Closed-Form Covariance Matrix for Portfolio Optimization: Theory and Empirical Evidence Under a Multidimensional Black–Scholes Model with Time-Varying Parameters

Abstract

This paper develops a model-driven analytical framework for portfolio optimization under a multidimensional Black–Scholes model with time-varying parameters, where both the drift and volatility functions evolve linearly over time. Within this framework, explicit closed-form expressions are derived for the covariance matrix of normalized asset prices and subsequently incorporated into the classical Markowitz mean–variance framework to obtain analytical representations of the global minimum-variance portfolio, the mean–variance efficient portfolio, and the corresponding efficient frontier. The proposed methodology establishes a direct connection between continuous-time stochastic asset-price modeling and portfolio optimization through a model-implied covariance structure. Its practical implementation is investigated through both numerical experiments and an empirical study using daily stock price data from 20 constituents of the S&P 500 index over the period 2020–2024. Monte Carlo simulations demonstrate the finite-sample sensitivity of portfolio optimization to covariance estimation, while the empirical analysis illustrates how the estimated model parameters, obtained using the maximum likelihood framework of Aït-Sahalia for discretely sampled diffusion processes, can be incorporated into the analytical covariance matrix for constructing efficient frontiers under realistic market conditions. Overall, the proposed framework provides an analytically tractable methodology that integrates continuous-time asset pricing models with classical mean–variance portfolio optimization, offering a coherent model-based covariance representation for portfolio selection under time-varying market environments.