Exploring non-analytical affine jump-diffusion models for path-dependent interest rate derivatives

Abstract

This study introduces an adapted Fourier-cosine series (COS) method that focuses on numerically solving characteristic functions linked to interest rate processes. The adaptation extends to encompass models within the affine jump-diffusion niche to assess the impact of different probability distributions on path-dependent option prices, with emphasis on the influence of stochastic volatility models on skewness and kurtosis. This study leverages the COS method, modified to numerically address characteristic functions linked to interest rate processes, to calculate the price of path-dependent derivatives. It investigates diverse models within the affine jump-diffusion framework, encompassing elements such as stochastic volatility, jumps, and correlated Brownian motion. An innovative approach is introduced, wherein the characteristic function is generated from the integral of the interest rate, as opposed to the interest rate itself. The research generated notable findings, highlighting the adaptability and effectiveness of the modified COS method. This significantly expands the range of applicable models for those with analytically unsolved characteristic functions. Remarkably, even in cases with analytically solvable characteristic functions, an unexpectedly low number of terms can accurately priced options. This study introduces original contributions by adapting the COS method to address the characteristic functions associated with interest rate processes. The distinct approach of generating the characteristic function from the interest rate integral, rather than the interest rate itself, is a substantial original contribution. The application of Kibble’s bivariate gamma probability distribution to correlate interest rates and volatility jump sizes further enhances the originality of this research.