INDEX OPTIONS AND VOLATILITY DERIVATIVES IN A GAUSSIAN RANDOM FIELD RISK-NEUTRAL DENSITY MODEL

Abstract

We propose a risk-neutral forward density model using Gaussian random fields to capture different aspects of market information from European options and volatility derivatives of a market index. The well-structured model is built in the framework of the Heath–Jarrow–Morton philosophy and the Musiela parametrization with a user-friendly arbitrage-free condition. It reduces to the popular geometric Brownian motion model for the spot price of the market index and can be intuitively visualized to have a better view of the market trend. In addition, we develop theorems to show how the model drives local volatility and variance swap rates. Hence, volatility futures and options can be priced taking the forward density implied by European options as the initialization input. The model can be accordingly calibrated to the market prices of these volatility derivatives. An efficient algorithm is developed for both simulating and pricing, and a numerical study is conducted using real market data.