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You could solve this by constructing a binomial tree with the stock price ex-dividend. Also keep in mind that you have to adjust your volatility by muliplying with S/(S-PV(D)).


you don't need $ud=1.$ In fact, there are now about 30 binomial trees which converge to Black--Scholes in the large step limit. Most of them do not have $ud=1.$ All you need is $$ d < e^{r \Delta t} < u $$ The tree recombines provided $u$ and $d$ don't change from step to step. See my book More Mathematical Finance for a comprehensive review and ...

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