期权期货和其他衍生品-英文第9版-Chap08-Securitization and the Cre
期权期货与其他衍生产品第九版课后习题与答案Chapter (33)
The final fixed payment is in millions of dollars: [(4 10415 4) 10415 4] 10415 4 170238 The final floating payment assuming forward rates are realized is [(405 1041 405) 1041 405] 1041 405 172238 The value of the swap is therefore 02000 (1044 ) 01710 or –$171,000. Problem 33.4. What is the value of a five-year swap where LIBOR is paid in the usual way and in return LIBOR compounded at LIBOR is received on the other side? The principal on both sides is $100 million. Payment dates on the pay side and compounding dates on the receive side are every six months and the LIBOR zero curve is flat at 5% with semiannual compounding (and is used for discounting). The value is zero. The receive side is the same as the pay side with the cash flows compounded forward at LIBOR. Compounding cash flows forward at LIBOR does not change their value. Problem 33.5. Explain carefully why a bank might choose to discount cash flows on a currency swap at a rate slightly different from LIBOR. In theory, a new floating-for-floating swap should involve exchanging LIBOR in one currency for LIBOR in another currency (with no spreads added). In practice, macroeconomic effects give rise to spreads. Financial institutions often adjust the discount rates they use to allow for this. Suppose that USD LIBOR is always exchanged Swiss franc LIBOR plus 15 basis points. Financial institutions would discount USD cash flows at USD LIBOR and Swiss franc cash flows at LIBOR plus 15 basis points. This would ensure that the floating-for-floating swap is valued consistently with the market. Problem 33.6. Calculate the total convexity/timing adjustment in Example 33.3 of Section 33.4 if all cap volatilities are 18% instead of 20% and volatilities for all options on five-year swaps are 13% instead of 15%. What should the five year swap rate in three years’ time be assumed for the purpose of valuing the swap? What is the value of the swap? In this case, yi 005 , y i 013 , i 05 , Fi 005 , F i 018 , and i 07 for all
HullOFOD9eSolutionsCh19第九版期权、期货及其他衍生品课后答案
ln( S0 K ) (01 0252 2)05 03712 025 05 The delta of the option is N (d1 ) or 0.64. d1
Problem 19.4. What does it mean to assert that the theta of an option position is −0.1 when time is measured in yt neither a stock price nor its implied volatility will change, what type of option position is appropriate?
A theta of 01 means that if t units of time pass with no change in either the stock price or its volatility, the value of the option declines by 01t . A trader who feels that neither the stock price nor its implied volatility will change should write an option with as high a negative theta as possible. Relatively short-life at-the-money options have the most negative thetas. Problem 19.5. What is meant by the gamma of an option position? What are the risks in the situation where the gamma of a position is large and negative and the delta is zero? The gamma of an option position is the rate of change of the delta of the position with respect to the asset price. For example, a gamma of 0.1 would indicate that when the asset price increases by a certain small amount delta increases by 0.1 of this amount. When the gamma of an option writer’s position is large and negative and the delta is zero, the option writer will lose significant amounts of money if there is a large movement (either an increase or a decrease) in the asset price. Problem 19.6. “The procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position.” Explain this statement. To hedge an option position it is necessary to create the opposite option position synthetically. For example, to hedge a long position in a put it is necessary to create a short position in a put synthetically. It follows that the procedure for creating an option position synthetically is the reverse of the procedure for hedging the option position. Problem 19.7. Why did portfolio insurance not work well on October 19, 1987? Portfolio insurance involves creating a put option synthetically. It assumes that as soon as a portfolio’s value declines by a small amount the portfolio manager’s position is rebalanced by either (a) selling part of the portfolio, or (b) selling index futures. On October 19, 1987, the market declined so quickly that the sort of rebalancing anticipated in portfolio insurance schemes could not be accomplished. Problem 19.8. The Black-Scholes-Merton price of an out-of-the-money call option with an exercise price of $40 is $4. A trader who has written the option plans to use a stop-loss strategy. The trader’s plan is to buy at $40.10 and to sell at $39.90. Estimate the expected number of times the stock will be bought or sold. The strategy costs the trader 010 each time the stock is bought or sold. The total expected cost of the strategy, in present value terms, must be $4. This means that the expected number of times the stock will be bought or sold is approximately 40. The expected number of times it will be bought is approximately 20 and the expected number of times it will be sold is also approximately 20. The buy and sell transactions can take place at any time during the life of the option. The above numbers are therefore only approximately correct because of the effects of discounting. Also the estimate is of the number of times the stock is bought or sold in the risk-neutral world, not the real world.
HullOFOD9eSolutionsCh17第九版期权、期货及其他衍生品课后答案
ln( S0 K ) (rB rA 2 2)T T and rA and rB are the risk-free rates in currencies A and B, respectively. The value of the d2
option is measured in units of currency B. Defining S0 1 S0 and K 1 K
250 05946e003025 250 05594e010025
or 11.15. Problem 17.7. Calculate the value of an eight-month European put option on a currency with a strike price of 0.50. The current exchange rate is 0.52, the volatility of the exchange rate is 12%, the domestic risk-free interest rate is 4% per annum, and the foreign risk-free interest rate is 8% per annum. In this case S0 052 , K 050 , r 004 , rf 008 , 012 , T 06667 , and
n( S0 K ) (rA rB 2 2)T T
ln( S0 K ) (rA rB 2 2)T T This shows that put option is equivalent to KS0 call options to buy 1 unit of currency A for 1 K units of currency B. In this case the value of the option is measured in units of currency A. To obtain the call option value in units of currency B (thesame units as the value of the put option was measured in) we must divide by S 0 . This proves the result. d2 d1
期权期货与其他衍生产品第九版课后习题与答案Chapter (30)
G( y )
01 22 2 (1 y) (1 y)3
02 66 3 (1 y) (1 y)4 It follows that G(01) 17355 and G(01) 46582 and the convexity adjustment that must be made for the two-year swap- rate is 46582 05 012 022 5 000268 17355 We can therefore value the instrument on the assumption that the swap rate will be 10.268% in five years. The value of the instrument is 0268 0167 115 or $0.167. G( y )
interest rate between years five and six and a correlation of 0.95 with the forward interest rate between years five and seven. In this case we have to make a timing adjustment as well as a convexity adjustment to the forward swap rate. For (a) equation (30.4) shows that the timing adjustment involves multiplying the swap rate by 08 020 020 01 5 exp 09856 1 01 so that it becomes 10268 09856 10120 . The value of the instrument is 0120 0068 116 or $0.068. For (b) equation (30.4) shows that the timing adjustment involves multiplying the swap rate by 095 02 02 01 2 5 exp 09660 1 01 so that it becomes 10268 0966 9919 . The value of the instrument is now 0081 0042 117 or –$0.042. Problem 30.6. The price of a bond at time T , measured in terms of its yield, is G( yT ) . Assume geometric Brownian motion for the forward bond yield, y , in a world that is forward risk neutral with respect to a bond maturing at time T . Suppose that the growth rate of the forward bond yield is and its volatility y . (a) Use Ito’s lemma to calculate the process for the forward bond price in terms of , y , y , and G( y) . (b) The forward bond price should follow a martingale in the world considered. Use this fact to calculate an expression for . (c) Show that the expression for is, to a first approximation, consistent with equation (30.1). (a) The process for y is
