Chebyshev reduced basis function applied to option valuation

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.

A pseudospectral method for Option Pricing with Transaction Costs under Exponential Utility

“…In [4] the authors use an adaptive method with Chebyshev polynomials coupled with a dynamic programing procedure for contracts with early exercise features. A spectral procedure coupled with a reduced basis method is used in [12] to calibrate a high dimensional GARCH model. In [16] a very efficient procedure for asian options defined on arithmetic averages has been proposed.…”

A pseudospectral method for option pricing with transaction costs under exponential utility

“…In [12] a dynamic Chebyshev method is employed for pricing American options. Chebyshev interpolation is also employed for option pricing in both [10] and [11]. In [18] a very efficient procedure for Asian options defined on arithmetic averages has been proposed and in [15] a Fourier cosine method is employed to solve backward stochastic differential equations.…”