ECTS : 6
Volume horaire : 30
Description du contenu de l'enseignement :
Course outline:
I. Discrete time modelling
I.1. Financial assets
I.2. The No arbitrage condition and martingale measures (FTAP)
I.3. Pricing and hedging of European options; market completeness and 2nd FTAP
I.4. Pricing and hedging of American options (in a complete market)
II. Continuous time modelling
II.1. Financial assets as Itô processes : general theory
II.2. Markovian models : PDE pricing, delta-hedging (European options, barrier options, American options)
II.3. Local volatility models and Dupire's formula
II.4. Stochastic volatility models : how to deal with market incompleteness; (semi-)static hedging; specific models and their properties
Compétence à acquérir :
The lecture starts with discrete time models which can be viewed as a proxy for continuous settings, and for which we present in detail the theory of arbitrage pricing. We then develop on the theory of continuous time models. We start with a general Itô-type framework and then specialize to different situations: Markovian models, local and stochastic volatility models. For each of them, we discuss the valuation and the hedging of different types of options : plain Vanilla and barrier options, American options, options on realized variance, etc. Finally, we present several specific volatility models (Heston, CEV, SABR,...) and discuss their specificities.
Bibliographie, lectures recommandées :
Bouchard B. et Chassagneux J.F., Fundamentals and advanced Techniques in derivatives hedging, Springer, 2016.
Lamberton D. et B. Lapeyre, Introduction au calcul stochastique appliqué à la finance, Ellipses, Paris, 1999.