Abstract: The futures price is the core element of the futures trading, and also reflects the state of market operation. Reasonable and effective futures price can play a leading action and make up for the deficiency of spot prices hysteresis. The importance of exchange-traded natural gas grows the necessity to find accurate pricing models for the different contracts.This lack of economical transportation and the limited storability of natural gas makes its supply unable to change in view of seasonal variations of demand, therefore natural gas prices are strongly seasonal.Natural gas as a commodity, its weak liquidity has led to the incompleteness of the futures market. Taking into account of the incompleteness of the futures market and the stochastic seasonality of natural gas prices, a novel pricing model of natural gas futures is proposed by using the basic theory of the stochastic discount factor. In this paper, focuses are put on the degree of incompleteness of the natural gas futures market generated by short-term deviation, middle-term deviation and the seasonal factor. The differential equations of short-term deviation, middle-term deviation , long term equilibrium and the seasonal factor are taken as a breakthrough point, the volatilities caused by four factors are divided into complete and incomplete part, the transition equation of state space is built, the parameters are estimated by the Kalman filter and maximum likelihood estimator. The data sets used in this paper consist of daily observations of Henry Hub natural gas futures prices traded at NYMEX. In order to account for structural changes in the natural gas price dynamics and estimate properly the relationship between the three non-seasonal factors (short-term deviation, middle-term deviation and long term equilibrium), it is desirable to consider different data sets with different sample periods, that is to say, futures contracts with short-term, middle-term and long-term maturities are also necessary to estimate properly the parameters of the non-seasonal factors. It is expected that the seasonal factor in the particular models has one year period, futures contracts with more than one year to maturity are needed to account for it. The results indicate that the volatility on the natural gas futures market caused by the incompleteness should be attributed to theshort-term deviation, middle-term deviation and the seasonal factor; the natural gas price is seasonal, and the seasonal period is one year; There is strong mean-reversion in the short-term and middle-term deviations, the expected appreciation of futures prices is positive; the Sharpe Ratio of the natural gas futures market is about 1.5 times of complete market price of risk, so risk compensation and the incompleteness of the market should be considered when natural gas futures pricing. By comparing with the five-factor model of Garcia, the proposed model has better goodness of fit and forecasting ability. The researches of natural gas futures pricing not only solve the problem of natural gas futures pricing, but also provide the valuable reference information for hedging and investment decisions, which has an important significance on the theory and practice.

Key words: natural gas futures, kalman filter, seasonality, stochastic dynamic model