Options on Stock Indices and Currencies

Transcription

1 Options on Stock Indices and Currencies 12.1 European Options on Stocks Paying Dividend Yields We get the same probability distribution for the stock price at time if either the stock 12.2 Chapter 12 or 1. Starts at price S and provides a dividend yield q 2. Starts at price S e q and provides no income BS with continuous dividends Adjusting the initial stock price S S S = S e -q q >, g < r, Real world q =, g = r, BS world r = g + q r= total return g=capital gains yield q=dividend yield t 12.3 BS with continuous dividends Adjusting the stock price cont d Suppose S grows to S in the real world of dividend-paying stocks his gives a yield r per annum Question: What is the initial stock price in the BS world that will generate the same final stock price S? Answer: S e q 12.4 hus, from the Figure, we can price E- options with the BS model by reducing the initial stock price to S = S e q and behaving as though there were no dividend 12.5 Exercise 1: Adjusting the stock price Genentec s stock s current price is $52 It has a continuously compounded annual yield of 3% he volatility of Genentec s stock price is estimated to be 2% per annum he strike price on a 3-month E-put on Genentec stock is $56 he safe rate of interest is 12% What is the price of Genentec s put? 12.6

2 Parity and lower bounds If you have Derivagem (Hull or Option! (Kolb then check you get the same results with S and the dividend S and no dividend Assume in using the software that the number of days in a quarter is 365/4=91.25 We need make no changes to the formulae we have for Put-call parity Lower bounds on options prices We simply redefine S as S e -q hen the formula overleaf apply as before Put-call parity and lower bounds restated Lower Bound for calls: c Lower Bound for puts S e r 12.9 BS pricing formula: continuous dividends Again, no adjustment is necessary other than the redefinition of S Put Call Parity p e r S c+ e r = p+ S he Binomial Model S ƒ p (1 p S u ƒ u S d ƒ d f=e -r [pf u +(1 pf d ] he Binomial Model continued In a risk-neutral world when there is a dividend yield at rate q the stock price grows at g=r-q rather than at r he probability, p, of an up movement must therefore satisfy ps u+(1 ps d=s e g so that g e d p = u d 12.12

3 12.13 Index Options his means that the pricing procedure using the BO model goes through exactly as before he only difference is that we have to redefine r as g=r-q he most popular underlying indices in the U.S. are he Dow Jones Index times.1 (DJ he Nasdaq 1 Index (ND he Russell 2 Index (RU he S&P 1 Index (OE he S&P 5 Index (SP Contracts are on: 1 times index settled in cash OE is American and the rest are European. LEAPS Exercise 2: Index Option Leaps are options on stock indices that last up to 3 years have December expiration dates are on 1 times the index Leaps also trade on some individual stocks Consider a call option on an index with a strike price of 56 Suppose 1 contract is exercised when the index level is 58 What is the payoff? Portfolio insurance he idea is based on that of a protective put see diagram overleaf his consists of a portfolio of 1 unit of stock 1 put option Formally equivalent to a long call Its function is to limit the downside risk Leave upside potential Figure 1: Stock index insurance: Graphics of a protected put max(, S Payoff@ maturityto onestock + onee - put: S + p = max(, S = insuredvalue S 12.18

4 he portfolio has payoffs at maturity of S + p = max(, S = insured value - where we have used the put-call parity result derived in an earlier chapter (see Figure 1 hus we can guarantee the portfolio it yields at least Example: Choosing the right Put for insurance Suppose s available for the E-puts in the market are 1,11,12. We need to insure that our share doesn t fall below 15 hen we buy an E-put for 11 his Does the trick But of course places a higher lower bound than we strictly need on the value of the share 12.2 Exercise 3: Protective puts I wish to insure that the value of my non-dividend paying Cashco stock does not fall below 4 in the next 3 months. Its current price is 43. he safe rate of interest over the 3 months is 5% per annum continuous he prices of 3 month E-puts and calls on Cashco stock are: Put and call data ($ on Cashco stock c p Exercise cont d Question: What is the best option strategy for me to achieve my objective? Example: Index options Suppose the FSE1 stands at 3,5 units (S price of one unit of the index is 5 (Q portfolio to be insured ( target portfolio is worth 4m (V strike price of a put on the index maturing in 3 months is 4, units ( Question: What matching portfolio will insure the current value of the target? 12.24

5 Data: = 4k V = 4m Answer S = 3.5 k index units index units Q = 5=.5k Number of puts required on the index (index-puts: N = V / Q = 4m /(.5k(4k = 2 index -puts Value of the matching index-portfolio at is then: V = NQ( p + S V = max(, S = (1k max(4k,3k = 4m In this case we have engineered the numbers to get an exact match: V =V In practice the value of the matched portfolio will generally be above that of the original portfolio See next Figure here we assume for simplicity that Q= Optimum matched Portfolio minimum value with 2 indexputs Insured portfolio 3 Optimum portfolio V (2 = 2max(, Matched Portfolio with too small minimum Value using 1 index-put S 2 (1 V = max(, S V (3 V = 3max(, S 2S Portfolio insurance: General case V 2 > V S Simplified diagram: Figure assumes Q=1. Required insurance V =N gives N=2 S Optimum matched Portfolio minimum value Insured portfolio 2 V Value of matched portfolio (2 V = 2max(, S 2S ( 2 V > 2 V Simplified diagram: Same as last one with only the optimum shown. S Using Index Options for Portfolio Insurance 12.3 he value of the index is S ; strike price is If a portfolio has a β of 1., portfolio insurance is obtained by buying 1 put option contract on the index for each QS = 1S dollars held If the β is not 1., the portfolio manager buys β put options for each 1S dollars held

6 In both cases, is chosen to give the appropriate insurance level Recall from the CAPM that Note that R R = β ( R RF P F M β = 1 R P = R M hus if the portfolio to be insured is well-diversified, the return on that portfolio will be the same as that on the index his is what we have assumed in the first example