Nizar Touzi

Professor, Applied Mathematics


 

 

Centre de Mathématiques Appliquées

Ecole Polytechnique

UMR CNRS 7641

91128 Palaiseau Cedex

FRANCE

 

Téléphone:

33 (0)1-69-33-46-12

Fax:

33 (0)1-69-33-30-11

e-mail:

nizar.touzi@polytechniqueDOTedu


Actualités

 

Second European Summer in Financial Mathematics

Bachelier Seminar, Institut Henri Poincaré, Paris

Lecture Notes

Research Topics

Recent Papers

Accepted Papers


LECTURE NOTES

 

 

Stochastic Calculus in Finance, Ecole Polytechnique, 3ème année, PA Mathématiques Appliquées (pdf file)

 

Optimisation dynamique,  ENSAE 2ème année (pdf file)

 

Stochastic control and application to Finance, Scuola Normale Superiore, Pisa. Special Research Semester on Financial Mathematics, April 29-July 15 2002 (pdf file).


RESEARCH TOPICS

1. Deterministic and stochastic control problems related to finance and insurance :

- Convex duality methods in the utility maximization problem with market imperfection, non-smooth utility function...

- Pricing and hedging by utility indifference.

- Super-replication and its extension to the stochastic target problem, duality and direct characterization.

- Connection with FBSDE’s and geometric flows in the field of differential geometry.

 

 2. Monte Carlo methods in finance :

- Variance reduction methods.

- Malliavin calculus for the computation of Greeks.

- Malliavin calculus for optimal stopping problems, i.e. American options. 

- BSDEs and Probabilistic numerical methods for nonlinear PDEs


RECENT PAPERS

 

[63] M. Soner, J. Zhang, and N. Touzi, Dual Formulation of Second Order Target Problems. (pdf file)

[62] B. Bouchard and N. Touzi, Weak Dynamic Programming Principle for Viscosity Solutions. (pdf file)

[61] Fahim A, N. Touzi. and X. Warin, A Probabilistic Numerical Scheme for Fully Nonlinear PDEs. (pdf file)

[60] Aïd R, A. Porchet A. and N. Touzi, Vertical integration and risk management in competitive markets of non-storable goods. (pdf file)

[59] I. Ben Tahar, M. Soner and N. Touzi, Modelling continuous-time financial markets with capital gains taxes. (ps file)

[58] D. Crisan, K. Monalorakis et N. Touzi, On the Monte Carlo simulation of Backward SDES: an improvement on the Malliavin weights (pdf file)


ACCEPTED PAPERS

[57] B. Bouchard, R. Elie and N. Touzi, Stochastic Target Problems with Controlled Loss. SIAM Journal on Control and Optimization, 48, 5, pp. 3123-3150. (pdf file)

[56] R. Aïd, L. Campi, A. Nguyen Huu, and N. Touzi, A Structural Risk Neutral Model of Electricity Prices, International Journal of Theoretical and Applied Finance 12, 7 (2009), 925-947. (pdf file)

[55] M. Soner and N. Touzi, The dynamic programming equation for second order stochastic target problems, SIAM Journal on Control and Optimization 48, 4, 2344-2365. (pdf file)

[54] U. Cetin, M. Soner and N. Touzi, Option hedging under liquidity costs, Finance and Stochastics, to appear. (pdf file)

[53] Elie R. and N. Touzi, Optimal lifetime consumption and investment under drawdown constraint. To Appear in Finance and Stochastics 12, 3 (July 2008), pp. 299-330. (pdf file)

[52] Porchet A., N. Touzi and X. Warin, Valuation of a powerplant under production constraints and markets incompleteness. Mathematical Methods of Operations research, Volume 70, Issue 1 (2009), Page 47-75 (2009). (pdf file)

[51] E. Jouini, W. Schachermayer and N. Touzi, Optimal risk sharing for law invariant monetary utility functions. Mathematical Finance 18, 2 (2008), pp. 269-292. (pdf file)

[50] I. Ben Tahar, M. Soner and N. Touzi, The dynamic programming equation for the problem of optimal investment under capital gains taxes. SIAM Journal on Control and Optimization 46, 5 (2007), pp. 1779-1801. (pdf file)

[49] R. Elie, J.-D. Fermanian and N. Touzi, Kernel estimation of Greek weights by parameter randomization. Annals of Applied Probability 17, 4 (2007), 1399-1423. (pdf file)

