American options: theory and numerical methods

V. Lemaire

This course takes place from January to February, 3 hours per week.

American options and optimal stopping theory

  • Optimal stopping in continuous time (regular case): reminders on essential supremums, surmartingales and martingales in continuous time (regularization, Doob-Meyer decomposition…). Snell envelope, characterization of optimal stopping times, smallest and largest optimal stopping time, dual formulation of the Snell envelope.
  • Valuation of American options in a complete market (Brownian market with multidimensional assets in the form of Itô processes): Link to continuous time optimal stopping, replication portfolio, hedging strategy.
  • Dual formulations (Rogers 2002; Haugh-Kogan 2002; Jamshidian 2005).
  • Analytical study of the American option price in the framework of the Black-Scholes model: continuity property, monotonicity, convexity, variational inequalities, free frontier, semi-closed formula, smooth-fit.
  • Study of examples.

Numerical methods (elements)

Description and brief analysis of some numerical methods of valuation and hedging for American options via Bermudan approximations.

  • Iteration on value functions: non-parametric regression (Carrière 1996), random mesh (Broadie-Glasserman 1997), optimal quantization (Bally-Pagès 2001), Malliavin calculation (Lions-Régnier 2001).
  • Iteration on stopping times: approximation of the continuation value by $L^2$ projection (Longstaff-Schwartz 2001).
  • Computation of coverages: flow method (Piterbarg 2002), projection/regression methods.