Lec 15a Options on Stock Indices

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Lec 15a Options on Stock Indices 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 dfdf

Lec 15a Options on Stock Indices 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 dfdf

Lec 15a Options on Stock Indices Cash Flow Analysis at T = Expiration. Possible scenarios: 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. ST Exercise? Call Value: CT Profit/Loss on Call Profit/Loss on OEF ➀ $60 No -$8,000 -$40,000 ➁ $65 -$20,000 ➂ $70 No/Yes ➃ $75 Yes $20,000 +$12,000 +$20,000 ➄ $80 $40,000 +$32,000 +$40,000 Lec 15a Options on Stock Indices dfdf

Lec 15a Options on Stock Indices 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 dfdf

Lec 15a Options on Stock Indices Cash Flow Analysis at T = Expiration. Possible scenarios: 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) 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 ➃ 1,000 -3% 485,000 ➄ 1,040 4% 5% 525,000 Lec 15a Options on Stock Indices dfdf

Lec 15a Options on Stock Indices 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 dfdf

Lec 15a Options on Stock Indices Dividend Total $ # of new Total # of Day Index Per Share Dividend Shares Shares owned 0 $50 100 1 52 $0.0208 $2.08 0.04 100.04 2 53 $0.0212 $2.12 0.04 100.08 3 48 … 100.12 . ... ... ... 365 52.67 $0.0211 $2.11 0.04627 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 Lec 15a Options on Stock Indices dfdf

Lec 15a Options on Stock Indices 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 Lec 15a Options on Stock Indices dfdf

Lec 15a Options on Stock Indices 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 Lec 15a Options on Stock Indices

Lec 15a Options on Stock Indices Thank You (A Favara)