Chebyshev Reduced Basis Function applied to Option Valuation

TL;DR

This paper introduces a Chebyshev-based reduced basis method for efficient, accurate valuation of complex financial derivatives with many variables, significantly speeding up computations in model calibration.

Contribution

It presents a novel polynomial interpolation approach using hierarchical orthogonalization to reduce degrees of freedom in high-dimensional derivative pricing models.

Findings

Enables fast valuation of large sets of contracts across parameters.

Achieves accurate results with reduced computational time.

Applicable to various models beyond the GARCH example.

Abstract

We present a numerical method for the frequent pricing of financial derivatives that depends on a large number of variables. The method is based on the construction of a polynomial basis to interpolate the value function of the problem by means of a hierarchical orthogonalization process that allows to reduce the number of degrees of freedom needed to have an accurate representation of the value function. In the paper we consider, as an example, a GARCH model that depends on eight parameters and show that a very large number of contracts for different maturities and asset and parameters values can be valued in a small computational time with the proposed procedure. In particular the method is applied to the problem of model calibration. The method is easily generalizable to be used with other models or problems.