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american option and cash dividends

Can someoe help with this : What is the precise arbitrage argument demonstrating that the price of an american option should be continuous around an ex-dividend date? Thanks
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Monte Carlo American Option Pricing under GARCH(1,1) volatitliy

I am attempting to price a couple of at-the-money American option using the LSM algorithm and GARCH(1,1) volatility. The LSM code I have works correctly for constant volatility, however, when I switch ...