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Abstract
An alternative to Duan's formulation of the approximating NGARCH(1,1) model is proposed and justified. The diffusion limit of this new model is obtained and the asymptotic limit of fast mean reversion is examined using the procedure first performed by Fouque, Papanicolaou and Sircar on the Ornstein-Uhlenbeck process. With this, the leading term behaviour of the long-maturity asymptotics of the implied volatility surface can be obtained. Inversely, it is hypothesized that with additional correction terms the GARCH parameters can be obtained from data-generated implied volatility surfaces.
For the purpose of comparison with its diffusion limit, it is necessary to generate implied volatility surfaces for NGARCH(1,1). Previously, this was possible only with slow-converging Monte Carlo methods. A computationally efficient scheme that applies the FFT to a discrete Markov chain approximation of NGARCH(1,1) is presented. The results suggest that a spectral analysis of NGARCH(1,1) would be advantageous.
