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Suppose that we have a one step binomial tree model for a company. Lets say that the time per step is T, and that price of the stock can go up to $p_1$ or go down to $p_2$. Suppose a T-month European call on the company, with strike price $X$ trades at price $Y$ while a T-month European put with strike price $W$ has a price of $Z$. How can we find the arbitrage-free price of an exotic option with payoff equal to the square root of the stock price at T=1/4?

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Let's $p_0$ be the initial stock price, $q$ be the risk free probability of $p_0$ ends up at $p_1$, assuming risk free rate is 0 then $$ p_0 = q p_1 + (1-q) p_2$$ so $$ q = \frac{(p_0 - p_2)}{(p_1-p_2)}$$.

So the exotic price after one step is $$ q \sqrt{p_1} + (1-q) \sqrt{p_2} $$.

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