The analysis of the convergence of tree methods for pricing barrier and lookback options has been the subject of numerous publications aimed at describing, quantifying and improving the slow and oscillatory convergence in such methods. For barrier and lookback options, we find path-independent options whose price is exactly that of the original path-dependent option. The usual binomial models converge at a speed of order 1=pn to the Black–Scholes price. Our new path-independent approach yields a convergence of order 1=n. Further, we derive a closed-form formula for the coefficient of 1=n in the expansion of the error of our path-independent pricing when the underlying is approximated by the Cox, Ross and Rubinstein (CRR) model. Using this, we obtain a corrected model with a convergence of order n-3=2 to the price of barrier and lookback options in the Black–Scholes model. Our results are supported and illustrated by numerical examples.

中文翻译：

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奇异期权的路径独立性和二项式近似的收敛

对定价障碍和回溯选项的树方法收敛性的分析已成为许多出版物的主题，旨在描述、量化和改进这些方法中缓慢和振荡收敛。对于障碍期权和回顾期权，我们找到了价格与原始路径相关期权完全相同的路径独立期权。通常的二项式模型以 1=pn 阶的速度收敛到 Black–Scholes 价格。我们新的与路径无关的方法产生 1=n 阶收敛。此外，当底层证券通过 Cox、Ross 和 Rubinstein (CRR) 模型近似时，我们推导出了系数 1=n 的封闭式公式，以扩展我们的路径无关定价的误差。使用这个，我们获得了一个修正模型，其阶数为 n-3=2 收敛到 Black–Scholes 模型中的障碍和回顾期权的价格。我们的结果得到了数值例子的支持和说明。