Stochastic methods for virtual asset pricing and risk management in Kenya

Abstract

The increasing adoption of Bitcoin (BTC) and Ethereum (ETH) in global financial markets has raised critical questions regarding their valuation, volatility, liquidity, and regulatory oversight. This study investigates the effectiveness of stochastic models including Geometric Brownian Motion (GBM), Heston, Ornstein-Uhlenbeck (O-U), and Jump-Diffusion in capturing the unique price dynamics and volatility patterns of BTC and ETH. Using historical market data, the research applies these models alongside Auto-regressive Conditional Heteroskedasticity (ARCH/GARCH) models to analyze volatility persistence and risk characteristics. The findings indicate that while GBM provides a basic framework for price evolution, it fails to account for volatility clustering and market shocks. The Heston model captures stochastic volatility, whereas Jump-Diffusion models effectively incorporate sudden price jumps. GARCH (1,1) models confirm significant volatility clustering in both BTC and ETH. To assess risk exposure, the study computes Value at Risk (VaR) and Conditional VaR (CVaR) at 95% and 99% confidence levels. Results show that ETH exhibits higher tail risk than BTC, implying greater vulnerability to extreme losses. Furthermore, liquidity analysis, measured through market depth reveals that BTC has stronger liquidity and lower relative volatility risk compared to ETH. The research also applies Monte Carlo simulations to price BTC and ETH derivatives, demonstrating that stochastic models significantly influence option valuation by incorporating market uncertainties. Stress-testing scenarios highlight vulnerabilities in price stability, underscoring the need for margin requirements, volatility controls, and liquidity monitoring to mitigate systemic risks. The study’s findings contribute to the growing discussions on risk management and policy formulation of the VA ecosystem, offering recommendations to enhance market stability while fostering innovation.

Keywords: Bitcoin, Ethereum, Volatility, ARCH/GARCH, VaR, CVaR, Liquidity, Derivatives, Risk Management, Regulation, Kenya