Abstract
We provide some simple and effective correction schemes to reduce the bias arising from an estimation of volatility matrix by the nonparametric Fourier transform method, proposed by Malliavin and Mancino (2002, 2009). The dynamic volatility matrix is defined in a multivariate diffusion process. Correction schemes include a linear method and an affine method. Simulation studies demonstrate effectiveness of these correction methods. For applications, we apply the corrected Fourier transform method to three empirical studies. We find multiple time scales of volatility given different frequencies of data. The linearity between VIX square and instantaneous variance is confirmed. Moreover, we estimate default probabilities for a portfolio of S&P 500 index and CDX through the basic Monte Carlo method and an importance sampling scheme. We find that the importance sampling scheme performs better in this application.