赫尔期权、期货及其他衍生产品(第9版)笔记和课后习题详解
赫尔《期权、期货及其他衍生产品》（第9版）笔记和课后习题详解目录第1章引言1.1复习笔记1.2课后习题详解第2章期货市场的运作机制2.1复习笔记2.2课后习题详解第3章利用期货的对冲策略3.1复习笔记3.2课后习题详解第4章利率4.1复习笔记4.2课后习题详解第5章如何确定远期和期货价格5.1复习笔记5.2课后习题详解第6章利率期货6.1复习笔记6.2课后习题详解第7章互换7.1复习笔记7.2课后习题详解第8章证券化与2007年信用危机8.1复习笔记8.2课后习题详解第9章OIS贴现、信用以及资金费用9.1复习笔记9.2课后习题详解第10章期权市场机制10.1复习笔记10.2课后习题详解第11章股票期权的性质11.1复习笔记11.2课后习题详解第12章期权交易策略12.1复习笔记12.2课后习题详解第13章二叉树13.1复习笔记13.2课后习题详解第14章维纳过程和伊藤引理14.1复习笔记14.2课后习题详解第15章布莱克-斯科尔斯-默顿模型15.1复习笔记15.2课后习题详解第16章雇员股票期权16.1复习笔记16.2课后习题详解第17章股指期权与货币期权17.1复习笔记17.2课后习题详解第18章期货期权18.1复习笔记18.2课后习题详解第19章希腊值19.1复习笔记19.2课后习题详解第20章波动率微笑20.1复习笔记20.2课后习题详解第21章基本数值方法21.1复习笔记21.2课后习题详解第22章风险价值度22.1复习笔记22.2课后习题详解第23章估计波动率和相关系数23.1复习笔记23.2课后习题详解第24章信用风险24.1复习笔记24.2课后习题详解第25章信用衍生产品25.1复习笔记25.2课后习题详解第26章特种期权26.1复习笔记26.2课后习题详解第27章再谈模型和数值算法27.1复习笔记27.2课后习题详解第28章鞅与测度28.1复习笔记28.2课后习题详解第29章利率衍生产品：标准市场模型29.1复习笔记29.2课后习题详解第30章曲率、时间与Quanto调整30.1复习笔记30.2课后习题详解第31章利率衍生产品：短期利率模型31.1复习笔记31.2课后习题详解第32章HJM，LMM模型以及多种零息曲线32.1复习笔记32.2课后习题详解第33章再谈互换33.1复习笔记33.2课后习题详解第34章能源与商品衍生产品34.1复习笔记34.2课后习题详解第35章章实物期权35.1复习笔记35.2课后习题详解第36章重大金融损失与借鉴36.1复习笔记36.2课后习题详解赫尔的《期权、期货及其他衍生产品》是世界上流行的证券学教材之一。
期权、期货及其他衍生产品第9版-赫尔】Ch(27)
h
3
Principal Protected Notes continued
Chapter 12 Trading Strategies Involving Options
h
1
Strategies to be Considered
Bond plus option to create principal protected note
Stock plus option
Two or more options of the same type (a spread)
Two or more options of different types (a combination)
h
2
Principal Protected Note
Allows investor to take a risky position without risking any principal Example: $1000 instrument consisting of
Viability depends on
Level of dividends Level of interest rates Volatility of the portfolio
Variations on standard product
Out of the money strike price Caps on investor return Knock outs, averaging features, etc
HullOFOD9eSolutionsCh01第九版期权期货及其他衍生品课后答案
CHAPTER 1IntroductionPractice QuestionsProblem 1.1.What is the difference between a long forward position and a short forward position?When a trader enters into a long forward contract, she is agreeing to buy the underlying asset for a certain price at a certain time in the future. When a trader enters into a short forward contract, she is agreeing to sell the underlying asset for a certain price at a certain time in the future.Problem 1.2.Explain carefully the difference between hedging, speculation, and arbitrage.A trader is hedging when she has an exposure to the price of an asset and takes a position in a derivative to offset the exposure. In a speculation the trader has no exposure to offset. She is betting on the future movements in the price of the asset. Arbitrage involves taking a position in two or more different markets to lock in a profit.Problem 1.3.What is the difference between entering into a long forward contract when the forward price is $50 and taking a long position in a call option with a strike price of $50?In the first case the trader is obligated to buy the asset for $50. (The trader does not have a choice.) In the second case the trader has an option to buy the asset for $50. (The trader does not have to exercise the option.)Problem 1.4.Explain carefully the difference between selling a call option and buying a put option.Selling a call option involves giving someone else the right to buy an asset from you. It gives you a payoff ofmax(0)min(0)T T S K K S --,=-, Buying a put option involves buying an option from someone else. It gives a payoff of max(0)T K S -,In both cases the potential payoff is T K S -. When you write a call option, the payoff is negative or zero. (This is because the counterparty chooses whether to exercise.) When you buy a put option, the payoff is zero or positive. (This is because you choose whether to exercise.)Problem 1.5.An investor enters into a short forward contract to sell 100,000 British pounds for US dollars at an exchange rate of 1.5000 US dollars per pound. How much does the investor gain or lose if the exchange rate at the end of the contract is (a) 1.4900 and (b) 1.5200?(a)The investor is obligated to sell pounds for 1.5000 when they are worth 1.4900. Thegain is (1.5000−1.4900) ×100,000 = $1,000.(b)The investor is obligated to sell pounds for 1.5000 when they are worth 1.5200. Theloss is (1.5200−1.5000)×100,000 = $2,000Problem 1.6.A trader enters into a short cotton futures contract when the futures price is 50 cents per pound. The contract is for the delivery of 50,000 pounds. How much does the trader gain or lose if the cotton price at the end of the contract is (a) 48.20 cents per pound; (b) 51.30 cents per pound?(a)The trader sells for 50 cents per pound something that is worth 48.20 cents per pound.Gain ($05000$04820)50000$900=.-.⨯,=.(b)The trader sells for 50 cents per pound something that is worth 51.30 cents per pound.Loss ($05130$05000)50000$650=.-.⨯,=.Problem 1.7.Suppose that you write a put contract with a strike price of $40 and an expiration date in three months. The current stock price is $41 and the contract is on 100 shares. What have you committed yourself to? How much could you gain or lose?You have sold a put option. You have agreed to buy 100 shares for $40 per share if the party on the other side of the contract chooses to exercise the right to sell for this price. The option will be exercised only when the price of stock is below $40. Suppose, for example, that the option is exercised when the price is $30. You have to buy at $40 shares that are worth $30; you lose $10 per share, or $1,000 in total. If the option is exercised when the price is $20, you lose $20 per share, or $2,000 in total. The worst that can happen is that the price of the stock declines to almost zero during the three-month period. This highly unlikely event would cost you $4,000. In return for the possible future losses, you receive the price of the option from the purchaser.Problem 1.8.What is the difference between the over-the-counter market and the exchange-traded market? What are the bid and offer quotes of a market maker in the over-the-counter market?The over-the-counter market is a telephone- and computer-linked network of financial institutions, fund managers, and corporate treasurers where two participants can enter into any mutually acceptable contract. An exchange-traded market is a market organized by an exchange where the contracts that can be traded have been defined by the exchange. When a market maker quotes a bid and an offer, the bid is the price at which the market maker is prepared to buy and the offer is the price at which the market maker is prepared to sell.Problem 1.9.You would like to speculate on a rise in the price of a certain stock. The current stock price is $29, and a three-month call with a strike of $30 costs $2.90. You have $5,800 to invest.Identify two alternative strategies, one involving an investment in the stock and the other involving investment in the option. What are the potential gains and losses from each?One strategy would be to buy 200 shares. Another would be to buy 2,000 options. If the share price does well the second strategy will give rise to greater gains. For example, if the share price goes up to $40 you gain [2000($40$30)]$5800$14200,⨯--,=,from the second strategy and only 200($40$29)$2200⨯-=,from the first strategy. However, if the share price does badly, the second strategy gives greater losses. For example, if the share price goes down to $25, the first strategy leads to a loss of 200($29$25)$800⨯-=,whereas the second strategy leads to a loss of the whole $5,800 investment. This example shows that options contain built in leverage.Problem 1.10.Suppose you own 5,000 shares that are worth $25 each. How can put options be used to provide you with insurance against a decline in the value of your holding over the next four months?You