Consider a European call option on a stock that is currently traded at $$20.$ The option has a time to maturity of $12$ months and a strike of $$21.$ Assume the stock volatility of $25\%$ p$.$a$.$ and a risk$-$free rate of $4\%$ p$.$a$.,$ continuously compounded. Show all workings.

a$)$ Consider an up and out European barrier option on the same underlying stock.

An up and out barrier option is active from the beginning but expires immediately and is worth nothing if the underlying stock hits the barrier value.

Use a $3-$step binomial model to price an up and out European barrier call option with a barrier at $$25.$

b$)$ Consider an up and in European barrier call option on the same underlying stock.

With an up and in barrier option the buyer gets an option that becomes active if and when the underlying hits a given barrier value. If the underlying never reaches this value, the option will expire without a value. What would be the price of an up and in European barrier call option on the same stock with a barrier at $$25?$ Why? Do not price the option on a tree.

Note: The same assumptions apply in part a$)$ and b$)$

Please do not use chatgpt and need either a hand$-$written answer or from excel. Please provide the necessary formulaes as well