
Nizar Touzi
Professor, Applied Mathematics

Centre de Mathématiques Appliquées
Ecole Polytechnique
UMR CNRS 7641
91128 Palaiseau Cedex
FRANCE
Actualités
* New
Advances in Financial Mathematics, 4 avril 2013
*
Bachelier Seminar, Institut Henri Poincaré, Paris
Lecture Notes
Distinctions
Recent
Papers
Accepted Papers
LECTURE NOTES
Chaînes de Markov et martingales en temps discret, Ecole
Polytechnique, 2ème année, PA Mathématiques Appliquées (pdf file)
Stochastic Calculus in Finance, Ecole Polytechnique, 3ème année, PA
Mathématiques Appliquées (pdf file)
Deterministic and
Stochastic Control, Application to Finance, Master Probabilité et Finance Ecole Polytechnique – Université Paris 6 (pdf file)
Optimal Stochastic
Control, Stochastic Target Problems, and Backward SDEs, Lecture Notes of my lectures at the Fields
Institute, AprilJune 2010 (pdf
file).
DISTINCTIONS
ICM 2010 (Hyderabad) Invited Session Speaker (pdf file)
ERC
Advanced Grant 2012
French
Academy of Science – Natixis Foudation, Louis Bachelier Prize 2012
The
University of Toronto Dean’s Distinguished Visitor Chair, Fields Institute, AprilJune 2010
Best
Young Researcher in Finance Award 2007 of the Europlace Institute of Finance
EDITORIAL
ACTIVITIES
Finance
& Stochastics,
Coeditor since January ’07, and Associate Editor June ’00  December 06
Mathematical Finance, Associate Editor since Novembre ’03
Advances in Calculus of Variations, Associate Editor, January ’18 — December
‘22
Annals of Applied Probability, Associate Editor since January ’13
Stochastic Processes and their Applications, Associate Editor since January ’16
Stochastics: an International Journal of
Probability and Stochastic Processes,
Associate Editor since January ’16
Journal of Optimization Theory and
Applications, Associate Editor since
January ’14
Mathematical Control and Related Fields, Associate Editor since September ’12
ParisPrinceton Lectures in Mathematical Finance, Founder and CoEditor
Springer Briefs in Mathematical Finance, Associate Editor
Past editorial
activities:
Electronic Journal of
Probability / Electronic Communications in Probability, Associate Editor June ’04  December ’14
SIAM Journal on Financial Mathematics, Associate Editor January ’09  December ’12
Journal of Financial Econometrics, Associate Editor January ’01  December ’06
Applied Mathematics Research eXpress (AMRX), Associate Editor January ’04  December ’07
PhD Students
Present : Heythem Farhat, Kaitong
Hu, Hadrien De March.
Past: Gaoyue Guo (PostDoc, Oxford
University), Zhenjie Ren (Assistant Professor, Université Paris Dauphine), Guillaume Royer (Bank of America,
London), Emilie Fabre (Société Générale, Paris), Xiaolu Tan (Assistant Professor, Université Paris
Dauphine), Dylan Possamaï (Columbia University), GillesEdouard Espinosa (BNPParibas), Arash Fahim (Florida State
University), Arnaud Porchet (Goldman Sachs, London), Romuald Elie (Professor,
Université Marne la Vallée), Fabian Astic (Moody’s NY), Imen Ben Tahar (Assistant
Professor, Université Paris Dauphine), Amina Zeghal (Université Dauphine Tunis), Moez Mrad (Calyon, London),
Bruno Bouchard (Professor,
Université Paris Dauphine)
RECENT
PAPERS
[107] O. El
Euch, T. Mastrolia, M. Rosenbaum, Nizar Touzi, N. Touzi, Optimal maketake fees for market making
regulation, (pdf file).
[106] Y. Lin,
Z. Ren, N. Touzi, J. Yang, Second order backward SDE with random terminal time, arXiv:1802.02260.
[105] P. HenryLabordère, N. Touzi, Branching
diffusion representation for nonlinear Cauchy problems and Monte Carlo
approximation, (pdf file).
[104] H. De
March, N. Touzi, Irreducible convex paving for decomposition of multidimensional
martingale transport plans, arXiv:1702.08298
[103] Z. Ren, N. Touzi, and J. Zhang, Comparison of
Viscosity Solutions of Semilinear PathDependent PDEs, arXiv:1410.7281.
