Question $4$: Binomial Trees for Option Pricing $(4/10)$ Shares in XYZ Corporation

sell today for $$100.$ The risk$-$free rate is $10\%($continuously compounded, annual$).$ In the

next $1$ month, XYZ shares will either increase in price by $25\%,$ or decrease in price by $20\%,$

with the same probability. XYZ pays no dividends.

One$-$Step Binomial Tree

$($i$)$ What is the price of a European call with strike price $$120$ and expiration in $1$ month?

Calculate the price explicitly using a binomial tree and the hedging$/$replicating method.

$($ii$)$ If the call of part $($i$)$ currently trades for $$2,$ is there an arbitrage? If so$,$ describe one.

If not, why not?

Risk$-$Neutral Pricing

$($iii$)$ What is the price of a European put with strike price $$120$ and expiration in $1$ month?

Calculate the price explicitly using risk$-$neutral probabilities $($the risk$-$neutral pricing

method$),$ and verify that put$-$call parity holds, together with your solution for $($i$).$

Multi$-$Step Binomial Tree

$(4)$ Suppose for each month, XYZ shares will either increase in price by $25\%,$ or decrease

in price by $20\%,$ with the same probability. What is the price of a European put

with strike price $$120$ and expiration in $3$ months? You can calculate the price using

multi$-$step binomial tree by hand or use the excel sheet "BinomialTree.xls$"($put the

screen shots of your excel Sheet $1$ and Sheet $2$ in your submission if you use the excel

sheet$).$