An application of wavelet analysis to pricing and hedging derivative securities (Q2772007)

Let \(Y\) be a martingale over a filtration generated by a Brownian motion \(W\), \((\varphi^n)_{n\in \mathbb{N}}\) an ONB of \(L^2[0,1]\) and \(\sigma_n\) the \(\sigma\)-field generated by the random variables \(\int_0^1 z(s) dW_s\) with \(z\) ranging through the linear subspace generated by \(\varphi^1, \dots, \varphi^n\). The main result of this paper is a representation in terms of \(Y\) of the martingale \(Y^n\) with final value \(Y^n_1 = E[Y_1 \mid \sigma_n]\), and also of the integrand \(y_n\) from the Itô representation of \(Y^n\). In mathematical finance, this has applications for (approximate) pricing and hedging of contingent claims in the context of the Black-Scholes model of geometric Brownian motion. The main tools used in the proof are results from wavelet analysis.