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 (μ), symmetry (δ), volatility (σ), shape (α), and scale (θ); 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 Γ(α,θ) Ornstein-Uhlenbeck type process; the associated Lévy density of the VG model is a KoBoL family of order ν=0, intensity α, and steepness parameters δσ2−δ2σ4+2θσ2 and δσ2+δ2σ4+2θσ2; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean (μ+αθδ) and variance α(θ2δ2+σ2θ). 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.