On page 339 in the Derivatives book (exercise 1) there is a small exercise on interest rate cap: [question removed by moderator] The answer is that the payoff is 17500 in period 1 and 5000 in period 4. 0 In the other two periods.

My question is why are these payoffs not discounted? Because I have read also in the book: " Interest rate options pay one period after exercise" so should not these payoffs be discounted 0,25 year?

Becease you are calculating the PAYOFFS and not the actual value of the caplets/cap. To find the actual value of the caplets/cap you would need to take the PV of the payoffs. Its a very subtle distinction.