On the nature of ill-posedness of an inverse problem arising in option pricing

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