Problem $2(20$ pts$).$ Given an $N-$period Binomial Asset Pricing Model and a stock price process $(S_n),n=0,1,$dotsN, the Max$-$to$-$date process is defined $M_n=ma{x}_k=0^n{S}_k.$

We assume now that $N=3,S_0=8,u=2,d=\frac{1}{2}$ and that the risk$-$free rate is $r=\frac{1}{4}.$

Compute $V_0,$ the Risk$-$neutral at time $0$ of the Lookback option with payoff at time $3$ given by $V_3=M_3-S_3.$

Consider now the $2-$dimensional adapted process $\{(S_n,M_n),n=0,1,$dotsN$\},$ where $S_n$ is stock price process at time $n$ and $M_n$ the Max$-$to$-$date process at at time $n.$ Prove that this $2-$dimensional adapted process is a Markov process.