ACCEPTED
PAPERS
[102] A.M.G.
Cox, J. Obloj and N. Touzi, The Root solution to the multimarginal embedding problem: an optimal
stopping and timereversal approach, to appear in Probability Theory and Related Fields. arXiv:1505.03169
[101] P. HenryLabordère,
N. Oudjane, X. Tan, N. Touzi, and X. Warin. Branching diffusion representation of
semilinear PDEs and Monte Carlo approximation, to appear in Annales de
l'Institut Henri Poincaré (B) Probabilités et Statistiques. (pdf
file).
arXiv:1603.01727 [math.PR]
[100] J.
Cvitanic, D. Possamaï and N. Touzi, Dynamic programming approach to PrincipalAgent
problems. Finance
and Stochastics, to appear. arXiv:1510.07111
[99] Z. Ren,
N. Touzi, and J. Zhang, Comparison of viscosity solutions of fully nonlinear degenerate
parabolic Pathdependent PDEs, SIAM Journal on Mathematical Analysis, 49(5),
40934116 (2017). (pdf file), arXiv: 1511.05910.
[98] A. Cosso, S. Federico, F. Gozzi, M.
Rosestolato, and N. Touzi, Pathdependent equations and viscosity solutions in infinite
dimension. Annals
of Probability, Volume 46, Number 1 (2018), 126174. (pdf file)
[97] P. HenryLabordère, X. Tan and N.
Touzi, Unbiased
simulation of stochastic differential equations. The Annals of Applied Probability Vol.
27, No. 6, 1–37 (2017). (pdf file) arXiv:1504.06107 [math.PR]
[96] M. Beiglböck, P. HenryLabordère and
N. Touzi, Monotone
Martingale Transport Plans and Skorohod Embedding. To appear in Stochastic
Processes and their Applications. Volume 127, Issue 9, Pages 30053013 (September 2017). (pdf
file)
[95] G. Guo,
X. Tan and N. Touzi, On the monotonicity principle of optimal Skorokhod embedding problem, SIAM Journal on Control and
Optimization 545 (2016), pp. 24782489. (pdf file)
[94] G. Guo, X. Tan and N. Touzi, Tightness and
duality of martingale transport on the Skorokhod space. To appear in Stochastic
Processes and their Applications, 127(3):927956 (2017).(pdf file)
[93]
G. Guo, X. Tan and N. Touzi, Optimal
Skorokhod embedding under finitelymany marginal constraints, SIAM Journal on Control and Optimization 544 (2016), pp. 21742201.(pdf file)
[92]
M. Beiglböck, M. Nutz and N. Touzi,
Complete Duality for Martingale Optimal Transport on the Line. Annals of Probability 2017, Vol. 45, No. 5, 3038–3074. DOI: 10.1214/16AOP1131.
(pdf file) arXiv:1507.00671
[91]
J. Cvitanic, D. Possamaï and N. Touzi,
Moral hazard in dynamic risk management, Management
Science, 63(10):33283346.
arXiv:1406.5852. (pdf file)
[90]
P. HenryLabordère, X. Tan and N. Touzi, An
Explicit Version of the Onedimensional Brenier’s Theorem with Full Marginals
Constraint. Stochastic Processes and their Applications, 126(9):28002834, 2016. (pdf file)
[89]
S. Kallblad, X. Tan and N. Touzi,
Optimal Skorokhod embedding given full marginals and AzémaYor peacocks. Annals of Applied Probability, 2017, Vol. 27, No. 2, 686719. (pdf file)
[88] E.
Fabre, G. Royer et N. Touzi. Liquidation of an indivisible asset with independent investment. Mathematical
Finance, Volume 28, Issue 1 January 2018, pages 153–176 (pdf
file).
[87]
P. HenryLabordère and N. Touzi, An
Explicit Martingale Version of Brenier's Theorem. Finance and
Stochastics, July 2016, Volume
20, Issue 3, pp
635–668. (pdf file)
[86]
J. Ma, Z. Ren, N. Touzi and J. Zhang,
Large Deviations for NonMarkovian Diffusions and a PathDependent Eikonal
Equation. Annales de l’IHP: Probabilités et Statistique, 11961216 (2016). (pdf file)
[85]
I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully
Nonlinear Parabolic Path Dependent PDEs: Part II, Annals of
Probability, Vol. 44, 4 (2016),
25072553. arXiv:1210.0007v2[math.PR].
