Computational Challenges for Value-at-Risk and Expected Shortfall: Chebyshev Interpolation to the Rescue?

Abstract

Computational challenges associated with calculating risk measures are inherent to many applications in financial institutions. An example is the need to revalue portfolios of trading positions hundreds or thousands of times to determine the future distribution of their present values. Established techniques to address this task are pricing approximations by means of Taylor expansion and the pre-calculation of grids of portfolio values, as a basis for interpolation. In recent years, the use of “smart grids” based on Chebyshev interpolation has been popularised. This paper reports on an exploratory study in which Chebyshev grids are compared to standard (uniform) grids as well as Taylor expansion when applied to computing a portfolio’s Value-at-Risk and Expected Shortfall. Using a variety of example portfolios it is illustrated that Chebyshev interpolation tends to perform better than standard grids but that it is also subject to similar drawbacks, such as difficult error control and the curse of dimensionality. In spite of advantageous properties Chebyshev grids are hence not yet seen to lead to a quantum leap in determining risk measures. Ongoing research focussing on sparse grids and their approximation quality, however, remains promising.

Keywords: Risk Measurement, Market Risk, Value-at-Risk, Expected Shortfall, Interpolation, Chebyshev

JEL Classification: C10, G13, G18