of the competition between markets. For example when modeling the market microstructure, one would like to find conditions that allow or exclude round trips. Beyond this, even if round trips are excluded, it can happen that an optimal selling strategy is made with large intermediate buy trades, which is unlikely and may lead to market instability.

We are interested in finding synthetic market models in which we can describe and solve the optimal execution problem. A. Alfonsi and A. Schied (Mannheim University) have already proposed a simple Limit Order Book model (LOB) in which an explicit solution can be found for the optimal execution problem. We are now interested in considering more sophisticated models that take into account realistic features of the market such as short memory or stochastic LOB. This is mid term objective. At a long term perspective one would like to bridge these models to the different agent behaviors, in order to understand the effect of the different quotation mechanisms (transaction costs for limit orders, tick size, etc.) on the market stability.

After the recent financial crisis, systemic risk has emerged as one of the major research topics in mathematical finance. The scope is to understand and model how the bankruptcy of a bank (or a large company) may or not induce other bankruptcies. By contrast with the traditional approach in risk management, the focus is no longer on modeling the risks faced by a single financial institution, but on modeling the complex interrelations between financial institutions and the mechanisms of distress propagation among these. Ideally, one would like to be able to find capital requirements (such as the one proposed by the Basel committee) that ensure that the probability of multiple defaults is below some level.

The mathematical modeling of default contagion, by which an economic shock causing initial losses and default of a few institutions is amplified due to complex linkages, leading to large scale defaults, can be addressed by various techniques, such as
network approaches (see in particular
R. Cont et al. and A. Minca ) or mean field interaction models (Garnier-Papanicolaou-Yang ).
The recent approach in seems very promising. It describes the
financial network approach as a weighted directed graph, in which nodes represent financial institutions and
edges the exposures between them. Distress propagation in a financial system
may be modeled as an epidemics on this graph. In the case of incomplete
information on the structure of the interbank network, cascade dynamics may be
reduced to the evolution of a multi-dimensional Markov chain that corresponds
to a sequential discovery of exposures and determines at any time the size of
contagion.
Little has been done so far on the *control* of such systems in order to
reduce the systemic risk and we aim to contribute to this domain.

The financial crisis has caused an increased interest in mathematical finance studies which take into account the market incompleteness issue and the default risk modeling, the interplay between information and performance, the model uncertainty and the associated robustness questions, and various nonlinearities. We address these questions by further developing the theory of stochastic control in a broad sense, including stochastic optimization, nonlinear expectations, Malliavin calculus, stochastic differential games and various aspects of optimal stopping.

The theory of American option pricing has been an incite for a number of research articles about optimal stopping. Our recent contributions in this field concern optimal stopping in models with jumps, irregular obstacles, free boundary analysis, reflected BSDEs.

.

Effective numerical methods are crucial in the pricing and hedging of derivative securities. The need for more complex models leads to stochastic differential equations which cannot be solved explicitly, and the development of discretization techniques is essential in the treatment of these models. The project MathRisk addresses fundamental mathematical questions as well as numerical issues in the following (non exhaustive) list of topics: Multidimensional stochastic differential equations, High order discretization schemes, Singular stochastic differential equations, Backward stochastic differential equations.

Monte-Carlo methods is a very useful tool to evaluate prices
especially for complex models or options. We carry on research
on *adaptive variance reduction methods* and to use *Monte-Carlo methods for calibration* of advanced models.

This activity in the MathRisk team is strongly related to the development of the Premia software.

The original Stochastic Calculus of Variations, now called the Malliavin calculus, was developed by Paul Malliavin in 1976 . 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 field of stochastic calculus followed. Several monographs and lecture notes (for example D. Nualart , D. Bell D. Ocone , B. Øksendal ) give expositions of the subject. See also V. Bally for an introduction to Malliavin calculus.

From the beginning of the nineties, applications of the Malliavin calculus in finance 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 .

Since then, the Malliavin calculus has raised increasing interest and subsequently many other applications to finance have been found , 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.

