Stochastic Modeling of Energy Commodity Spot Price Processes with Delay in Volatility

Employing basic economic principles, we systematically develop both deterministic and stochastic dynamic models for the log-spot price process of energy commodity. Furthermore, treating a diffusion coefficient parameter in the non-seasonal log-spot price dynamic system as a stochastic volatility functional of log-spot price, an interconnected system of stochastic model for log-spot price, expected log-spot price and hereditary volatility process is developed. By outlining the risk-neutral dynamics and pricing, sufficient conditions are given to guarantee that the risk-neutral dynamic model is equivalent to the developed model. Furthermore, it is shown that the expectation of the square of volatility under the risk-neutral measure is a deterministic continuous-time delay differential equation. The presented oscillatory and non-oscillatory results exhibit the hereditary effects on the mean-square volatility process. Using a numerical scheme, a time-series model is developed to estimate the system parameters by applying the Least Square optimization and Maximum Likelihood techniques. In fact, the developed time-series model includes the extended GARCH model as a special case.