Before revision: Deregulation of energy markets in the 90s boosted the interest in energy derivatives. Over the last two decades, more and more complex financial instruments were developed. Pricing exotic derivatives often involves Monte Carlo simulations, which rely on stochastic processes to model the underlyings: it is thus critical to choose appropriate models and precisely calibrate them, so that they reflect the market scenario. Several models have been proposed in the literature , from the simple geometric Brownian motion to more complex mean-reverting, multi-factor models. To enable their calibration against listed vanilla options, it is required to compute the variance of their states. This paper presents a simple and general method to compute the covariance matrix of the state though a matrix Lyapunov differential equation , and discusses its numerical and analytical solutions .
After revision: A general method is presented to compute the variance of a linear stochastic process through a matrix Lyapunov differential equation. This approach, adopted from control theory, is alternative and easier with respect to the classical arguments found in quantitative finance literature and can be readily applied to high-dimensional models. Both analytical and numerical methods to solve the Lyapunov equation are discussed and compared in terms of computational efficiency. A practical application is presented, where numerical and analytical solutions .