The Complexity of GARCH Option Pricing Models

When using trees to price options, the standard practice is to increase the number of partitions per day, n, to improve accuracy. But increasing n incurs computational overhead. In fact, raising n makes the popular Ritchken-Trevor tree under non-linear GARCH (NGARCH) grow exponentially when n exceeds a typically small threshold. Worse, when this happens, the tree cannot grow beyond a certain maturity because of the impossibility of finding valid probabilities. Lyuu and Wu prove the results under NGARCH. They also prove that, by making the tree track the mean value, valid probabilities can always be found if n does not exceed some threshold; furthermore, the growth rate of the tree’s size is only quadratic in n. This paper completes that line of research by proving that LGARCH, AGARCH, GJR-GARCH, TS-GARCH and TGARCH share the same properties as NGARCH. The theoretical results are verified by numerical experiments.