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1、CHAPTER 23OPTION CONTRACTSAnswers to Questions1.A long straddle consists of a long call and a long put on the same stock and profits from dramatic price movement by the stock. A short straddle involves the sale of a call and a put on the same stock and profits from little or no stock price change. I
2、nvestors going long would anticipate volatility in excess of that discounted by the options prices while investors going short would expect volatility below that already discounted. Since volatility enhances option prices, long straddles would tend to pay higher premiums for more volatile options, w
3、hereas short straddles must accept lower premiums for less volatile options.2.A range forward is actually an option strategy that combines a long call and a short put (or vice versa) through a costless transaction. Because the options will not have the same striking price, the combination is classif
4、ied as a range forward as opposed to an actual forward, created by combining long and short options with the same striking price. It is fair to view actual forwards as a special case (or zero-cost version) of range forwards.3. CFA Examination II (1993)Call options give the owner the right, but not t
5、he obligation, to purchase SFrs for a pre-specified amount of domestic currency. Purchasing an at-the-money call option would guarantee the current exchange rate over the life of the option. If the SFr declines in value, the call will not be exercised since francs can be purchased more cheaply in th
6、e open market and redeeming the bond issue will be less costly.Contrasting characteristics: (1) Currency options are traded worldwide and enjoy a liquid market. (2) Exchange-traded currency option contracts have standard amounts, maturities, etc. (3) Over-the-counter options could be tailored to mee
7、t Michelles needs. (4) The initial cash outflow would be the premium. (5) The use of options preserves the ability to profit. (6) No counterparty credit risk. (7) Must roll to match year obligation.Currency forward contracts commit the seller to deliver the specified amount of currency to the buyer
8、on a specified future date at a fixed price. A short position in a forward contract requires delivery.Contrasting characteristics: (1) The market for forward contracts is over-the-counter and sometimes may not be as liquid as option or futures market. (2) Forward contracts may be customdesigned for
9、specific applications. (3) Cash does not change hands until a forward contract is settled. (4) Counterparty credit risk. (5) Can best match 5-year obligation.Currency futures are like forward contracts except the gain or loss on the contract is settled daily under the supervision of an organized exc
10、hange. A short position in the futures requires either offset or delivery at expiration.Contrasting characteristics: (1) Futures are traded in standardized contracts and are highly liquid. (2) Cash is required for daily settlement. (3) A margin account is required. (4) Management and administration
11、costs are higher than with a forward or option contract. (5) No counterparty credit risk. (6) Must roll to match year obligation.4. CFA Examination III (1991)The other three factors affecting the value of call options and the ways that changes in them affect value are:(1). Increases in underlying st
12、ock volatility. A call cannot be worth less than zero no matter how far the stock price falls, but rising stock prices can increase the calls value without limit. Therefore, the wider the range within which a stocks price can fluctuate (i.e. the greater its volatility), the greater the chance that t
13、he option will expire in-the-money, the higher the expected payoff from owning it, and the higher its value. A wider range of probable future prices on the underlying stock increases the probability of higher payoffs in general but because the calls value cannot decline below zero does not symmetric
14、ally increase the probability of lower payoffs.(2). The risk-free interest rate. Call value increase with increases in interest rates (given constant stock prices) because higher interest rates make the ownership of call options more attractive. The call owner does not pay for the stock until the op
15、tion is exercised; its owner can, therefore, take advantage of the time value of money by investing free interest rate increases the time value benefit to the call owner, increasing the value of the call option.(3). The exercise price of the option. Call values decrease with increases in the exercis