could buy 50 put option contracts (each on 100 shares) with a strike price of $25 and an expiration date in four months. If at the end of four months the stock price proves to be less than $25, you can exercise the options and sell the shares for $25 each.Problem 1.11.When first issued, a stock provides funds for a company. Is the same true of anexchange-traded stock option? Discuss.An exchange-traded stock option provides no funds for the company. It is a security sold by one investor to another. The company is not involved. By contrast, a stock when it is first issued is sold by the company to investors and does provide funds for the company.Problem 1.12.Explain why a futures contract can be used for either speculation or hedging.If an investor has an exposure to the price of an asset, he or she can hedge with futures contracts. If the investor will gain when the price decreases and lose when the price increases, a long futures position will hedge the risk. If the investor will lose when the price decreases and gain when the price increases, a short futures position will hedge the risk. Thus either a long or a short futures position can be entered into for hedging purposes.If the investor has no exposure to the price of the underlying asset, entering into a futures contract is speculation. If the investor takes a long position, he or she gains when the asset’s price increases and loses when it decreases. If the investor takes a short position, he or she loses when the asset’s price increases and gains when it decreases.Problem 1.13.Suppose that a March call option to buy a share for $50 costs $2.50 and is held until March. Under what circumstances will the holder of the option make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit on a long position in the option depends on the stock price at the maturity of the option.The holder of the option will gain if the price of the stock is above $52.50 in March. (This ignores the time value of money.) The option will be exercised if the price of the stock isabove $50.00 in March. The profit as a function of the stock price is shown in Figure S1.1.Figure S1.1:Profit from long position in Problem 1.13Problem 1.14.Suppose that a June put option to sell a share for $60 costs $4 and is held until June. Under what circumstances will the seller of the option (i.e., the party with a short position) make a profit? Under what circumstances will the option be exercised? Draw a diagram showing how the profit from a short position in the option depends on the stock price at the maturity of the option.The seller of the option will lose money if the price of the stock is below $56.00 in June. (This ignores the time value of money.) The option will be exercised if the price of the stock is below $60.00 in June. The profit as a function of the stock price is shown in Figure S1.2.Figure S1.2:Profit from short position in Problem 1.14Problem 1.15.It is May and a trader writes a September call option with a strike price of $20. The stock price is $18, and the option price is $2. Describe the investor’s cash flo ws if the option is held until September and the stock price is $25 at this time.The trader has an inflow of $2 in May and an outflow of $5 in September. The $2 is the cash received from the sale of the option. The $5 is the result of the option being exercised. The investor has to buy the stock for $25 in September and sell it to the purchaser of the optionfor $20.Problem 1.16.A trader writes a December put option with a strike price of $30. The price of the option is $4. Under what circumstances does the trader make a gain?The trader makes a gain if the price of the stock is above $26 at the time of exercise. (This ignores the time value of money.)Problem 1.17.A company knows that it is due to receive a certain amount of a foreign currency in four months. What type of option contract is appropriate for hedging?A long position in a four-month put option can provide insurance against the exchange rate falling below the strike price. It ensures that the foreign currency can be sold for at least the strike price.Problem 1.18.A US company expects to have to pay 1 million Canadian dollars in six months. Explain how the exchange rate risk can be hedged using (a) a forward contract and (b) an option.The company could enter into a long forward contract to buy 1 million Canadian dollars in six months. This would have the effect of locking in an exchange rate equal to the current forward exchange rate. Alternatively the company could buy a call option giving it the right (but not the obligation) to purchase 1 million Canadian dollars at a certain exchange rate in six months. This would provide insurance against a strong Canadian dollar in six months while still allowing the company to benefit from a weak Canadian dollar at that time. Problem 1.19.A trader enters into a short forward contract on 100 million yen. The forward exchange rate is $0.0090 per yen. How much does the trader gain or lose if the exchange rate at the end of the contract is (a) $0.0084 per yen; (b) $0.0101 per yen?a)The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0084⨯.millions of dollars or $60,000.per yen. The gain is 10000006b)The trader sells 100 million yen for $0.0090 per yen when the exchange rate is $0.0101⨯.millions of dollars or $110,000.per yen. The loss is 10000011Problem 1.20.The CME Group offers a futures contract on long-term Treasury bonds. Characterize the investors likely to use this contract.Most investors will use the contract because they want to do one of the following: a) Hedge an exposure to long-term interest rates.b) Speculate on the future direction of long-term interest rates.c) Arbitrage between the spot and futures markets for Treasury bonds.This contract is discussed in Chapter 6.Problem 1.21.“Options and futures are zero -sum games.” What do you think is meant by this statement?The statement means that the gain (loss) to the party with the short position is equal to the loss (gain) to the party with the long position. In aggregate, the net gain to all parties is zero.Problem 1.22.Describe the profit from the following portfolio: a long forward contract on an asset and a long European put option on the asset with the same maturity as the forward contract and a strike price that is equal to the forward price of the asset at the time the portfolio is set up.The terminal value of the long forward contract is:0T S F -where T S is the price of the asset at maturity and 0F is the delivery price, which is the same as the forward price of the asset at the time the portfolio is set up). The terminal value of the put option is:0max (0)T F S -,The terminal value of the portfolio is therefore00max (0)T T S F F S -+-,0max (0]T S F =,-This is the same as the terminal value of a European call option with the same maturity as the forward contract and a strike price equal to 0F . This result is illustrated in the Figure S1.3. The profit equals the terminal value of the call option less the amount paid for the put option. (It does not cost anything to enter into the forward contract.Figure S1.3: Profit from portfolio in Problem 1.22Problem 1.23.In the 1980s, Bankers Trust developed index currency option notes (ICONs). These are bonds in which the amount received by the holder at maturity varies with a foreign exchange rate. One example was its trade with the Long Term Credit Bank of Japan. The ICON specified that if the yen –U.S. dollar exchange rate,T S , is greater than 169 yen per dollar at maturity(in 1995), the holder of the bond receives $1,000. If it is less than 169 yen per dollar, the amount received by the holder of the bond is 1691000max 010001T S ⎡⎤⎛⎫,-,,-⎢⎥ ⎪⎝⎭⎣⎦ When the exchange rate is below 84.5, nothing is received by the holder at maturity. Show that this ICON is a combination of a regular bond and two options.Suppose that the yen exchange rate (yen per dollar) at maturity of the ICON is T S . The payofffrom the ICON is1000if 169169100010001if 8451690if 845T T T T S S S S ,>⎛⎫,-,-.