# Arbitrage strategy using binomial tree

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?

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 $$q \sqrt{p_1} + (1-q) \sqrt{p_2}$$.