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 δ σ 2 − δ 2 σ 4 + 2 θ σ 2 and δ σ 2 + δ 2 σ 4 + 2 θ σ 2 ; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean ( μ + α θ δ ) and variance α ( θ 2 δ 2 + σ 2 θ ) . 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.