ECTS : 6
Description du contenu de l'enseignement :
Generalities on Monte-Carlo methods
1. Generalities on the convergence of moment estimators
2. Generators of uniform law
3. Simulation of other laws (rejection method, transformation, …)
4. Low discrepancy sequences
Variance reduction
1. Antithetical control
2. Payoff regularization
3. Control Variable
4. Importance sampling
Process simulation and payoff discretization
1. Black-Scholes model
2. Discretisation of SDEs
3. Diffusion’s bridges and applications to Asian, barrier and lookback options.
Calculation of sensitivities (greeks)
1. Finite differences
2. Greeks in the Black-Scholes model
3. Tangent process and Greeks
4. Malliavin calculus, Greeks, conditional expectations and pricing of American options
Calculation of conditional expectations and valuation of American options.
1. Nested Monte Carlo approach
2. Regression Methods (Tsitsiklis Van Roy, Longstaff Schwartz)
3. Rogers’ Duality
Finite difference methods: the linear case
1. Construction of classical schemes (explicit, implicit, theta-scheme)
2. Conditions for stability and convergence
Finite difference methods: the non-linear case
1. Monotonous schemes: General conditions of stability and convergence
2. Examples of numerical schemes: variational problems, Hamilton-Jacobi-Bellman equations.
Compétence à acquérir :
This course provides an in-depth presentation of the main techniques for the evaluating of options using Monte Carlo techniques.