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We observe that a European Call option with strike $L > K$ can be seen as aCall option with strike $L-K$ on a Call option with strike $K$. Under noarbitrage assumptions, this yields immediately that the prices of the twocontracts are the same, in full generality. We study in detail the relativepricing function which gives the price of the Call on Call option as a functionof its underlying Call option, and provide quasi-closed formula for those newpricing functions in the Carr-Pelts-Tehranchi family [Carr and Pelts, Duality,Deltas, and Derivatives Pricing, 2015] and [Tehranchi, A Black-Scholesinequality: applications and generalisations, Finance Stoch, 2020] thatincludes the Black-Scholes model as a particular case. We also study theproperties of the function that maps the price normalized by the underlier,viewed as a function of the moneyness, to the normalized relative price, whichallows us to produce several new closed formulas. In connection to the symmetrytransformation of a smile, we build a lift of the relative pricing function inthe case of an underlier that does not vanish. We finally provide someproperties of the implied volatility smiles of Calls on Calls and lifted Callson Calls in the Black-Scholes model.

Options are also options on options: how to smile with Black-Scholes