“Master's” vs. “master”
Master's is an abbreviated form of master's degree, and it is considered an informal term by several dictionaries (e.g., Merriam-Webster Learner's Dictionary, MacMillan Dictionary).

Master is just an even more abbreviated form, which is usually not reported in the dictionaries with that meaning (but see note below).

In a formal letter I would use the full form master's degree, but should you choose to use one of the two abbreviated forms, try to be consistent and don't use both forms in the same text. Inconsistencies can be interpreted as a sign of sloppiness, and you don't want to give this impression while applying for a degree or a position. 

Thus, I would write:


  I am writing this letter to apply for XXX Master's degree in XXXX.


and


  I am absolutely sure that I will complete the Master's degree program in XXX.


Check also what is the denomination used in the university documentation: how do they refer to their own programs?

Note: The Merriam-Webster Learner's Dictionary reports for master also the following meaning (meaning n. 5 in the linked definition):


  5 [count] : a person who has received a master's degree