Inverse problems in option pricing are frequently regarded as simple and resolved if a formula of Black–Scholes type defines the forward operator. However, precisely because the structure of such problems is straightforward, they may serve as benchmark problems for studying the nature of ill-posedness and the impact of data smoothness and no arbitrage on solution properties. In this paper, we analyse the inverse problem (IP) of calibrating a purely time-dependent volatility function from a term-structure of option prices by solving an ill-posed nonlinear operator equation in spaces of continuous and power-integrable functions over a finite interval. The forward operator of the IP under consideration is decomposed into an inner linear convolution operator and an outer nonlinear Nemytskii operator given by a Black–Scholes function. The inversion of the outer operator leads to an ill-posedness effect localized at small times, whereas the inner differentiation problem is ill posed in a global manner. Several aspects of regularization and their properties are discussed. In particular, a detailed analysis of local ill-posedness and Tikhonov regularization of the complete IP including convergence rates is given in a Hilbert space setting. A brief numerical case study on synthetic data illustrates and completes the paper.