Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process

The paper builds a Variance-Gamma (VG) model with five parameters: location (mu), symmetry (delta), volatility (sigma), shape (alpha), and scale (theta); and studies its application to the pricing of European options. The results of our analysis show that the five-parameter VG model is a stochastic volatility model with a & UGamma;(alpha,theta) Ornstein-Uhlenbeck type process; the associated Levy density of the VG model is a KoBoL family of order nu=0, intensity alpha, and steepness parameters delta/sigma(2)- root delta(2)/sigma(4 )+ 2/theta sigma(2) and delta/sigma(2 )+ root delta(2)/sigma(4 )+ 2/theta sigma(2); and the VG process converges asymptotically in distribution to a Levy process driven by a normal distribution with mean (mu+alpha theta delta) and variance alpha(theta(2)delta(2 )+ sigma(2)theta). The data used for empirical analysis were obtained by fitting the five-parameter Variance-Gamma (VG) model to the underlying distribution of the daily SPY ETF data. Regarding the application of the five-parameter VG model, the twelve-point rule Composite Newton-Cotes Quadrature and Fractional Fast Fourier (FRFT) algorithms were implemented to compute the European option price. Compared to the Black-Scholes (BS) model, empirical evidence shows that the VG option price is underpriced for out-of-the-money (OTM) options and overpriced for in-the-money (ITM) options. Both models produce almost the same option pricing results for deep out-of-the-money (OTM) and deep-in-the-money (ITM) options.