## Section: New Results

### American Options

Participants : A. Alfonsi, E. Chevalier, D. lamberton, B. Jourdain, A. Zanette.

The exercise boundary of American options near maturity has been studied, in the classical Black-Scholes setting with dividends, by D. Lamberton and S. Villeneuve [91] . Their results have been extended to local volatility models by E. Chevalier in his thesis [30] . Chevalier has results for multi-dimensional models as well and has obtained error estimates for the approximation of the free boundary when an American option is approximated by a Bermudean option (i.e. with a finite number of exercise dates) [29] .

Lamberton has started a collaboration with Michalis Zervos (King's College, London) on optimal stopping problems for one dimensional diffusions.

B.Jourdain and A. Zanette [50] have developped a new binomial lattice method (MSM) consistent with the Black-Scholes model in the limit of an infinite step number and such that the strike Kis equal to one of the final nodes of the tree. They have obtained asymptotic expansions for the MSM European Put price and delta which motivate the use of Richardson extrapolation. Moreover they have made a numerical comparison between the MSM approach and the best lattice based numerical methods known in literature.

It is well-known that in models with time-homogeneous local volatility functions
(
x) and constant interest and dividend rates, the European Put prices are transformed into European Call prices by the simultaneous exchanges of the interest and dividend rates and of the strike and the spot price of the underlying. Aurélien Alfonsi and Benjamin
Jourdain have investigated such a Call Put duality for perpetual American options. It turns out that the perpetual American Put price
P(
x,
y) for the spot price
xand the strike
yis equal to the perpetual American Call price in a model where, in addition to the previous exchanges between the spot price and the strike and between the interest and dividend rates, the local volatility function is modified. The equality of the dual volatility
functions only holds in the standard Black-Scholes model with constant volatility. These duality results lead to a theoretical calibration procedure of the local volatility function from the perpetual Call and Put prices for a fixed spot price
x_{0} . The knowledge of the Put (resp. Call) prices for all strikes enables us to recover the local volatility function on the interval
(0,
x_{0}) (resp.
(
x_{0}, +
) ).