The Black Model on Options on Futures, the S & P 500 (a spot index) is presently at 1,860 and the 0.25 expiration futures contract is trading at 1,851.65. Suppose further that the exercise price is 1,860, the continuously compounded Rf is 0.2%, time to expiration is 0.25, volatility is 15%, and the dividend yield is 2.0%. Based on this information, the following results are obtained for options on the futures contract.

Calls                                  Puts
N(d1) = 0.491                   N(-d1) = 0.509
N(d2) = 0.461                   N(-d2) = 0.539
C = US$51.41                   P = US$59.76

To value a European call using the Black model, what values would the present value of exercise price and present value of futures price be multiplied with?

To value a European put using the Black model, what values would the present value of exercise price and present value of futures price be multiplied with?

What are the underlying and exercise prices to use in the Black futures options model?

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