16、e price. When a call option is exercised, the payoff is the difference between the stock price at the time of exercise and the exercise (or strike) price. A higher exercise price decreases the expected payoff from the call, thus reducing the options value.5.Put-call parity indicates that a long posi
17、tion in a stock combined with being short a call and long a put (with the same strike price) is a risk-free investment. In other words, no matter what the stock price at expiration, the payoff will be the same. Consequently, any investment in this portfolio should earn the risk-free return. The thre
18、e-step process for valuing options is to(1).Determine a distribution of future stock prices,(2).Calculate the cash flows from the option at the future prices, and(3).Discount these expected cash flows to the present at the risk-free rate.It is this final step that is relevant. Cash flows can be disc
19、ounted at the risk-free rate because of the riskless replicating portfolio strategy.6.The Black-Scholes model is derived by showing how a portfolio of the underlying asset and risk-free bonds can be created that exactly mimics the price of an option. This involves taking a long position in the under
20、lying asset to replicate a call option. For currency options, the underlying asset can be thought of as risk-free deposits (bonds) in the foreign currency. So, just as stocks pay dividends, foreign deposits pay interest. Therefore, we can just substitute the foreign risk-free rate for the dividend y
21、ield when pricing options on currency.7.If there are no transaction costs, it is only rational to exercise a call option early, immediately before a dividend payment. This is because it will always be more profitable to sell the option and buy the stock in the open market rather than exercise the op
22、tion, unless there is a dividend. When a firm pays a dividend, the price of the stock usually declines by approximately the amount of the dividend. This has two impacts on the option holder. First, the decrease in the stock price decreases the value of his option. Second, the investor does not get t
23、he dividend payment unless he actually has exercised the option and owns the stock. Consequently, if the value of the dividend is greater than C+K-S, it will be beneficial to exercise early. Note, this is most likely to happen for every in-the-money options.For put options, it may be the case that t
24、he interest that could be earned on the proceeds from early exercise is greater than the intrinsic option value (P+S-K), so early proceeds from early exercise is optimal. Using similar logic as above, puts should always be exercised immediately after a dividend (on a dividend paying stock) because t
25、he stocks price declines after a dividend, thus making the put more valuable.8.In the Black-Scholes model, the expected future value of a stock is a function of the risk-free interest rate and the dividend yield. As long as the risk-free rate is greater than the dividend yield, the future expected v
26、alue will be greater than todays price. The longer the time period, the higher the expected price. So, as time to expiration increases, there are two opposing forces on the value of a European put. First, the increased time to expiration increases the chances of the option being more in-the-money. T
27、his increases put value. Second, the higher expected price at expiration decreases the expected value of the puts payoff at expiration and, therefore, decreases the put value. Depending on which of these two effects is larger, the put may increase or decrease in price with an increase in time to exp
28、iration. For a European call option, these two effects work in the same direction, since an increase in expected future price increases the value of a call. Hence, an increase in the time to expiration always increases the value of a European call.9.Since the price of an option is positively related
29、 to volatility, “buying low vol and selling high vol” is the same as the idea of “buy low, sell high” for any risky asset if the other parameters that affect option prices are fixed. The other factors that affect option prices can be effectively neutralized by holding the appropriate portfolio of op
30、tions and the underlying asset (beyond the scope of this question). If this is the case, then the risky asset is volatile itself, not the underlying foreign currency.10.A decrease in security volatility will cause an increase in both the call and put option values. For example, when the volatility o
31、f the underlying assets price decreases, the call option becomes less valuable since this decreases the probability that the option will be deeper in the money at expiration (a similar scenario is also true for the put option).11. On October 19, 1987, implied volatilities sky-rocketed. The jump in i
