Question

Suppose that the spot price of the Canadian dollar is U.S. $0.95 and that the Canadian dollar/U.S. dollar exchange rate has a volatility of 8% per annum. The risk-free rates of interest in Canada and the United States are 3% and 6% per annum, respectively. Calculate the value of a European call option to buy one Canadian dollar for U.S. $0.95 in nine months. Use put-call parity to calculate the price of a European put option to sell one Canadian dollar for U.S. $0.95 in nine months.

What is the CAD-denominated price of a European call option to buy one U.S. dollar for Canadian $1.0526(=1/0.95) in nine months?

Answer 1

Using Black Scholes model, we have:

S (current rate) = 0.95; K (strike rate) = 0.95; r (US risk free rate) = 6%; rf (Canadian risk free rate) = 3%; s (volatility)= 8%; t (time to expiry)= 9/12 = 0.75

d1 = {ln(S/K) + (r -rf +s^2/2)t}/(s(t^0.5))

= {ln(0.95/0.95) + (6% - 3% + (8%^2)/2)*0.75}/(8%*0.75^0.5)

= 0.3594

d2 = d1 - (s(t^0.5))

= 0.3594 - (8%*0.75^0.5) = 0.2901

N(d1) = 0.6404; N(d2) = 0.6141

Value of call option C = S*e^(-rf*t)*N(d1) - N(d2)*K*(e^(-r*t))

C = (0.95*e^(-3%*0.75)*0.6404) - (0.6141*0.95*e^(-6%*0.75))

= US$ 0.0370

Using put-call parity, we have:

P + S*e^(-rf*t) = C + K*e^(-r*t)

P = 0.0370 + (0.95*exp^(-6%*0.75)) - (0.95*exp^(-3%*0.75))

= US$ 0.0164

CAD denominated European call option: S = 1/0.95 = 1.0526; k = 1.0526; r = 3%; rf = 6%;s = 8%;  t = 0.75

d1 = {ln(S/K) + (r -rf +s^2/2)t}/(s(t^0.5))

= {ln(1.0526/1.0526) + (3% - 6% + (8%^2)/2)*0.75}/(8%*0.75^0.5)

= -0.2901

d2 = d1 - (s(t^0.5))

= -0.2901 - (8%*0.75^0.5) = -0.3594

N(d1) = 0.3859; N(d2) = 0.3596

Value of call option C = S*e^(-rf*t)*N(d1) - N(d2)*K*(e^(-r*t))

C = (1.0526*e^(-6%*0.75)*0.3859) - (0.3596*1.0526*e^(-3%*0.75))

= CAD 0.0181

(Note: Price of a call option to buy $0.95 with 1CAD will be the put option value to buy 1CAD with $0.95, so its value will be $0.0164)