Chebyshev Tensors

Chebyshev Tensors are based on a key theorem by Bernstein which states that, for analytical functions*, Chebyshev Spectral projections and interpolants converge exponentially on the number of points where we know the value of the function. This result is remarkable; typically used approximation frameworks tend to converge much slowly.

This means that we can approximate any analytical function, using Chebyshev, with extreme accuracy (e.g. 1e-5) with knowing the value of the function in very few points (e.g. 10 points). Once the approximating objects are created in a Chebyshev Tensor, evaluating them is very efficient in no time.

These results are mathematically robust, so they pass all model validation and regulatory constraints.

This theory can be extended to multidimensional functions, as well as to the derivatives of the functions (i.e. its sensitivities).

This is another remarkable property of Chebyshev Tensors: the derivative of the tensor converges also exponentially to the derivative of the function being approximated.

Furthermore, Chebyshev Tensors offer a way, via Fast Fourier Transforms, to estimate the error of the approximation ex-ante.

Pricing functions are all analytic or piece-wise analytic*. As a result, they are a strong candidate to build objects that are very fast-to-compute versions of any analytical function*.

The potential limitation that Chebyshev Tensors have comes from their curse of dimensionality, which is strong (exponential). However, the combination if these tensors with Tensors Extension Algorithms and other techniques enable their sound use in several risk calculations to over 1,000 dimensions. This is explained in detail in some of our papers and book.

More resources on this fascinating topic can be found in

• Papers
• Book
• YouTube channel
• Our software library
• The excellent monograph on the subject “Approximation Theory and Approximation Practice” (SIAM) by Prof. Trefethen
• The Chebfun project
• Many very good papers by Kathryn Glau

(*) By analytic it is meant in a mathematical sense. That is, that its Taylor expansion converges to the function in every point. Sometimes in finance the term “analytic” is used in the context of computational evaluation, meaning that its evaluation can be done via a close-form formula, as opposed to numerical methods like Monte Carlo. Chebyshev Tensors work the same if the computational method is based in closed-form or via Monte Carlo.