1. Options on Stock Indices, Currencies, and Futures Chapter 13

2. STOCK INDEXES (INDICES)
A STOCK INDEX IS A SINGLE NUMBER BASED ON INFORMATION ASSOCIATED WITH A
PORTFOILO OF STOCKS.
A STOCK INDEX IS SOME KIND OF AN AVERAGE OF THE PRICES AND THE QUANTITIES OF THE SHARES OF THE STOCKS THAT ARE INCLUDED IN THE PORTFOLIO THAT UNDERLYING THE INDEX.

3. STOCK INDEXES (INDICES)
THE MOST USED INDEXES ARE
A SIMPLE PRICE AVERAGE
AND
A VALUE WEIGHTED AVERAGE.

4. STOCK INDEXES - THE CASH MARKET
A. AVERAGE PRICE INDEXES: DJIA, MMI:
N = The number of stocks in the index
P = Stock market price
D = Divisor
INITIALLY, D = N AND THE INDEX IS SET AT A GIVEN LEVEL. TO ASSURE INDEX CONTINUITY, THE DIVISOR IS ADJUSTED OVER TIME.

5. EXAMPLES OF INDEX ADJUSMENTS
STOCK SPLITS: 2 FOR 1: 1. 2.
Before the split:
( ) /5 = I = 40 and D = 5.
An instant later:
( )/D = The new divisor is D = 4.5

6. CHANGE OF STOCKS IN THE INDEX1. 2.
Before the change:
( )/4.5 = 40
I = 40 and D =4.5.
An instant later:
( )/D = 40
The new divisor is D = 7.75

7. A STOCK DIVIDEND DISTRIBUTION
Firm 4 distributes 66 2/3% stock dividend. Before the distribution:
( )/7.75 =
D = 7.75.
An instant later:
( )/D =
The new divisor is D =

8. STOCK # 2 SPLIT 3 FOR 1.
Before the split:
( )/ =
An instant later:
( )/D =
The new Divisor is D =

9. ADDITIONAL STOCKS 1. 2.
Before the stock addition:
( )/ =
An instant later:
( )/D =
D =

10. As a result, the new divisor for the DJIA became:
A price adjustment of Altria Group Inc. (MO), (due to a distribution of Kraft Foods Inc. (KFT) shares,) was effective for the open of trade on trade date April 2, 2007.
As a result, the new divisor for the DJIA became: 
D =

11. VALUE WEIGHTED INDEXES
S & P500, NIKKEI 225, VALUE LINE
B = SOME BASE TIME PERIOD
INITIALLY t = B THUS, THE INITIAL INDEX VALUE IS SOME ARBITRARILY CHOSEN VALUE: M.

12. The S&P500 index base period was 1941-1943
and its initial value was set at
M = 10.
The NYSE index base period was
Dec. 31, 1965
M = 50.

13. The rate of return on the index:
The return on a value weighted index in any period t, is the weighted average of the individual stock returns; the weights are the dollar value of the stock as a proportion of the entire index value.

14. THE RATE OF RETURN ON THE INDEX

16. THE BETA OF A PORTFOLIO
THEOREM:
A PORTFOLIO’S BETA IS THE WEIGHTED AVERAGE OF THE BETAS OF THE STOCKS THAT COMPRISE THE PORTFOLIO.
THE WEIGHTS ARE THE DOLLAR VALUE WEIGHTS OF THE STOCKS IN THE PORTFOLIO.

17. Assume that the index is a well diversified
THE BETA OF A PORTFOLIO
Proof:
Assume that the index is a well diversified
portfolio. I.e., the index represents the market portfolio.
Let: P denote the portfolio underlying the
Index, I.
j denote the individual stock; j = 1, 2, …,N.

