We intend to price options on a binomial tree. We assume that the current spot price of the security is $$100.$ One year later, the security price can either go up to $$120,$ or go down to $$83.$

$($a$)$ Based on the above assumption, price a one$-$year put option with a strike of $$110.$

$($b$)$If the market quote for the put option is $$12,$ is there an arbitrage based on your model assumption? How do you do the arbitrage trading to profit from it$?$