1. 15.1 Options on Stock Indices and Currencies Chapter 15

2. 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. 15.3 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. 15.5 Using Index Options for Portfolio Insurance Suppose the value of the index is S 0 and the strike price is K If a portfolio has a  of 1.0, the portfolio insurance is obtained by buying 1 put option contract on the index for each 100 S 0 dollars held If the  is not 1.0, the portfolio manager buys  put options for each 100 S 0 dollars held In both cases, K is chosen to give the appropriate insurance level

6. 15.6 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. 15.7 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. 15.9 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) Calculating Relation Between Index Level and Portfolio Value in 3 months

10. 15.10 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. 15.11 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. 15.12 European Options on Stocks Paying Dividend Yields We get the same probability distribution for the stock price at time T in each of the following cases: 1.The stock starts at price S 0 and provides a dividend yield = q 2.The stock starts at price S 0 e –q T and provides no income

13. 15.13 European Options on Stocks Paying Dividend Yield continued We can value European options by reducing the stock price to S 0 e –q T and then behaving as though there is no dividend

14. 15.14 Extension of Chapter 10 Results Lower Bound for calls: Lower Bound for puts Put Call Parity

15. 15.15 Extension of Chapter 13 Results

16. 15.16 The Binomial Model (p is the sole parameter that changes) S0u ƒuS0u ƒu S0d ƒdS0d ƒd S0 ƒS0 ƒ p (1  – p ) f = e -rT [pf u +(1 – p)f d ]

17. 15.17 The Binomial Model continued In a risk-neutral world the stock price grows at r – q rather than at r when there is a dividend yield at rate q The probability, p, of an up movement must therefore satisfy pS 0 u+(1 – p)S 0 d=S 0 e (r-q)T so that

18. 15.18 Valuing European Index Options We can use the formula for an option on a stock paying a continuous dividend yield Set S 0 = current index level Set q = average dividend yield expected during the life of the option

19. 15.19 The Foreign Interest Rate We denote the foreign interest rate by r f When a U.S. company buys one unit of the foreign currency it has an investment of S 0 dollars The return from investing at the foreign rate is r f S 0 dollars This shows that the foreign currency provides a “dividend yield” at rate r f

20. 15.20 Valuing European Currency Options A foreign currency is an asset that provides a continuous “dividend yield” equal to r f We can use the formula for an option on a stock paying a continuous dividend yield : Set S 0 = current exchange rate Set q = r ƒ

21. 15.21 Formulas for European Currency Options

22. 15.22 Alternative Formulas (combine cost-of-carry model with previous model) Using