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+Chapter 34: Futures and Futures Options 677 
+Thus, if one is long 8 calls with a delta of 0. 75, then that position has an EFP of 
+6 (8 x 0.75). This means that being long those 8 calls is the same as being long 6 
+futures contracts. 
+Note that in the case of stocks, the equivalent stock position formula has anoth­
+er factor shares per option. That concept does not apply to futures options, since 
+they are always options on one futures contract. 
+MATHEMATICAL CONSIDERATIONS 
+This brief section discusses modeling considerations for futures options and options 
+on physicals. 
+Futures Options. The Black model (see Chapter 33 on mathematical consider­
+ations for index options) is used to price futures options. Recall that futures don't pay 
+dividends, so there is no dividend adjustment necessary for the model. In addition, 
+there is no carrying cost involved with futures, so the only adjustment that one needs 
+to make is to use 0% as the interest rate input to the Black-Scholes model. This is an 
+oversimplification, especially for deeply in-the-money options. One is tying up some 
+money in order to buy an option. Hence, the Black model will discount the price 
+from the Black-Scholes model price. Therefore, the actual pricing model to be used 
+for theoretical evaluation of futures options is the Black model, which is merely the 
+Black-Scholes model, using 0% as the interest rate, and then discounted: 
+Call Theoretical Price = e-rt x Black-Scholes formula [r = O] 
+Recall that it was stated above that: 
+Futures call = Futures put + Future price - Strike price 
+The actual relationship is: ~ 
+Futures call= Futures put+ e-rt (Futures price - Strike price) 
+where 
+r = the short-term interest rate, 
+t = the time to expiration in years, and 
+e-rt = the discounting factor. 
+The short-term interest rate has to be used here because when one pays a debit 
+for an option, he is theoretically losing the interest that he could earn if he had that 
+money in the bank instead, earning money at the short-term interest rate. 
+The difference between these two formulae is so small for nearby options that 
+are not deeply in-the-money that it is normally less than the bid-asked spread in the 
+options, and the first equation can be used.