List the arguments (variables) of which a FX call or put option model price is a function.  How does the call and put premium change with respect to a change in the arguments?

Answer:  Both put and call options are functions of just six variables:  S_t, E, r_i, r_us, T ands.  While all else remains similar, the price of a European FX call (put) option will raise:

a) The larger (smaller) is S,

b) The smaller (larger) is E,

c) The smaller (larger) is r_i,

d) The larger (smaller) is r_us,

e) The larger (smaller) r_us is relative to r_i, and

f) The greater is s.

While r_us and r_i are not very much different in size, a European FX call and put will increase in price while the option term-to-maturity increases.  Though, when r_us is extremely much larger than r_i, a European FX call will increase in price, although the put premium will decrease, while the option term-to-maturity increases.  The opposed is true while r_i is very much greater as compared to r_us.  For American FX options the analysis is less complex.  As a longer term American option can be exercised on any date which a shorter term option can be exercised, or a some later date, it follows that the all else remaining similar, the longer term American option will sell at a price as a minimum as large as the shorter term option.