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

Paper: arXiv 2601.14272

Authors: Ekleen Kaur

Abstract

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.

Complexity vs Empirical Score

• Math Complexity: 4.0/10
• Empirical Rigor: 3.5/10
• Quadrant: Philosophers — conceptual discussion, limited math and data

Why this score: The paper serves primarily as a baseline exercise applying standard textbook Geometric Brownian Motion and Monte Carlo simulations to a crypto portfolio to demonstrate the well-known failure of lognormal assumptions in fat-tailed assets. While mathematically sound at an introductory level, it lacks methodological novelty, statistical depth, and rigorous backtesting details.

Research Flowchart

  flowchart TD
    A["Research Goal: Assess GBM Model & VaR Limits for Crypto vs. Equity Portfolios"] --> B["Data Inputs: Historical Prices (Crypto: XRP, SOL, ADA | Equity: AAPL, TSLA, NVDA)"]
    B --> C["Portfolio Optimization: Minimum Variance Portfolio (MVP) Construction"]
    C --> D["Parameter Estimation: Maximum Likelihood Estimation (MLE)"]
    D --> E["Simulation Engine: Correlated Monte Carlo via Cholesky Decomposition"]
    E --> F["Risk Quantification: Value-at-Risk (VaR) Calculation at 5% Significance"]
    F --> G["Key Finding: GBM fails in heavy-tailed crypto environments but serves as a baseline for future models"]

Cited References (16)

1. Fitting Latent Non-Gaussian Models Using Variational Bayes and Laplace Approximations (2022) — link
2. A bibliometric review of cryptocurrencies: how have they grown? (2022) — link
3. The impact of the shutdown policy on the asymmetric interdependence structure and risk transmission of cryptocurrency and China’s financial market (2021) — link
4. Volatility models for cryptocurrencies and applications in the options market (2021) — link
5. A bibliometric review of cryptocurrencies as a financial asset (2021) — link
6. Connectedness between cryptocurrencies and foreign exchange markets: Implication for risk management (2020) — link
7. Blockchain-based multi-organization taxonomy for smart cities (2020) — link
8. Modelling volatility of cryptocurrencies using Markov-Switching GARCH models (2019) — link

0 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.