[84]
D. Possamaï, M. Soner and N. Touzi, Homogenization
and asymptotics for small transaction costs: the multidimensional case. Communications in Partial Differential
Equations, vol. 40 (11), pp 20052046, 2015. arXiv:1212.6275
[83]
J. Obloj, P. Spoida, and N. Touzi, Martingale Inequalities for the Maximum
via Pathwise Arguments, Séminaires
de Probabilités XLVII, Lecture Notes in Mathematics 2137, Springer, pp 227248. (pdf file)
[82]
N. Touzi, Martingale Inequalities,
Optimal Martingale Transport, and Robust Superhedging, ESAIM:
Proceedings and Surveys, Vol. 45
(September 2014), Congrès SMAI 2013. (pdf file)
[81]
Z. Ren, N. Touzi and J. Zhang, An
Overview of Viscosity Solutions of Path Dependent PDEs, In
Stochastic Analysis and Applications 2014, in Honour of Terry Lyons, Springer
Proceedings in Mathematics and Statisctics, Volume 100, pp 397454. (pdf file)
[80]
P. HenryLabordère, J. Obloj, P. Spoida, and N. Touzi, Maximum Maximum of
Martingales given Marginals, Annals of Applied Probability, 2016, Vol. 26, No. 1, 1–44. (pdf file)
[79]
I. Ekren, N. Touzi and J. Zhang, Viscosity Solutions of Fully
Nonlinear Parabolic Path Dependent PDEs: Part I, Annals of
Probability, 2016, Vol. 44, No. 2,
12121253. arXiv:1210.0006v2[math.PR].
[78]
D. Possamaï, G. Royer and N. Touzi, On
the robust superhedging of measurable claims. Electronic
Communications in Probability,
18(95):113, arXiv: 1302.1850v2. (pdf file)
[77]
P. HenryLabordère, X. Tan and N. Touzi, A
numerical algorithm for a class of BSDE via branching process. Stochastic Processes and their Applications 124:11121140, 2014. arXiv:1302.4624 [math.NA](pdf file)
[76]
M. Soner and N. Touzi, Homogenization
and asymptotics for small transaction costs. SIAM Journal on
Control and Optimization, 51(4), 2893–2921. (2013) (pdf file)
[75]
I. Ekren, N. Touzi and J. Zhang, Optimal
Stopping under Nonlinear Expectation, Stochastic
Processes and Their Applications, 124 (2014), 32773311. arXiv:1209.6601v2[math.PR].
[74]
A. Galichon, P. HenryLabordère and N. Touzi,
A stochastic control approach to noarbitrage bounds given marginals, with an
application to Lookback options. Annals of Applied
Probability, Volume 24, Number 1
(2014), 312336. (pdf
file)
[73]
G.E. Espinosa and N. Touzi, Optimal
Investment under Relative Performance Concerns.
Mathematical
Finance, Volume 25, Issue 2 April 2015, Pages 221–257. (pdf file)
[72]
X. Tan and N. Touzi, Optimal
Transportation under Controlled Stochastic Dynamics, Annals of
Probability, 2013,
Vol. 41, No. 5, 3201–324. (pdf file)
[71]
I. Ekren, C. Keller, N. Touzi and J. Zhang, On
Viscosity Solutions of Path Dependent PDEs, Annals of
Probability, Volume 42, Number 1
(2014), 204236. (pdf file)
[70]
G.E. Espinosa and N. Touzi, Detecting
the Maximum of a MeanReverting Scalar Diffusion, SIAM
Journal on Control and Optimization,
Vol. 50, No. 5, pp. 2543–2572 (2012). (pdf file)
[69]
R. Carmona, F. Delarue, G.E. Espinosa and N. Touzi, Singular
forwardbackward stochastic differential equations and emissions derivatives. Annals of Applied Probability, Volume
23, Number 3 (2013), 10861128. (pdf file)
[68]
M. Soner, N. Touzi and J. Zhang, Dual
Formulation of Second Order Target Problems, Annals of
Applied Probability 23(1), 308347
(2013), arXiv:1003.6050. (pdf file)
[67]
D. Possamaï, M. Soner, and N. Touzi, Large
liquidity expansion of superhedging costs, Asymptotic
Analysis, to appear. (pdf file)
[66]
M. Soner, N. Touzi and J. Zhang, Quasisure
Stochastic Analysis through Aggregation, Electronic
Journal of Probability 16, 18441879 (2011), arXiv:1003.4431v1. (pdf file)
[65]
Aïd R, G. Chemla, A. Porchet A. and N. Touzi (2011),
Hedging and vertical integration in electricity markets. Management Science, Vol. 57, 8, 14381452, ISSN:00251909. (pdf file)
[64]
B. Bouchard and N. Touzi, Weak Dynamic
Programming Principle for Viscosity Solutions.
