This paper explores the calculation of risk-minimization hedging strategies, specifically local and global risk minimization strategies for contingent claims under affine and non-affine GARCH models with the known closed forms of its first four moments across times under the physical measure. A unified and efficient willow tree method is introduced for various GARCH models. Unlike methods that provide option values and hedging ratios solely at the inception time, the proposed willow tree method generates a comprehensive table of option values and hedging ratios at each discrete time step across possible asset prices. Additionally, the method showcases robust performance in hedging at lower frequencies than the underlying asset’s modeling frequency (e.g., weekly or monthly hedging using a daily GARCH model). Lastly, the willow tree method outperforms the Monte Carlo method, offering greater efficiency, accuracy, and flexibility in solving risk-minimization hedging problems.

中文翻译：

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广义自回归条件异方差性模型下风险最小化对冲的新晶格方法


本文探讨了风险最小化对冲策略的计算，特别是仿射和非仿射 GARCH 模型下或有索赔的局部和全局风险最小化策略，在物理测度下其前四个矩在时间上的已知闭合形式。为各种 GARCH 模型引入了一种统一高效的柳树方法。与仅在开始时提供期权价值和对冲比率的方法不同，拟议的柳树方法在可能的资产价格的每个离散时间步长生成一个全面的期权价值和对冲比率表。此外，该方法在低于标的资产建模频率（例如，使用每日 GARCH 模型进行每周或每月对冲）的对冲中展示了稳健的性能。最后，柳树方法优于蒙特卡洛方法，在解决风险最小化对冲问题方面提供了更高的效率、准确性和灵活性。