Analysis and Application of Jump-Diffusion Processes in Financial Models

Specialeforsvar: Emil Ellegaard

Titel: Analysis and Application of Jump-Diffusion Processes in Financial Models
 A Comparative Study of the Black-Scholes, Merton, Heston, & Bates models

Abstract: This thesis investigates the application of jump diffusion models in option pricing, with a primary focus on the Merton and Bates models. These models extend the classical Black-Scholes and Heston framework by incorporating jumps, providing a more comprehensive representation of asset price dynamics and addressing the empirical observation of fat tails in return distributions.
Theoretical derivations for the Merton and Bates models are presented, including detailed discussions of their underlying assumptions and mathematical formulations. Using historical S&P-500 data, the models are calibrated under different loss functions—SSE, MAE, Huber, and Log-loss and under different weights, using bid-ask spreads. The models are analyzed from multiple different angles including calibration-stability and implied volatility.
The results are compared to findings in the existing literature. Notably, the analysis highlights the effects when including a jump-term and the challenges in fitting stochastic volatility parameters and underscores the impact of loss function selection on model calibration. Furthermore the thesis present the trade-off between computational efficiency and accuracy in model implementation.
This research contributes to the field of financial modeling by systematically comparing jump diffusion models and providing valuable insights into their practical implementation and performance. The findings highlight a significant improvement in accuracy when incorporating jumps compared to the Black-Scholes model. However, adding jumps to models with stochastic
volatility results in marginal effects and presents greater challenges in parameter estimation.
While the Bates model demonstrates strong capabilities in capturing market dynamics, the analysis identifies potential over-parameterization issues in its volatility equation. Simpler models, such as the Merton model, may often prove more advantageous due to their balance of complexity and performance.

Vejleder: Rolf Poulsen
Censor:   Thomas Kokholm, Aarhus Universitet