Options on Stock Indices, Currencies, and Futures Chapter 13

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.

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

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.

EXAMPLES OF INDEX ADJUSMENTS STOCK SPLITS: 2 FOR 1: 1. 2. Before the split: (30 + 40 + 50 + 60 + 20) /5 = 40 I = 40 and D = 5. An instant later: (30 + 20 + 50 + 60 + 20)/D = 40 The new divisor is D = 4.5

CHANGE OF STOCKS IN THE INDEX 1. 2. Before the change: (31 + 19 + 53 + 59 + 18)/4.5 = 40 I = 40 and D =4.5. An instant later: (31 + 149 + 53 + 59 + 18)/D = 40 The new divisor is D = 7.75

A STOCK DIVIDEND DISTRIBUTION Firm 4 distributes 66 2/3% stock dividend. Before the distribution: (32 + 113 + 52 + 58 + 25)/7.75 = 36.129 D = 7.75. An instant later: (32 + 113 + 52 + 34.8 + 25)/D = 36.129 The new divisor is D = 7.107857587.

STOCK # 2 SPLIT 3 FOR 1. Before the split: (31 + 111 + 54 + 35 + 23)/7.107857587 = 35.7351 An instant later: (31 + 37 + 54 + 35 + 23)/D = 35.73507 The new Divisor is D = 5.0370644.

ADDITIONAL STOCKS 1. 2. Before the stock addition: (30 + 39 + 55 + 33 + 21)/5.0370644 = 35.338 An instant later: (30 + 39 + 55 + 33 + 21 + 35)/D = 35.338 D = 6.0275.

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 = 0.123051408.  

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.

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.

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.

THE RATE OF RETURN ON THE INDEX

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.

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.

By definition:

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

A STOCK PORTFOLIO BETA STOCK NAME PRICE SHARES VALUE WEIGHT BETA P = .122(.95) + .187(1.1) + .203(.85) + .048(1.15) + .059(1.15) + .076(1.0) + .263(.85) + .042(.75) = .95

Sources of calculated Betas and calculation inputs Example: ß(GE) 6/20/00 Source ß(GE) Index Data Horizon Value Line Investment Survey 1.25 NYSECI Weekly Price 5 yrs (Monthly) Bloomberg 1.21 S&P500I Weekly Price 2 yrs (Weekly) Bridge Information Systems 1.13 S&P500I Daily Price 2 yrs (daily) Nasdaq Stock Exchange 1.14 Media General Fin. Svcs. (MGFS) S&P500I Monthly P ice 3 (5) yrs Quicken.Excite.com 1.23 MSN Money Central 1.20 DailyStock.com 1.21 Standard & Poors Compustat Svcs S&P500I Monthly Price 5 yrs (Monthly) S&P Personal Wealth 1.2287 S&P Company Report) 1.23 Charles Schwab Equity Report Card 1.20 S&P Stock Report 1.23 AArgus Company Report 1.12 S&P500I Daily Price 5 yrs (Daily) Market Guide S&P500I Monthly Price 5 yrs (Monthly) YYahoo!Finance 1.23 Motley Fool 1.23

(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

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%.

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

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 = .08137. Thus, MoneyGone’s promise is equivalent

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) = $.040685A. Therefore, the investor’s initial deposit is only 95.9315% of A. Investing $.959315A and receiving $A in six months, yields a guaranteed return of: = 8.3%

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.

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

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)

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:

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 :

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:

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.

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.

1. The number of puts is:

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

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)

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

Substitute for n:

To substitute for V1 we solve the equation:

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.

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.

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:

Solution: Purchase

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

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.

Example: Portfolio insurance for 3 months Solution: Purchase

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

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.

Table 13.2 ΒP = 2. V0 = 500,000. 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

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:

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

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

Dynamic Portfolio Insurance