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
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
– 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)