A two-step binomial tree is used to value an option on the Australian dollar (AUD). The strike price is 1.00 USD per AUD and the expiration date is in 6 months. Each step is 3 months. The current price of one AUD is 1.04 USD. The US risk-free rate is 2.0%, and the AUD risk-free rate is 2.5%. The exchange rate has a volatility of 6% per annum.
a. What is the proportional up movement, u, for the currency?
b. What is the probability of an up movement, p?
c. What is the price of an American call option on the currency?

a.) The proportional up movement , u, for the currency can be calculated using the following formula:

u = eStd Dev * Square root of t

u = e0.06*square root of 0.25

u = 1.0305

b.) Probability of up movement, p , = (a - d) / (u - d)

where   a = ert where r = 0.025, t = 0.25

a = e0.025*0.25 = 1.0063

d = 1 / u = 1 / 1.3050 = 0.7663

p = (1.0063-0.7663) / (1.3050-0.7663)

p = 0.46

1-p = 1-0.46 = 0.54

c)  Price of an American Call Option on the currency : we use binomial tree for that , as follows: The amounts below line indicate the option price and figures above line indicate the underlying asset price which is 0.55555

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