Question

Suppose that the continuously compounded risk-free interest rates for dollars and pounds are 0.04 and 0.06, respectively. A 6-month dollar-denominated European call option on pounds with strike price 1.45 costs $0.05, and a 6-month dollar-denominated European put option on pounds with strike price 1.45 costs $0.02.

a) Find the spot (current) exchange rate

b) Find the 6-month forward exchange rate on pounds (in dollars per pound).

Answer 1

(a) Strike Price of the Put and Call Option = $ 1.45 / pound = K, Call Premium (price) = $ 0.05 and Put Premium (Price) = $ 0.02

Let the current spot price of the underlying asset (underlying asset is pound in this case) be $ S / pound = Spot Exchange Rate

$ Interest Rate = 0.04

Using Put-Call Parity we have:

Present Value of S(current underlying asset price) discounted at the foreign currency's risk-free rate + Put Premium = Call Premium + Present Value of Option Strike Price discounted at domestic currency's risk-free rate

S / EXP(0.06 x 0.5) + 0.02 = 0.05 + 1.45 / EXP(0.04 x 0.5) (where 0.5 is the tenure of the option in years and 0.04 is the $ interest rate)

0.05 - 0.02 = S / EXP(0.06 x 0.5) + 1.45 / EXP(0.04 x 0.5)

0.03 = S / EXP(0.06 x 0.5) - 1.4213

S = EXP(0.06 x 0.5) x [0.03+1.4213]

S ~ $ 1.495 / pound

(b) Let the 6-month forward exchange rate be $ F/pound

Therefore, S x [EXP(0.04 x 0.5) / EXP(0.06 x 0.5)] = F

F = 1.495 x [EXP(0.04 x 0.5) / EXP(0.06 x 0.5)] = $ 1.481 / pound