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How to price a contract that pays only 1 at the first stock price drop? The stock follows an infinite binomial with no arbitrage $d<R<u$ condition.

So the probability of the price going down is $p = (u-R)/(u-d)$

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Risk neutral valuation tells you to discount the expected payoff in the risk free world. Therefore, it should be $\sum_{i=0}^{N-1}R^{i+1}(1-p)^i p$, where $N$ is the number of periods.

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