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3

If I understand well, you have a market with 3 states: up, flat or down. You have 3 instruments: The stock The risk-free rate (50%) The option If you can create a portfolio today with these 3 instruments that can replicate de payoff of the option you have to price, then the law of one price tells you that the price of the option should be the price of ...

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You are correct that showing the self-financing condition for the BS-portfolio is not as straightforward as one may think: A portfolio $V_t(\alpha_t,\beta_t)$ (for stock $S_t$ and zerobond $B_t$) is self-financing iff: $$V_t=\alpha_tS_t+\beta_t B_t$$ It further implies $$dV_t=\alpha_tdS_t+\beta_tdB_t$$ To replicate a derivative $C(S_t,t)$ by a ...

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if you let the implied vol depend on K you get two terms the first is $N(d_2)$ but you get a correction term which is the slope times the vega $$\frac{\partial C}{\partial \sigma} \frac{\partial \sigma}{\partial K}.$$ (see eg my book)

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