We give below 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

An important feature is that one has a relatively explicit expression for the weight

Let us now look at one of the main consequences of the integration by parts formula.
If one considers the *Dirac* function *Heaviside* function and the above
integration by parts formula reads *Greeks*) or conditional
expectations, which appear in the pricing of American options
by the dynamic programming).
See the papers by Fournié et al
and and the papers by Bally et al., Benhamou, Bermin et al., Bernis et al., Cvitanic et al., Talay and Zheng and Temam
in .

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 field. Significant results have been obtained up to dimension 5 and the corresponding algorithms have been implemented in the Premia software.

Moreover, there is an increasing interest in considering jump components in the financial models, especially motivated by calibration reasons. Algorithms based on the integration by parts formulas have been developed 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 locally smooth law - represents a large field of research, also for applications to credit risk problems.

The Malliavin calculus is also used in models of insider trading. The "enlargement of filtration" 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 filtration enlargement is possible. See the paper by P. Imkeller in ). 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 find 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 ).

A. Kohatsu Higa and
A. Sulem have studied a controlled stochastic system whose state
is described by a stochastic differential equation with anticipating
coefficients. 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 . This methodology has
been applied for utility maximization with insiders.

Risk management, Quantitative finance, Computational Finance, Market Microstructure analysis, Systemic risk, Portfolio optimization, Risk modeling, Option pricing and hedging in incomplete markets, insurance.

Conference in honor of Vlad Bally for his 60th birthday, Le Mans, October 6-9 2015
http://

J. Reygner received the 2014 Jacques Neveu prize for his thesis entitled "Longtime behaviour of particle systems : applications in physics, finance and PDEs" co-supervised by B.Jourdain and L. Zambotti

Keywords: Computational finance - Option pricing

Scientific Description

Premia is a software designed for option pricing, hedging and financial model calibration. It is provided with it's C/C++ source code and an extensive scientific documentation. The Premia project keeps track of the most recent advances in the field of computational finance 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 financial teams and academic researchers, 3 - to provide a useful teaching support for Master and PhD students in mathematical finance.

Functional Description

Participants: Mathrisk project team and contributors

Partners: Inria - Ecole des Ponts ParisTech - Université Paris-Est - Consortium Premia

Contact: Agnès Sulem

URL: http://

AMS: 91B28;65Cxx;65Fxx;65Lxx;65Pxx

License: Licence Propriétaire (genuine license for the Consortium Premia)

Type of human computer interaction: Console, interface in Nsp, Web interface

OS/Middelware: Linux, Mac OS X, Windows

APP: The development of Premia started in 1999 and 16 are released up to now and registered at the APP agency. Premia 16 has been registered on 0303/2015 under the number IDDN.FR.001.190010.013.S.C.2001.000.31000

Programming language: C/C++ librairie Gtk

Documentation: the PNL library is interfaced via doxygen

Size of the software: 280580 lines for the Src part only, that is 11 Mbyte of code, 130400 lines for PNL, 105 Mbyte of PDF files of documentation.

interfaces : Nsp for Windows/Linux/Mac, Excel, binding Python, and a Web interface.

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.

**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.

For high dimensional American options, Premia provides the most recent Monte-Carlo algorithms: Longstaff-Schwartz, Barraquand-Martineau, Tsitsklis-Van Roy, Broadie-Glassermann, quantization methods and 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.

**Interest rate derivatives**

The following models are considered:

HJM and Libor Market Models (LMM): affine models, Hull-White, CIR

Premia provides a calibration toolbox for Libor Market model using a database of swaptions and caps implied volatilities.

**Credit derivatives: Credit default swaps (CDS), Collateralized debt obligations (CDO) **

Reduced form models and copula models are considered.

Premia provides a toolbox for pricing CDOs using the most recent algorithms (Hull-White, Laurent-Gregory, El Karoui-Jiao, Yang-Zhang, Schönbucher)

**Hybrid products**

A PDE solver for pricing derivatives on hybrid products like options on inflation and interest or change rates is implemented.

**Energy derivatives: swing options**

Mean reverting and jump models are considered.

Premia provides a toolbox for pricing swing options using finite differences, Monte-Carlo Malliavin-based approach and quantization algorithms.

