Question

3) Suppose S.(SVC) is 1.10 and in the next year it either goes up to 1.20 or down to 1.05. What is the price of a call option on 1000 USD running one year with strike price 1.10, if the anual rikss free rate of the USD is 2% and 0.5% for the EURO? 4) Take all the numbers from above (Problem3) and price a 1 year Future option with the strike price of F.(S/E) if the market is arbitrage free.

Answer 1

3) S₀ = 1.10, Strike price is 1.10, 50% probability of going up to 1.20 (S₊​​​​​​) and 50% probability of going down to 1.05 (S_-​​​​​​)

Since, the call option is on USD, the buyer of the option would want the exchange rate to go up, i.e exchange more USD per Euro

The payoff if the exchange rate goes up = max(0, 1.20 - 1.10) = 0.10

The payoff if the exchange rate goes down = max(0, 1.05 - 1.10) = 0

In the second case the buyer will let the option expire worthless.

Value of call option in 1 year = 50% * 0.10 + 50% * 0 = 0.05 + 0 = 0.05

We need to discount the value of call option at the risk free rate of 2%

Value of call option = 0.05/1.02 = 0.049

4) S₀ = 1.10 ($/€). Here the base currency is Euro and the price currency is USD.

Thus, risk free rate of USD (price currency) = R_PC = 2%

Risk free rate of Euro (base currency) = R_BC = 0.5%

T is time period in days but since here it is 1 year future option T=1

Future currency rate (F_T​​​​​​) = S₀ * (1+R_PC​​​​​​) ^T/(1+R_BC)^T​​​​​​

F_T = 1.10 * (1 + 2%)¹/(1 + 0.5%)¹ = 1.10 * (1.02)/(1.005) = 1.10 * 1.0149 = 1.1164

Future currency rate = 1.1164 = 1.12 ($/€)