2-1 - A Further Analysis of the Convergence Rates and...

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Unformatted text preview: A Further Analysis of the Convergence Rates and Patterns of the Binomial Models San-Lin Chung Department of Finance, National Taiwan University No. 85, Section 4, Roosevelt Road, Taipei 106, Taiwan, R.O.C. Tel: 886-2-33661084 Email: chungs@management.ntu.edu.tw Pai-Ta Shih Department of Economics, National Dong Hwa University No. 1, Section 2, Da Hsueh Road, Shoufeng, Hualien 974, Taiwan, R.O.C. Tel: 886- 3-8635541 Email: bdshih@mail.ndhu.edu.tw Abstract This paper extends the generalized Cox-Ross-Rubinstein (hereafter GCRR) model of Chung and Shih (2007). We provide a further analysis of the convergence rates and patterns based on various GCRR models. The numerical results indicate that the GCRR- XPC model and the GCRR-JR ( 1 2 p = ) model (defined in Table 1) outperform the other GCRR models for pricing European calls and American puts. Our results confirm that the node positioning and the selection of the tree structures (mainly the asymptotic behavior of the risk-neutral probability) are important factors for determining the convergence rates and patterns of the binomial models. JEL Subject Classification: G13. Key words: binomial model, rate of convergence, monotonic convergence 1 1. Brief Review of the Binomial Models Ever since the seminal works of Cox, Ross, and Rubinstein (1979, hereafter CRR) and Rendleman and Bartter (1979), many articles have extended the binomial option pricing models in many aspects. One stream of the literature modifies the lattice or tree type to improve the accuracy and computational efficiency of binomial option prices. The pricing errors in the discrete-time models are mainly due to distribution error and nonlinearity error (see Figlewski and Gao (1999) for thorough discussions). Within the literature, there are many proposed solutions that reduce the distribution error and/or nonlinearity error. It is generally difficult to reduce the distribution error embedded in the binomial model when the number of time steps ( n ) is small, probably due to the nature of the problem. Nevertheless there are two important works which might shed some light on understanding or dealing with distribution error. First, Omberg (1988) developed a family of efficient multinomial models by applying the Gauss-Hermite quadrature 1 to the integration problem (e.g. ) ( 1 d N in the Black-Scholes formula) presented in the option pricing formulae. Second, Leisen and Reimer (1996) modified the sizes of up- and down- movements by applying various normal approximations (e.g. the Camp-Paulson inversion formula) to the binomial distribution derived in the mathematical literature. Concerning the techniques to reduce nonlinearity error, there are least three important works (methods) worth noting in the literature. First, Ritchken (1995) and Tian (1999) showed that one can improve the numerical accuracy of the binomial option prices by allocating one of the final nodes on the strike price or one layer of the nodes on the barrier price. Second, Figlewski and Gao (1999) proposed the so-called adaptive mesh barrier price....
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2-1 - A Further Analysis of the Convergence Rates and...

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