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## Financial market modeling

**Responsable :** N. Touzi, Professor at Ecole Polytechnique.

- Members
- Research activity
- Path-dependent partial differential equations
- Martingale optimal transport
- Approximation and simulation of non-linear systems
- Control of stochastic systems and applications to finance
- Asymptotic methods and applications
- Financial engineering
- Statistics of continuous-time processes and order books analysis

**Funded projects of this team**: ANR Isotace, ERC Advanced Grant ROFIRM 2012, Chaire Financial Risks, Chaire Finance and Sustainable Development, Chaire "Marchés en mutation"

**Permanent members**

Ankush Agarwal, post-doctoral fellow

Emmanuel Bacry, Chargé de recherche CNRS

Jocelyne Bion-Nadal, Chargée de recherche CNRS

Julien Claisse, post-doctoral fellow

Stefano De Marco, Assistant Professor

Emmanuel Gobet, Professor

Yiqing Lin, post-doctoral fellow

Thibaut Mastrolia, Assistant Professor

Marcello Rambaldi, post-doctoral fellow

Mathieu Rosenbaum, Professor

Mauro Rosestolato, post-doctoral fellow

Nizar Touzi, Professor

Junjian Yang, post-doctoral fellow

**Associate researchers**

Laurent Denis (University of Le Mans)

Céline Labart (University of Chambery)

Pierre Henry-Labordère (Société Générale)

Jérome Lelong (University of Grenoble)

Anis Matoussi (University of Le Mans)

Jean-François Muzy (University of Corsica)

Denis Talay (INRIA Sophia-Antipolis)

**PhD students (current)**

Hadrien De March (supervisor: Nizar Touzi). Martingale optimal transport.

Kaitong Hu (supervisor: N. Touzi).

Omar El Euch (supervisor: M. Rosenbaum). Derivative pricing under rough volatility.

Gustav Matulewicz (supervisors: Emmanuel Gobet et Stéphane Gaiffas). Random graph models.

Saad Mouti (supervisors: N. El Karoui et M. Rosenbaum). Derivative hedging from an insurer’s perspective.

Pamela Saliba (supervisor: M. Rosenbaum). High frequency trading: statistical analysis and regulation.

Uladzislau Stazhynski (supervisor: E. Gobet).

**PhD students (last five years)**

Khalil Al Dayri, Market Microstructure and Modeling of the Trading Flow. Supervisor: E. Bacry and M. Hoffman, defended in 2012. Khalil is presently quant at Antares Technologies.

Romain Bompis. Stochastic expansion for diffusions and applications. Supervisor: Emmanuel Gobet. Romain is presently quant at CA-CIB.

Tarik Ben Zineb, Efficient Valuation of GMWB Contrats. Supervisor: Emmanuel Gobet, defended December 2012. Tarik is presently quant at Thomson-Reuters.

Emilie Fabre, Some Contributions to Control and Backward Equations in finance. Supervisor: Nizar Touzi, defended 02/12. Emilie is presently quant at Société Générale.

Gaoyue Guo: Continuous-time martingale optimal transport and PCOC. Supervisor: Nizar Touzi, defended in 2016. Gaoyue is currently post-doctoral fellow at Oxford University.

Thibault Jaisson: Statistical problems arising from market microstructure. Supervisors: Emmanuel Bacry and Mathieu Rosenbaum, defended in 2015.

Nabil Kazi-Tani, Dynamic Risk Measures and Second Order Backward SDEs. Supervisors: Nicole El Karoui and Jocelyne Bion-Nadal, defended 12/12. Nabil is presently Assistant Professor at Lyon University.

Nicolas Landon, Risk Management in Energy Markets. Supervisor: Emmanuel Gobet, defended February 2013. Nicolas is presently quant at Dexia.

Gang Liu, Rare event simulation. Supervisor: Emmanuel Gobet, defended en 2016. Gang is presently quant at Jump Trading.

Trung-Lap Nguyen, Financial Risk in Insurance and Long term risk. Supervisor: Nicole El Karoui, defended 09/12.

Guillaume Royer, Martingale Optimal Transport. Supervisor: Nizar Touzi. Guillaume is now quant at Bank of America Merrill Lynch.

Chao Zhou, Model Uncertainty and Second Order Backward SDEs. Supervisor: Anis Matoussi, defended 12/12. Chao is presently Assistant Professor at NUS (Singapoor).

Nicole El Karoui (Professor until 08/09) retired from Polytechnique.

Caroline Hillairet (Assistant Professor until 08/15), presently Professor at ENSAE.

Peter Tankov (Assistant Professor until 08/11), presently Professor at ENSAE.

