The trinomial-tree GARCH option pricing algorithms of Ritchken and Trevor (1999) and Cakici and Topyan (2000) are claimed to be efficient and accurate. However, this thesis proves that both algorithms generate trees that explode exponentially when the number of partitions per day, n, exceeds a typically small number determined by the model parameters. Worse, when explosion happens, the tree cannot grow beyond a certain maturity, making it useless for pricing derivatives with a longer maturity. Meanwhile, a small n has accuracy problems and does not prevent explosion. This thesis then presents a trinomial-tree GARCH option pricing algorithm that solves the above problems.The algorithm provably does not have the short-maturity problem. Furthermore, the tree size is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This result for the first time puts a tree-based GARCH option pricing algorithm in the same complexity class as binomial and trinomial trees under the Black-Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the accuracy of the algorithm.