Chapter 06 - R 1 R 2 R 3 R 4 R 5 R 6 CHAPTER 6 REVIEW...

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Unformatted text preview: R. 1. R. 2. R. 3. R. 4. R. 5. R. 6. CHAPTER 6 REVIEW QUESTIONS A financial option is a contract offering the holder the right to either accept a transaction or to decline a transaction. The terms of the option usually include a specific transaction price and a date of expiration. Call options on common stock are options that give the holder the opportunity to purchase shares of a particular stock at a specific transactions price before a certain period of time. Put options on common stock are options that give the holder the right to sell shares of a particular stock at a specific transactions price before a certain period of time. Shareholders own the firm’s assets subject to the claims of the debtholders. We can therefore say that shareholders have a call option on the assets of the firm. If the firm’s assets turn out to be worth more than the value of the debtholder’s claim, the shareholders will exercise their option by paying the debtholders their promised payments and taking ownership of the assets. If the firm’s assets turn out to be worth less than the debtholder’s claim, the shareholders will walk away from the firm by declaring bankruptcy, allowing their option to the ownership of the firm’s assets to expire worthless. As discussed in question 1 above, call options on common stock are options that give the holder the opportunity to purchase shares of a particular stock at a specific transactions price before a certain period of time. The value of call options rise as the value of the common stock rises. However, call options offer limited losses (their value can never go below zero) as the holder of the option can allow the option to expire worthless. The seller of an option is the person who obligates himself to make promised payments to the option buyer. Every dollar of gain to the option buyer represents a dollar of loss to the option seller. Thus, the potential of unlimited gains to buyers of options means the potential for unlimited losses to option sellers. Option sellers enter these agreements because they are paid a fee, known as the option premium, for taking the risks. Option sellers hope that the options will expire worthless such that their profit will be equal to the premium collected. The Black Scholes model estimates the value of a call option based on five variables; (1) the price of the underlying asset, (2) the option’s exercise price, (3) the standard deviation of the underlying assets, (4) the option’s time to expiration, and (5) the risk free rate of interest. The option’s exercise price is one of the five variables in the Black Scholes option pricing model. For call options, the value of the option is inversely related to the option’s exercise price. The call option becomes profitable as the stock price exceeds the option’s exercise price and the option 29 R. 7. R. 8. is in—the~money. In situations where the exercise price is high relative to the stock price, the stock price must rise by a large amount for the option to be in—the—money. When the exercise price is low relative to the stock price, the option is worth more. The standard deviation of the underlying asset is one of the five variables in the Black Scholes option pricing model. The Black Scholes model illustrates that call option prices rise as the standard deviation of the underlying asset rises. An option is a claim on an asset which permits the holder to have large or unlimited profit potential with limited loss potential. If the standard deviation is high, the option’s upside potential will be increased while the downside potential is limited. Therefore, an increase in the standard deviation of the underlying asset benefits the call option holder. The option’s time to maturity is one of the five variables in the Black Scholes option pricing model. Since most options allow the holder to exercise the option at any time during the option’s lifetime, a long term option is worth more than a short term option. Longer time to expiration allows more time for the underlying asset to make large price movements. The benefit to more time is similar to the benefit of more volatility as discussed in Review Question 7. 30 CHAPTER 6 PROBLEMS 1. a. The coupon .is the call option. b. The current value of a one—week stay, the variance of the value of a one—week stay, the price at which the coupon allows the holder to purchase a stay, the risk—free rate and the time to the coupon’s expiration. 2. a. In the money options have exercise prices less than the underlying assets current market value. These include both the September and October expiration dates for both the $40 and $45 exercise (strike) prices. The out of the money options are the September and October call options with an exercise or strike price of $50 (since it exceeds the $46 market value). b. The value of this call option, like all out of the money call options, results from the possibility that the underlying asset’s market value will rise above the exercise or strike price before the option expires. When the option is far out of the money and there is little time remaining, the option price will be very small to reflect the small probability that the option will expire with value. c. The October call options have an additional month before expiration and therefore have more value to the option holder (buyer) since there is more time for the asset underlying the option to rise and increase the value of the option. 