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Options on Stock Indices and Currencies Chapter 13

Options on Stock Indices and Currencies Chapter 13. Valuing European Index Options. We can use the formula for an option on a stock paying a dividend yield Set S 0 = current index level Set q = average dividend yield expected during the life of the option.

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Options on Stock Indices and Currencies Chapter 13

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  1. Options onStock Indices and CurrenciesChapter 13

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

  3. European Options on StocksProviding a Dividend Yield We get the same probability distribution for the stock price at time T in each of the following cases: 1. The stock grows from S0 to ST and provides a dividend yield = q 2. The stock grows from S0 to STeqTprovides no dividend yield. The stock grows from S0e–qTto ST and provides no income

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

  5. Extension of Chapter 9 Results Lower Bound for calls: Lower Bound for puts Put Call Parity (both portfolios are worth max (ST, K at time T)

  6. Extension of Chapter 12 Results

  7. Index Options • Option contracts are on 100 times the index • The most popular underlying indices are • the Dow Jones Industrial (European) DJX • the S&P 100 (American) OEX • the S&P 500 (European) SPX • Contracts are settled in cash • Example on page 292 (old), 304 (new text)

  8. 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?

  9. 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

  10. Example 1 • Portfolio has a beta of 1.0 • It is currently worth $500,000 • The index currently stands at 1000 • Assuming a strike price of $900, what trade is necessary to provide insurance against the portfolio value falling below $450,000? • If index drops to 880, portfolio will be worth $440,000 with the payoff from the options being (900-880) x 100 x 5 = $10,000 • Total value of the portfolio is $450,000 (or $446,760 when the cost of the options is accounted for)

  11. Example 2 • If the b is not 1.0, the portfolio manager buys b put options for each 100S0 dollars held • Portfolio has a beta of 2.0 • It is currently worth $500,000 and index stands at 1000 • The risk-free rate is 12% per annum • The dividend yield on both the portfolio and the index is 4% • How many put option contracts should be purchased for portfolio insurance for 3 months?

  12. Calculating Relation Between Index Level and Portfolio Value in 3 months • Index rises to 1040 • Provides a 40/1000 or 4% return in 3 months • Total return (incl dividends) = 4% + 4%*(0.25) = 5% • Risk free rate = 12% * 0.25 = 3% • Excess return of index over risk-free rate = (5% - 3%) = 2% • Excess return for portfolio = 4% • Return from portfolio = 3% + 4% = 7% • Dividends on portfolio = 4% * 0.25 = 1% • Increase in Portfolio Value = 7% - 1% = 6% • Portfolio value in three months = $530,000

  13. Currency Options • Primarily traded over-the-counter • Currency options are used by corporations to buy insurance when they have an FX exposure

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

  15. Formulas for European Currency Options (Equations 13.9 and 13.10)

  16. Questions 5th and 6th edition: 13.1, 13.6, 13.7, 13.10, 13.11

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