M2MO: Modélisation Aléatoire, Finance et Data Science

Master en statistique, probabilités et finance - Université Paris 7 - Paris Diderot

Courses Core courses Derivatives modeling

Derivatives modeling

Lecturers S. Crépey 
Tutorials M.C. Quenez 
Périod Terms 1 and  2
Nombre de crédits: 3+3
Hourly Volume 

3 hours per week  + tutorials 


This course bears on the fundamentals of financial derivatives modeling, with volatility as a core underlying concept. These are  applied to market models of FX, interest rates, credit, and volatility derivatives. 

  1. Revision week: European and American options in the Cox-Ross-Rubinstein model 


1.  The probabilistic characterizations of no arbitrage and completeness (FTAPs):

in a finite setup (with full proofs) 

in continuous time (the easy parts) 


2.  Model-free results in continuous time:

Q-prices of options

Forwards and call-put parity

Call-put bound and shape constraints

The Breeden-Litzenberger formula


3. The Black-Scholes model:

  Calls and puts pricing and Greeking formulas

Dynamic hedging and the Black-Scholes equation

Black-Scholes  Implied volatility and the Black-Scholes model in practice

Binomial approximation to Black-Scholes 


4.  The local volatility model:

     Dupire equation and formula 

     Gatheral equations and formulas,  Berestycki-Busca-Florent short maturity asymptotics

     Local volatility in stochastic volatility models: Gyönfi’s formula

    Static arbitrage and Kellerer’s theorem

     Pricing with a local volatility: implicit finite differences scheme vs trinomial approximation

     Extracting the local volatility from an SVI interpolation of the implied total variance


5.  Affine models:

   The Heston model

   The Merton Model 

    Fourier pricing schemes in affine models


6.  Multivariate continuous Itô processes market model:

     Change of numéraire 

    Exchange and FX options


7. Libor Market Model (LMM) of interest-rate derivatives:

Short Rate (CIR/Vasicek) and HJM Models in a Nutshell

      Caps/floors and swaptions in the LMM

Pricing a swaption by Monte Carlo simulation in the Libor market model


8.  Local stochastic volatility models:

The SABR model

A SABR/Bergomi-Type Model of Rough Volatility

Volatility derivatives: variance and volatility swaps, VIX options


 9.  One-factor Gaussian copula model of portfolio credit risk:

  CDSs, CDOs

Implied correlation.



10.  Nonparametric model calibration techniques:

     Tikhonov regularization 

     Introduction to martingale optimal transport and applications to calibration

Case studies: (the yield curve, risk-neutral density,) local volatility, stochastic local    volatility,  the VIX/SPX joint calibration problem, calibration to path-dependent options



Main reference

Crépey, S. (2022-23), Derivatives Modeling, M2MO Lecture Notes.


Bouchard, B. and J.-F. Chassagneux (2016), Fundamentals and Advanced Techniques in Derivatives Hedging, Springer Universitext. 

Gatheral, J. (2011), The volatility surface: a practitioner's guide, Wiley.

Hull, J. (2021), Options, Futures, and Other Derivative Securities, 11e edition, Pearson [il existe une version française mais préférer la version anglaise].

Karatzas, I. and S. Shreve (1991), Brownian Motion and Stochastic Calculus, Springer Graduate Texts in Mathematics.

Lamberton, D. and Lapeyre B. (2013), Introduction au calcul stochastique appliqué à la finance.  3ème édition, Ellipses.

Shreve, S.: Stochastic Calculus for Finance II: Continuous-Time Models, Springer, 2004 or later.

Tankov, P. (2015 et 2008), Mathématiques financières et Calibration de modèles, polycopiés M2MO.