Exercise $1$

Consider a binomial pricing market model over $N=3$ periods with a stock price $S_0=100$ and a risk$-$free interest rate $r=0\mathrm{.}25.$ The sizes of the up and down factors are $u=2$ and $d=0\mathrm{.}5.($These values are of course highly unrealistic, but make the calculations reasonably nice.$)$

On this market consider the following derivative

$($C$)$ European call option with the strike price $K=100.$

Draw a binomial tree and denote the stock prices $S_0,S_1,S_2$ and $S_3$ in it$.$

Find the payoff $V_3({}_1{}_2{}_3)$ of the derivative for all states of the world $($coin toss sequences$){}_1{}_2{}_3.$

What are the risk neutral probabilities? Use them to compute the the value of the derivative backwards in time, that is compute $V_2(HH),V_2(HT),V_2(TH),V_2(TT),V_1(H),V_1(T)$ and $V_0.$

Replicate the payoff of the derivative with a portfolio. How many units of the stock need to be held at each point in time?

Try doing the replication backwards in time: start at the time $N-1=2$ and find a portfolio replicating the payoff $V_3,$ for each node $(HH,HT,TH,TT).$ Then move to the time $N-2=1$ and there find a portfolio replicating the values $x_2,$ and so on$.$