32、mplied volatility increased the value of call options more than enough to offset the negative impact of the in the index level.CHAPTER 23Answers to Problems1. CFA Examination III (1987)1(a). Cu - Cd 20 - 0Hedge ratio = = = 0.5 uS dS 120 80 (1 + r) - d (1 + 0.1) - 0.0Implied probability (p) = = = 0.7
33、5 u d 1.2 0.8 pCu + (1 p)Cd .75(20)+.25(0)Call value = = = 13.6 1 + r 1 + .01ORStep 1Set up binomial tree and calculate the option values at expiration for each ending stock price.Step 2Solve for the amount to invest in the stock and the amount to borrow in order to replicate the option given its va
34、lue in the up state and its value in the down state. Solve these equations simultaneously.Step 3Use the values derived in Step 2 to solve for the value of the option at the beginning of the period.ORStep 1 120 (C=20) 100 80 (C=0)Step 220 = 120 x D 1.1 x B 0 = 80 x D 1.1 x B -1.1(20) - (-1.1)(0) -22D
35、 = = = 0.5 120(-1.1) - 80(-1.1) -44C = 80(.5) 1.1 x B0 = 80(.5) 1.1 x BB = 40/1.1 36.4Step 3C = 50 - 36.4 = 13.61(b). The binomial option pricing model is a discrete version of the continuous time Black-Scholes option pricing model. As the number of intervals in the binomial model approaches infinit
36、y, the option value derived from this model approaches the option value derived from the Black-Scholes model.The binomial model is more flexible than the Black-Scholes model because it does not require one to assume constant interest rates and constant variance throughout the horizon. These values c
37、an be changed at any of the nodes in the binomial tree. However, the binomial model is more cumbersome to use since accuracy requires that the tree include many nodes.2. CFA Examination III (1992)2(a). An individual put will hedge an amount of the underlying stock index equal to the underlying value
38、 of the put, which in each case here is the respective stock index times $100.Also, due to the differing betas, the portfolio and the stock indexes are expected to produce gains or losses in proportion to their respective betas. Thus the number of puts required to hedge the portfolio must be adjuste
39、d for the betas of the respective stock indexes relative to the beta of the portfolio.The number and cost of protective puts could be calculated indirectly or directly.Indirect method:(bport)(Vport ) = (bGAC)(VGAC) = (bopt)(Vopt)The indirect method is based on the fact that an overall portfolios bet
40、a is equal to the weighted average of its component betas. Since we are seeking a zero beta portfolio, the left-hand side of the equation can be set to zero.For the S&P 100: 0 = -(1.05)(7,761,700) = (0.95)(Vopt)Vopt = -(1.05/0.95)(7,761,700)Vopt = -8,578,721Number of puts = 8,578,721/(365 x 100) = 2
41、35Cost of puts = 235 x $10.25 x 100 = $240,875For the S&P 500: 0 = -(1.05)(7,761,700) = (1.00)(Vopt)Vopt = -(1.05/1.00)(7,761,700) Vopt = -8,149,785Number of puts = 8,149,785/(390 x 100) = 209Cost of puts = 209 x $11.00 x 100 = $229,900For the NYSE: 0 = (1.05)(7,761,700) = (1.03)(Vopt)Vopt = -(1.05/
42、1.03)(7,761,700)Vopt = -7,912,413Number of puts = 7,912,413/(215 x 100) = 368Cost of puts = 368 x $6.25 x 100 = $230,000Direct method:The direct method calculates the number of puts by utilizing the hedge ratio (HR).HR = volatility of hedged security / volatility of hedging instrumentBeta, which mea
43、sures the movement in a security relative to the S&P 500, serves as a measure of volatility.For the S&P 100:Number of puts = (1.05/0.95)(7,761,700)/(365 x 100) = 235Cost of puts = 235 x $10.25 x 100 = $240,875For the S&P 500:Number of puts = (1.05/1.00)(7,761,700)/(390 x 100) = 209Cost of puts = 209
44、 x $11.00 x 100 = $229,900For the NYSE:Number of puts = (1.05/1.03)(7,761,700)/(215 x 100) = 368Cost of puts = 368 x $6.25 x 100 = $230,0002(b). The cost of the S&P 500 and NYSE puts are essentially the same and less than the cost of the S&P 100 puts; other relevant factors in the decision are corre
45、lation and liquidity.While the hedge calculated in Part A is intended to protect GACs portfolio from a decline, the portfolio does not replicate any of the indices. The hedge could be less than perfect if the price movement of GACs portfolio does not track the index movement. In this context, the in
46、dex which has the highest correlation with GACs portfolio (the S&P 500, 0.95) would be most preferred, while the index exhibiting the lowest correlation with GAC (the S&P 100, 0.86) would be least preferred.The hedge should be implemented with a minimum of market impact thus another consideration is liquidity. If the hedge transaction represents a disproportionate share of average daily trading volume, the market
