The current exchange rate is AU$1 buys US$0.83. In your answers, you must always

Question

The current exchange rate is AU$1 buys US$0.83. In your answers, you must always specify if the $ are AU or US.

(a) Express this currency exchange in direct form (that is, how many Australian dollars can US$1 buy?).

(b) Using Coss–Ross–Rubinstein notation, the initial exchange rate X is your solution from part (a), the up factor is u = 1.04 and the down factor is d = 0.98. Construct a three-step binomial pricing tree for this exchange rate using u, d and X.

(c) Construct three-step binomial pricing trees for the following options, using the exchange rate in part (b) as the underlying asset. The domestic return over each time step is 1.060 and the US return over each time step is 1.055:

(i) A European call with strike rate k = 1.25 (here, use face value F = US$1).
(ii) A European put with strike rate k = 1.25 (here, use face value F = US$1).

Explanation / Answer

a) Given is 1AUD=0.83 USD, It means with 1 AUD we can buy 0.83 USD.

So, simply to find out how many AUD we can buy from 1 USD we will divide 1 by 0.83

i.e. 1/0.83=1.2048 AUD. So it means we can buy 1.2048 AUD from 1 USD.

b) So, as the spot rate we have found out in point (a) is 1.2048 it can either go up by 1.04 or go down by 0.98. So it means that the Forward rate for Period 1(t=1) will be like this:-

F=1.2048*1.04=1.253 OR F=1.2048*0.98=1.181

So at t=1 the Rates will be 1.253 and 1.181.

With same logic we can also calculate the rates at t=2 where there will be 4 rates (This is the 2nd part of Binomial Tree).

At t=2,

When U=1.04, F1=1.253*1.04=1.30312, F2=1.181*1.04=1.22824

When d=0.98, F3=1.253*0.98=1.2279, F4=1.181*0.98=1.1574

Similarly with this 4 rates we also find out the forward rates at 3 rd step of Binomial Tree, there will be 8 rates.

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