[48] G. Carlier, I. Ekeland and N. Touzi, Optimal derivatives design for mean-variance agents under adverse selection. Mathematics and Financial Economics, 1, 1 (April 2007), pp. 57-80. (pdf file)

[47] F. Astic and N. Touzi, No arbitrage conditions and liquidity. Journal of Mathematical Economics, to appear. (ps file)

[46] P. Cheridito, M. Soner, N. Touzi and Nicolas Victoir, Second Order Backward Stochastic Differential Equations and Fully Non-Linear Parabolic PDEs. Communications in Pure and Applied Mathematics, to appear. (pdf file)

[45] E. Jouini, W. Schachermayer and N. Touzi, Law Invariant Risk Measures have the Fatou Property. Advances in Mathematical Economics 9, 49-72 (2006). (pdf file)

[44] M. Mrad, N. Touzi, and A. Zeghal, Monte Carlo estimation of a joint density using Malliavin calculus. Computational Economics 27, 4, 497-531 (2006).  (pdf file)

[43] M. Soner and N. Touzi, Hedging under gamma constraints by optimal stopping and face-lifting. Mathematical Finance 17, 1, 59-80 (2007). (pdf file)

[42] R. Carmona and N. Touzi, Optimal multiple stopping and valuation of swing options. Mathematical Finance 18, 2 (April 2008) pp. 239-268 (pdf file)

[41] B. Bouchard, N. El Karoui and N. Touzi, Maturity randomisation for stochastic control problems. Annals of Applied Probability, 15, 4, 2575-2605 (2005).

(pdf file)

[40] P. Cheridito, M. Soner and N. Touzi, Small time path behavior of double stochastic integrals and applications to stochastic control. Annals of Applied Probability 15, 4, 2472-2495 (2005). (pdf file)

[39] P. Cheridito, M. Soner and N. Touzi, The multi-dimensional super-replication problem under Gamma constraints, Annales de l’Institut Henri Poincaré, Série C: Analyse Non-Linéaire 22, 633-666 (2005). (pdf file)

[38] A. Bensoussan, N. Touzi and J.-L. Menaldi, Penalty approximation and analytical characterization of the problem of super-replication under portfolio constraints, Asymptotic Analysis 41, 311-330 (2005). (pdf file)

[37] B. Bouchard and N. Touzi, Discrete-time approximation and Monte Carlo simulation of backward stochastic differential equations, Stochastic Processes and their Applications, 111, 175-206 (2004). (pdf file)

[36] B. Bouchard, I. Ekeland and N. Touzi, On the Malliavin approach to Monte Carlo approximation of conditional expectations, Finance and Stochastics, 8, 45-71 (2004).  (pdf file)

[35] E. Jouini, M. Meddeb and N. Touzi, Vector-valued measure of risk, Finance and Stochastics, 8, 531-552. (pdf file)

[34] B. Bouchard, N. Touzi and A. Zeghal, Dual Formulation of the Utility Maximization Problem : the case of Nonsmooth Utility. Annals of Applied Probability, 14, 678-717 (2004). (pdf file)

[33] H.M. Soner and N. Touzi, The problem of super-replication under constraints, to appear in Paris-Princeton Lectures in Mathematical Finance, Lecture Notes in Mathematics, Springer-Verlag. (pdf file)

[32] H.M. Soner and N. Touzi, Stochastic representation of mean curvature type geometric flows, Annals of Probability, 31, 1145-1165 (2003). (pdf file)

[31] H.M. Soner and N. Touzi, Level set characterization of stochastic target problems, Communications in PDE’s, 27, 2031-2053. (pdf file)

[30]  B. Bouchard, Y. Kabanov and N. Touzi, Option pricing by large risk aversion utility under transaction costs, Decision in Economics and Finance, 24, 127-136 (2001).

[29] H.M. Soner and N. Touzi, Dynamic programming for stochastic target problems and geometric flows, Journal of the European Mathematical Society, 4, 201-236 (2002). (pdf file)


[28]
G. Deelstra, H. Pham and N. Touzi, Dual formulation of the utility maximization problem under transaction costs, Annals of Applied Probability, 11 (4), 1353-1383 (2002).


[27]
N. Touzi and N. Vieille, Continuous-time Dynkin games with mixed strategies, SIAM Journal on Control and Optimization 41, 1073-1088 (2002). (pdf file)


[26]
H.M. Soner and N. Touzi, Stochastic target problems, dynamic programming and viscosity solutions, SIAM Journal on Control and Optimization,  41, 404-424 (2002). (pdf file)


[25]
N. Touzi, Super-replication under proportional transaction costs: from discrete to continuous-time models, Mathematical Methods of Operations Research 50, 297-320 (1999).


