We introduce a new forward CPI model that is based on a multi-factor volatility structure and leads to SABR-like dynamics for forward inflation rates. Our approach is the first in the financial literature to reconcile zero-coupon and year-on-year quotes, granting, at the same time, a both fast and accurate calibration to market data. Explicit formulas for year-on-year caps/floors as well as for zero-coupon options are then derived in terms of the SABR volatility form. An example of calibration to market data is finally provided.