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You can think about them as noise traders in the sense of Glostem and Milgrom (1985). It it is a fairly wide used assumption that there is someone out there that soaks up residual supply/demand. Usually one thinks about this guys as large mutual funds or pension funds.


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time value "appears" from two sources a) convexity of payoff function max(S-K,0) b) settlement of stock in future (at option expiry). if u recall put-call parity: C-P=Forward and consider statement max(S-K,0) [this is call]-max(K-S,0)[this is put]=S-K. you see that call has time value (convexity), put has time value (convexity), but C-P does not have time ...


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you need a positive dividend rate or a negative interest rate. Without these, it is a model-free result that early exercise is never optimal for a call option.


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What if you write $$ P[R_{n+1} = d|F_n] = 1 - P[R_{n+1} = u|F_n] ? $$ Let us write $P(u) = P[R_{n+1} = u|F_n]$ Then the part to show is $$ u \bar{S}_n P(u) + d \bar{S}_n (1-P(u)) $$ and this $$ \bar{S}_n \left(d +(u-d)P(u) \right), $$ where we just expanded terms and then extracted the coefficients.



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