In the temperature derivatives market, there is a problem about how to choose an arbitrage free price. This thesis mathematically defines different models of the temperature index, and builds models for temperature derivatives. After the careful analysis of seasonal model, this thesis proposes an Ornstein-Uhlenbeck process with dynamic volatility and jump for the temperature evolution in time, and fits this model with the transition density function maximum likelihood estimation method to data observed in Macau, China. The proposed model has been proved that, in Lemma 3, satisfies the crucial condition that the expectation must equal to seasonal model. Using the maximum likelihood estimation method with the transition density function, the evaluation result of parameters is fast, accurate and trusted. Futures price for contracts on CAT index is derived. In the Monte Carlo experiment, generation method of jump is significantly optimized, thus the simulation is more realistic and more stable.