3. The option is out of the money since the price of the underlying asset is below the strike price or exercise price. 4. $20. Only the $20 price is possible since the value of the underlying asset ($53) already exceeds the exercise price by $18. Generally speaking, a call option is worth at least as much as the amount that the price of the underlying asset exceeds the strike price. 5. a. For a $0 sales price in Reno stock, an investment in 1 share would lose $50. For a $50 sales price in Reno stock, an investment in 1 share would lose $0. For a $60 sales price in Reno stock, an investment in 1 share makes a $10 profit. For a $100 sales price in Reno stock, an investment in 1 share makes a $50 profit. 31 Profit/Loss 6. a. P refit/Loss Profit/Loss of Stock in Reno (10 40 20 —60 l 60 Reno Stock Price 40 80 120 For a $0 price in Reno stock, a $10 investment in a call option would lose $10. For a $50 price in Reno stock, a $10 investment in a call option would lose $10. For a $60 price in Reno stock, a $10 investment in a call option would break even. For a $100 price in Reno stock, a $10 investment in a call option makes $40 profit Profit/Loss of $10 Call Option on Reno 40 20 / I 60 Reno Stock Price 20 40 120 32 c. The stock investment performs better by $10 for all Reno stock price values of greater than $50. However, the call option performs much better for the lowest outcomes in Reno. The call option exposes the investor to a smaller dollar loss since the largest possible loss is $10. However, this reduced loss potential is offset by the reduced profits when Reno goes up or stays the same. 7. a. For a $0 price in Reno stock, writing a $10 call option makes a $10 profit. For a $50 price in Reno stock, writing a $10 call option makes a $10 profit. For a $60 price in Reno stock, writing a $10 call option breaks even. For a $100 price in Reno stock, writing a $10 call option loses $40. Profit/Loss of Writing $10 Call Option on Reno 60 Profit/Loss —20 " _60 i l t I l 0 20 40 60 80 100 120 Reno Stock Price c. The graph in 7(b) is the mirror image to the graph in 6(b). This reflects the fact that an option is a contract between two people and therefore any profit or loss to one of the participants must be offset by an equal and opposite effect to the other participant. Note that option buying offers limited loss potential with, in theory, unlimited profit potential. Call option writing offers limited profit potential with virtually unlimited loss potential. 8. a. $12 b. $2 c. $0 d. $0 The call option at expiration will be worth the stock price minus the strike price or $0, whichever is greater. 33 9. a. $2 b. greater than $2 c. $0 01. greater than $0 e. $0 f. greater than $0 The call option at expiration will be worth the stock price minus the strike price or $0, whichever is greater. Prior to expiration, the call prices will be greater. 10. Value of Call Option at Expiration 330 ~ * m» $25 a /} $20 A l/ $15 ‘ Option Price $10 A ss- 1 so: I u I 1 /1 I i 4— t 25 30 35 40 45 50 55 60 65 70 75 Stock Price 11.a. Up b. Up c. Up d. Up 6. Down f. Nothing (beta is in Chapter 12) 12. a. $100. The most a call option can be worth is the price of the underlying asset. b. $0. The least a call option can be worth is zero. 13. The owner of the call option hopes that the underlying stock price will rise. The writer of the call option would hope that the price falls (or at least does not rise). 14. a — call option — on the the ~ stockholders - wish to firm’s — debtholders — by paying them firrn’s — debt ~ which represents or — strike price — of the option. option’s — time to expiration — The as a — call option price — and it firm’s — price of underlying assets - or the -— variance of the firm’s underlying assets — to 34 15. a. Underlying Asset Strike Price Time Std. Dev. Risk—free Rate $16 $20 0.25 0.40 10.00% From formula as in appendix: d1= —0.8907 d2= — 1.0907 From Table 6.2: (and interpolating when appropriate) N(d1)= 0.1865 N(d2)= 0.1377 From Formula A61: exp(——rt) = 0.97531 CALL PRICE = $0.30 b. Underlying Asset Strike Price Time Std. Dev. Risk—free Rate $18 $20 0.25 0.40 10.00% d1= ——0.3018 d2= —0.5018 N(d1)= 0.3814 N(d2)= 0.3079 exp(—rt) = 0.97531 CALL PRICE = $0.86 c. Underlying Asset Strike Price Time Std. Dev. Risk—free Rate $20 $20 0.25 0.40 10.00% d1 = 0.2250 d2= 0.0250 N(d1)= 0.5890 N(d2)= 0.5100 exp(—rt) = 0.97531 CALL PRICE = $1.83 (I. Underlying Asset Strike Price Time Std. Dev. Risk—free Rate $22 $20 0.25 0.40 10.00% d1= 0.7016 d2= 0.5016 N(d1)= 0.7585 N(d2)= 0.6920 exp(—rt) = 0.97531 CALL PRICE = $3.19 e. Underlying Asset Strike Price Time Std. Dev. Risk-free Rate $24 $20 0.25 0.40 10.00% d1= 1.1366 d2= 0.9366 N(d1)= 0.8721 N(d2)= 0.8255 exp(—rt) = 0.97531 CALL PRICE = $4.83 35 16‘ Value of Call Option $6 $5— / $2“ Option Price ‘1“ /I/ so I l I I I 16 18 20 22 24 Stock Price 17. This problem must be solved by trial and error. In other words, various values of the volatility must be inserted into the option price model until the volatility is found that will generate the option price ($2.00). We start with a volatility of 0.20 and compute the option price: Underlying Asset Strike Price Time Volatility Risk—free Rate $20 $20 0.25 0.20 10.00% d1 = 0.3000 d2: 0.2000 N(d1)= 0.6179 N(d2)= 0.5793 exp(—rt) = 0.97531 CALL PRICE = $1.06 As a shortcut, we can note that problem 15 c found that the call option price for a volatility of 0.4 was $1.83. Thus by increasing the volatility by 0.2 (from 0.2 to 0.4), the call option price increased from $1.06 to $1.83, or by $0.77. In order to increase the call option price another $0.17, a volatility of .44 appears reasonable: Underlying Asset Strike Price Time Volatility Risk—free Rate $20 $20 0.25 0.44 10.00% d1: 0.2236 d2: 0.0036 N(d1)= 0.5885 N(d2)= 0.5015 exp(—rt) = 0.97531 CALL PRICE = $1.99 We are interpolating and rounding the values from Table 6.2 and therefore it is reasonable to assume that 0.44 produces an adequate approximation. The precise answer is 0.443033. 