of option prices, hedging, calibration, etc... , describes the price with an horizon of months or years.

In contrast, a very active topic now is concerned with models for tick prices or order books. The

structure of the price at the microscopic level is very di_erent from the structure of the usual (often

continuous) semimartingales used at a macroscopic level. In particular the microscopic prices evolves

on the tick grid, usually going up or down by one tick only. Our aim is to see how it is possible to

reconcile the two viewpoints, using a scaling limit of tick-level price models. We will see that this

question (going back to the thesis of Bachelier, in a sense) raises a number of non trivial questions

if we want a reasonably simple microscopic model, together with a macroscopic model exhibiting

stochastic volatility or jumps or a drift.

(Joint work with Yacine Ait-Sahalia).