Stochastic Modeling of Energy Commodity Spot Price Processes with Delay in Volatility
Olusegun M. Otunuga, Gangaram S. Ladde

Abstract
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, a 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.

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