The investigation Long-term relationship between oil and natural gas spot price in formwork of stochastic differential equations

Many studies about modeling of oil and natural gas price were done in formwork of stochastic differential equations (derivatives pricing model). The most of these studies have examined the short-term relationship in formwork of commodity pricing model، but they have not significantly examined long-term relationship between oil and natural gas price. Therefore، in this study we try to add the long-term relationship between oil and natural gas spot price to the commodity pricing models (stochastic differential equations). Existence the long-term relationship between spot price of underling commodities induces that the spot price deviate from its long term level. Therefore at extended model، there are two factors that induce deviation of long term level. One of these factors is the convenience yield and other factors related to the long-term relationship (long-term error) among the spot price of the commodities. Our data consist of monthly futures of crude oil and natural gas. This monthly data are obtained through the NYMEX for period from 1994 to 2011. The time to maturity ranges from 1 month to 9 months. We have used contract futures with maturity 1، 3، 5 and 9 months. For estimation of parameter and examination of the extended model efficiency، we have used the Kalman Filter via the MATLAB soft. The results of this study show that there is the long term relationship between spot price of two commodities. Therefore، in modeling by stochastic differential equations (pricing of energy derivatives for example the options)، we must consider existence co-movement or the long term relationship between spot price of two commodities.