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 .
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 for the variance of a two-factor mean-reverting model are embedded into the Black pricing framework and market calibration of model parameters is performed. It is shown that the availability of an analytical formula for the variance makes the calibration 14 times faster, thus proving the practical value of tractable and general methods to derive it .
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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
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
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. The availability of an analytical solution paves the way to an efficient
for the variance of a two-factor mean-reverting model are embedded into the Black pricing framework and
R
. 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
is performed. It is shown that the availability of an analytical formula for the variance makes the calibration 14 times faster, thus proving the practical value of tractable and general methods to derive it