Multivariate Geometric Brownian Motion (MGBM) is a stochastic process used in quantitative finance to model the simultaneous price evolution of multiple correlated assets over time. It extends the univariate Geometric Brownian Motion by incorporating a covariance matrix that captures the interdependencies between asset returns. This model assumes that asset prices follow a random walk with a constant drift and volatility, allowing for the simulation of complex market dynamics. It is a cornerstone for multi-asset derivative pricing.
Assumption
Key assumptions of the MGBM model include log-normally distributed asset prices, constant drift rates, and constant volatility for each asset. Crucially, it assumes that the correlations between assets remain constant over the modeling horizon. While these assumptions simplify mathematical tractability, they may not always hold true in real-world markets, particularly during periods of extreme volatility or regime shifts. Acknowledging these limitations is vital for model calibration.
Application
MGBM finds extensive application in the pricing of multi-asset options and structured financial products, such as basket options, rainbow options, and correlation swaps. It provides a framework for Monte Carlo simulations to estimate option payoffs under various scenarios, considering the stochastic interaction of underlying assets. In cryptocurrency derivatives, it can be adapted to model the co-movement of different digital assets for portfolio risk management and exotic option valuation.
Meaning ⎊ Cross-asset correlation is a critical parameter for pricing multi-asset derivatives and accurately assessing portfolio risk, particularly in high-volatility environments where correlations dynamically shift during market stress.
