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

Abstract

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 $\fracδ{σ^2} - \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$ and $\fracδ{σ^2}+ \sqrt{\frac{δ^2}{σ^4}+\frac{2}{θσ^2}}$; and the VG process converges asymptotically in distribution to a Lévy process driven by a normal distribution with mean $(μ+ αθδ)$ and variance $α(θ^2δ^2 + σ^2θ)$. 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