Corrections to Fourier Transform Method for Nonparametric Estimation of Volatility with Applications in Risk Management

Abstract

This thesis consists of two parts. In the first part, we aim to estimate parameters arising from stochastic volatility models by means of the nonparametric Fourier transform method (Malliavin and Mancino, 2002, 2009). Under the assumption that data satisfy the continuous semimartingale property, this Fourier transform method is based on integration of the time series rather than on their differentiation. Due to some boundary deficiency in numerical approximation (Reno, 2008), we propose some correction methods including model-free and model-dependent approaches to the Fourier estimation. In the second part, the Fourier transform method is applied to VaR (Value at Risk) and CVaR (Conditional Value at Risk) estimation under stochastic volatility models. Through Monte Carlo simulations with importance sampling, we test the performance of VaR with our corrected Fourier transform method using some foreign exchange and the S&P 500 index data. We find that our corrected Fourier transform method under stochastic volatility models outperforms other VaR measurements from historical simulation, RiskMetrics, and GARCH(1,1) model.