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 . In terms of computational speed, the latter is found to be 30 to 40 times faster . The availability of an analytical solution paves the way to an efficient market calibration of model parameters. As case studies, EEX German electricity and TTF Dutch gas markets were considered. Two different single-factor models and a two-factor one were calibrated against market prices: out-of-sample validation showed that a two-factor model outperforms the other two approaches.
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 . The availability of an analytical solution paves the way to an efficient market calibration of model parameters. As case studies, EEX German electricity and TTF Dutch gas markets were considered. Two different single-factor models and a two-factor one were calibrated against market prices: out-of-sample validation showed that a two-factor model outperforms the other two approaches.
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