Lec 15A: Options on Stock Indices
(Hull, Ch. 15)
Index Options are traded on the CBOE. Most popular:
European Calls and Puts on
▸ SP500 Index (SPX), and SP100(OEX) (multiplier for both = 100)
▸ DJ Index (DJX)
▸ NASDAQ 100 Index (NDX)
All settled in cash
Lec 15a Options on Stock Indices
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Example 1: Speculation with Index Options.
▸ Suppose you expect good earnings for many stocks at the end of
the current quarter.
▸ You may gamble on this outlook by going long either
(A) iShares SP100 (ticker symbol OEF, this is an ETF).
current price is $70/ share
➟ 4,000 shares × $70 = $280,000.
Or
(B) 40 Dec OEF at-the-money calls priced at $2
➟ 40 contracts × $2 × 100 = $8,000
Lec 15a Options on Stock Indices
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Cash Flow Analysis at T = Expiration. Possible scenarios:
ST
Exercise?
Call Value: CT
Profit/Loss
on Call
Profit/Loss on
OEF
➀ $60
No
0
-$8,000
-$40,000
➁ $65
No
0
-$8,000
-$20,000
➂ $70
No/Yes
0
-$8,000
0
➃ $75
Yes
$20,000
+$12,000
+$20,000
➄ $80
Yes
$40,000
+$32,000
+$40,000
So, what do we learn from this exercise?
▸ The Option requires only $8,000 (vs. $280,000 for the ETF)
▸ It may be easier to gamble on a large portfolio of stocks than
an individual stock.
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Example 2: (Risk Management)
Use a Put Option to Hedge Portfolio risk. (Hull, 15.1)
Assume at t=0 (now)
▸ SP500 = 1,000 and the SMF portfolio value = $500,000
▸ Suppose the portfolio managers can tolerate a max loss of 11%.
➟ Min Portfolio Value = $445,000
▸ To hedge downside risk,
➟ Buy Puts on the SP500. Key Questions: How many Puts? K = ?
▸ Assume β for the SMF Portfolio = 2 .
Div yield = 0 for both the SP and SMF.
Rf = 0.03 over next 3 months
▸ Use CAPM: RORSMF = Rf +( RORSP - Rf )β,
Lec 15a Options on Stock Indices
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Cash Flow Analysis at T = Expiration. Possible scenarios:
SP500
IndexT
RORSP
RORSMF
SMF Value
CF from +PT
Insured Value =
SMF ValueT + PT
➀ 880
-12%
-27%
$365,000
$80,000
$445,000
➁ 920
-8%
-19%
405,000
40,000
445,000
➂ 960
-4%
-11%
455,000
0
445,000
➃ 1,000
0
-3%
485,000
0
485,000
➄ 1,040
4%
5%
525,000
0
525,000
To find Nputs and K must solve 2 Eqs for 2 unknowns:
➀ $365,000 + NPuts × 100 × (K-880) = $445,000
➁ $405,000 + NPuts × 100 × (K-920) = $445,000
➟ K= 960, NPuts = 10 Contracts (buy 10 Puts)
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Stock Indexes and Dividends
Stock indexes pay dividends.
We may assume that the index pays a “daily dividend”
and this dividend is invested back into the index.
(i.e., Dividend is used to buy new shares)
Example:
▸ Suppose you start with 100 shares when the index is at S0=50.
▸ Assume a dividend rate of q=14.6%/year; dividend paid daily.
▸ Suppose the Index evolves as follows:
Lec 15a Options on Stock Indices
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Day
0
1
2
3
.
365
Index
$50
52
53
48
...
52.67
Dividend Total $
Per Share Dividend
$0.0208
$0.0212
…
...
$0.0211
$2.08
$2.12
$2.11
# of new
Shares
100
0.04
0.04
0.04627
Total # of
Shares owned
100.04
100.08
100.12
...
115.7162
Computations:
Day 0: buy 100 shares at $50.
Day 1: Daily dividend/sh = $52*(0.146/365) = $0.0208/sh
Total $ Dividend = 100 sh * 0.0208 = $2.08.
Buy new shares=$2.08/$52 = 0.04000
Total number of shares = 100.04
Day 2: Dividend = $53*(0.146)/365 = $0.0212
Total $ Div = 100.04*0.0212 = $2.1208.
Buy new shares = $2.1208/$53 = 0.040016.
Total number of shares = 100.08
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What do we learn? Regardless of the price path,
Total number of shares evolves as
(1+q/365)365T ≈ eqT, T is fraction of one year.
For example,
after 6 months, the Total number of shares = 100 e0.146(1/2) = 107.5731
and after 1 year, the Total number of shares = 100 e0.146 = 115.7196
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BOPM for a Stock Index (p. 4)
Example:
▸ Consider a C(K = $50, T=1 year) on a stock index.
▸ The index pays an annual dividend rate of q=14.6% (paid daily),
▸ all dividends are re-invested in the index.
▸ r = 25%/year (c.c.), and the stock Index prices (ex-dividend) are:
Stock Price Tree
Call Values
Repliction Values .
t=0
T=1
t=0
T=1
t=0
T=1
100( SU=2S0)
50
(Δeq)100+Be0.25=50
S0=50
C0 = ?
ΔS0+B
25 (SD = ½S0)
0
(Δeq)25+Be0.25 = 0
Δ*= e-0.146{50-0}/{100-25}=0.5761, B*= -[(0.5761 e0.146)25]e-0.25= -12.98
Replicating portfolio:
ΔS0+B
and the call price is: C0 = (0.5761)50 - 12.98 = $15.83
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