A new ADI framework for oil option pricing with cubic B-spline and Gauss-Hermite integration

Abstract

This paper investigates the pricing of European options within the commodity market, specifically focusing on the dynamics of oil spot prices. We employ a long memory stochastic volatility model to effectively capture and predict the behavior of oil prices. Initially, we utilize statistical tests such as the Augmented Dickey-Fuller (ADF) test and other relevant techniques to extract essential characteristics from oil spot price data, thereby establishing a foundation for our analysis. To model these features, we propose a two-factor financial model grounded in a mixed fractional Brownian motion process. This approach is crucial, as it recognizes that the increments of the process exhibit dependence and retain long memory characteristics for values of H within the range $(\frac{3}{4},1)$. However, this complexity poses challenges in deriving an explicit formula for forward contract pricing in commodity markets due to the inherent non-normal behavior of the underlying asset. To address this issue, we implement an advanced Monte Carlo method to approximate the prices of the contracts. This approach enables us to simulate a large number of potential price paths for oil, thereby facilitating more accurate pricing under the observed stochastic volatility conditions. Additionally, we incorporate stochastic calculus techniques alongside a delta hedging strategy to derive the partial differential equation (PDE) governing the option price. To solve this equation, we employ the Alternating Direction Implicit (ADI) method, known for its efficiency and stability in managing high-dimensional problems. Unlike explicit methods, which can suffer from stability issues and require very small time steps, the ADI method allows for larger time steps while maintaining stability, making it computationally efficient. This is particularly important in option pricing, where the financial markets can exhibit rapid changes. By leveraging these methodologies, the project aims to enhance the understanding of option pricing dynamics in the commodity market, providing valuable insights for traders and financial analysts involved in oil market operations.