SIAM Journal
on Control and Optimization, 49, 3, 948962 (2001). (pdf file)
[63]
M. Soner, N. Touzi and J. Zhang, Wellposedness
of second order backward SDEs, Probability Theory and Related
Fields, 153, 149–190. (pdf file)
[62]
M. Soner, N. Touzi and J. Zhang, Martingale
Representation Theorem for the GExpectation, Stochastic
Processes and their Applications 121, 265287. (pdf file)
[61]
R. Aïd, O. Féron, N. Touzi and C. Vialas, An
arbitragefree interest rate model consistent with economic constraints for
longterm asset liability management, Bankers, Markets
and Investors, to appear. (pdf file)
[60]
Fahim A, N. Touzi. and X. Warin, A
Probabilistic Numerical Scheme for Fully Nonlinear PDEs, Annals of Applied Probability 21, 4, 13221364. (pdf file)
[59]
I. Ben Tahar, M. Soner and N. Touzi, Merton
problem with taxes: characterization, computation and approximation, SIAM Journal on Financial Mathematics 1, 366395, (2010). (ps file)
[58]
D. Crisan, K. Manolarakis and N. Touzi,
On the Monte Carlo simulation of Backward SDES: an improvement on the Malliavin
weights, Stochastic Processes and Their Applications 120 (2010), no. 7, 11331158. (pdf file)
[57]
B. Bouchard, R. Elie and N. Touzi, Stochastic
Target Problems with Controlled Loss. SIAM Journal on
Control and Optimization, 48, 5,
pp. 31233150. (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), 925947. (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,
23442365.
(pdf file)
[54]
U. Cetin, M. Soner and N. Touzi, Option
hedging under liquidity costs, Finance and Stochastics, 14, 317341. (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. 299330. (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 4775 (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. 269292. (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.
17791801. (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), 13991423.
(pdf file)
[48]
G. Carlier, I. Ekeland and N. Touzi, Optimal
derivatives design for meanvariance agents under adverse selection. Mathematics and Financial Economics, 1, 1 (April 2007), pp. 5780. (pdf file)
[47]
F. Astic and N. Touzi, No arbitrage
conditions and liquidity. Journal of Mathematical Economics 43
(2007), 692708. (ps file)
[46]
P. Cheridito, M. Soner, N. Touzi and Nicolas
Victoir, Second Order Backward Stochastic Differential Equations and Fully
NonLinear Parabolic PDEs. Communications in Pure and Applied
Mathematics 60 (7): 10811110 (2007). (pdf file)
[45]
E. Jouini, W. Schachermayer and N. Touzi, Law
Invariant Risk Measures have the Fatou Property. Advances in
Mathematical Economics 9, 4972 (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, 497531 (2006). (pdf file)
[43]
M. Soner and N. Touzi, Hedging under
gamma constraints by optimal stopping and facelifting. Mathematical
Finance 17, 1, 5980 (2007). (pdf file)
[42]
R. Carmona and N. Touzi, Optimal
multiple stopping and valuation of swing options.
