## Team : mathfi

## Section: Scientific Foundations

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

## Numerical methods for option pricing and hedging and calibration of financial assets models

**Participants:**V. Bally, E. Clément, B. Jourdain, A. Kohatsu Higa, D. Lamberton, B. Lapeyre, J. Printems, D. Pommier, A. Sulem, E. Temam, A. Zanette.

Efficient computations of prices and hedges for derivative products is a major issue for financial institutions. Although this research activity exists for more that fifteen years at both academy and bank levels, it remains a lot of challenging questions especially for exotic products pricing on interest rates and portfolio optimization with constraints.

This activity in the Mathfi team is strongly related to the development of the Premia software. It also motivates theoretical researches both on Monte–Carlo methods and numerical analysis of (integro) partial differential equations : Kolmogorov equation, Hamilton-Jacobi-Bellman equations, variational and quasi–variational inequalities (see [75].

### MonteCarlo methods.

The main issues concern numerical pricing and hedging of European and American derivatives and sensibility analysis. Financial modelling is generally based on diffusion processes of large dimension (greater than 10), often degenerate or on Lévy processes. Therefore, efficient numerical methods are required. Monte-Carlo simulations are widely used because of their implementation simplicity. Nevertheless, efficiency issues rely on tricky mathematical problems such as accurate approximation of functionals of Brownian motion (e.g. for exotic options), use of low discrepancy sequences for nonsmooth functions .... Speeding up the algorithms is a major issue in the developement of MonteCarlo simulation (see the thesis of A. Kbaier). We develop Montecarlo algorithms based on quantization trees and Malliavin calculus. V. Bally, G. Pagès and J. Printems have developed quantization methods especially for the computation of American options [53][52], [2], [16][26]. Recently, G. Pagès and J. Printems showed that functional quantization may be efficient for path-dependent options (such that Asian options) and for European options in some stochastic volatility models [32].

### Approximation of stochastic differential equations.

In the diffusion models, the implementation of Monte-Carlo methods generally
requires the approximation of a stochastic differential equation,
the most common being the Euler scheme. The error can then be controlled
either by the L_{P}-norm or the probability transitions.

### PDE-based methods.

We are concerned with the numerical analysis of degenerate parabolic partial differential equations, variational and quasivariational inequalities, Hamilton-Jacobi-Bellman equations especially in the case when the discrete maximum principle is not valid and in the case of an integral term coming from possible jumps in the dynamics of the underlying processes. In large dimension, we start to investigate sparse grid methods.

### Model calibration.

While option pricing theory deals with valuation of derivative instruments given a stochastic process for the underlying asset, model calibration is about identifying the (unknown) stochastic process of the underlying asset given information about prices of options. It is generally an ill-posed inverse problem which leads to optimisation under constraints.