36 CHAPTER 6 DISCUSSION QUESTIONS 1. Call options do offer virtually unlimited potential gains and losses are limited to the amount invested, however, there are no "free lunches" in life and there is a cost to the benefits of owning a call option. If the price of the asset underlying the option remains approximately the same, the call option will fall in value. Thus, generally speaking, call options lose money whenever the price of the underlying asset falls or remains level. A call option will only be a winner of the price of the underlying asset rises substantially, in general. Thus, call options tend to offer small probabilities of large gains and large probabilities of losses. 2. The chapter discussion assumes that the firm’s assets become riskier but that the value of the assets remains constant — for example if a company could costlessly trade their low risk assets for a group of high risk assets with identical total values. In such a case, their would be no net increase or net decrease in total value. However, there would be a wealth transfer from the firm’s bondholders to the firm’s stockholders. The bondholders would lose because the probability that they will be repaid in full has fallen due to the higher risk of the assets. The stockholders will gain because the added risk increases their potential profits without an offsetting increase in their potential losses since bankruptcy limits the amount that the stockholders can lose. 3. No. Since a call option gives its owner a right to buy something rather than an obligation, the worst thing that can happen is that the owner allows the option to expire without using it. 4. Option trading may or not be speculation depending upon how the particular option trader uses the options and whether the risks of the options add or hedge away the total risks that the trader faces. The question of whether or not it should be legal is helped by an analysis of Chapter 3. Briefly, some people would argue that option trading is mostly or completely speculation, that speculation can hurt people and therefore it should be illegal. Others argue that speculators help an economy by providing liquidity and more accurate prices. Still others argue that people have the right to do whatever they want with their money including speculation. Speculation and government attempts to limit or prevent speculation have existed for centuries. Time has shown that speculation that was vehemently opposed when begun has produced enormous benefits to society when allowed to continue. The trading of agricultural futures contracts in the U. S. is an excellent example. 5. It is generally true that options expire, the transaction is completed and therefore options force the recognition of losses (and profits) on tax returns and other accounting statements. Alternatively, stocks do not in general force similar recognition. However, it is important to realize that market prices are usually the best indication of true value and therefore when a stock price falls the investor 37 has lost money whether or not the stock is sold. The idea that money is not lost until the stock is sold is an extremely commonly held but very dangerous viewpoint. If you wreck your sportscar into a tree is it reasonable to concloude that no loss has taken place unless you are forced to sell it? There is no reason to believe that a stock that has fallen greatly in value recently is any more likely to rise in the future than any other equally volatile stock. . Both statements are correct. Whether options or stocks are riskier depends upon the basis for comparison. In terms of equal dollar investments, options are riskier than stocks because options fluctuate more in percentage terms. Thus it is riskier to purchase $10,000 of call options on IBM rather than $10,000 of IBM common stock. In terms of equal units of invesments, options are less risky than stocks because options fluctuate fewer dollars per unit than do the underlying shares. Thus, it is generally less risky to purchase 2 call options (on 200 shares of IBM common stock) than it is to purchase 200 shares of IBM stock. Note that call options are generally priced much lower than stocks. Thus, each unit requires a relatively small investment and therefore exposes the investor to less risk. However, for equal dollar investments, call options are almost always much riskier than the underlying stocks. 38 ...
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Chapter 06 - R 1 R 2 R 3 R 4 R 5 R 6 CHAPTER 6 REVIEW...

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