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.