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Options on Stock Indices and Currencies

Options on Stock Indices and Currencies. Chapter 15. Index Options. The most popular underlying indices in the U.S. are The S&P 100 Index (OEX and XEO) The S&P 500 Index (SPX) The Dow Jones Index times 0.01 (DJX) The Nasdaq 100 Index (NDX)

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Options on Stock Indices and Currencies

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  1. Options on Stock Indices and Currencies Chapter 15

  2. Index Options • The most popular underlying indices in the U.S. are • The S&P 100 Index (OEX and XEO) • The S&P 500 Index (SPX) • The Dow Jones Index times 0.01 (DJX) • The Nasdaq 100 Index (NDX) • Contracts are on 100 times index; they are settled in cash; OEX is American; the XEO and all other options are European.

  3. Index Option Example • Consider a call option on an index with a strike price of 560 • Suppose 1 contract is exercised when the index level is 580 • What is the payoff? (580–560)100=2000

  4. Portfolio Insurance with Index Puts Type of Protective Put (but applied to portfolios) Problem: Portfolio not the same as index Must determine #puts and K # of puts = betaX(Vport/(100xVindex)) Strike price, K: Derived from Vprotect and SML (security market line) Rindex & Rport each split into 2 components: capital gains & dividends

  5. Using Index Options for Portfolio Insurance • Suppose the value of the index is S0 and the strike price is K • If a portfolio has a b of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100S0 dollars held • If the b is not 1.0, the portfolio manager buys b put options for each 100S0 dollars held • In both cases, Kis chosen to give the appropriate insurance level

  6. Example 1 • Portfolio has a beta of 1.0 • It is currently worth $500,000 • The index currently stands at 1000 • Portfolio & index: same dividend yield of 4% q.c. • What trade is necessary to provide insurance against the portfolio value falling below $450,000 by end of quarter? • Risk-free rate = 12% with quarterly compound. (compounding period matches time horizon) • Buys 5 puts with K = 900

  7. Example 2 (same as Example 1 except for higher beta) • Portfolio has a beta of 2.0 • It is currently worth $500,000 and index stands at 1000 • Interest rates quoted: quarterly compounding • The risk-free rate is 12% per annum • The dividend yield on both the portfolio and the index is 4% • Portfolio insurance: #puts = 10; K = 960

  8. Effect of beta increase on portfolio insurance #puts increases. Why? K, strike price, also increases. Why? Increase in beta means greater probability mass below the threshold (Vprotect). Thus, require higher indemnity (payoff from the insurance policy). Ergo, more insurance contracts, i.e. puts.

  9. Calculating Relation Between Index Level and Portfolio Value in 3 months • If index rises to 1040, it provides a 40/1000 or 4% return in 3 months • Total return (incl. dividends)=5% • Excess return of index over risk-free rate= 2% = (5-3)% • Excess return for portfolio=4%= 2(2%) • Increase in Portfolio Value=4–1+3=6% • Portfolio value= $530,000 = $500,000(1.06)

  10. Determining the Strike Price:nexus between Vprotect & K is SML An option with a strike price of 960 will provide protection against a 10% decline in the portfolio value from $500,000.

  11. Currency Options • Currency options trade on the Philadelphia Exchange (PHLX) • There also exists an active over-the-counter (OTC) market • Currency options are used by corporations to buy insurance when they have an FX exposure

  12. European Options on StocksPaying Dividend Yields We get the same probability distribution for the stock price at time Tin each of the following cases: 1. The stock starts at price S0 and provides a dividend yield = q 2. The stock starts at price S0e–qTand provides no income

  13. European Options on StocksPaying Dividend Yieldcontinued We can value European options by reducing the stock price to S0e–qTand then behaving as though there is no dividend

  14. Extension of Chapter 10 Results Lower Bound for calls: Lower Bound for puts Put Call Parity

  15. Extension of Chapter 13 Results

  16. The Binomial Model (p is the sole parameter that changes) S0u ƒu p S0 ƒ S0d ƒd (1– p ) • f = e-rT[pfu+(1– p)fd ]

  17. The Binomial Modelcontinued • In a risk-neutral world the stock price grows at r – qrather than at rwhen there is a dividend yield at rate q • The probability, p, of an up movement must therefore satisfy pS0u+(1 – p)S0d=S0e(r-q)T so that

  18. Valuing European Index Options We can use the formula for an option on a stock paying a continuous dividend yield Set S0 = current index level Set q = average dividend yield expected during the life of the option

  19. The Foreign Interest Rate • We denote the foreign interest rate by rf • When a U.S. company buys one unit of the foreign currency it has an investment of S0dollars • The return from investing at the foreign rate is rf S0dollars • This shows that the foreign currency provides a “dividend yield” at rate rf

  20. Valuing European Currency Options • A foreign currency is an asset that provides a continuous “dividend yield” equal to rf • We can use the formula for an option on a stock paying a continuous dividend yield : Set S0= current exchange rate Set q = rƒ

  21. Formulas for European Currency Options

  22. Alternative Formulas(combine cost-of-carry model with previous model) Using

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