Unreasonably Effective Function Approximation With Novel Use of Derivatives

5 minute video overview recorded for Bloomberg’s BBQ seminar 28th May 2020

Brian Huge and I just posted a working paper following six months of research and development on function approximation by artificial intelligence (AI) in Danske Bank. One major finding was that training machine learning (ML) models for regression (i.e. prediction of values, not classes) may be massively improved when the gradients of training labels wrt training inputs are available. Given those differential labels, we can write simple, yet unreasonably effective training algorithms, capable of learning accurate function approximations with remarkable speed and accuracy from small datasets, in a…

Video Tutorial with Application in Machine Learning and Finance

I have given many lectures and professional presentations on automatic differentiation (AAD) in the past five years. I even wrote a book about it. What many students, academics, and professionals found most useful is a 15min tutorial recorded in Bloomberg’s London offices in November 2019.

The challenge was to explain the main ideas of AAD and its application in machine learning and finance in less than 1/4 hour. I wanted to give more than an overview, an actual peek at the key mathematical and technical notions.

I somehow managed to cover the main notions in such a short time:

It is considered best practice in financial Monte-Carlo simulations to apply quasi-random numbers generated by Sobol’s algorithm in place of (pseudo-)random numbers. Sobol numbers offer a lower discrepancy (they fill the space of possibilities more evenly) resulting in a faster convergence and more stable estimates of financial product values and risk sensitivities.

left: random, right: Sobol — Modern Computational Finance (Antoine Savine, Wiley, 2018), page 205

The sequence was invented in USSR in 1967 and its computation was massively improved in 1979 by Antonov and Saleev, resulting in a remarkably efficient generation of the numbers with just a few low level bit-wise operations.

Sobol’s sequence was published in Press and al’s famous Numerical Recipes…

Jesper Andreasen and I (Antoine Savine) are writing a book on scripting and applications to risk management and counterparty valuation management (CVA) and, more generally, regulatory calculations like FRTB (fundamental review of the trading book) or CCR (counterparty credit risk):


As an introduction, Jesper compiled a personal history of working with scripting, reproduced below.

I also posted a short presentation with some of the main themes covered in the book:


My life in scripting, by Jesper Andreasen

Antoine came to General Re Financial Products in London in 1998 with a lot of youthful spirit and many refreshing ideas…

I finally put all my years of teaching and professionally developing generic, parallel financial Monte-Carlo libraries and automatic adjoint differentiation (AAD) in a book published by Wiley on November 13th, 2018:


I was fortunate to get excellent early reviews by leading academics and practitioners like Bruno Dupire, Paul Glasserman or Vladimir Piterbarg. Read them here: https://www.amazon.com/Modern-Computational-Finance-Parallel-Simulations-dp-1119539455/dp/1119539455

The book explains in deep detail the key technologies Jesper Andreasen, my colleagues in Danske Bank and myself implemented to earn the In-House System of the Year 2015 Risk award.

It comes with a complete, professional C++ library freely available on GitHub: watch the…

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)…

Do you have what it takes to be a quant? This article on quora lists two of the most commonly asked interview questions and answers in mathematical finance (quants also need increasingly strong computational skills, which this post does not address).

The first question (Feynman-Kac’s theorem) focuses on knowledge and the manipulation of the mathematics that quants apply every day. The skills required to answer are the same needed to follow financial modeling literature and conferences. The second question (model-free valuation of binary options) focuses on agility and reasoning, and tests the skills required to apply mathematics to resolve modeling problems.

Antoine Savine

volatility, correlation and principal deformations, historical and model implied (historical calibration here)

This introductory presentation of Interest Rate Models, posted on slideShare, is designed for a professional audience of quantitative analysts, developers, traders and risk managers. It introduces the fundamental concepts of arbitrage, risk premium and risk factors and focuses on the Heath-Jarrow-Morton approach and the Markov (Cheyette) family of models.

The lectures were given internally in Danske Bank in Q1 2018 and received an excellent feedback from the audience. It is hoped that other professionals may find the notes useful. While the lectures don’t shy away from the mathematics, they are meant as an introduction. …

To export calculation C++ code to Excel is extremely practical. Excel, with all its flexibility and convenience, basically becomes the front end of your C++ application for free. Unfortunately, this is not a particularly intuitive and painless process. Books have been written on the topic, including Dalton’s excellent 600 pager:

Financial Applications using Excel Development

For the benefit of my students and colleagues, I wrote a quick and painless tutorial:

See my tutorial and download necessary files here

I have had excellent feedback from people who found the tutorial particularly helpful, and decided to publish it for everybody’s benefit.

Comments and suggestions are most welcome.

Antoine Savine

Antoine Savine

Quantitative Finance Practitioner (Danske Bank), Academic (Copenhagen University) and Author (Modern Computational Finance, Wiley)

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