ENSAE Paris - École d'ingénieurs pour l'économie, la data science, la finance et l'actuariat

Pricing and hedging of financial derivatives



Département : Finance


The goal of this course is to review the most important methods for pricing and hedging derivatives in financial markets, from theoretical, practical and numerical point of view. The focus of the course will be on continuous time models for a single asset, more complex settings like credit derivatives and interest rate models being the topic of other courses. We shall also discuss the role of quantitative models in finance, and the basic principles of model selection, calibration and validation. We shall also discuss the numerical implementation of the proposed models, using Jupyter Notebooks. 

The evaluation will be based on a written exam


1. Introduction 
   - Derivative markets and products
   - Role of quantitative models in investment banks and finance in general
   - Lessons from the 2009 subprime crisis
   - The principles of derivative pricing  
2.  Black-Scholes option pricing model
   - Multidimensional Black-Scholes-Samuelson model
   - Risk-neutral pricing of contingent claims
   - PDE approach to option pricing 
   - Black-Scholes formula and the greeks. Implied volatility
   - Unhedged risks in the Black-Scholes model: discretization error,
   transaction costs, stochastic volatility
   - Change of num'eraire and generalized Black-Scholes formula
   - Numerical methods for option pricing and hedging: finite difference methods
   - Numerical methods for option pricing and hedging: Monte Carlo methods
3. Volatility
   - Local volatility and Dupire's formula
   - Volatility derivatives and model-free pricing and hedging 
   - Stochastic volatility and stochastic variance curve models


- Detailed lecture notes will be provided to students

As a general reference for the course, we recommend

Lamberton, D. and B. Lapeyre, Introduction to stochastic calculus applied to finance, 2nd edition, Chapman and Hall / CRC (2008)

As a reference more focused on volatility models, we recommend

Gatheral, J., Volatility surface: a practitioner's guide, John Wiley & Sons (2011)