# Infinite Binomial Pricing no arbitrage

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 condition.

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

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.