Question

Suppose the current exchange rate is $ 1.84 divided by pound, the interest rate in the United States is 5.24 %, the interest rate in the United Kingdom is 3.81 %, and the volatility of the $/£ exchange rate is 10.3 %. Use the Black-Scholes formula to determine the price of a six-month European call option on the British pound with a strike price of $ 1.84 divided by pound.

Answer 1

We use Black-Scholes Model to calculate the price of the currency call option.

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

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

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 in years

σ = volatility of underlying currency

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(1.84 / 1.84). We input the same formula into Excel, i.e. =LN(1.84 / 1.84)
• (r_d - r_f + σ²/2)*T = (0.0524 - 0.0381 + (0.103²/2)*0.50
• σ√T = 0.103 * √0.50

d₁ = 0.1346

d₂ = 0.0618

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) = 0.5535

N(d₂) = 0.5246

Now, we calculate the price of the call option as below:

C = (S₀ * e^-rf*T)*N(d₁) - (K * e^-rd*T)*N(d₂) , which is (1.84 * e^{(-0.0381 * 0.50)})*(0.5535) - (1.84 * e^{(-0.0524 * 0.50)})*(0.5246)    ==> $0.0589

Price of call option is $0.0589