Consider a discrete-time market ((,F,F,P),S)Tn=0 with one stock S and a fixed (continuously compounded interest rate of r>0 per period.

Part 1:

Provide a theoretical proof that for an American call option on S, there is no benefit for early exercise and the price of this option is the same as the price of its European counterpart.

Part 2:

Show that the same does not hold generally for an American put option. First, show where the theoretical proof from part 1 breaks down in this case. Then, give a numerical example with Snell envelope, optimal exercise strategy, Doob's decomposition, and hedging portfolio in the Binomial tree model with T=2 and r=10. Keep only two decimal points in all calculations.

Part 3:

Describe and price the so-called perpetual American put option within the Binomial tree model. Begin with short theory, then proceed with pricing the example from part 2 with the same r and K.