≤≤ ⎪⎝⎭<.When 845169T S .≤≤ the payoff can be written 1690002000TS ,,-The payoff from an ICON is the payoff from:(a) A regular bond(b) A short position in call options to buy 169,000 yen with an exercise price of 1/169 (c) A long position in call options to buy 169,000 yen with an exercise price of 1/84.5 This is demonstrated by the following table, which shows the terminal value of the various components of the positionProblem 1.24.On July 1, 2011, a company enters into a forward contract to buy 10 million Japanese yen on January 1, 2012. On September 1, 2011, it enters into a forward contract to sell 10 million Japanese yen on January 1, 2012. Describe the payoff from this strategy.Suppose that the forward price for the contract entered into on July 1, 2011 is 1F and thatthe forward price for the contract entered into on September 1, 2011 is 2F with both 1F and 2F being measured as dollars per yen. If the value of one Japanese yen (measured in USdollars) is T S on January 1, 2012, then the value of the first contract (in millions of dollars)at that time is110()T S F -while the value of the second contract at that time is:210()T F S -The total payoff from the two contracts is therefore122110()10()10()T T S F F S F F -+-=-Thus if the forward price for delivery on January 1, 2012 increased between July 1, 2011 and September 1, 2011 the company will make a profit. (Note that the yen/USD exchange rate is usually expressed as the number of yen per USD not as the number of USD per yen)Problem 1.25.Suppose that USD-sterling spot and forward exchange rates are as follows :What opportunities are open to an arbitrageur in the following situations?(a) A 180-day European call option to buy £1 for $1.52 costs 2 cents.(b) A 90-day European put option to sell £1 for $1.59 costs 2 cents.Note that there is a typo in the problem in the book. 1.42 and 1.49 should be 1.52 and 1.59 in the last two lines of the problem s(a) The arbitrageur buys a 180-day call option and takes a short position in a 180-day forward contract. If T S is the terminal spot rate, the profit from the call option is 02.0)0,52.1max(--T SThe profit from the short forward contract isT S -5518.1The profit from the strategy is thereforeT T S S -+--5518.102.0)0,52.1max(orT T S S -+-5318.1)0,52.1max(This is1.5318−S T when S T <1.520.0118 when S T >1.52This shows that the profit is always positive. The time value of money has been ignored in these calculations. However, when it is taken into account the strategy is still likely to be profitable in all circumstances. (We would require an extremely high interest rate for $0.0118 interest to be required on an outlay of $0.02 over a 180-day period.)(b) The trader buys 90-day put options and takes a long position in a 90 day forwardcontract. If T S is the terminal spot rate, the profit from the put option is02.0)0,59.1max(--T SThe profit from the long forward contract isS T −1.5556The profit from this strategy is therefore5556.102.0)0,59.1max(-+--T T S Sor5756.1)0,59.1max(-+-T T S SThis isS T −1.5756 when S T >1.590.0144 when S T <1.59The profit is therefore always positive. Again, the time value of money has been ignored but is unlikely to affect the overall profitability of the strategy. (We would require interest rates to be extremely high for $0.0144 interest to be required on an outlay of $0.02 over a 90-day period.)Problem 1.26.A trader buys a call option with a strike price of $30 for $3. Does the trader ever exercise the option and lose money on the trade. Explain.If the stock price is between $30 and $33 at option maturity the trader will exercise the option, but lose money on the trade. Consider the situation where the stock price is $31. If the trader exercises, she loses $2 on the trade. If she does not exercise she loses $3 on the trade. It is clearly better to exercise than not exercise.Problem 1.27.A trader sells a put option with a strike price of $40 for $5. What is the trader's maximum gain and maximum loss? How does your answer change if it is a call option?The trader’s maximum gain from the put option is $5. The maximum loss is $35,corresponding to the situation where the option is exercised and the price of the underlying asset is zero. If the option were a call, the trader’s maxim um gain would still be $5, but there would be no bound to the loss as there is in theory no limit to how high the asset price could rise.Problem 1.28.``Buying a put option on a stock when the stock is owned is a form of insurance.'' Explain this statement.If the stock price declines below the strike price of the put option, the stock can be sold for the strike price.Further QuestionsProblem 1.29.On May 8, 2013, as indicated in Table 1.2, the spot offer price of Google stock is $871.37 and the offer price of a call option with a strike price of $880 and a maturity date ofSeptember is $41.60. A trader is considering two alternatives: buy 100 shares of the stock and buy 100 September call options. For each alternative, what is (a) the upfront cost, (b)the total gain if the stock price in September is $950, and (c) the total loss if the stockprice in September is $800. Assume that the option is not exercised before September andif stock is purchased it is sold in September.a)The upfront cost for the stock alternative is $87,137. The upfront cost for the optionalternative is $4,160.b)The gain from the stock alternative is $95,000−$87,137=$7,863. The total gain fromthe option alternative is ($950-$880)×100−$4,160=$2,840.c)The loss from the stock alternative is $87,137−$80,000=$7,137. The loss from theoption alternative is $4,160.Problem 1.30.What is arbitrage? Explain the arbitrage opportunity when the price of a dually listed mining company stock is $50 (USD) on the New York Stock Exchange and $52 (CAD) on the Toronto Stock Exchange. Assume that the exchange rate is such that 1 USD equals 1.01 CAD. Explain what is likely to happen to prices as traders take advantage of this opportunity. Arbitrage involves carrying out two or more different trades to lock in a profit. In this case, traders can buy shares on the NYSE and sell them on the TSX to lock in a USD profit of52/1.01−50=1.485 per share. As they do this the NYSE price will rise and the TSX price will fall so that the arbitrage opportunity disappearsProblem 1.31 (Excel file)Trader A enters into a forward contract to buy an asset for $1000 in one year. Trader B buys a call option to buy the asset for $1000 in one year. The cost of the option is $100. What is the difference between the positions of the traders? Show the profit as a function of the price of the asset in one year for the two traders.Trader A makes a profit of S T 1000 and Trader B makes a profit of max (S T 1000, 0) –100 where S T is the price of the asset in one year. Trader A does better if S T is above $900 as indicated in Figure S1.4.Figure S1.4: Profit to Trader A and Trader B in Problem 1.31Problem 1.32.In March, a US investor instructs a broker to sell one July put option contract on a stock. The stock price is $42 and the strike price is $40. The option price is $3. Explain what the investor has agreed to. Under what circumstances will the trade prove to be profitable? What are the risks?The investor has agreed to buy 100 shares of the stock for $40 in July (or earlier) if the party on the other side of the transaction chooses to sell. The trade will prove profitable if the option is not exercised or if the stock price is above $37 at the time of exercise. The risk to the investor is that the stock price plunges to a low level. For example, if the stock price drops to $1 by July , the investor loses $3,600. This is because the put options are exercised and $40 is paid for 100 shares when the value per share is $1. This leads to a loss of $3,900 which is only a little offset by the premium of $300 received for the options.Problem 1.33.A US company knows it will have to pay 3 million euros in three months. The current exchange rate is 