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Let's say an asset has a continuous dividend yield of 5% (and assume interest rate is 0%). If I want to price an American call option on such an asset, I take each time step individually and construct a replicating portfolio, such that the linear combination of the asset and the riskless asset is equal to the payoff at the children nodes. When finding the effective payoff at the parent node, I would need to multiply the number of units of the asset held by $e^{-0.05 * dt}$, since I would only need to hold that many units of the asset at time t - 1 to get the same payoff at t. Back-stepping through the tree, you get the option price (checking whether exercise is more optimal at each node). Doesn't this method give you the no-arbitrage price for a call option on $e^{0.05 * T}$ units of the asset, since a continuous dividend payment would decrease the price at each node by $e^{-0.05 * dt}$?

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