Question

Suppose the 120-day futures price on gold is $110 per ounce and the volatility 20%. Interest rates are 3.5%. What is the price of a $115 strike call futures option that expires in 120 days?

ANSWER: $2.99

Answer 1

The futures price of a commodity depends on the interest rates, time and current price (spot price)

Futures Price = Spot price* (1+ r* time/ Annual Base)

r= interest rate

The annual base is the number of days in a year, 365.

Futures price = 110

days = 120

t = days/annual base = 120/365 = 0.3287

Spot price = Futures Price/ (1+ r* time/ Annual Base)

Spot Price = 110/(1+ 3.5%*120/365)

Spot Price = S=$108.7486

Call Option Price = S*N(d1) - K*exp(-rt)*N(d2)

where d1 = ln(S/K) - (r+volatility^2/2)*t / volatility*sqrt(t)

d2=d1- volatility*sqrt(t)

d1 = ln(108.7486/115) - (3.5%+20%^2/2)/ 20%*sqrt(0.3287)

d1 = -0.32972

d2= -0.32972-20%*sqrt(0.3287)

N(d1) = 0.37

N(d2) = 0.3283

call option price = 108.74*0.37 - 0.3283*115*exp(-3.5%*0.3287)

=0.2993