To facilitate contributions, a standardized numerical library (PNL) has been developed by J. Lelong under the LGPL since 2009, which offers a wide variety of high level numerical methods for dealing with linear algebra, numerical integration, optimization, random number generators, Fourier and Laplace transforms, and much more. Everyone who wishes to contribute is encouraged to base its code on PNL and providing such a unified numerical library has considerably eased the development of new algorithms which have become over the releases more and more sophisticated. J. Ph Chancelier, B. Lapeyre and J. Lelong are using Premia and Nsp for Constructing a Risk Management Benchmark for Testing Parallel Architecture .

**Development of the PNL in 2015 (J. Lelong) .** Release 1.70 and 1.71, PNL Library (http://

Release 1.72. of the *PNL* library (http://

Addition of a CMake module to include the library in other projects.

Improvement of the `pnl_basis` module.

Addition of the non central chi squared distribution to the random number generation toolbox.

Addition of new functions in the linear algebra toolbox to build views.

Premia 17 has been delivered to the consortium members in March 2015.

It contains the following new algorithms:

Variables Annuities GLWB pricing in the Heston and Black-Scholes/Hull-White models with finite difference techniques.

Variables Annuities GMAB, GMDB, GMMB pricing with Fourier-cosine techniques.

A numerical scheme for the impulse control formulation for pricing variable annuities with a Guaranteed Minimum Withdrawal Benefit (GMWB) Z.Chen P.Forsyth

*Numerische Mathematik 109, 2008*

Managing Gap Risks in iCPPI for life insurance companies: A risk/return/cost analysis. A.Kalife S.Mouti L.Goudenege

*Insurance Markets and Companies: Analyses and Actuarial
Computations, Issue 2 2014*

Simulating CVA on American Options. L. Abbas Turki, M.Mikou

Being particular about calibration. J.Guyon and P. Henry-Labordère.

*Risk magazine, Jan 2012.*

The Heston Stochastic-Local Volatility Model: Efficient Monte Carlo Simulation. A.W. van der Stoepb, L. A. Grzelakb, C. W. Oosterlee

*International Journal of Theoretical and Applied Finance, to appear*

On the Heston model with stochastic interest rates. L. Grzelak C.W.Oosterlee

*SIAM J. Fin. Math. 2,2011*.

Alternating direction implicit finite difference schemes for the Heston Hull-White partial differential equation.

*The Journal of Computational
Finance Volume 16/Number 1, Fall 2012*

Pricing American options in the Heston Hull-White and Hull-White2d Models: an hybrid tree-finite difference approach. M.Briani, L.Caramellino, A.Zanette

Efficient pricing of Asian options under Lévy processes based on Fourier cosine expansions. Part I: European-style products. B.Zhang C.W.Oosterlee.

*SIAM J. Financial Math.*, 4(1)

Low-bias simulation scheme for the Heston model by Inverse Gaussian approximation. S. T. Tse J. W. L. Wan.

*Quantitative Finance, Volume 13, Issue 6, 2013*

Simple Simulation Scheme for CIR and Wishart Processes P. Baldi, C.Pisani

*International Journal of Theoretical and Applied Finance Vol. 16,
No. 08, 2013*

Importance sampling for jump processes and applications to finance. L. Badouraly Kassim, J. Lelong and I. Loumrhari.

*Journal of Computational Finance, to appear*

A Wiener-Hopf Monte Carlo simulation technique for Lévy process. A. Kuznetsov, A.E.Kyprianou J. C. Pardo and K. van Schaik.

*The Annals of Applied Probability, Volume 21, Number
6, 2011.*

A Wiener-Hopf Monte Carlo simulation approach for pricing path-dependent options under Lévy process. O. Kudryavtsev

*Preprint.*

An Efficient Binomial Lattice Method for Step Double Barrier Options. E.Appolloni, M.Gaudenzi A.Zanette.

*International Journal of
Applied and Theoretical Finance Vol.17, Issue No. 6, 2014*.

The algorithms

“Pricing American-Style Options by Monte Carlo Simulation : Alternatives to Ordinary Least Squares” by Stathis Tompaidis and Chunyu Yang

“Value Function Approximation or Stopping Time Approximation : A comparison of Two Recent Numerical Methods for American Option Pricing using Simulation and Regression” by Lars Stentoft

implemented in 2015 by CélineLabart will be included in the following release.

