Question

The current price of gold is $1688 per ounce. The volatility of gold price is 20% per annum. The continuously-compounded risk-free rate is 5% per annum. What is the value of a 3-month call option on an ounce of gold with a strike price of $1750 according to the BSM model?

Answer 1

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

The value of a call option is:

C = (S0 * N(d1)) - (Ke-rT * N(d2))

where :

S0 = current spot price

K = strike price

N(x) is the cumulative normal distribution function

r = risk-free interest rate

T is the time to expiry in years

d1 = (ln(S0 / K) + (r + σ2/2)*T) / σ√T

d2 = d1 - σ√T

σ = standard deviation of underlying stock returns

First, we calculate d1 and d2 as below :

· ln(S0 / K) = ln(1688 / 1750). We input the same formula into Excel, i.e. =LN(1688/1750)

· (r + σ2/2)*T = (0.05 + (0.202/2)*0.25

· σ√T = 0.20 * √0.25

d1 = -0.1857

d2 = -0.2857

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) = 0.4263

N(d2) = 0.3875

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

C = (S0 * N(d1))   - (Ke-rT * N(d2)), which is (1688 * 0.4263) - (1750 * e(-0.05 * 0.25))*(0.3875)    ==> $49.8675

Value of call option is $49.8675