Comparison of Basis Functions

The choice of basis functions is more a matter of practical application than mathematical accuracy. Different basis functions have different applications when it comes to portfolio management and understanding how market forces affect different components. This chapter shows the implications of different basis functions and how they can be used for trading or portfolio management and how to transform from one set of basis functions to another. It examines how one can transform Chebyshev basis functions (CBFs) to polynomial basis functions (PBFs). The orthogonal basis functions (OBFs) can be constructed by an iterative process, similar to the derivation of Chebyshev polynomials. This is accomplished by requiring that every basis function is orthogonal to all the lower order basis. The CBF volatility falls steadily for each successive component. This is another very attractive property of CBF for risk measurement and risk management.