Moreover, Jérome Lelong has performed the following tasks:

Add an importance sampling based code for jump diffusion models.

Improve the internal enumeration mechanism (PremiaEnum).

Update gnuplot files generation for reports.

Everyday maintenance to fix various bugs.

Clean the generation of the documentation process.

A. Alfonsi and P. Blanc are working on the optimal execution problem when many large traders who modify the market prices. In a previous study, they have developed a price impact model that takes into account an exogeneous flow of market orders, in which the optimal execution strategy is known explicitly. This year,they have worked on the practical implementation of this model. Namely, they have proposed an estimation procedure to estimate the model parameters (decay kernel of the price impact and Hawkes kernel for the self excitation of the order flow). They have run this estimation on market data and backtested the optimal execution strategy.

R. Dumitrescu, M.C. Quenez and A. Sulem have provided a weak dynamic principle for Combined Optimal Stopping/Stochastic Control with

In the recent paper , they study game options in an imperfect market
with default.
They extend the results obtained by Kifer in a perfect market model to the case of imperfections taken into account via the nonlinearity of the wealth dynamics. In this framework, the pricing system is expressed as a nonlinear *generalized* Dynkin game expressed in terms of the *mixed generalized* Dynkin game.
They prove the existence of a cancellation time and a trading strategy which allows the seller to be super-hedged, whatever the model is. This study is introduced by the analysis of the simpler case of American options.

In collaboration with Jane Bielagt (Humbold Univ.) and Gona̧lo Dos Reis (Univ. of Edimburgh), Arnaud Lionnet investigates in the effects of the social interactions of a finite set of agents on an equilibrium pricing mechanism. They consider an incomplete market where agents invest so as to minimize their risk measure. Here, agents assess risk using convex dynamic risk measures expressed by Backward Stochastic Differential Equations (BSDE). Beside the risk associated with their own economic activity, the agents compare their trading gains to that of the others, and factor this relative performance in the evaluation of their risk/satisfaction. When a derivative product is introduced to complete the market and allow agents to trade a non-financial risk factor (such as temperature), the risk of each agent is lowered, as expected. However, agents then find it in their interest to be more concerned with their relative performance. This leads them to behave more like a herd and this destabilizes the previously stable, purely financial market.

Moreover R. Elie obtained a CVRS PEPS grant on systemic risk modeling with graphs in collaboration with the Inria team COATI and the economic department of Université de Nice.

A.Alfonsi with A. Kebaier and C. Rey have studied the Maximum Likelihood Estimator for the Wishart processes and in particular its convergence in the ergodic and in some non ergodic cases. In the non ergodic cases, their analysis rely on refined results on the Laplace transform for Wishart processes. This work also extends a recent paper by Ben Alaya and Kebaier on the maximum likelihood estimation for the CIR process.

A. Alfonsi, E. Palidda and A. Ahdida extend the Linear Gaussian Model (LGM) by replacing the constant convariation matrix by some Wishart dynamics. This extension allows them to generate smile while keeping the affine structure of the model. They have obtained a price expansion around the LGM for Caplet and Swaption prices. They also present a second order discretization scheme that allow them to compute exotic prices within this model.

A. Alfonsi with A. Kohatsu-Higa and M. Hayashi are investigating how to apply the parametric method recently proposed by V. Bally and A. Kohatsu-Higa for reflected SDEs. This method allows them to obtain an unbiased estimator for expectations of general functions of the process.

With J. Corbetta (postdoc financed by the chair financial risks), A. Alfonsi and B. Jourdain study a general formula for the time-derivative of the wasserstein distance between the time-marginals of two Markov processes. They have checked the validity of this formula for pure-jump Markov processes with a bounded intensity of jumps. They now study the extension to piecewise deterministic Markov processes.

B. Jourdain and J. Reygner study multitype sticky particle systems which can be obtained as vanishing noise limits of multitype rank-based diffusions. Rank-based diffusion processes and their multitype generalization permit to reproduce empirical features of stock markets. B. Jourdain and J. Reygner have obtained the optimal rate of convergence as the number of particles grows to infinity of the approximate solution to a diagonal system of hyperbolic conservation laws based on multitype sticky particles.

