On accurate and provably efficient GARCH option pricing algorithms

Yuh-Dauh Lyuu and Chi-Ning Wu

Quantitative Finance, 2005, vol. 5, issue 2, 181-198

Abstract: The GARCH model has been very successful in capturing the serial correlation of asset return volatilities. As a result, applying the model to options pricing attracts a lot of attention. However, previous tree-based GARCH option 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 (Ritchken, P. and Trevor, R., Pricing options under generalized GARCH and stochastic volatility processes. J. Finance, 1999, 54(1), 377-402.) and Cakici and Topyan (Cakici, N. and Topyan, K., The GARCH option pricing model: a lattice approach. J. Comput. Finance, 2000, 3(4), 71-85.) explode exponentially when the number of partitions per day, n, exceeds a threshold determined by the GARCH parameters. 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.

Date: 2005
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DOI: 10.1080/14697680500040157

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