35 lines
2.3 KiB
Plaintext
35 lines
2.3 KiB
Plaintext
Chapter 33: Mathematical Considerations for Index Products 64S
|
||
idend is subtracted from the index price and the model is evaluated using that adjust
|
||
ed stock price. With stock options, there was a second alternative - shortening the
|
||
time to expiration to be equal to the ex-date - but that is not viable with index options
|
||
since there are numerous ex-dates.
|
||
Let's look at an example using the same fictional dividend information and index
|
||
that were used in Chapter 30 on stock index hedging strategies.
|
||
Example: Assume that we have a capitalization-weighted index composed of three
|
||
stocks: AAA, BBB, and CCC. The following table gives the pertinent information
|
||
regarding the dividends and floats of these three stocks:
|
||
Dividend Days until
|
||
Stock Amount Dividend Float
|
||
AAA 1.00 35 50,000,000
|
||
BBB 0.25 60 35,000,000
|
||
CCC 0.60 8 120,000,000
|
||
Divisor: 150,000,000
|
||
One first computes the present worth of each stock's dividend, multiplies that
|
||
amount by the float, and then divides by the index divisor. The sum of these compu
|
||
tations for each stock gives the total dividend for the index. The present worth of the
|
||
dividend for this index is $0.8667.
|
||
Assume that the index is currently trading at 175.63 and that we want to evalu
|
||
ate the theoretical value of the July 175 call. Then, using the Black-Scholes model,
|
||
we would perform the following calculations:
|
||
1. Subtract the present worth of the dividend, 0.8667, from the current index price
|
||
of 175.63, giving an adjusted index price of 174.7633.
|
||
2. Evaluate the call's fair value using 17 4. 7633 as the stock price. All other variables
|
||
are as they are for stocks, including the risk-free interest rate at its actual value
|
||
(10%, for example).
|
||
The theoretical value for puts is computed in the same way as for equity
|
||
options, by using the arbitrage model. This is sufficient for cash-based index options
|
||
because it is possible - albeit difficult to hedge these options by buying or selling
|
||
the entire index. Thus, the options should reflect the potential for such arbitrage.
|
||
The put value should, of course, reflect the potential for dividend arbitrage with the
|
||
index. The arbitrage valuation model p"resented in Chapter 28 on modeling called for
|
||
the dividend to be used. For these index puts, one would use the present worth of |