Damien Lamberton with M. Pistorius has worked on the approximation of American options by Canadian options, which originated from the work of Peter Carr. This lead them to revise old results on the binomial approximation of the American put. D. Lamberton is also working with M. Zervos on American options involving the maximum of the underlying.

(V. Bally and PhD student C. Rey )

The main issue was to consider very general approximation schemes and to estimate the approximation error for test functions which are just measurable and bounded. It is worth to mention that the input of noise in the approximation schemes is allowed to be quite general, while in the standard approximation schemes for diffusion processes one considers Gaussian input only. In some sense this means that we treat invariance principles as well. We also considered approximation schemes of higher order, as the Victoir Nynomia scheme for example. An important ingredient is an abstract Malliavin calculus for general random variables (which has been settled in previous papers of V. Bally and Lucia Caramellino.

(V. Bally and PhD student V.Rabiet ).

PDMP processes are very popular in many practical fields as biology, chemistry or fiability theory. The main idea is that such a model may present different scales: slow ones and rapid onces. And from a numerical point of view it is extremely difficult to implement algorithms which take care of rapid scales in details. Then the idea is to average the rapid scales (in the spirit of the Central Limit Theorem) and consequently to replace small (and rapid) jumps by a Brownien component. This procedure is already widely used by practitioners. Our work was to derive estimates of the error which is done by this procedure.

(V. Bally with Lucia Caramellino and Guillaume Poly)

In the classical theory, the convergence which has already been studied is the convergence in total variation (measurable test functions). The main result is the theorem of Prohorov, in the fifties. We have proved that under similar hypothesis (with more finite moments however) one may obtain a much more accurate estimate of the error, in some norms which are close to distribution norms. As a remarkable consequence, we obtained a CLT for the zeros of trigonometric polynomials with random coefficients.

Consortium PREMIA, Natixis - Inria

Consortium PREMIA, Crédit Agricole CIB - Inria

Chaire X-ENPC-UPMC-Société Générale "Financial Risks" of the Risk fondation : A. Alfonsi, B. Jourdain, B. Lapeyre

A. Sulem: Research Grant for the project "*Stochastic Control of Systemic Risk*", awarded by the scientific council and Professional Fellows of
Institut Europlace de Finance (EIF) and Labex Louis Bachelier with A. Minca (Cornell University).

R. Elie with Arthur Charpentier:

Chaire with COVEA on digital impact in actuarial activities (2015-2018).

ANR Stab 2013-2016, Participant : B. Jourdain, Partners : Lyon 1, Paris-Dauphine

ANR Cosmos 2015-2018, Participant: B. Jourdain ; Partners : Ecole des Ponts, Telecom, INIRIA Rennes and IBPC

**Pôle Finance Innovation.**

Center of Excellence program in Mathematics and Life Sciences at the Department of Mathematics, University of Oslo, Norway, (B. Øksendal).

Department of Mathematics, University of Manchester (Tusheng Zhang, currently in charge of an EU-ITN program on BSDEs and Applications).

Kensas University (Yaozhong Hu)

Mannheim University (Alexander Schied, Chair of Mathematics in Business and Economics, Department of Mathematics)

Roma Tor Vergata University (Lucia Caramellino)

Ritsumeikan University (A. Kohatsu-Higa).

Oleg Kudryavtsev, Rostov University (Russia), 2 months

Xiao Wei, Beijing university, 2 months

Houzhi Li (April to July 2015): Study and implementation in Premia of the 4/2 stochastic volatility model proposed by M. Grasselli; adviser A. Alfonsi

A. Alfonsi IPAM, UC Los Angeles , invited by René Carmona (April 13- 23)

A. Sulem: Participation to the "Stochastics in Environmental and Financial Economics" program, Centre of Advanced Studies of the Norwegian Academy of Sciences and Letters, Oslo,Spring 2015.

