Paul Gassiat

Maître de Conférence, Université Paris-Dauphine

Evaluation de dérivés et calcul stochastique 2

Les objectifs du cours

The aim of this lecture is to present the theory of derivative asset pricing as well as the main models and techniques used in practice. The lecture starts with discrete time models which can be viewed as a proxy for continuous settings. 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, constant volatility models, local and stochastic volatility models. For each of them, we discuss their calibration, and the valuation and the hedging of different types of options (plain Vanilla and barrier options, contracts on future, American options, options on foreign markets, options on realized variance,...).

Plan du cours

I. Introduction to derivatives markets and products.

II. Discrete time modelling

II.1. Financial assets

  • General setting
  • Tree markets

II.2. The absence of arbitrage

II.3. Pricing and hedging of European options

  • The super-hedging problem
  • The complete market case : example of the CRR model
  • Approximate hedging in incomplete markets and selection of a pricing measure (example of the trinomial tree model)

II.4. Pricing and hedging of American options

II.5. Discrete time models as a proxy for continuous time models

II.6. Introduction and discussion of some imperfections on the market.

III. Continuous time modelling

III.1. Financial assets as Itô processes

  • Discussion of the Absence of arbitrage opportunity
  • Complete and incomplete markets
  • The general pricing and hedging principle for European and American claims
  • Impact of the dividends and non-constant interest rates - forward risk neutral measure)

III.2. Markovian models in complete markets

  • PDE valuation (plain vanilla, barrier, Asian, American options
  • Greeks and hedging
  • Tracking error and convexity

III.3. Model studies

  • Closed form prices in the Black and Scholes type model(Plain Vanilla options, Barrier options, Futures in the Black model, Options on foreign markets and currency derivatives
  • Merton's model
  • Local volatily models (Dupire Formula and calibration, CEV model and FFT valuation method)
  • Stochastic volatility (Completion of the market with options : general principal, Approximate static hedging : example of the variance swap hedging problem, The Heston model, SABR model, Jumps in the volatility (Bates model), Variance swap market models and options on realized variance)

Bibliographie

Cont R. et P. Tankov, Financial Modelling with Jump Processes, Chapman and Hall, 2004.

Hull J., Options, Futures and other Derivatives Securities, Prentic-Hall, 2002.

Lamberton D. et B. Lapeyre, Introduction au calcul stochastique appliqué à la finance, Ellipses, Paris, 1999.

Musiela M. et M. Rutkowski, Martingale methods in financial modelling, Springer, 1997.

Examen

1 final exam.