Automatic Implicit Function Theorem

Abstract

The Implicit Function Theorem, or IFT, is a powerful tool for calculating derivatives of functions that solve inverse, i.e. calibration, problems prevalent in financial applications. It is commonly believed that a degree of manual intervention is required to enable financial code to take advantage of the IFT even when using Automatic Adjoint Differentiation (AAD). In this note we explain in mathematical terms, and demonstrate on a simple example with Python code, how the Automatic IFT, a special version of the IFT, enables fully-automated differentiation of exact-fit calibration routines. We show that the Automatic IFT gives an approximate solution to nearly-exact calibration problems typical in practice, where we also derive numerical stability estimates. Furthermore, we extend the approach to the general best-fit calibration set-up.

We provide links to self-contained Python and C++/QuantLib code as working examples.

Keywords: AAD, Automatic Adjoint Differentiation, Algorithmic Differentiation, Calibration, Implicit Function Theorem, IFT, AIFT, non-linear least-squares

JEL Classification: C15, C18, C61, C63, G12, G13