A. Alfonsi: Co-organizer of the working group seminar of MathRisk “Méthodes stochastiques et finance”.
http://

R. Dumitrescu: Co-organizer of the young researchers in Mathematics Seminar of Université Paris Dauphine.

B. Jourdain (with B. Bouchard and E. Gobet): organization of the 2015-2016 thematic semester on Monte Carlo methods financed by the Institute Louis Bachelier.

R. Elie

Associate editor of *SIAM Journal on Financial Mathematics (SIFIN)* (since November 2014)

D. Lamberton

Associate editor of

Mathematical Finance,

Associate editor of ESAIM Probability & Statistics

A. Sulem

Associate editor of

2011- Present: *Journal of Mathematical Analysis and Applications (JMAA)*

2009- Present: *International Journal of Stochastic Analysis (IJSA)*

2008- Present: *SIAM Journal on Financial Mathematics (SIFIN)*

The members of the team reviewed numerous papers for numerous journals.

A. Alfonsi

- "Dynamic optimal execution in a mixed–market–impact Hawkes price model", Workshop on "The Mathematics of High Frequency Financial Markets", IPAM, Los Angeles, April 13 2015:

- "Optimal transport bounds between the time-marginals of a multidimensional diffusion and its Euler scheme", AMS-EMS-SPM conference, Porto, June 10 2015:

- "Dynamic optimal execution in a mixed–market–impact Hawkes price model", séminaire de la chaire Risques financiers, June 12 2015:

- "Optimal execution in a Hawkes price model", London-Paris Bachelier Workshop on Mathematical Finance 2015, Sept 26 2015

R. Dumitrescu

- Financial Mathematics Seminar, Univ Marne-la-Vallée, ENPC and Inria, 12 January

- Stochastic Analysis Seminar, Imperial College London, 17 March

- Financial Mathematics Seminar, Univ Evry, 26 March.

B. Jourdain

- Conference in honor of Vlad Bally, le Mans, 6-9 October 2015 : Strong convergence properties of the Ninomiya Victoir scheme and applications to multilevel Monte Carlo

- Workshop on NASPDE 2015, Inria Sophia, 22-23 September 2015 : Multitype sticky particles and diagonal hyperbolic systems of PDEs

- Workshop BIRS Free-energy calculations, a mathematical perspective, Oaxaca 19-24 july 2015 : Analysis of discrete space versions of the self-healing umbrella sampling and well-tempered metadynamics algorithms

- Workshop Probabilistic Numerical Methods for Non-Linear PDEs, Imperial College London , 29 june-1st july 2015 : On a stochastic particle approximation of the Keller-Segel equation - Workshop Numerical Probability and Applications to Finance, Enit Tunis, 30 april 2015 : Strong convergence properties of the Ninomiya Victoir scheme and applications to multilevel Monte Carlo

- Maxwell Colloquium, Edimbourg, 6 february 2015 : Multitype sticky particles and diagonal hyperbolic systems of PDEs

C. Labart : **MCM2015**, Linz, Austria, July 2015.

J. Lelong : **MCM2015**, Linz, Austria, July 2015.

D. Lamberton:

- Workshop Numerical Probability and Applications to Finance, Tunis, April 2015. "On the Canadian approximation of the American put".

- Workshop in honor of Vlad Bally, Stochastic Calculus, Monte Carlo Methods and Mathematical Finance, Le Mans October 2015. "On the binomial approximation of the American put".

R. Elie

- 7th general AMAMEF and Swissquote conference, Lausanne, 2015

- Workshop Advanced methods in Mathematical Finance, Angers, France, 2015

- Workshop on current challenges in financial mathematics and economics, London, 2015

- Workshop on smart energy and stochastic optimization, ENPC, Paris, 2015

- South East Asia conference in Mathematical Finance, Siem Reap, Cambodia, 2015

- 9^{th} Bachelier Colloquium, Metabief, France, 2015

- seminar, Imperial College, London, 2015

A. Lionnet - MS-EMS-SPM International Meeting, special session on Stochastic Numerical Methods for Non-linear Equations. Porto, June 10-13.

- 2nd IMA conference on mathematics in finance. Manchester, 18–19 June.

- Workshop on Probabilistic numerical methods for nonlinear PDEs at Imperial College. London, June 29 – July 1.

