M2MO: Modélisation Aléatoire, Finance et Data Science

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

Courses Group Quantitative Finance Derivatives modeling

Derivatives modeling

Lecturers S. Crépey and S. Scotti
Tutorials S. Scotti
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. Applications to interest rates modeling, counterparty risk analysis, and the case of credit derivatives, are developed in optional courses.


0. The investment banking universe, the regulation of derivatives, and the role of mathematics, including The 2007-2008 financial crisis and its regulatory implications

I Financial derivatives

  1. Forwards and futures
  2. Introduction to options
  3.  Principles of option valuation

II. The Black-Scholes paradigm and beyond

  1. The Black-Scholes model
  2. Building the replication formula
  3. The Black-Scholes formula
  4. The Greeks
  5. Robustness of the BS formula
  6. Implied volatility
  7. Multidimensional Black-Scholes model
  8. Risk-neutral valuation

III. Local volatility

  1. Local volatility model
  2. Dupire equation and formula
  3. Gatheral formula
  4. Berestycki-Busca-Florent short maturity asymptotics

 IV. Stochastic volatility modeling

  1. Heston model
  2. General affine models
  3. Fourier transform methods
  4. Large strikes analysis
  5. Is volatility rough?

V.  Valuation of exotic options

  1. Change of numéraire
  2. Barrier options, volatility derivatives
  3. Model-free approaches: static replication, robust hedging

VI.  Interest rates modeling, an introduction

  1. Short rate models (Vacicek and CIR)
  2. The HJM approach

Bibliography: lecture notes to be provided during the course. Additional book references:

Bjork, T., Arbitrage Theory in Continuous Time, 3rd edition, Oxford University Press (2009)

Chesney M., M. Jeanblanc et M. Yor, Mathematical Methods for Financial Markets, Springer Finance Textbooks (2009).

Crépey, S., Financial Modeling, Springer Finance Textbooks (2013) [chapitres 1 à 9].

Hull, J., Options, Futures, and Other Derivative Securities, 10th edition, Pearson (2018) [il existe une version française mais préférer la version anglaise !]

Lamberton, D. and Lapeyre P., Introduction au calcul stochastique appliqué à la finance3ème édition, Ellipses (2013).

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

Prior knowledge:  Stochastic calculus and mathematical finance at a good MSc level. 

Assessment:  Written examination.