An inverse parabolic problem arising in finance

We derive a quasilinear degenerate parabolic partial differential equation linking the implied and local volatility in the extended Black-Scholes mode l. Using this equation we establish a closed-form asymptotic formula for th e implied volatility near expiry. This in turn leads us to propose a new fo rmulation of the calibration problem near expiry which we show to be well-p osed. (C) 2000 Academie des sciences/Editions scientifiques et medicales El sevier SAS.