- The 2015 conference on Stochastic Processes and Applications. Oxford, July 13–17.

A. Sulem

- International Conference "Stochastics and Computational Finance 2015- from academia to industry", July 6-10, 2015, ISEG - University of Lisbon, keynote speaker,
http://

- Conference on Mathematical Finance and Partial Differential Equations, Rutgers University, New Brunswick, New Jersey, May 2015, Plenary speaker,
http://

- Second conference on " Stochastics of Environmental and Financial Economics", April 20-24, 2015, Academy of Science in Oslo, Plenary speaker,

- Conference in honor of Prof. V. Bally, "Stochastic Calculus and Malliavin Calculus, Monte-Carlo Methods and Mathematical Finance", Université du Mans, 6-9 Octobre 2015.
http://

A. Alfonsi: In charge of the Master “Finance and Application” at the Ecole des Ponts.

V. Bally: Member of the scientific committee of Université Paris-Est Marne-la-Vallée; member of the scientific committee and responsible of the theme "Stochastic equations and PDEs" of ; Responsible of the Probability team of LAMA (until April) and organizer of the seminar of the LAMA laboratory. Labex Bezout;

R. Dumitrescu : creation of the ALUMNI of *Fondation des Sciences Mathematiques de Paris*

B. Jourdain: Head of the doctoral school MSTIC, University Paris-Est

D. Lamberton: Vice-president for research at Université Paris-Est Marne-la-Vallée

A. Sulem: member of the Committee for technology development , Inria Paris-Rocquencourt

**Undergraduate programs**

A. Alfonsi: “Modéliser, Programmer et Simuler”, second year course at the Ecole des Ponts.

R. Dumitrescu, Applied courses (Travaux Dirigés) "introduction to finance", L2, Université Paris Dauphine

R. Elie

- Machine learning for actuarial sciences , Institut des actuaires français

- Arbitrage et valorisation d'options, ENSAE

- Introduction to Python, TD L2 Université Paris-Est

B. Jourdain : - course "Probability theory and statistics", first year ENPC

- "Introduction to probability theory", 1st year, Ecole Polytechnique

- "Stochastic numerical methods", 3rd year, Ecole Polytechnique

- projects in finance and numerical methods, 3rd year, Ecole Polytechnique

A. Lionnet

*Stochastic Processes*, ESIEA, Paris , 18h .

**Graduate programs**

A. Alfonsi:

- “Calibration, Volatilité Locale et Stochastique”, third-year course at ENSTA (Master with University Paris I).

- “Traitement des données de marché : aspects statistiques et calibration”, lecture for the Master at UPEMLV.

- A. Alfonsi: “Mesures de risque”, Master course of UPEMLV and Paris VI.

V. Bally,

- Advanced Probability (Master 1 Recherche Mathématiques et Application, University of Marne-la-Vallée

- "The Malliavin calculus and applications in finance", 30h, Master 2 Mathematical Finance, Université Marne la Vallée

- V. Bally, " Interest Rates", 20h, Master 2 Mathematical Finance , Université Marne la Vallée

- V. Bally, " Risk methodes in actuarial science", 36h, Master IMIS, Université Marne la Vallée

R. Dumitrescu, Applied courses (Travaux Dirigés) in " Asset pricing by absence of arbitrage opportunity", Université Paris Dauphine

R. Elie - Imperfect markets modeling M2 Master MASEF (Paris-Dauphine)

- Machine learning and applications, Master 2 Mathematics, , ENPC

- Portfolio management and risk measures, Master Actuariat, Université Paris-Est

- Arbitrage, volatility and portfolio management, Master 2 Math. Finance, Université Paris-Est

B. Jourdain, B. Lapeyre "Monte-Carlo methods", 3rd year ENPC and Master Recherche Mathématiques et Application, University of Marne-la-Vallée

A. Sulem

- "Finite difference for PDEs in Finance", Master 2 MASEF, Université Paris IX-Dauphine, Département Mathématiques et Informatique de la Décision et des Organisations (MIDO), 18h.

- Master of Mathematics, University of Luxembourg, 22 h lectures and responsible of the module "Numerical Methods in Finance".

