## Simultaneous identification of volatility and interest rate functions-a two-parameter regularization approach

Christopher Hofmann, Bernd Hofmann, and Alois Pichler

### Abstract

This paper investigates a specific ill-posed nonlinear inverse problem that arises in financial markets. Precisely, as a benchmark problem in the context of volatility surface calibration, we consider the simultaneous recovery of implied volatility and interest rate functions over a finite time interval from corresponding call- and put-price functions for idealized continuous families of European vanilla options over the same maturity interval. We prove identifiability of the pair of functions to be identified by showing injectivity of the forward operator in $L^2$-spaces. To overcome the ill-posedness we employ a two-parameter Tikhonov regularization with heuristic parameter choice rules and demonstrate chances and limitations by means of numerical case studies using synthetic data.

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### Key words

inverse option pricing, simultaneous identification, volatility, interest rate, regularization

### AMS subject classifications

65J20, 91G60, 47H30, 47A52

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