From Model to Market Risks: The Implicit Function Theorem (IFT) Demystified

Abstract

One persisting conundrum in the theory and practice of quantitative risk management models is the relationship of model risks (the risk sensitivities of a transaction or set of transactions to the parameters of the model, for example, in a Dupire (1992) model, the local volatility surface) and market risks (the sensitivities to the market variables, for example, the implied volatility surface). Model parameters are typically calibrated to market variables, sometimes analytically (see Dupire’s formula expressing a local volatility as a function of the implied volatilities) but mostly numerically, where the model parameters are iteratively set to minimize the (generally, squared) error to market instruments. In machine learning lingo, the model learns its parameters by calibration to market instruments (underlying assets and European options) and applies them to off-market instruments (exotics). The value of a transaction is an explicit (although, most of the time, numerical) function of the model parameters, so the model risks are easily computed by finite difference or AAD (automatic adjoint differentiation, see for instance Savine’s textbook, Wiley, 2018). It is however, in general, not advisable, for speed, memory footprint or accuracy, to proceed in the same manner for market risks. To differentiate through a numerical calibration is an inefficient process that may lead to unstable or wrong results. The solution, presented by a few authors (Giles, Capriotti, Naumann, Huge-Flyger-Savine), is the application of a multi-dimensional version of the implicit function theorem (IFT) to deduce market risks from model risks. This document explains, and hopefully demystifies the application of the IFT in this contexts, and outlines the steps of an efficient and accurate algorithm for the determination of the market risks, in particular in the context of AAD.

Keywords: Antoine Savine, Automatic Differentiation, AAD, Risk Management, Market Risk, Quantitative Finance, Volatility, Finance, Options