Options and mathematics:

The number of points assigned to the tasks in this part will be based on the amount of details provided in the solution. Use your own words and explain each step in the proof, in details. Do not skip calculations!

4. Let U₁ be the call stock option with strike K₁ and maturity T and U₂ be the physically-settled digital call option on the same stock with strike K₂ and maturity T. Decide whether the following statements are true or false in an arbitrage-free market and explain your answer:

a) If K₂ K₁, the value of U₂ is greater or equal than the value of U₁ for all t < T

b) If K₂ > K₁, the value of U₁ is greater or equal than the value of U₂ for all t < T