18. By definition:

19. STOCK PORTFOLIO BETA
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
P = (.044)(1.00) + (.152)(.8) + (.046)(.5) + (.061)(.7)
+ (.147)(1.1) + (.178)(1.1) + (.144)(1.4) + (.227)(1.2)
= 1.06

20. A STOCK PORTFOLIO BETA
STOCK NAME PRICE SHARES VALUE WEIGHT BETA
P = .122(.95) (1.1) (.85) (1.15) (1.15) (1.0)
+ .263(.85) (.75)
= .95

21. Sources of calculated Betas and calculation inputs
Example: ß(GE) 6/20/00
Source ß(GE) Index Data Horizon
Value Line Investment Survey NYSECI Weekly Price yrs (Monthly)
Bloomberg S&P500I Weekly Price 2 yrs (Weekly)
Bridge Information Systems S&P500I Daily Price 2 yrs (daily)
Nasdaq Stock Exchange
Media General Fin. Svcs. (MGFS) S&P500I Monthly P ice 3 (5) yrs Quicken.Excite.com
MSN Money Central
DailyStock.com
Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly)
S&P Personal Wealth
S&P Company Report)
Charles Schwab Equity Report Card 1.20
S&P Stock Report
AArgus Company Report S&P500I Daily Price 5 yrs (Daily)
Market Guide S&P500I Monthly Price 5 yrs (Monthly)
YYahoo!Finance
Motley Fool

22. (INDEX VALUE)($MULTIPLIER) ACCOUNTS ARE SETTLED BY CASH
STOCK INDEX OPTIONS
ONE CONTRACT VALUE =
(INDEX VALUE)($MULTIPLIER)
One contract = (I)($m)
ACCOUNTS ARE SETTLED BY CASH

23. EXAMPLE: Options on a stock index
MoneyGone, a financial institution, offers its clients the following deal:
Invest $A ≥ $1,000,000 for 6 months. In 6 months you receive a guaranteed return:
The Greater of {0%, or 50% of the return on the SP500I during these 6 months.}
For comparison purposes: The annual risk-free rate is 8%. The SP500I dividend payout ratio is q = 3% and its annual VOL σ= 25%.

24. MoneyGone offer:
Deposit: $A now.
Receive: $AMax{0, .5RI} in 6 months. Denote the date in six month = T.
Rewrite MoneyGone offer at T:

25. Calculate this options value based on:
The expression:
is equivalent to the at-expiration cash flow of an at-the money European call option on the index, if you notice that K = I0.
Calculate this options value based on:
S0 = K = I0; T – t = .5; r = .08; q = .03 and σ = .25. Using DerivaGem:
c =
Thus, MoneyGone’s promise is equivalent

26. Therefore, the investor’s initial deposit is only 95.9315% of A.
to giving the client NOW, at time 0, a value of: (.5)(.08137)($A) = $ A.
Therefore, the investor’s initial deposit is only % of A.
Investing $ A and receiving $A in six months, yields a guaranteed return of:
= 8.3%

27. STOCK INDEX OPTIONS FOR STATIC PORTFOLIO INSURANCE Decisions:
How many puts to buy?
Which exercise price will guarantee
a desired level of protection?
The answers are not easy because the
index underlying the puts is not the same
as portfolio to be protected.

28. The protective put with a single stock:
STRATEGY
ICF
AT EXPIRATION
ST < K
ST ≥ K
Hold the stock
Buy put
-St -p ST
K - ST
TOTAL
-St – p K

29. The protective put consists of holding the portfolio and purchasing n puts.
STRATEGY
ICF (t = 0)
AT EXPIRATION (T = 1)
I1 < K
I1 ≥ K
Hold the portfolio
Buy n puts
-V0
-n P($m) V1
n(K- I1)($m)
TOTAL
-V0 –nP($m)
V1+n($m)(K- I1)

30. THE CAPITAL ASSET PRICING MODEL.
WE USE
THE CAPITAL ASSET PRICING MODEL.
For any security i,the expected excess return on the security and the expected excess return on the market portfolio are linearly related by their beta:

31. THE INDEX TO BE USED IN THE STRATEGY, IS TAKEN TO BE A PROXY FOR THE MARKET PORTFOLIO, M. FIRST, REWRITE THE ABOVE EQUATION FOR THE INDEX I AND ANY PORTFOLIO P :

32. Second, rewrite the CAPM result, with actual returns:
In a more refined way, using V and I for the portfolio and index market values, respectively:

33. The ratio V1/ V0 indicates the portfolio required protection ratio.
NEXT, use the ratio Dp/V0 as the portfolio’s annual dividend payout ratio qP and DI/I0 the index annual dividend payout ratio, qI.
The ratio V1/ V0 indicates the portfolio required protection ratio.