Anis Al Gerbi (started november 2013) "Discretization of stochastic differential equations and systemic risk modeling", supervised by B. Jourdain and E. Clément

Pierre Blanc, "Price impact on marker orders and limit order books (from Nov. 2012), Ecole des Ponts, Adviser : A. Alfonsi defended on October 9th 2015.

Rui Chen (Fondation Sciences Mathématiques de Paris grant), "Stochastic Control of mean field systems and applications to systemic risk, from September 2014, Université Paris-Dauphine, Adviser A. Sulem

Roxana Dumitrescu, (Fondation Sciences Mathématiques de Paris grant, RDMath Ile de France),
*Contributions au contrôle stochastique avec des espérances non linéaires et aux
équations stochastiques rŕetrogrades *,
Advisers: A. Sulem with B. Bouchard and R. Elie, defended on
September 28 2015, Université Paris-Dauphine.

Antoine Ly, CIFRE agreement, Partner: Miliman, advisers: R. Elie and Arthur Charpentier, Applications du machine learning en Actuariat, started 2015, Advisers: R. Elie and Arthur Charpentier

Sebastien Mollaret, CIFRE agreement, Partner: Natixis, Valorisation et couverture dans les modèles à changement de régime started 2015, Advisers: R. Elie.

Ernesto Palidda, Advisers: A. Alfonsi and Bernard Lapeyre, Ecole des Ponts, defended on May 29 2015.

- Paulo Pigato, "Lower bounds for the density of the solution of SDE's under the weak Hörmander condition, and applications in finance", Advisers: V. Bally and A. Dai Pra, University of Padova, defended October 2015, Université Paris-Est Marne la Vallée,

Victor Rabiet : "On a class of jump type stochastic equations", Université Paris-Est Marne la Vallée, Advisers: V. Bally (75 %) and E. Locherbach, defended June 2015.

Clément Rey (from Oct. 2012), " High order discretization schemes for singular diffusions", Ecole des Ponts, Advisers : A. Alfonsi and Vlad Bally, defended on December 4th 2015.

Giulia Terenzi (from October 2015) "American options in complex financial models", Advisers : D. Lamberton and Lucia Caramellino, from University Tor Vergata, Rome.

Alexandre Zhou (started November 2015) "Analysis of stochastic particle methods applied to finance", supervised by B.Jourdain

B.Jourdain

- Head of the HCERES visit committee of the Laboratoire de Mathématiques Raphael Salem, Rouen, November 20 2015

- PhD of Victor Rabiet, defended on June 23 2015, University Paris-Est

- PhD of Thi Quynh Giang NGUYEN, defended on October 19 2015, University Paris-Est

A. Sulem

- PhD Roxana Dumitrescu, September 2015, Université Paris-Dauphine

- Referee (Rapporteur) HdR *"Quelques contributions en finance mathématique, risque de liquidité et finance d'entreprise"*, Ly Vath Vathana, Université Evry Val d'Essonne, November 23 2015

- Faculty opponent, PhD thesis "*Aspects of Waiting and Contracting in Game Theory*", Peter Helgesson,
Chalmers University of Technology and University of Gothenburg, Sweden, June 5th 2015.

- Sulem : Committe for the recrutment of a Professor of Insurance Mathematics, ETH Zurich January 2015

- Committe for the recrutment of a Professor ( in Financial Mathematics and numerical probability, Laboratoire de probabilités, Université Paris-Diderot, May 2015

- Committe for the recrutment of a researcher (chargé de recherche) Inria Paris-Rocquencourt, June 2015

A. Lionnet:

- article on Financial Mathematics published in the online popularization magazine Interstices
https://

- Conference given for middle-schoolers at University Paris 8 during the "Week of Mathematics 2015".

- Participation to the colloquium "Science and You", Nancy, June 1–6.

- The two entries of A. Lionnet to the contest SIAM Math Matters, Apply It (https://

A. Sulem:

- "Applications of stochastic analysis", (2015), invited contribution for the *Princeton Companion to Applied Mathematics*, http://

- Responsible for the partnership of Inria with Institut Henri Poincaré for the creation of a museum of Mathematics.