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This study investigates the convergence patterns and the rates of convergence of binomial Greeks for the CRR model and several smooth price convergence models in the literature, including the binomial Black-Scholes (BBS) model of Broadie M and Detemple J (1996), the flexible binomial model (FB) of Tian YS (1999), the smoothed payoff (SPF) approach of Heston S and Zhou G (2000), the GCRR-XPC models of Chung SL and Shih PT (2007), the modified FB-XPC model, and the modified GCRR-FT model. We prove that the rate of convergence of the CRR model for computing deltas and gammas is of order O(1/n), with a quadratic error term relating to the position of the final nodes around the strike price. Moreover, most smooth price convergence models generate deltas and gammas with monotonic and smooth convergence with order O(1/n). Thus, one can apply an extrapolation formula to enhance their accuracy. The numerical results show that placing the strike price at the center of the tree seems to On the Rate of Convergence of Binomial Greeks 563Journal of Futures Markets DOI: 10.1002/fut enhance the accuracy substantially. Among all the binomial models considered in this study, the FB-XPC and the GCRR-XPC model with a two-point extrapolation are the most efficient methods to compute Greeks.

This paper first reviews empirical evidence and estimation methods of structural credit risk models. Next, an empirical investigation of the performance of default prediction under the down-and-out barrier option framework is provided. In the literature review, a brief overview of the structural credit risk models is provided. Empirical investigations in extant literature papers are described in some detail, and their results are summarized in terms of subject and estimation method adopted in each paper. Current estimation methods and their drawbacks are discussed in detail. In our empirical investigation, we adopt the Maximum Likelihood Estimation method proposed by Duan [Mathematical Finance 10 (1994) 461-462]. This method has been shown by Ericsson and Reneby [Journal of Business 78 (2005) 707-735] through simulation experiments to be superior to the volatility restriction approach commonly adopted in the literature. Our empirical results surprisingly show that the simple Merton model outperforms the Brockman and Turtle [Journal of Financial Economics 67 (2003) 511-529] model in default prediction. The inferior performance of the Brockman and Turtle model may be the result of its unreasonable assumption of the flat barrier.

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In this paper, we use three structural models to investigate a country's credit risk by applying it to a sovereign balance sheet. The transformed-data maximum likelihood estimation method and the maximization-maximization algorithm are adopted for model calibration. The derived probability of default over time for four sample countries matched well with the events and economic conditions that occurred during the sample period. Our empirical analyses show that structural models can be used to determine with high accuracy whether the credit of a sovereign country is in a precarious situation. We then illustrate how the structural approach can be an effective tool to monitor the sovereign credit risk.

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