33 lines
2.0 KiB
Plaintext
33 lines
2.0 KiB
Plaintext
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- |