1.3500 dollars per euro. Discuss how forward and options contracts can be used by the company to hedge its exposure.The company could enter into a forward contract obligating it to buy 3 million euros in three months for a fixed price (the forward price). The forward price will be close to but not exactly the same as the current spot price of 1.3500. An alternative would be to buy a call option giving the company the right but not the obligation to buy 3 million euros for a particular exchange rate (the strike price) in three months. The use of a forward contract locks in, at no cost, the exchange rate that will apply in three months. The use of a call option provides, at a cost, insurance against the exchange rate being higher than the strike price. Problem 1.34. (Excel file)A stock price is $29. An investor buys one call option contract on the stock with a strike price of $30 and sells a call option contract on the stock with a strike price of $32.50. The market prices of the options are $2.75 and $1.50, respectively. The options have the same maturity date. Describe the investor's position.This is known as a bull spread (see Chapter 12). The profit is shown in Figure S1.5.Figure S1.5: Profit in Problem 1.34Problem 1.35.The price of gold is currently $1,400 per ounce. The forward price for delivery in one year is $1,500. An arbitrageur can borrow money at 4% per annum. What should the arbitrageur do? Assume that the cost of storing gold is zero and that gold provides no income.The arbitrageur should borrow money to buy a certain number of ounces of gold today and short forward contracts on the same number of ounces of gold for delivery in one year. This means that gold is purchased for $1,400 per ounce and sold for $1,500 per ounce. Interest on the borrowed funds will be 0.04×$1400 or $56 per ounce. A profit of $44 per ounce will therefore be made.Problem 1.36.The current price of a stock is $94, and three-month call options with a strike price of $95 currently sell for $4.70. An investor who feels that the price of the stock will increase istrying to decide between buying 100 shares and buying 2,000 call options (20 contracts). Both strategies involve an investment of $9,400. What advice would you give? How high does the stock price have to rise for the option strategy to be more profitable?The investment in call options entails higher risks but can lead to higher returns. If the stock price stays at $94, an investor who buys call options loses $9,400 whereas an investor who buys shares neither gains nor loses anything. If the stock price rises to $120, the investor who buys call options gains⨯--=,$2000(12095)940040600An investor who buys shares gains$⨯-=,100(12094)2600The strategies are equally profitable if the stock price rises to a level, S, where⨯-=--100(94)2000(95)9400S SorS=100The option strategy is therefore more profitable if the stock price rises above $100.Problem 1.37.On May 8, 2013, an investor owns 100 Google shares. As indicated in Table 1.3, the share price is about $871 and a December put option with a strike price $820 costs $37.50. The investor is comparing two alternatives to limit downside risk. The first involves buying one December put option contract with a strike price of $820. The second involves instructing a broker to sell the 100 shares as so on as Google’s price reaches $820. Discuss the advantages and disadvantages of the two strategies.The second alternative involves what is known as a stop or stop-loss order. It costs nothing and ensures that $82,000, or close to $82,000, is realized for the holding in the event the stock price ever falls to $820. The put option costs $3,750 and guarantees that the holding can be sold for $8,200 any time up to December. If the stock price falls marginally below $820 and then rises the option will not be exercised, but the stop-loss order will lead to the holding being liquidated. There are some circumstances where the put option alternative leads to a better outcome and some circumstances where the stop-loss order leads to a better outcome.If the stock price ends up below $820, the stop-loss order alternative leads to a better outcome because the cost of the option is avoided. If the stock price falls to $800 in November and then rises to $850 by December, the put option alternative leads to a betteroutcome. The investor is paying $3,750 for the chance to benefit from this second type of outcome.Problem 1.38.A bond issued by Standard Oil some time ago worked as follows. The holder received no interest. At the bond’s maturity the company promised to pay $1,000 plus an additionalamount based on the price of oil at that time. The additional amount was equal to the product of 170 and the excess (if any) of the price of a barrel of oil at maturity over $25. The maximum additional amount paid was $2,550 (which corresponds to a price of $40 per barrel). Show that the bond is a combination of a regular bond, a long position in call options on oil with a strike price of $25, and a short position in call options on oil with a strike price of $40.Suppose T S is the price of oil at the bond’s maturity. In addition to $1000 the Standard Oilbond pays:$250$40$25170(25)$402550T T T T S S S S <:>>:->:,This is the payoff from 170 call options on oil with a strike price of 25 less the payoff from 170 call options on oil with a strike price of 40. The bond is therefore equivalent to a regular bond plus a long position in 170 call options on oil with a strike price of $25 plus a short position in 170 call options on oil with a strike price of $40. The investor has what is termed a bull spread on oil. This is discussed in Chapter 12.Problem 1.39.Suppose that in the situation of Table 1.1 a cor porate treasurer said: “I will have £1 million to sell in six months. If the exchange rate is less than 1.52, I want you to give me 1.52. If it is greater than 1.58 I will accept 1.58. If the exchange rate is between 1.52 and 1.58, I will sell the sterling for the exchange rate.” How could you use options to satisfy the treasurer?You sell the treasurer a put option on GBP with a strike price of 1.52 and buy from the treasurer a call option on GBP with a strike price of 1.58. Both options are on one million pounds and have a maturity of six months. This is known as a range forward contract and is discussed in Chapter 17.Problem 1.40.Describe how foreign currency options can be used for hedging in the situation considered in Section 1.7 so that (a) ImportCo is guaranteed that its exchange rate will be less than 1.5700, and (b) ExportCo is guaranteed that its exchange rate will be at least 1.5300. Use DerivaGem to calculate the cost of setting up the hedge in each case assuming that the exchange rate volatility is 12%, interest rates in the United States are 5% and interest rates in Britain are 5.7%. Assume that the current exchange rate is the average of the bid and offer in Table 1.1.ImportCo should buy three-month call options on $10 million with a strike price of 1.5700. ExportCo should buy three-month put options on $10 million with a strike price of 1.5300. In this case the spot foreign exchange rate is 1.5543 (the average of the bid and offer quotes in。
HullOFOD9eSolutionsCh13第九版期权、期货及其他衍生品课后答案
