Question

2. Given the following information, determine the value of a currency call option and currency put option that expire in the next 6 months. (use the Table attached to the back of this exam)

- Spot price the Australian Dollar is $0.45

- The strike price of the option is $0.50

- The volatility of the Australian Dollar is 40%

- The expiration is 6 months

- The simple interest rate in the U.S. is 3%

- The simple interest rate in the U.K. is 4%

Answer 1

We use Black-Scholes Model to calculate the value of the currency call and put options.

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

C = (S₀ * e^-rfT)*N(d₁) - (K * e^-rdT)*N(d₂)

P = (K * e^-rdT)*N(-d₂) - (S₀ * e^-rfT)*N(-d₁)

where :

S₀ = current spot rate

K = strike price

N(x) is the cumulative normal distribution function

r_d = domestic risk-free simple interest rate

r_f = foreign risk-free simple interest rate

T is the time to maturity

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

d₂ = d₁ - σ√T

First, we calculate d₁ and d₂ as below :

• ln(S₀ / K) = ln(0.45 / 0.50). We input the same formula into Excel, i.e. =LN ln(0.45 / 0.50)
• (r_d - r_f + σ²/2)*T = (0.04 - 0.03 + (0.40²/2)*0.50
• σ√T = 0.40 * √0.50

d₁ = -0.2134

d₂ = -0.4962

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

N(d₁) = 0.4155

N(d₂) = 0.3099

N(-d₁) = 0.5845

N(-d₂) = 0.6901

Now, we calculate the values of the call and put options as below:

C = (S₀ * e^-rfT)*N(d₁) - (K * e^-rdT)*N(d₂) , which is (0.45 * e^{(-0.03 * 0.50)})*(0.4155) - (0.50 * e^{(-0.04 * 0.50)})*(0.3099)    ==> $0.0323

P = (K * e^-rdT)*N(-d₂) - (S₀ * e^-rfT)*N(-d₁), which is (0.50 * e^{(-0.04 * 0.50)})*(0.6901) - (0.45 * e^{(-0.03 * 0.50)})*(0.5845) ==> $0.0791

Value of call option is $0.0323

Value of put option is $0.0791