Bayesian Inference in Mathematical Finance : Jump-Diffusion Model

Bagchi, Sumit (2025) Bayesian Inference in Mathematical Finance : Jump-Diffusion Model. Masters thesis, Indian Institute of Science Education and Research Kolkata.

Text (MS Dissertation of Sumit Bagchi (20MS001)) 20MS001_Thesis_file.pdf Restricted to Repository staff only Download (715kB)

Abstract

This thesis develops a Bayesian framework for parameter estimation in jumpdiffusion models, which are widely used in mathematical finance to capture both continuous fluctuations and discontinuous jumps in asset prices. We extend Merton’s classical jump-diffusion model [4] by incorporating Gammadistributed jumps and allowing for both positive and negative jump impacts. The proposed Bayesian approach leverages conjugate priors and derives full conditional distributions for key parameters—drift (μ), volatility (σ2), and jump sensitivity (ρ)—enabling efficient posterior sampling via Markov Chain Monte Carlo (MCMC) methods. To address the challenges posed by latent jump variables, we introduce a Transdimensional Transformation-based MCMC (TTMCMC) algorithm as described in [3], which jointly estimates model parameters and jump components while maintaining computational tractability. The methodology is validated through simulations, demonstrating robust recovery of true parameter values and outperforming traditional estimation techniques. Additionally, we explore advanced inference techniques for non-Gaussian Ornstein-Uhlenbeck processes discussed in [2], highlighting their potential for stochastic volatility modeling. Our results contribute to the broader literature on financial time-series analysis by providing a flexible, computationally efficient framework for jumpdiffusion models, with applications to risk management, option pricing, and portfolio optimization. The thesis bridges theoretical rigor with practical implementation, offering insights into both methodological advancements and empirical performance.

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