HullOFOD9eSolutionsCh13第九版期权、期货及其他衍生品课后答案CHAPTER 13 Binomial TreesPractice QuestionsProblem 13.1.A stock price is currently $40. It is known that at the end of one month it will be either $42 or $38. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-month European call option with a strike price of $39?Consider a portfolio consisting of 1-: Call option +?: Shares If the stock price rises to $42, the portfolio is worth 423?-. If the stock price falls to $38, it is worth 38?. These are the same when42338?-=? or 075?=.. The value of the portfolio in one month is 28.5 for both stock prices. Its value today must be the present value of 28.5, or 0080083332852831e -.?..=.. This means that 402831f -+?=.where f is the call price. Because 075?=., the call price is 400752831$169?.-.=.. As an alternative approach, we can calculate the probability, p , of an up movement in a risk-neutral world. This must satisfy: 0080083334238(1)40p p e .?.+-= so that 00800833344038p e .?.=-or 05669p =.. The value of the option is then its expected payoff discounted at the risk-free rate: 008008333[305669004331]169e -.?.?.+?.=. or $1.69. This agrees with the previous calculation.Problem 13.2.Explain the no-arbitrage and risk-neutral valuationapproaches to valuing a European option using a one-step binomial tree.In the no-arbitrage approach, we set up a riskless portfolio consisting of a position in the option and a position in the stock. By setting the return on the portfolio equal to the risk-free interest rate, we are able to value the option. When we use risk-neutral valuation, we first choose probabilities for the branches of the tree so that the expected return on the stock equals the risk-free interest rate. We then value the option by calculating its expected payoff and discounting this expected payoff at the risk-free interest rate.Problem 13.3.What is meant by the delta of a stock option?The delta of a stock option measures the sensitivity of the option price to the price of the stock when small changes are considered. Specifically, it is the ratio of the change in the price of the stock option to the change in the price of the underlying stock.Problem 13.4.A stock price is currently $50. It is known that at the end of six months it will be either $45 or $55. The risk-free interest rate is 10% per annum with continuous compounding. What is the value of a six-month European put option with a strike price of $50?Consider a portfolio consisting of 1-: Put option +?: Shares If the stock price rises to $55, this is worth 55?. If the stock price falls to $45, the portfolio is worth 455?-. These are the same when 45555?-=?or 050?=-.. The value of the portfolio in six months is 275-. for both stock prices. Its value today must be the present valueof 275-., or 010********e -.?.-.=-.. This means that 502616f -+?=-.where f is the put price. Because 050?=-., the put price is $1.16. As an alternative approach we can calculate the probability, p , of an up movement in a risk-neutral world. This must satisfy: 01055545(1)50p p e .?.+-= so that 010*******p e .?.=- or 07564p =.. The value of the option is then its expected payoff discounted at the risk-free rate: 0105[007564502436]116e -.?.?.+?.=. or $1.16. This agrees with the previous calculation.Problem 13.5.A stock price is currently $100. Over each of the next two six-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 8% per annum with continuous compounding. What is the value of a one-year European call option with a strike price of $100?In this case 110u =., 090d =., 05t ?=., and 008r =., so that00805090***********e p .?.-.==..-.The tree for stock price movements is shown in Figure S13.1. We can work back from the end of the tree to the beginning, as indicated in the diagram, to give the value of the option as $9.61. The option value can also be calculated directly from equation (13.10): 22200805[0704121207041029590029590]961e -?.?..?+?.?.?+.?=. or $9.61.Figure S13.1: Tree for Problem 13.5Problem 13.6.For the situation considered in Problem 13.5, what is the value of a one-year European put option with a strike price of $100? Verify that the European call and European put prices satisfy put –call parity.Figure S13.2 shows how we can value the put option using the same tree as in Problem 13.5. The value of the option is $1.92. The option value can also be calculated directly from equation (13.10): 20080522[0704102070410295910295919]192e -?.?..?+?.?.?+.?=.or $1.92. The stock price plus the put price is 10019210192$+.=.. The present value of the strike price plus the call price is 008110096110192e $-.?+.=.. These are the same, verifyingthat put –call parity holds.Figure S13.2: Tree for Problem 13.6Problem 13.7.What are the formulas for u and d in terms of volatility?u e =and d e -=Problem 13.8.Consider the situation in which stock price movements during the life of a European option are governed by a two-step binomial tree. Explain why it is not possible to set up a position in the stock and the option that remains riskless for the whole of the life of the option.The riskless portfolio consists of a short position in the option and a long position in ? shares. Because ? changes during the life of the option, this riskless portfolio must also change.Problem 13.9.A stock price is currently $50. It is known that at the end of two months it will be either $53 or $48. The risk-free interest rate is 10% per annum with continuous compounding. What is thevalue of a two-month European call option with a strikeprice of $49? Use no-arbitrage arguments.At the end of two months the value of the option will be either $4 (if the stock price is $53) or $0 (if the stock price is $48). Consider a portfolio consisting of:shares1option+?:-:The value of the portfolio is either 48? or 534?- in two months. If48534?=?- i.e.,08?=. the value of the portfolio is certain to be 38.4. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is: 0850f .?-where f is the value of the option. Since the portfolio must earn the risk-free rate of interest010212(0850)384f e .?/.?-=.i.e.,223f =.The value of the option is therefore $2.23.This can also be calculated directly from equations (13.2) and (13.3). 106u =., 096d =. so that01021209605681106096e p .?/-.==..-. and010212056814223f e -.?/=?.?=.Problem 13.10.A stock price is currently $80. It is known that at the end of four months it will be either $75or $85. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a four-monthEuropean put option with a strike price of $80? Use no-arbitrage arguments.At the end of four months the value of the option will be either $5 (if the stock price is $75) or $0 (if the stock price is $85). Consider a portfolio consisting of:shares1option-?:+:(Note: The delta, ? of a put option is negative. We have constructed the portfolio so that it is +1 option and -? shares rather than 1- option and +? shares so that the initial investment is positive.)The value of the portfolio is either 85-? or 755-?+ in four months. If 85755-?=-?+ i.e.,05?=-. the value of the portfolio is certain to be 42.5. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is: 0580f .?+where f is the value of the option. Since the portfolio is riskless005412(0580)425f e .?/.?+=.i.e.,180f =.The value of the option is therefore $1.80.This can also be calculated directly from equations (13.2) and (13.3). 10625u =., 09375d =. so that00541209375063451062509375e p .?/-.==..-. 103655p -=. and005412036555180f e -.?/=?.?=.Problem 13.11.A stock price is currently $40. It is known that at the end ofthree months it will be either $45 or $35. The risk-free rate of interest with quarterly compounding is 8% per annum. Calculate the value of a three-month European put option on the stock with an exercise price of $40. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.At the end of three months the value of the option is either $5 (if the stock price is $35) or $0 (if the stock price is $45).Consider a portfolio consisting of:shares1option-?:+:(Note: The delta, ?, of a put option is negative. We have constructed the portfolio so that it is +1 option and -? shares rather than 1- option and +? shares so that the initial investment is positive.)The value of the portfolio is either 355-?+ or 45-?. If:35545-?+=-?i.e.,05?=-.the value of the portfolio is certain to be 22.5. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is 40f -?+where f is the value of the option. Since the portfolio must earn the risk-free rate of interest (4005)102225f ?.+?.=. Hence 206f =. i.e., the value of the option is $2.06.This can also be calculated using risk-neutral valuation. Suppose that p is the probability of an upward stock price movement in a risk-neutral world. We must have 4535(1)40102p p +-=?. i.e., 1058p =. or: 058p =.The expected value of the option in a risk-neutral world is:00585042210?.