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I have a call option with expiry in two years. In my case the option is bermudan style with first 9 months w/o ability to exercise (i.e. European) and after exercise at any time (i.e. American), but I believe this may not matter and will accept answer for purely american options. This option starts with let's say 100 shares as underlying, and then every quarter the "notional" increases by say 20 shares. So Q1 = 120, Q2 = 140 ... etc. How can I implement this in a binomial tree? Should I price each of the increases as a forward starting option? I.e. value the first increase as 21 month option with 6 month lock-out period and discount? Any other ideas? I know the strike at inception of the option.

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