Sigrid Kallblad, post-doctoral fellow (13-16)

Christian Litterer, post-doctoral fellow (09/14 to 08/15)

Stefano Pagliarani, post-doctoral fellow (14-15)

Plamen Turkedjiev, post-doctoral fellow (09/13 to 08/15)

### Research activity

Our research is motivated by applications to financial mathematics which opens news perspectives and raises new problems in stochastic analysis, numerical approximation techniques, and statistical methods. Therefore, our topics of interest range from applied problems in financial engineering, requiring sophisticated mathematical techniques, to theoretical problems in stochastic analysis and stochastic control theory, with an emphasis on the related numerical and statistical aspects.

**Path-dependent PDEs and backward SDEs.**
Motivated by applications in stochastic control and financial mathematics, the theory of backward stochastic differential equations (BSDE) has experienced a huge development since the early nineties. In the Markov framework, a BSDE is associated to a semi-linear parabolic second order partial differential equation (PDE), and the solution of a BSDE is in fact a specific Sobolev solution of the corresponding PDE. For this reason, BSDEs can be viewed as a Sobolev solution of a semilinear path-dependent PDE. A first stream of contributions of our team to this area is the extension to the context where the underlying noise is a martingale in a general probability space. Such developments achieved in the work of Bion-Nadal, are motivated by continuous-time dynamic risk measurement. Another important contribution to the theory of BSDEs was achieved in the work of Touzi and co-authors with the extension to second order BSDEs, which can then be viewed as a Sobolev solution of a fully nonlinear parabolic second order path-dependent PDE. In particular, similar to Hamilton-Jacobi-Bellman equations, second order BSDEs provide a unique characterization of the value function of a stochastic control problem in a possibly non-Markov framework. Recently, a new point of view was developed in subsequent work of Touzi and co-authors (Ekren, Keller and Zhang), which develop a notion of viscosity solutions for path-dependent parabolic second order PDEs. The main difficulty is to by-pass the local compactness requirement of the underlying state space in the standard theory of viscosity solutions. This is achieved by replacing the pointwise tangency condition by a tangency condition in the sense of an optimal stopping problem under a nonlinear expectation.

**Martingale optimal transport.**
The robust management of risks requires to weaken as much as possible the assumptions on the underlying model. In this context, an interesting ramification of the theory of optimal transport was introduced in recent work by Galichon, Henry-Labordère and Touzi. Assume that the financial market obeys to the no-arbitrage condition, and that there exists a linear continuous pricing rule. Assume further that the prices of European calls of all strikes, and with some given maturities, are available for trading. Then, the problem of robust hedging of derivatives can be identified to the Kantorovitch dual of an optimal transport problem on the space of paths with a martingale restriction on the set of coupling measures. The transport cost is determined by the nature of the derivative contract under consideration. Moreover, this new optimal transport problem is closely connected with the Skorohod embedding problem, see work by Henry-Labordère and Touzi with Obloj and Spoida, and the newly developed theory of PCOC *(Processus croissants pour l’ordre convexe*). Another recent contribution is the analogue of the Brenier theorem in the martingale context. This was achieved in the one-dimensional case, in recent work by Henry-Labordère and Touzi, and the general context is an ongoing project.

This active new topic is funded by the ANR Project ISOTACE, and by an ERC grant (Advanced grant ROFIRM).

**Approximation and simulation of non-linear systems.**
Motivated by applications in stochastic control and optimal stopping problems, an important activity of our team is on the design of new probabilistic algorithms for the approximation of the solution of a forward-backward SDE and, more generally, of a second-order BSDE. Another more involved motivation is the problem of optimal transportation under controlled dynamics whose dual formulation reduces to optimization of BSDEs, as it appears in work by Tan and Touzi, paving the way for new numerical methods for optimal transport problems.
Gobet introduced an optimal time-discretization of BSDEs whose specification is made according to the fractional smoothness (on Wiener space) of the terminal condition (see work of Gobet and Makhlouf and Gobet and Geiss). The numerical approximation of second order BSDEs is addressed in \cite*ftouw* where a bound on the rate of convergence is obtained by adapting techniques from the theory of viscosity solutions. On the other hand, new fully-implementable schemes for BSDEs are designed, by using marked branching processes as in Henry-Labordère, Tan and Touzi, or empirical least-squares regressions based on statistical learning tools as in the work of Gobet and Turkedjiev. These methods are developed and analyzed under rather great generality, to provide efficient and robust algorithms and to handle multi-dimensional situations. Designing parallel algorithms is also an important concern for us, which we intend to further develop in the future.

This research topic benefits from the financial support of the *(Initiative de Recherche "Méthodes numériques pour le contrôle stochastique")* supported by the FiME laboratory.

