  We observe that a European Call option with strike $L > K$ can be seen as a
Call option with strike $L-K$ on a Call option with strike $K$. Under no
arbitrage assumptions, this yields immediately that the prices of the two
contracts are the same, in full generality. We study in detail the relative
pricing function which gives the price of the Call on Call option as a function
of its underlying Call option, and provide quasi-closed formula for those new
pricing functions in the Carr-Pelts-Tehranchi family [Carr and Pelts, Duality,
Deltas, and Derivatives Pricing, 2015] and [Tehranchi, A Black-Scholes
inequality: applications and generalisations, Finance Stoch, 2020] that
includes the Black-Scholes model as a particular case. We also study the
properties of the function that maps the price normalized by the underlier,
viewed as a function of the moneyness, to the normalized relative price, which
allows us to produce several new closed formulas. In connection to the symmetry
transformation of a smile, we build a lift of the relative pricing function in
the case of an underlier that does not vanish. We finally provide some
properties of the implied volatility smiles of Calls on Calls and lifted Calls
on Calls in the Black-Scholes model.
