Question

Put Option Valuation.  Determine the value of a put option on the CAD that has the following characteristics: (a) it is of type European (b) it matures in 6 months (c) the strike price is USD 0.95. In the spot market the CAD is trading at USD 0.92. The US and Canadian interest rates are 4% and 6% respectively. The CAD has an annual standard deviation of 12%.

a) Write out fully the general form of the Black-Scholes formula for a European put.

b) Write the BS formula for the European put by plugging in for the specific parameter values of the European put given above.

c) Provide the value (price) for the European put given above.

d)       Write down the expression for the put-call parity relationship.

e) Use the put-call parity relationship to find the price of the corresponding call option, that is, the call option with similar parameters. Write down the put-call parity condition by plugging in for the specific parameter values of the European option given above with the only unknown variable being the call option price.

f) Provide the value (price) for the European call option given by the put-call parity condition.

Answer 1

a]

The Black-Scholes Model to calculate the value of a currency put option is :

The domestic currency value of a put option into the foreign currency is:

P = (K * e-rd*T)*N(-d2) - (S0 * e-rf*T)*N(-d1)

where :

S0 = current spot rate

K = strike price

N(x) is the cumulative normal distribution function

rd = domestic risk-free simple interest rate. This is the US interest rate of 4%, or 0.04.

rf = foreign risk-free simple interest rate. This is the Canada interest rate of 6%, or 0.06.

T is the time to maturity in years. This is (6/12) , or 0.50. (since it is a 6-month option).

σ = volatility of underlying currency. This is the standard deviation of 12%, or 0.12.

d1 = (ln(S0 / K) + (rd - rf + σ2/2)*T) / σ√T

d2 = d1 - σ√T

b]

First, we calculate d1 and d2 as below :

· ln(S0 / K) = ln(0.92 / 0.95). We input the same formula into Excel, i.e. =LN(0.92 / 0.95)

· (rd - rf + σ2/2)*T = (0.04 - 0.06 + (0.122/2)*0.50

· σ√T = 0.12 * √0.50

d1 = -0.4536

d2 = -0.5384

N(-d1) and N(-d2) are calculated in Excel using the NORMSDIST function and inputting the value of d1 and d2 into the function.

N(-d1) = 0.6749

N(-d2) = 0.7049

Now, we calculate the values of the put option as below:

P = (K * e-rd*T)*N(-d2) - (S0 * e-rf*T)*N(-d1), which is (0.95 * e(-0.04 * 0.50))*(0.7049) - (0.92 * e(-0.06 * 0.50))*(0.6749)  

c]

Value of put option is $0.0538

d]

As per the put-call parity equation, C + (K/(1 + r)t) = P + S,

where C = price of call option,

P = price of put option,

S = current stock price

K = strike price of option

r = risk free rate

t = time to expiration in years