Two$-$period binomial tree.

A stock has a price of $$30,$ and a standard deviation of $29\%.$ The continuous risk$-$free rate is $10\%.$ There are European and American call and put options with a strike price of $$33$ and time to expiration of $2$ years written on the stock. Using a two$-$step recombining CRR binomial tree, answer the following:

a$.$ What is the risk$-$neutral probability of the stock price going up in a single step?

Round your answer to two decimals.

b$.$ What is the theoretical value of the European call?

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Round your answer to the nearest cent.

c$.$ What is the theoretical value of the European put?

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Round your answer to the nearest cent.

d$.$ What is the theoretical value of the American put??

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Round your answer to the nearest cent.