**Control of stochastic systems and applications to finance.**
Optimal decision making in finance is usually formulated by stochastic control problems with specific features requiring new mathematical developments. An important contribution of our group is the control theory of stochastic target problems, see Touzi et al., which can be viewed as an extension of standard stochastic control problems, and which are intimately connected to geometric equations describing front propagation in differential geometry. A second important contribution to the general theory is the newly developed weak dynamic programming principle (in Bouchard and Touzi) which by-passes the heavy measurability issues in the classical dynamic programming principle. Also, motivated by the importance of path-dependency in finance, in particular in the measurement of risk (see Bion-Nadal), an extension of the dynamic programming approach was developed in the context of path-dependent stochastic control problems in the work of Touzi with Ekren and Zhang. Moreover, many applications to relevant problems in finance are developed. For instance, the problem of hedging and optimal investment under liquidity cost and optimal investment in jump-diffusion models, possibly under risk measure type of constraint (see Tankov), optimal investment under capital gains taxes, optimal decision making under partial information (see Hillairet).

**Asymptotic methods and applications.**
Similar to the standard engineering applications, asymptotic approximations are widely used in the financial industry. This is in particular very developed in the context of model calibration. A relevant point of view was developed in order to address systematically, and efficiently, the asymptotic approximation of valuation functionals in the work of Gobet, Miri and Benahmou and Gobet and Bompis. In addition, De Marco provides theoretical results on the exponential bounds on the density of diffusions in the context of non-Lipschitz coefficients, and in different works he derives representations or asymptotic approximations of the (implied- and local-) volatility surface so as to improve simulation and calibration techniques. Another important issue concerns the error associated to the discretization of a continuous hedging strategy: precisely, the question is to design a (random) time grid, representing the dates of discrete-time trading, which is optimal in the sense that it asymptotically minimizes the continuous-time hedging error. An elegant general solution of this problem was derived in the work of Gobet and his PhD student Landon. Finally, the asymptotic expansion of the problem of optimal investment under small transaction costs has been tackled in the research of Touzi and co-authors. A rigorous derivation of the first order expansion was open since the early nineties. In the context of a power utility function, this was solved in the previous literature by relying on the explicit structure of the problem. The general case solved in works of Soner, Touzi and Possamai adapts the modern viscosity solutions approach to homogenization to the present problem where the fast variable is determined endogenously by the optimization of the limiting system.

**Financial engineering.**
Many new problems have emerged from the recent financial crisis which require new developments based on stochastic analysis, optimization tools, numerical methods, and simulation techniques. El Karoui and Hillairet have explored new problems arising in longevity risk and the related derivatives. Tankov has analyzed gap-risk contracts, whose payoffs is contingent to a downward jump in the underlying asset price process. Motivated by financial contracts introduced by the French government, Hillairet has analyzed the relevance of public-private partnership. Tankov has provided an improvement of the known model-free bounds on the pricing of certain multi-asset derivatives. Touzi has studied the effect of illiquidity in options hedging and provided a characterization in terms of a degenerate fully nonlinear PDE, and analyzed the corresponding small-illiquidity expansion is analyzed, while Tankov has studied the problem of optimal portfolio allocation under illiquidity risk. Gobet and co-auteurs introduced new analytical approximation techniques suitable for model calibration, while discretization problems related to hedging error analysis have been considered in several works by Tankov. Finally, Touzi has studied the problem of management of CO2 emission derivatives, which gives rise to some singular FBSDEs.

**Statistics of continuous-time processes and order books analysis.**
With the increase of electronic trading and with the different crisis and krachs that took place in the last decade, accurate modeling of financial time-series has become a major challenge. This challenge must clearly be addressed at very various time-scales: from the ultra-high frequency (of the order of the ms, i.e., the time-scale of the order-book), to the microstructure time-scale (of the order of the second) down to the daily or the monthly scales where the price is clearly diffusive. This is the key to understanding the underlying dynamics of price formation and the key to many applications (risk control, regulation, market design, market making, high frequency hedging, optimal execution,...). The recent contributions of our team in this area are in two directions:

- Using the theory of invariance scaling phenomena, Bacry has developed some particularly parcimonious models that account for most stylized facts of price time-series on a wide time-scale (from 1 hour to several years). The so-called Multifractal Random Walk (MRW) model only uses 3 parameters and has been proved to be particularly elegant and useful for risk forecast. These models have now become standard multifractal models that are studied by many international academic teams.

- Microstructure models based on (multivariate) high frequency point processes are introduced in works of Bacry. These models reproduce very accurately microstructure effects such as signature plot, Epps effect... Moreover, their diffusive properties can be controlled easily since there are closed formulas that express the covariance diffusive matrix in terms of the microscopic parameters. In that sense, these models should be particularly useful for studying the systemic risk induced by high-frequency trading and understanding how it can be controlled through regulation. Theses models make extensive use of Hawkes processes and can be seen as building blocks that can be easily put together for addressing a particular issue. Moreover, recently Bacry and Muzy have developed a new non-parametric estimation algorithm for Hawkes processes that can make parameter estimation of any of these models particularly efficient.