The Limits of Lognormal: Assessing Cryptocurrency Volatility and VaR using Geometric Brownian Motion

The integration of cryptocurrencies into institutional portfolios necessitates the adoption of robust risk modeling frameworks. This study is a part of a series of subsequent works to fine-tune model risk analysis for cryptocurrencies. Through this first research work, we establish a foundational benchmark by applying the traditional industry-standard Geometric Brownian Motion (GBM) model. Popularly used for non-crypto financial assets, GBM assumes Lognormal return distributions for a multi-asset cryptocurrency portfolio (XRP, SOL, ADA). This work utilizes Maximum Likelihood Estimation and a correlated Monte Carlo Simulation incorporating the Cholesky decomposition of historical covariance. We present our stock portfolio model as a Minimum Variance Portfolio (MVP). We observe the model's structural shift within the heavy-tailed, non-Gaussian cryptocurrency environment. The results reveal limitations of the Lognormal assumption: the calculated Value-at-Risk at the 5% confidence level over the one-year horizon. For baselining our results, we also present a holistic comparative analysis with an equity portfolio (AAPL, TSLA, NVDA), demonstrating a significantly lower failure rate. This performance provides conclusive evidence that the GBM model is fundamentally the perfect benchmark for our subsequent works. Results from this novel work will be an indicator for the success criteria in our future model for crypto risk management, rigorously motivating the development and application of advanced models.