Extreme Risk Modeling in Market Finance
Simulation methods for rare event estimation and Value-at-Risk calculation in options portfolios
Abstract
This project focuses on the study and comparison of different simulation methods for estimating rare events in a financial context, particularly the calculation of tail loss probabilities and risk quantiles such as Value at Risk (VaR).
We presented several techniques, including the naive Monte Carlo method, importance sampling with mean and variance shifts, as well as splitting methods (with fixed and variable steps) and the last particle method. These approaches were evaluated across various thresholds by comparing estimated probabilities and relative errors.
The results indicate that advanced methods like importance sampling and splitting provide better accuracy and significantly reduced errors compared to naive Monte Carlo, especially for very rare events.
Additionally, Brownian motion trajectories satisfying high percentile quantiles were simulated using the Brownian bridge principle to illustrate the evolution of the common market factor affecting financial assets.
While this work offers a thorough study of simulation methods, a more critical analysis involving other portfolio and simulation scenarios remains an interesting direction for future research.