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

Pricing and hedging of financial derivatives



Department: Finance


The objective of the course is to provide an overview of the main methods for pricing and hedging risks on the financial markets. After a quick overview of the methods used in discrete-time models, we will mainly study continuous-time models. We will study these issues from a theoretical, practical and numerical point of view.


    1. Financial markets in discrete times.
    Reminders on the Cox-Ross-Rubinstein model.
    Valuation and hedging of European options.
    Valuation and hedging of American options, optimal exercise strategy.
    The Black-Scholes model and its approximation to the limit.
    2. Black Scholes model
    Arbitrage, Risk Neutral Probability and Cash Change
    Evaluation and coverage of European options (probabilistic and PDE approach).
    American options (probabilistic and PDE approach).
    Calculation of Greek
    Numerical probabilistic and associated deterministic methods
    3. Volatility
    Historical and Implied Volatility
    Local volatility and Dupire's formula
    Stochastic volatility models
    Main calibration techniques
    Static and semi-static replication

    4. Market Imperfections
    Valuation by indifference of utility
    Quantity and approximate coverage
    Portfolio or liquidity constraints
    Transaction costs


- 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)