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I will assume that the interest rate is 0. The price of a binary option is then the same as the risk-neutral probability that the event will occur $$\mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{S(T)\geq K}\right]=\mathbb{Q}\left[S(T) \geq K\right]$$ Denote the current spot price $s$. You need to find \mathbb{E}^{\mathbb{Q}}\left[\mathbb{1}_{0.9 s< S(T) <...

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I think you can simply construct a portfolio equivalent to the double digital option (let's call it $DO$) you want to price, that qualitatively will look like this (dotted lines): The replicating portfolio should contain: a zero-coupon bond expiring in one year (current value ($t=0$) = \\exp(-r \cdot (1 - t)\text{ years})\$); a shorted digital valid if S&...

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Lets assume the price of the underlying equals the strike at some point prior to expiry. Then the probability of the price being still greater or equal the strike at expiry is 0.5. So the probability of the European option paying out is exactly half of the probability for the American option.

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