On accurate and provably efficient GARCH option pricing algorithms
Resource
Quantitative Finance,5(2),181-198.
Journal
Quantitative Finance
Journal Volume
5
Journal Issue
2
Pages
181-198
Date Issued
2005
Date
2005-04
Author(s)
Chi-Ning, W.U.
Abstract
The GARCH model has been very successful in capturing the serial corre-lation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH op-tion pricing algorithms suffer from exponential running time, a cut-off maturity, inaccuracy, or some combination thereof. Specifically, this paper proves that the popular trinomial-tree option pricing algorithms of Ritchken and Trevor (1999) and Cakici and Topyan (2000) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH pa-rameters. Furthermore, when explosion happens, the tree cannot grow beyond a certain maturity date, making it unable to price derivatives with a longer maturity. As a result, the algorithms must be limited to using small n, which may have accuracy problems. The paper presents an alternative trinomial-tree GARCH option pricing algorithm. This algorithm provably does not have the short-maturity problem. Furthermore, the tree-size growth is guaranteed to be quadratic if n is less than a threshold easily determined by the model parameters. This level of efficiency makes the proposed algorithm practical. The surprising finding for the first time places a tree-based GARCH option pricing algorithm in the same complexity class as binomial trees under the Black-Scholes model. Extensive numerical evaluation is conducted to confirm the analytical results and the numerical accuracy of the proposed algorithm. Of independent interest is a simple and efficient technique to calculate the transition probabilities of a multinomial tree using generating functions.
Type
journal article