Chebyshev Spectral Decomposition

You won't believe how powerful it is

Chebyshev Spectral Decomposition and Chebyshev Interpolants 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 theorem is outstanding; typically used approximation frameworks tend to converge linearly.

This means that we can approximate any analytical function, using Chebyshev, with extreme accuracy (e.g. 10E-5) with knowing the value of the function in very few points (e.g. 10 points). Once the approximating objects are created, evaluating them is very efficient. Typically, one million pricings can be obtained in less than a second on a standard PC.

Importantly, these results are mathematically proven, 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).

As a result, it is an ideal mathematical framework to build methods that are very fast-to-compute versions of any analytical function.

Often, the computational bottleneck of risk calculations is calling the same portfolio pricing functions lots of times; pricing functions are always analytical (or piece-wise analytical) functions… and this is where the MoCaX story commences.

Resources

If you want to research this fascinating field by yourself, we can recommend

 

  • Our research papers in this field, with applications
  • A very good monograph on this topic, “Approximation Theory and Approximation Practice”, Siam, by Prof. L.N. Trefethen (Amazon link)
  • chebfun.org, an open source project in Matlab
  • Kathrin Glau research

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