["Journal of Futures Markets, EarlyView. ", "\nABSTRACT\nAmerican options are widely traded in financial markets, yet there is a scarcity of literature on hedging in incomplete markets. In this paper, we derive optimal hedging ratios and option values using Local Risk Minimization (LRM) and Global Risk Minimization (GRM) hedging strategies through dynamic programming. Then, by utilizing the willow tree structure, the ratios and values can be computed efficiently and accurately. Moreover, this method provides a comprehensive table of hedging strategies across possible asset prices and discrete times, rather than a single initial ratio. Finally, we examine the effectiveness of LRM, GRM, and Delta hedging, assuming the underlying asset follows two different GARCH models under both the physical (\n\n\nP\n\n\n ${\\mathbb{P}}$\n\n\n) and risk‐neutral (\n\n\nQ\n\n\n ${\\mathbb{Q}}$\n\n\n) measures. Our objective is to assess the added value of GRM over LRM and Delta hedging, the impact of using the \n\n\nP\n\n\n ${\\mathbb{P}}$\n\n\n versus \n\n\nQ\n\n\n ${\\mathbb{Q}}$\n\n\n measure for hedging, and the influence of hedging frequency on hedging errors.\n"]