648 Part V: Index Options and Futures 
Example: The following prices exist: 
ZYX Cash Index: 17 4.49 
ZYX December future: 177.00 
There are 80 days remaining until expiration, the volatility of ZYX is 15%, and 
the risk-free interest rate is 6%. 
In order to evaluate the theoretical value of a ZYX December 185 call, the fol­
lowing steps would be taken: 
l. Evaluate the regular Black-Scholes model using 185 as the strike, 177.00 as the 
stock price, 15% as the volatility, 0.22 as the time remaining (80/365), and 0% as 
the interest rate. Note that the futures price, not the index price, is input to the 
model as stock price. 
Suppose that this yields a result of 2.05. 
2. Discount the result from step l: 
Black Model call value = e-(.0 6 x 0-22) x 2.05 
= 2.02 
In this case, the difference between the Black model and the Black-Scholes 
model is small (3 cents). However, the discounting factor can be large for longer-term 
or deeply in-the-money options. 
The other items of a mathematical nature that were discussed in Chapter 28 on 
mathematical applications are applicable, without change, to index options. Expected 
return and implied volatility have the same meaning. Implied volatility can be calcu­
lated by using the Black-Scholes formulas as specified above. 
Neutral positioning retains its meaning as well. Recall that any of the above the­
oretical value computations gives the delta of the option as a by-product. These deltas 
can be used for cash-based and futures options just as they are used for stock options 
to maintain a neutral position. This is done, of course, by calculating the equivalent 
stock position (or equivalent "index" or "futures" position, in these cases). 
FOLLOW-UP ACTION 
The various types of follow-up action that were applicable to stock options are avail­
able for index options as well. In fact, when one has spread options on the same 
underlying index, these actions are virtually the same. However, when one is doing 
inter-index spreads, there is another type of follow-up picture that is useful. The rea-