Videos & Tutorials

See it with your own eyes
In this page, you will be able to access lots of videos & tutorials that we have prepared for you. They will help you better understand

  • Fundamentals of approximation theory
  • The power of the Chebyshev techniques; the maths behind them
  • How our software library works
  • Applications of the Chebyshev framework to enhance risk calculations


Who we are and about this video series

We introduce ourselves and explain what to expect from these videos

3 minutes

The Computational Challenge

We explain the computational challenge that financial institutions are facing and introduce the solution via Chebyshev Spectral Decomposition

19 minutes

Practical applications of our technology

Presentation at Bank of America

We were invited by Bank of America in London to present our research around risk calculation optimisation via Chebyshev and Machine Learning methods.





Vitamin B, Chebyshev, Homocysteine and... Dynamic Initial Margin

Chebyshev provides the tools for an accurate and ultra-fast stochastic simulation of dynamic sensitivities and Initial Margin. Find out how in this video

30 minutes

Chebyshev for IMA-FRTB

In this talk we present results obtained within the systems of a tier-1 bank for a capital calculation within FRTB IMA, using Chebyshev tensors to massively accelerate and economise the calculation while retaining a high level of accuracy required by the regulation.

This video is similar to the one titled “Machine Learning + Chebyshev for Risk calculations”, but making more emphasis on the IMA-FRTB results at at a Tier-1 bank.





Chebyshev Tensors & Machine Learning in Dynamic Initial Margin

In this presentation we see how Chebyshev Tensors and Machine Learning techniques can be used in the calculation of Dynamic Initial Margin (DIM). We start by giving an overview of the main mathematical properties behind Chebyshev Tensors. Then we see how these can be used to approximate pricing functions within risk calculations to alleviate the huge computational burden associated with them. Finally we explain how Chebyshev Tensors can be used in the calculation of DIM and present DIM calculations obtained with Chebyshev Tensors, Deep Neural Networks and other regression types.





Our software library

Machine Learning + Chebyshev for Risk calculations

We describe how to use Chebyshev Tensors in combination with Machine Learning techniques to compute risk calculations efficiently.


This video is similar to the one titled “Computational Challenge of IMA FRTB”, but making more emphasis on the Machine Learning part of the algorithms.

Downloading the library

Here you see how easy it is to download the MoCaX library

2 minutes

Installing the library

Learn how to install our library in your system

7 minutes


See what documentation comes with your download

4 minutes

Main functionality

Here you will learn how to start using our library and unleash the power of Chebyshev in your computer. We strongly suggest you watch this video

16 minutes

Function Derivatives

See how to compute the derivatives of a function with our library

9 minutes


Learn how to serialize and deserialize MoCax Objects

4 minutes

Algebra of MoCaX Objects

You can sum MoCaX Objects, as well as multiply them by a number. This creates an algebra of Objects. Learn how to use it here

7 minutes

Singularity Points & Splines

Our library allows you to deal with singularity points and build Splines within the Chebyshev approximation framework, easily. Learn how here

14 minutes

Slicing MoCaX Objects

See how you can slice MoCaX Objects of high dimensionality into simpler ones of lower dimensionality

5 minutes


Extrusion is the operation of MoCaX Objects by which you can expand its dimensionality. Learn how to do it here

5 minutes

Higher dimensions

Most of our examples are in only one dimension for illustrative purposes. Here you see how you can easily extend all features to higher dimensions

24 minutes


With our library, you can combine several MoCaX Objects into a single one via Slide Aggregation. In this way, problems of very high dimension can be tackled. Learn how to do it here

13 minutes

Maths and Methods – fundamentals

Fundamentals of Approximation Theory and of the Chebyshev Methods - Part I

This is the first part of our 101 sessions on approximation theory and the maths behind the Chebyshev Spectral Decomposition techniques

20 minutes

Fundamentals of Approximation Theory and of the Chebyshev Methods - Part II

This is the second part of our 101 sessions on approximation theory and the maths behind the Chebyshev Spectral Decomposition techniques

24 minutes

Evaluation of Chebyshev Objects - the Barycentric Interpolation formula

We describe here the way to evaluate Chebyshev Objects with minimum CPU load and no numerical instabilities

15 minutes

Stability of the Barycentric Interpolation formula

Learn about the numerical stability of the Barycentric Interpolation formula

8 minutes

The Clenshaw algorithm

Here we explain the connection between the Clenshaw algorithm and the Barycentric Interpolation formula

4 minutes

Maths and Methods – more advanced stuff

Accuracy & Error prediction

Remarkably, we can know ex-ante the accuracy of a Chebyshev Object. Here you can see how.

21 minutes

Singularity points & Splines (method)

It is very easy to build Splines of Chebyshev Objects.

7 minutes

Function Derivatives

See how to efficiently compute function derivatives with Chebyshev Objects

17 minutes

High Dimensional Chebyshevs

In this video we discuss different ways to extend the Chebyshev framework to High Dimensions

27 minutes

Algebra of Chebyshev Objects

We can define a (most convenient) algebra of Chebyshev Objects. See how in this video.

23 minutes