[24]
B. Bouchard and N. Touzi, Explicit solution of the multivariate super-replication problem under transaction costs, Annals of Applied Probability 10, 685-708 (2000).


[23]
N. Touzi, Direct characterization of the value of super-replication under stochastic volatility and portfolio constraints, Stochastic Processes and their Applications 88, 305-328 (2000).


[22] H.M. Soner and N. Touzi, Super-replication under Gamma constraint, Journal on Control and Optimization 39, 73-96 (2000).


[21]
P.-F. Koehl, H. Pham and N. Touzi, On super-replication under Transaction costs in general discrete-time models, Theory of Probability and its Applications 45, 783-788 (1999).


[20]
L. Carassus, H. Pham and N. Touzi, Arbitrage and domination cost in a discrete-time model with convex portfolio constraints, Mathematical Finance, to appear.


[19] E. Jouini, P.-F.
Koehl and N. Touzi, Optimal investment with taxes: an existence result, Journal of Mathematical Economics, 33, 373-388.

[18] E. Jouini, P.-F. Koehl and N. Touzi, Optimal investment with taxes~: an optimal control problem with endogeneous delay, Nonlinear Analysis~: Theory, Methods and Applications 37, 31-56 (1999).


[17]
N. Touzi, Optimal insurance demand under marked point processes shocks, Annals of Applied Probability 10, 283-312 (2000).


[16]
J. Cvitanic, H. Pham and N. Touzi, Super-replication in stochastic volatility models with portfolio constraints, Journal of Applied Probability 36, 523-545 (1999).


[15]
J. Cvitanic, H. Pham and N. Touzi, A closed-form solution for the problem of super-replication under transaction costs, Finance and Stochastics 3, 35-54 (1999).


[14]
E. Fournié, J.M. Lasry and N. Touzi, Monte Carlo Methods in Stochastic Volatility Models, in Numerical Methods in Financial  Mathematics, edited by C. Rogers et D. Talay, Cambridge University Press (1997).


[13] E. Fournié, J.M. Lasry, J. Lebuchoux, P.-L.
Lions and N. Touzi, Some applications of Malliavin calculus to Monte Carlo methods in finance, Finance and Stochastics 3, 391-412 (1999).


[12]
P.-F. Koehl, H. Pham and N. Touzi, Hedging in discrete-time  under Transaction costs and continuous-time limit, Journal of Applied Probability 36, 163-178 (1999).


[11]
H. Pham and N. Touzi, The fundamental theorem of asset pricing with cone constraints, Journal of Mathematical Economics, to appear.


[10]
N. Touzi, American options exercise boundary when the volatility changes randomly, Applied Mathematics and Optimization 39, 411-422 (1999).


[9] E. Fournié, J. Lebuchoux and N. Touzi, Small Noise Expansion and Importance Sampling, Asymptotic Analysis 14, 361-376 (1997).


[8] E. Jouini, P.-F.
Koehl and N. Touzi, Incomplete Markets, Transaction costs and Liquidity effects, European Journal of Finance 4, 325-348.


[7]
L.P. Hansen, J.A. Scheinkman and N. Touzi, Spectral Methods for Identifying Scalar Diffusions, Journal of Econometrics 86, 1-32.


[6]
J.-P. Florens, E. Renault and N. Touzi, Testing Embeddability by Stationary Reversible Continuous-Time Markov Processes, Econometric Theory 14.


[5]
C. Gouriéroux, E. Renault and N. Touzi, Calibration by Simulation for Small Sample Bias Correction, in Simulation Based Inference in Econometrics Methods, edited by J. Geweke and R. Mariano.


[4]
S. Pastorello, E. Renault and N. Touzi, Statistical Inference for Random Variance Option Pricing, Journal of Business and Economic Statistics, to appear.


[3]
M. Romano and N. Touzi, Contingent Claims and Market Completeness in a Stochastic Volatility Model, Mathematical Finance 7, 399-412 (1997).


[2]
H. Pham and N. Touzi, Equilibrium State Prices in a Stochastic Volatility Model, Mathematical Finance 6, 215-236 (1996).

[1] E. Renault and N. Touzi, Option Hedging and Implicit Volatilities, Mathematical Finance 6, 279-302 (1996).