+?.=. This has a present value of210206102.=..This is consistent with the no-arbitrage answer.Problem 13.12.A stock price is currently $50. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%. The risk-free interest rate is 5% per annum with continuous compounding. What is the value of a six-month European call option with a strike price of $51?A tree describing the behavior of the stock price is shown in Figure S13.3. The risk-neutral probability of an up move, p , is given by00531209505689106095e p .?/-.==..-. There is a payoff from the option of 561851518.-=. for the highest final node (which corresponds to two up moves) zero in all other cases. The value of the option is therefore 2005612518056891635e -.?/.?.?=.This can also be calculated by working back through the tree as indicated in Figure S13.3. The value of the call option is the lower number at each node in the figure.Figure S13.3:Tree for Problem 13.12Problem 13.13.For the situation considered in Problem 13.12, what is the value of a six-month European put option with a strike price of $51? Verify that the European call and European put prices satisfy put–call parity. If the put option were American, would it ever be optimal to exercise it early at any of the nodes on the tree?The tree for valuing the put option is shown in Figure S13.4. We get a payoff of-.=.if -.=.if the middle final node is reached and a payoff of 51451255875 515035065the lowest final node is reached. The value of the option is therefore2005612.??.?.+.?.=.(06520568904311587504311)1376e-.?/This can also be calculated by working back through the tree as indicated in Figure S13.4. The value of the put plus the stock price is.+=.137********The value of the call plus the present value of the strike price is005612e-.?/.+=.16355151376This verifies that put–call parity holdsTo test whether it worth exercising the option early wecompare the value calculated for the option at each node with the payoff from immediate exercise. At node C the payoff from -.=.. Because this is greater than 2.8664, the option should immediate exercise is 5147535be exercised at this node. The option should not be exercised at either node A or node B.Figure S13.4:Tree for Problem 13.13Problem 13.14.A stock price is currently $25. It is known that at the end of two months it will be either $23 or $27. The risk-free interest rate is 10% per annum with continuous compounding. Suppose T S is the stock price at the end of two months. What is the value of a derivative that pays off2T S at this time?At the end of two months the value of the derivative will be either 529 (if the stock price is 23) or 729 (if the stock price is 27). Consider a portfolio consisting of:shares1derivative+?:-:The value of the portfolio is either 27729?- or 23529?- in twomonths. If2772923529?-=?- i.e.,50?= the value of the portfolio is certain to be 621. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is: 5025f ?-where f is the value of the derivative. Since the portfolio must earn the risk-free rate of interest 010212(5025)621f e .?/?-= i.e., 6393f =. The value of the option is therefore $639.3.This can also be calculated directly from equations (13.2) and (13.3). 108u =., 092d =. so that01021209206050108092e p .?/-.==..-. and 010212(0605072903950529)6393f e -.?/=.?+.?=.Problem 13.15.Calculate u , d , and p when a binomial tree is constructed to value an option on a foreign currency. The tree step size is one month, the domestic interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.In this case (005008)11209975a e .-.?/==.010352u e .==.109660d u =/=.0997509660045531035209660p .-.==..-.Problem 13.16.The volatility of a non-dividend-paying stock whose price is $78, is 30%. The risk-free rate is 3% per annum (continuously compounded) for all maturities. Calculate values for u, d, and pwhen a two-month time step is used. What is the value of a four-month European call option with a strike price of $80 given by a two-step binomial tree. Suppose a trader sells 1,000 options (10 contracts). What position in the stock is necessary to hedge the trader’s position at the time of the trade?4898.08847.01303.18847.08847.0/11303.112/230.01667.030.0=--=====??e p u d e uThe tree is given in Figure S13.5. The value of the option is $4.67. The initial delta is 9.58/(88.16 –69.01) which is almost exactly 0.5 so that 500 shares should be purchased.Figure S13.5: Tree for Problem 13.16Problem 13.17.A stock index is currently 1,500. Its volatility is 18%. The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.5%. Calculate values for u, d, and p when a six-month time step is used. What is the value a 12-month American put option with a strike price of 1,480 given by a two-step binomial tree.4977.08805.01357.18805.08805.0/11357.15.0)025.004.0(5.018.0=--=====?-?e p u d e uThe tree is shown in Figure S13.6. The option is exercised at the lower node at the six-month point. It is worth 78.41.Figure S13.6: Tree for Problem 13.17Problem 13.18.The futures price of a commodity is $90. Use a three-step tree to value (a) a nine-month American call option with strike price $93 and (b) a nine-month American put option with strike price $93. The volatility is 28% and the risk-free rate (all maturities) is 3% with continuous compounding.4651.08694.01503.18694.018694.0/11503.125.028.0=--=====?u u d e u The tree for valuing the call is in Figure S13.7a and that for valuing the put is in Figure S13.7b. The values are 7.94 and 10.88, respectively.824637Figure S13.7a : CallFigure S13.7b : PutFurther QuestionsProblem 13.19.The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The risk-free rate is 4%. For a three-month time step: (a) What is the percentage up movement? (b) What is the percentage down movement?(c) What is the probability of an up movement in a risk-neutral world? (d) What is the probability of a down movementin a risk-neutral world?Use a two-step tree to value a six-month European call option and a six-month European put option. In both cases the strike price is $150.(a) 25.025.0?=e u = 1.1331. The percentage up movement is13.31% (b) d = 1/u = 0.8825. The percentage down movement is11.75%(c) The probability of an up movement is 5089.0)8825.1331.1/()8825.()25.004.0=--?e (d) The probability of a down movement is0.4911.The tree for valuing the call is in Figure S13.8a and that for valuing the put is in Figure S13.8b. The values are 7.56 and 14.58, respectively.Figure S13.8a : CallFigure S13.8b : PutProblem 13.20.In Problem 13.19, suppose that a trader sells 10,000 European call options. How many shares of the stock are needed to hedge the position for the first and second three-month period? For the second time period, consider both the case where the stock price moves up during the first period and the case where it moves down during the first period.The delta for the first period is 15/(158.64 – 123.55) = 0.4273. The trader should take a long position in 4,273 shares. If there is an up movement the delta for the second period is 29.76/(179.76 – 140) = 0.7485. The trader should increase the holding to 7,485 shares. If there is a down movement the trader should decrease the holding to zero.Problem 13.21.A stock price is currently $50. It is known that at the end of six months it will be either $60 or $42. The risk-free rate of interest with continuous compounding is 12% per annum. Calculate the value of a six-month European call option on the stock with an exercise price of $48. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.At the end of six months the value of the option will be either $12 (if the stock price is $60) or $0 (if the stock price is $42). Consider a portfolio consisting of:shares1option+?:-:The value of the portfolio is either 42? or 6012?- in sixmonths. If426012?=?- i.e.,06667?=. the value of the portfolio is certain to be 28. For this value of ? the portfolio is therefore riskless. The current value of the portfolio is: 0666750f .?