34. The manager wants V1, to drop down to
For example:
The manager wants V1, to drop down to
No less than 90% of the initial portfolio
market value, V0: V1 = V0(.9).
We denote this desired level by:
(V1/ V0)*.
This is the manager’s decision variable about the amount of protection to obtain by the protective puts.

35. 1. The number of puts is:

36. 2. The exercise price, K, is determined by substituting I1 = K and the required level, (V1/ V0)* into the equation:
and solving for K:

37. We rewrite the Profit/Loss table for the protective put strategy:
INITIAL CASH FLOW
AT EXPIRATION
I1 < K
I1 ≥ K
Hold the portfolio
Buy n puts
-V0
-n P($m) V1
n(K - I1)($m)
TOTAL
-V0 - nP($m)
V1+n($m)(K - I1)
We are now ready to calculate the floor level of the portfolio: V1+n($m)(K- I1)

38. Min portfolio value = V1+n($m)(K- I1)
We are now ready to calculate the floor level of the portfolio:
Min portfolio value = V1+n($m)(K- I1)
This is the lowest level that the portfolio value can attain. If the index falls below the exercise price and the portfolio value declines too, the protective puts will be exercised and the money gained may be invested in the portfolio and bring it to the value of:
V1+n($m)K- n($m)I1

39. Substitute for n:

40. To substitute for V1 we solve the equation:

41. 3. Substitution V1 into the equation for the Min portfolio value
The desired level of protection is made at time 0. This determines the exercise price and management can also calculate the minimum portfolio value.

42. The STATIC Portfolio Insurance strategy is to BUY puts
Conclusion
The STATIC Portfolio Insurance strategy is to BUY puts
and hold them
for the entire insurance period.

43. EXAMPLE: A portfolio manager expects the market to fall by 25% in the next six months. The current portfolio value is $25M. The manager decides on a 90% hedge by purchasing 6-month puts on the S&P500 index. The portfolio’s beta with the S&P500 index is 2.4. The S&P500 index stands at a level of 1,250 points and its dollar multiplier is $100. The annual risk-free rate is 10%, while the portfolio and the index annual dividend payout ratios are 5% and 6%, respectively. The data is summarized below:

44. Solution: Purchase

45. The exercise price of the puts is:
Solution: Purchase n = 480 six-months puts with exercise price K = 1,210.

47. CONCLUSION:
Hold the portfolio.
Purchase 480, 6-months protective puts on the S&P500 index,
with the exercise price K = 1210.
The portfolio value, currently $25M, will not fall below $22,505,000 during the insurance period of the next six months.

48. Example: Portfolio insurance for 3 months
Solution: Purchase

49. The exercise price of the puts is:
Solution: Purchase n = 10 three-months puts with exercise price K = 960.

51. CONCLUSION:
Hold the portfolio.
Purchase 10, 3-months protective puts on the S&P500 index, with the exercise price K = 960.
Insurance:
The portfolio value, currently $500,000, will not fall below $450,000 in three months.

52. Table 13.2
ΒP = V0 = 500, n = 10 puts
[V1/V0]* V1 K
1.14
570,000
1,080
1.06
530,000
1,040
.98
490,000
1,000
.90
450,000
960
.82
410,000
920
.74
370,000
880

53. A SPECIAL CASE:
In the case that
a) β = 1 and b) qP =qI,
the portfolio is statistically similar to the
index. In this case:

54. Assume that in the above example
βp = 1 and qP =qI, then:

55. Example: (page 295,6)
βp = 1 and qP =qI, then:

56. Dynamic Portfolio Insurance