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 ($\mu$), 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 $\Gamma(\alpha, \theta)$ Ornstein-Uhlenbeck type process; the associated L\'evy density of the VG model is a KoBoL family of order $\nu=0$, intensity $\alpha$, and steepness parameters $\frac{\delta}{\sigma^2} - \sqrt{\frac{\delta^2}{\sigma^4}+\frac{2}{\theta \sigma^2}}$ and $\frac{\delta}{\sigma^2}+ \sqrt{\frac{\delta^2}{\sigma^4}+\frac{2}{\theta \sigma^2}}$; and the VG process converges asymptotically in distribution to a L\'evy 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