Section: Overall Objectives

Numerical methods for option pricing and model calibration

Simulation of stochastic differential equations

Participants : Benjamin Jourdain, Aurélien Alfonsi, Damien Lamberton, Mohamed Sbai.

Most financial models are described by SDEs. Except in very special cases, no closed-form solution is available for such equations and one has to approximate the solution via time-discretization schemes in order to compute options prices and hedges by Monte Carlo simulations. Usually this is done by using the standard explicit Euler scheme since schemes with higher order of strong convergence involve multiple stochastic integrals which are difficult to simulate. In addition, the weak order of convergence of the explicit Euler scheme can be improved by using Romberg-Richardson's extrapolations. Nevertheless, some schemes with weak order of convergence two or more have been designed recently. The idea is either to replace the multiple Brownian integrals by discrete random variables which share their moments up to a given order or to integrate ordinary differential equations associated with the vector fields giving the coefficients of the Stochastic Differential Equation up to well-chosen random time-horizons. Another interesting new direction of investigation is the design of exact simulation schemes.

Three directions of research have been investigated in the Mathfi project. First, fine properties of the Euler scheme have been studied [74] , [69] , [77] . Secondly, concerning SDEs for which the Euler scheme is not feasible, A. Alfonsi [63] have proposed and analysed new schemes respectively for Cox-Ingersoll-Ross processes and for equations with locally but not globally Lipschitz continuous coefficients. Last, the team has contributed to the new directions of research described above. For CIR processes, A. Alfonsi has designed a scheme with weak order two even for large values of the volatility parameter. Adapting exact simulation ideas, B. Jourdain and M. Sbai [76] have proposed an unbiased Monte Carlo estimator for the price of arithmetic average Asian options in the Black-Scholes model.

Monte-Carlo simulations

Participants : Benjamin Jourdain, Aurélien Alfonsi, Damien Lamberton, Mohamed Sbai, Vlad Bally, Bernard Lapeyre, Ahmed Kebaier, Céline Labart, Jérôme Lelong, Sidi-Mohamed Ould-Aly, Lokmane Abbas-Turki, Abdelkoddousse Ahida, Antonino Zanette, El Hadj Aly Dia.

Efficient computations of prices and hedges for derivative products are major issues for financial institutions (see [80] ) . Monte-Carlo simulations are widely used because of their implementation simplicity and because closed formulas are usually not available. Speeding up the algorithms is a constant preoccupation in the development of Monte-Carlo simulations. The team is mainly concerned with adaptive versions which improve the Monte-Carlo estimator by relying only on stochastic simulations.

The team has also been active on numerical methods in models with jumps and large dimensional problems.

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

Model calibration

The modeling of the so called implied volatility smile which indicates that the Black-Scholes model with constant volatility does not provide a satisfactory explanation of the prices observed in the market has led to the appearance of a large variety of extensions of this model as the local volatility models (where the stock price volatility is a deterministic function of the price level and time), stochastic volatility models, models with jump, and so on. An essential step in using any such approach is the model calibration, that is, the reconstruction of model parameters from the prices of traded options. This is an inverse problem to that of option pricing and as such, typically ill-posed.

The calibration problem is yet more complex in the interest rate markets since in this case the empirical data that can be used includes a wider variety of financial products from standard obligations to swaptions (options on swaps). The underlying model may belong to the class of short rate models like Hull-White [75] , [67] , CIR [70] , Vasicek [89] etc. or to the popular class of LIBOR (London Interbank Offered Rates) market models like BGM [68] .

The choice of a particular model depends on the financial products available for calibration as well as on the problems in which the result of the calibration will be used.

The calibration problem is of particular interest for Mathfi project because due to its high numerical complexity, it is one of the domains of mathematical finance where efficient computational algorithms are most needed.