15 years of Adjoint Algorithmic Differentiation (AAD) in finance

Abstract

Following the seminal ‘Smoking Adjoint’ paper by Giles and Glasserman [Smoking adjoints: Fast monte carlo greeks. Risk, 2006, 19, 88–92], the development of Adjoint Algorithmic Differentiation (AAD) has revolutionized the way risk is computed in the financial industry. In this paper, we provide a tutorial of this technique, illustrate how it is immediately applicable for Monte Carlo and Partial Differential Equations applications, the two main numerical techniques used for option pricing, and review the most significant literature in quantitative finance of the past fifteen years.

Keywords:

• Algorithmic differentiation
• Monte Carlo simulations
• Partial differential equations
• Derivatives pricing
• Calibration of stochastic models

Acknowledgements

It is a pleasure to acknowledge a fruitful collaboration over several years with Jacky Lee, Adam Peacock, Matthew Peacock, Mark Stedman, Uwe Naumann, Alex Prideaux, Yupeng Jiang, and Andrea Macrina.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 To the best of our knowledge the acronym AAD was first used in (Capriotti 2008).

2 Provided that the state vector is a regular enough function of θ (Glasserman 2004).

3 Here for simplicity of exposition we omit the boundary term, see, e.g. Capriotti et al. (2015).

4 To keep the notation as light as possible, we denote the exact solution of the PDE (90(90) $\begin{array}{rl}& \frac{\mathrm{\partial }V}{\mathrm{\partial }t}+\mu (x,t,\theta )\frac{\mathrm{\partial }V}{\mathrm{\partial }x}\\ & +\frac{1}{2}{\sigma }^2(x,t,\theta )\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}-\nu (x,t,\theta )V=0,\end{array}$(90) ) and its finite-difference approximation with the same symbol.

5 A collection of useful results for generic matrix functions is contained in Goloubentsev et al. (2021a).