Consider a $6-$month American put option on a non$-$

dividend$-$paying stock with a strike price of $$21.$ The

current stock price is $$20$ per share. In each of the two $3-$

month periods in the life of the option, the stock price can

either go up by $10\%$ or go down by $10\%.$ The risk$-$free rate

is $12\%$ per annum $($continuously compounded$).$ What is the

value of this put option according to a $2-$step binomial tree

model?