The three-month S&P 500 futures contract is trading at a level of 1250. The rate of interest is 2%. The average rate of dividends for stocks in the index is 3%. Index volatility is 20%. What is the Black-Scholes price of a one-year at-the-money put option on the futures?
A) $97.34
B) $97.60
C) $98.33
D) $99.12

Question

Quizplus · Accepted Answer

First, we need to calculate the forward price of the S&P 500 index as:

F = S * e^(r - q) * T
where F is the forward price, S is the current price of the index (1250), r is the interest rate (2%), q is the average dividend yield (3%), and T is the time to expiration in years (1).

F = 1250 * e^(0.02 - 0.03) * 1 = 1225.77

Next, we need to estimate the option's implied volatility. We can do this using the Black-Scholes formula and the current market price of at-the-money call options on the S&P 500 index. Assuming that the implied volatility for the put option is the same as that for the call option, we can use an implied volatility of 20%.

Finally, we can use the Black-Scholes formula to calculate the price of the put option:

C = S * N(d1) - F * e^(-rT) * N(d2)
P = F * e^(-rT) * N(-d2) - S * N(-d1)

where C is the price of the call option, P is the price of the put option, N(d) is the standard normal cumulative distribution function, and d1 and d2 are given by:

d1 = [ln(S/F) + (r - q + σ^2/2)T] / (σ√T)
d2 = d1 - σ√T

Plugging in the appropriate values, we get:

d1 = [ln(1250/1225.77) + (0.02 - 0.03 + 0.2^2/2) * 1] / (0.2 * sqrt(1)) = 0.245
d2 = 0.245 - 0.2 * sqrt(1) = 0.045

N(d1) = 0.5987
N(d2) = 0.5167

P = 1225.77 * e^(-0.02 * 1) * 0.4833 - 1250 * 0.5987 = $97.60

Therefore, the answer is B.

The Three-Month S&P 500 Futures Contract Is Trading at a Level