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An American call option with exercise price $K = 90$ written on an asset where the asset prices in dollars are given below, the interest rate per period is zero, and a dividend of $5$ is paid between time $0$ and time $1$. \begin{array}{|c|c|c|} \hline \omega& S(0) & S(1) & S(2) \\ \hline \omega_1&100 & 115&145 \\ \hline \omega_2&100 &115 & 105\\ \hline \omega_3&100 &85 & 105\\ \hline \omega_4&100 &85 & 65\\ \hline \end{array}

My question is, when calculating risk neutral probabilities, does one add back the dividend to both $S(1)$ and $S(2)$ or just to $S(1)$ ?

\begin{matrix} & & & \\&& S_2=(145+5) & \\&S_1=(115+5)&& \\S_0=100 & & S_2=(105+5) & \\&S_1=(85+5)&& \\&& S_2=(65+5) & \end{matrix} or

\begin{matrix} & & & \\&& S_2=(145) & \\&S_1=(115+5)&& \\S_0=100 & & S_2=(105) & \\&S_1=(85+5)&& \\&& S_2=(65) & \end{matrix}

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    $\begingroup$ Both. Otherwise it looks like the 5 dollars disappeared between t=1 and t=2. $\endgroup$ – dm63 Apr 22 at 13:36

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