Ill-posedness versus ill-conditioning–an example from inverse option pricing

Abstract

The article considers a problem of inverse option pricing aimed at the identification of a not directly observable time-dependent volatility function from maturity-dependent option prices. In this situation, an important aspect is the calibration of the antiderivative of the squared volatility. This inverse problem leads to an operator equation with a forward operator of Nemytskii type generated by a monotone function of two variables. In recent literature, an analysis of this forward operator and several numerical case studies have been conducted which revealed certain instability effects. This article supplements these results by studying the nature of these instabilities. In this context, the focus is on the question whether the problem is well-posed or ill-posed, i.e. whether the inverse operator is continuous or not continuous in suitable Banach spaces. As the mentioned instabilities result in strongly oscillating approximate solutions, we finish by presenting a numerically effective algorithm which uses a priori information about the monotonicity of the searched antiderivative to compute a smooth approximate solution.