Stochastic Volatility Models for Natural Gas Markets: A Temperature-Driven Approach

This thesis develops a temperature-driven stochastic volatility model for energy markets, establishing a direct mathematical link between temperature fluctuations and price volatility. While the theoretical framework applies to both natural gas and power markets, the empirical implementation focuses on German electricity markets, which exhibit distinctive characteristics including non-storability, strong seasonal demand patterns driven by heating and cooling requirements and pronounced sensitivity to weather conditions that create unique modeling challenges for day-ahead pricing. The research introduces a novel framework where price volatility is explicitly connected to observable temperature deviations, departing from traditional stochastic volatility models that treat volatility as a hidden process. In classical approaches like the Heston model, volatility is treated as an unobservable endogenous variable calibrated from market data, resulting in incomplete markets. This work establishes volatility as a function of measurable meteorological factors through an explicit quadratic relationship with temperature anomalies. The temperature-volatility relationship reflects the fundamental economics of energy markets, where unexpected deviations from seasonal temperature norms drive demand fluctuations and corresponding price volatility. The framework decomposes actual temperature into a deterministic seasonal component and a mean-reverting Ornstein-Uhlenbeck process representing stochastic deviations. Market volatility is then constructed quadratically from temperature anomalies plus a baseline level, capturing the empirical observation that both unusually hot and cold conditions increase energy price uncertainty. Two comprehensive pricing models are developed: an additive specification with linear price dynamics and an exponential model ensuring price positivity. The mathematical framework derives the characteristic function involving confluent hypergeometric functions, enabling semi-analytical solutions through systems of Riccati differential equations. The theoretical relationships establish parameter constraints that reduce the calibration complexity from six unknowns to three market parameters, with temperature parameters estimated independently from meteorological data. The thesis derives explicit analytical expressions for forward and swap contract pricing under both model specifications. The option pricing methodology adapts the Carr-Madan Fourier transform approach to handle the additive model's potential for negative prices, transforming options on swap contracts into options on the stochastic component with appropriately modified strikes. This adaptation preserves computational advantages while addressing the unique characteristics of energy markets. Empirical implementation using German power market data demonstrates the framework's viability through proof-of-concept calibration. Temperature parameters are estimated from twenty-five years of meteorological data using maximum likelihood methods. Due to significant computational complexity involving confluent hypergeometric functions and nested numerical integrations, the calibration focused on a targeted single-day exercise, validating the theoretical framework while revealing economically meaningful relationships between temperature deviations and market volatility dynamics. This work contributes both theoretical advancement through rigorous integration of weather-dependent volatility with established mathematical finance principles and practical validation of temperature-driven price dynamics in energy markets, establishing foundations for future large-scale implementations once computational optimizations are achieved through parallel processing or alternative programming architectures.