Automatic Implicit Function Theorem

14 Pages Posted: 15 Dec 2021 Last revised: 31 May 2022

See all articles by Dmitri Goloubentsev

Dmitri Goloubentsev


Evgeny Lakshtanov

Matlogica Limited; Universidade de Aveiro

Vladimir Piterbarg

NatWest Markets; Imperial College London

Date Written: December 14, 2021


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

Suggested Citation

Goloubentsev, Dmitri and Lakshtanov, Evgeny and Piterbarg, Vladimir, Automatic Implicit Function Theorem (December 14, 2021). Available at SSRN: or

Dmitri Goloubentsev

Matlogica ( email )

411 Oxford Street, Office 1.01
London, W1C 2PE
United Kingdom

Evgeny Lakshtanov

Matlogica Limited ( email )

411 Oxford Street, Office 1.01
London, W1C 2PE
United Kingdom

Universidade de Aveiro ( email )

Rua Associação Humanitária Bombeiros de Aveiro
Aveiro, 3800

Vladimir Piterbarg (Contact Author)

NatWest Markets ( email )

250 Bishopsgate
London, EC2M 4AA
United Kingdom

Imperial College London ( email )

South Kensington Campus
Imperial College
United Kingdom

Do you have negative results from your research you’d like to share?

Paper statistics

Abstract Views
PlumX Metrics