A look at future exposures, through a 19th century lens

The calculation of forward sensitivities is a relatively unexplored area of derivatives research. That is partly because it isn’t strictly required, but chiefly due to the computational load, which can only be handled with mathematical techniques based on adjoint algorithmic differentiation (AAD).

At least so far. This very contemporary problem has now been tackled using so-called Chebyshev tensors, an idea dating back to the nineteenth century. 

In their paper, Tensoring dynamic sensitivities and dynamic initial margin, Mariano Zeron and Ignacio Ruiz argue the technique can yield results comparable to AAD. 

“What we propose in the paper is an easy-to-implement method to accurately and efficiently simulate forward sensitivities. We are not aware of anyone doing so outside of an AAD implementation,” says Ruiz.

An efficient technique for calculating forward sensitivities opens the way to several valuable applications, such as the computation of forward value-at-risk and margin valuation adjustments. The simulation of forward market risk and counterparty risk capital, which are currently not even considered, may also be possible with Chebyshev tensors.

The paper singles out one application in particular, the computation of dynamic initial margin, as a prime example of what can be achieved when the technique is applied to foreign exchange swaps and European spread options.

Chebyshev tensors are used to replicate functions – in this case, sensitivities – with polynomials. They work by interpolating a parsimonious number of points, the roots of the so-called Chebyshev polynomials, to approximate the given function, provided this holds certain properties.

The Chebyshev polynomials were discovered by the Russian mathematician Pafnuty Chebyshev more than a hundred years ago. Back then, in the pre-electricity era, Chebyshev’s discovery had no obvious practical applications.

“The interest at the time was purely theoretical and directed at finding whether continuous functions could be approximated by polynomials” says Zeron. “This and other numerical techniques developed at the end of the 19th century got lost and, because of the unfair bad reputation of polynomial interpolation methods, they have not resurfaced until very recently.”

What we propose in the paper is an easy-to-implement method to accurately and efficiently simulate forward sensitivities. We are not aware of anyone doing so outside of an AAD implementation

Ignacio Ruiz, MoCaX Intelligence

Zeron and Ruiz have spent about six years developing their approach at MoCaX Intelligence, a consultancy specialising in risk engine analytics. They say one of the advantages of using Chebyshev tensors is that it does not require an overhaul of banks’ data structure and model libraries.

“To implement AAD one needs to redevelop the whole pricing engine. It’s very challenging to run it on an existing infrastructure that has not been set up specifically for AAD,” says Ruiz. “Chebyshev tensors can be applied easily on existing risk engines and the performance is in practice comparable in terms of accuracy and computations cost.”

“It is a very simple technique,” adds Zeron. “Conceptually, understanding what ones needs to do is straightforward. The difficulty of implementing it within an existing risk system depends on the set-up of the engine.”

So, are Chebyshev tensors set to take centre stage in sensitivities computation? Not so fast, says Brian Huge, chief analyst at Danske Bank, who recently proposed a solution to this problem with his colleague Antoine Savine that combines AAD and neural networks.

Huge acknowledges that the rediscovery of old theories such as Chebyshev tensors can lead to new solutions to modern-day problems.

“The Chebyshev interpolation is a very interesting mathematical instrument,” he says. “When the interpolating points are placed in an optimal way, the approximation can be really good.” 

But he is not yet sold on its use for the computation of forward sensitivities.

“The authors use a smart technique to reduce the dimensionality of the problem. But despite that, I’m still sceptical that the idea can be used for this particular purpose, because the computation times are not yet satisfactory.”

Zeron and Ruiz are adamant their tests prove the technique is ready to be put into production. It shouldn’t take another century to find out if they’re right.