Question #$5$

A stock is selling for $$100.$ Each year it will either increase or decrease in value by $20\%)=1\mathrm{.}2,d=(0\mathrm{.}8.$ The continuously compounded riskless rate is such that $e^r=1\mathrm{.}10.$ A European put option on this stock with an exercise price of $$100$ matures in two years years$).$ In the binomial pricing, one period corresponds to one year.

Below is a binomial tree showing the evolution of the stock price.

$($i$)$ Calculate the option payoffs at all nodes at $t=2,$ i$.$e$.$ calculate $f_{},f_{ud}$ and $f_{dd}.$

$($ii$)$ Construct a replicating portfolio using the underlying stock and by borrowing$/$lending at the risk$-$free rate to replicate the options payoffs. Describe your portfolio at $t=0$ and also at $t=1$ in both the up and down nodes. Based on your replicating portfolio, determine the value of the option at $t=0$ and at $t=1$ at each node, i$.$e$.$ calculate $f_u,f_d$ and $f_0.$

$($iii$)$ What are the risk$-$neutral probabilities, $q_u$ and $q_d?$ Confirm you get the same price for the put option using these risk$-$neutral probabilities.