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First note that the price of binary call is related to the price of an ordinary call in any model by $$ BinC(T,K) = e^{-rT}\mathbb{E}^{\mathbb{Q}}[1_{S_T>K}] = - \frac{\partial}{\partial K}e^{-rT}\mathbb{E}^{\mathbb{Q}}[(S_T-K)_+] = - \frac{\partial}{\partial K}C(T,K) $$ Now the volatility smile is implicitly defined by $$ C(T,K) = ...


First, as far as I can tell, you are not taking into account dividends. Second, If you simply take the forward price of the SPX @ 5.5% which is what you are using, you get 1411 * e^(.055*2.99) = 1663. Given a strike of 1300, the call should have an intrinsic value of 1663-1300= 363. You have a price of 272. The price is less than the intrinsic value of the ...

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