Structural Breakpoints in Volatility in International Markets

Unlike Fourier series, wavelets are suitable building-block functions for signals
whose features change over time, and for non-smooth signals. A wavelet allows for
decomposing a signal into multi-resolution components: fine and coarse resolution
components.

There are father wavelets φ and mother wavelets ψ such that

∫ φ(t)dt = 1           ∫ ψ(t)dt = 0                                                (1)

Father wavelets are good at representing the smooth and low-frequency parts of a
signal, whereas mother wavelets are good at representing the detailed and high-frequency
parts of a signal. The most commonly used wavelets are the orthogonal ones (i.e., haar,
daublets, symmelets, and coiflets). In particular, the orthogonal wavelet series
approximation to a continuous signal f(t) is given by

f(t)≈∑s_J,kφ_J,k(t)+∑d_J,kψ_J,k(t)+∑d_J-1,kψ_J-1,k(t)+...+∑d_1,kψ_1,k(t)        (2)

kk   k     k

where J is the number of multi-resolution components or scales, and k ranges from 1 to the
number of coefficients in the corresponding component. The coefficients sJ,k, dJ,k,..., d1,k are
the wavelet transform coefficients, whereas the functions φj,k(t) and ψj,k(t) are the
approximating wavelet functions. These functions are generated from φ and ψ as follows

φj,k(t) = 2-^j'² φf ^t-2k )           ψ_ik(t) = 2^-j/² ψ[^ ^t--j^k 1                 (3)

The wavelet coefficients can be approximated by the following integrals
sJ,k≈∫φJ,k(t)f(t)dt           dj,k ≈∫ψj,k(t)f(t)dt, j=1, 2,..., J                      (4)

These coefficients are a measure of the contribution of the corresponding wavelet
function to the total signal. On the other hand, the approximating wavelet functions φj,k(t)

<<   3   4   5   6   7   8   9   >>

More intriguing information