The work requires python code and graphs.

In this part, you will evaluate an American put option. Assume that T = 1, S_0 = 10, $\mu$ = 5%, $\sigma$= 20%, and the risk-free rate r = 2%. Use N = 5000 and a strike K = 10.

(a) Implement the valuation and exercise boundary of the American put option using two methods: (i) with B as the numeraire, and (ii) with S as the numerarie.

i. Plot the exercise boundary as a function of t for both approaches. ii. Generate two sample paths where in sample path 1) the option is exercised early (say around t = 0.5)

path 2) the option is not exercised. iii. Along the two sample paths above, plot the hedging strategy that a trader would use to hedge the option as a function of time.

iv. Illustrate how the results in i,ii, and iii vary as volatility and risk-free rate change (pari-wise). [For example, $\sigma$ = 10%; 20%; 30% and r = 0%; 2%; 4%]