Mathematical
Finance 18,
2 (April 2008) pp. 239268 (pdf file)
[41]
B. Bouchard, N. El Karoui and N. Touzi, Maturity
randomisation for stochastic control problems. Annals of
Applied Probability, 15, 4, 25752605 (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, 24722495 (2005). (pdf file)
[39]
P. Cheridito, M. Soner and N. Touzi,
The multidimensional superreplication problem under Gamma constraints, Annales de l’Institut Henri Poincaré, Série C:
Analyse NonLinéaire 22, 633666
(2005). (pdf file)
[38]
A. Bensoussan, N. Touzi and J.L. Menaldi, Penalty
approximation and analytical characterization of the problem of
superreplication under portfolio constraints, Asymptotic
Analysis 41, 311330 (2005). (pdf file)
[37] B. Bouchard and N. Touzi, Discretetime
approximation and Monte Carlo simulation of backward stochastic differential
equations, Stochastic Processes and their Applications, 111, 175206 (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, 4571 (2004). (pdf file)
[35]
E. Jouini, M. Meddeb and N. Touzi,
Vectorvalued measure of risk, Finance and Stochastics, 8, 531552. (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, 678717 (2004). (pdf file)
[33]
H.M. Soner and N. Touzi, The problem of
superreplication under constraints, to appear
in ParisPrinceton
Lectures in Mathematical Finance, Lecture Notes in Mathematics, SpringerVerlag.
(pdf file)
[32]
H.M. Soner and N. Touzi, Stochastic
representation of mean curvature type geometric flows, Annals of
Probability, 31, 11451165 (2003). (pdf file)
[31] H.M. Soner and N. Touzi, Level set
characterization of stochastic target problems, Communications
in PDE’s, 27, 20312053. (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, 127136 (2001).
[29]
H.M. Soner and N. Touzi, Dynamic
programming for stochastic target problems and geometric flows, Journal of the European Mathematical Society, 4, 201236 (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), 13531383 (2002).
[27] N. Touzi and N. Vieille,
Continuoustime Dynkin games with mixed strategies, SIAM
Journal on Control and Optimization
41, 10731088 (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, 404424 (2002). (pdf file)
[25] N. Touzi, Superreplication under
proportional transaction costs: from discrete to continuoustime models, Mathematical Methods of Operations Research 50, 297320 (1999).
[24] B. Bouchard and N. Touzi, Explicit
solution of the multivariate superreplication problem under transaction costs,
Annals of Applied Probability 10, 685708 (2000).
[23] N. Touzi, Direct characterization
of the value of superreplication under stochastic volatility and portfolio
constraints, Stochastic Processes and their Applications 88, 305328 (2000).
[22] H.M. Soner and N. Touzi,
Superreplication under Gamma constraint, Journal on Control
and Optimization 39, 7396 (2000).
[21] P.F. Koehl, H. Pham and N. Touzi,
On superreplication under Transaction costs in general discretetime models, Theory of Probability and its Applications 45, 783788 (1999).
[20] L. Carassus, H. Pham and N. Touzi,
Arbitrage and domination cost in a discretetime 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,
373388.
[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, 3156 (1999).
[17] N. Touzi, Optimal insurance demand
under marked point processes shocks, Annals of Applied
Probability 10, 283312 (2000).
[16] J. Cvitanic, H. Pham and N. Touzi,
Superreplication in stochastic volatility models with portfolio constraints, Journal of Applied Probability 36, 523545 (1999).
[15] J. Cvitanic, H. Pham and N. Touzi,
A closedform solution for the problem of superreplication under transaction
costs, Finance and Stochastics 3, 3554 (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, 391412 (1999).
[12] P.F. Koehl, H. Pham and N. Touzi,
Hedging in discretetime under Transaction costs and continuoustime
limit, Journal of Applied Probability 36, 163178 (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,
411422 (1999).
[9] E. Fournié, J. Lebuchoux and N. Touzi,
Small Noise Expansion and Importance Sampling, Asymptotic
Analysis 14, 361376 (1997).
[8] E. Jouini, P.F. Koehl and N. Touzi,
Incomplete Markets, Transaction costs and Liquidity effects, European Journal of Finance 4, 325348.
[7] L.P. Hansen, J.A. Scheinkman and N. Touzi,
Spectral Methods for Identifying Scalar Diffusions, Journal of
Econometrics 86, 132.
[6] J.P. Florens, E. Renault and N. Touzi,
Testing Embeddability by Stationary Reversible ContinuousTime 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, 399412 (1997).
[2] H. Pham and N. Touzi, Equilibrium
State Prices in a Stochastic Volatility Model, Mathematical
Finance 6, 215236 (1996).
[1]
E. Renault and N. Touzi, Option Hedging
and Implicit Volatilities, Mathematical Finance 6, 279302 (1996).