## Section: Scientific Foundations

Keywords : Monte-Carlo, Euler schemes, approximation of SDE, tree methods, quantization, Malliavin calculus, finite difference.

### Numerical methods for option pricing and hedging

Participant : all the Mathfi team.

Efficient computations of prices and hedges for derivative products is a major issue for financial institutions.

Monte-Carlo simulations are widely used because of their implementation simplicity and because closed formulas are usually not available. Nevertheless, efficiency relies on difficult mathematical problems such as accurate approximation of functionals of Brownian motion (e.g. for exotic options), use of low discrepancy sequences for nonsmooth functions, quantization methods etc. Speeding up the algorithms is a constant preoccupation in the developement of Monte-Carlo simulations. Another approach is the numerical analysis of the (integro) partial differential equations which arise in finance: parabolic degenerate Kolmogorov equation, Hamilton-Jacobi-Bellman equations, variational and quasiâ€“variational inequalities (see [74] ).

This activity in the MathFi team is strongly related to the development of the Premia software.