-where f is the value of the option. Since the portfolio must earn the risk-free rate of interest01205(0666750)28f e .?..?-=i.e.,696f =.The value of the option is therefore $6.96.This can also be calculated using risk-neutral valuation. Suppose that p is the probability of an upward stock price movement in a risk-neutral world. We must have 0066042(1)50p p e .+-=? i.e., 181109p =. or: 06161p =.The expected value of the option in a risk-neutral world is: 120616100383973932?.+?.=. This has a present value of 00673932696e -..=.Hence the above answer is consistent with risk-neutral valuation.Problem 13.22.A stock price is currently $40. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. The risk-free interest rate is 12% per annum with continuous compounding.a. What is the value of a six-month European put option witha strike price of $42?b. What is the value of a six-month American put option with a strike price of $42?a. A tree describing the behavior of the stock price is shownin Figure S13.9. The risk-neutral probability of an up move, p , is given by012312090065231109e p .?/-.==..-.Calculating the expected payoff and discounting, we obtain the value of the option as 2012612[24206523034779603477]2118e -.?/.??.?.+.?.=.The value of the European option is 2.118. This can also be calculated by working back through the tree as shown in Figure S13.9. The second number at each node is the value of the European option.b. The value of the American option is shown as the third number at each node on the tree. It is 2.537. This is greater than the value of the European option because it is optimal to exercise early at node C.40.0002.1182.53744.000 0.8100.81036.0004.7596.00048.4000.0000.00039.6002.4002.40032.4009.6009.600ABCFigure S13.9: Tree to evaluate European and American put options in Problem 13.22. At each node, upper number is the stock price, the next number is the European put price, and the final number is the American put priceProblem 13.23.Using a “trial -and-error” approach, estimate how high the strike price has to be in Problem 13.17 for it to be optimal toexercise the option immediately.Trial and error shows that immediate early exercise is optimal when the strike price is above 43.2. This can be also shown to be true algebraically. Suppose the strike price increases by a relatively small amount q . This increases the value of being at node C by q and the value of being at node B by 0030347703374e q q -..=.. It therefore increases the value of being at node A by 003(065230337403477)0551q q e q -..?.+.=.For early exercise at node A we require 253705512q q .+.<+ or 1196q >.. This corresponds to the strike price being greater than 43.196.Problem 13.24.A stock price is currently $30. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%. The risk-free interest rate is 5%. Use a two-step tree to calculate the value of a derivative that pays off 2[max(300)]T S -, whereT S is the stock price in four months? If the derivative is American-style, should it be exercised early?This type of option is known as a power option. A tree describing the behavior of the stock price is shown in Figure S13.10. The risk-neutral probability of an up move, p , is given by 005212090602010809e p .?/-.==..-. Calculating the expected payoff and discounting, we obtain the value of the option as393.5]3980.049.323980.06020.027056.0[12/405.02=?+-eThe value of the European option is 5.393. This can also be calculated by working back through the tree as shown in Figure S13.10. The second number at each node is the value of theEuropean option. Early exercise at node C would give 9.0 which is less than 13.2435. The option should therefore not be exercised early if it is American.Figure S13.10: Tree to evaluate European power option in Problem 13.24. At each node, upper number is the stock price and the next number is the option priceProblem 13.25.Consider a European call option on a non-dividend-paying stock where the stock price is $40, the strike price is $40, the risk-free rate is 4% per annum, the volatility is 30% per annum, and the time to maturity is six months.a. Calculate u , d , and p for a two step treeb. Value the option using a two step tree.c. Verify that DerivaGem gives the same answerd. Use DerivaGem to value the option with 5, 50, 100, and 500 time steps.0.0000 29.1600 0.705624.3000 32.4900 F(a) This problem is based on the material in Section 13.8. In this case 025t ?=.so that03011618u e .==., 108607d u =/=., and00402508607049591161808607e p .?.-.==..-.(b) and (c) The value of the option using a two-step tree as given by DerivaGem is shown in Figure S13.11 to be 3.3739. To use DerivaGem choose the first worksheet, select Equity as the underlying type, and select Binomial European as the Option Type. After carrying out the calculations select Display Tree.(d) With 5, 50, 100, and 500 time steps the value of the option is 3.9229, 3.7394, 3.7478, and 3.7545, respectively.Figure S13.11: Tree produced by DerivaGem to evaluate European option in Problem 13.25Problem 13.26.Repeat Problem 13.25 for an American put option on a futures contract. The strike price and the futures price are $50, the risk-free rate is 10%, the time to maturity is six months, and the volatility is 40% per annum.(a) In this case 025t ?=.and 04012214u e .==., 108187d u =/=., and0102508187045021221408187e p .?.-.==..-.(b) and (c) The value of the option using a two-step tree is4.8604.(d) With 5, 50, 100, and 500 time steps the value of the option is 5.6858, 5.3869, 5.3981, and 5.4072, respectively.Problem 13.27.Footnote 1 shows that the correct discount rate to use for the real world expected payoff inAt each node:Upper v alue = Underlying Asset PriceLower v alue = Option Price Values in red are a result of early exercise.Strike price = 40Discount factor per step = 0.9900Time step, dt = 0.2500 years, 91.25 daysGrowth factor per step, a = 1.0101Probability of up mov e, p = 0.4959Up step size, u = 1.16180.00000.25000.5000the case of the call option considered in Figure 13.1 is 42.6%. Show that if the option is a put rather than a call the discount rate is –52.5%. Explain why the two real-world discount rates are so different.The value of the put option is012312.?+.?=.*************)10123e-.?/The expected payoff in the real world is.?+.?=.*************)08877The discount rate R that should be used in the real world is therefore given by solving025.=.1012308877Re-.-.or 52.5%.The solution to this is 0525The underlying stock has positive systematic risk because it expected return is higher than the risk free rate. This means that the stock will tend to do well when the market does well. The call option has a high positive systematic risk because it tends to do very well when the market does well. As a result a high discount rate is appropriate for its expected payoff. The put option is in the opposite position. It tends to provide a high return when the market does badly. As a result it is appropriate to use a highly negative discount rate for its expected payoff.Problem 13.28.A stock index is currently 990, the risk-free rate is 5%, and the dividend yield on theindex is 2%. Use a three-step tree to value an 18-month American put option with astrike price of 1,000 when the volatility is 20% per annum. How much does the option holder gain by being able to exercise early? When is the gain made?The tree is shown in Figure S13.12. The value of the option is 87.51. It is optimal to exercise at the lowest node at time one year. If early exercise were not possible the value at this node would be 236.63. The gain made at the one year point is therefore 253.90 – 236.63= 17.27.Figure 13.12: Tree for Problem 13.28Problem 13.29.Calculate the value of nine-month American call option on a foreign currency using athree-step binomial tree. The current exchange rate is 0.79 and the strike price is 0.80 (both expressed as dollars per unit of the foreign currency). The volatility of the exchange rate is 12% per annum. The domestic and foreign risk-free rates are 2% and 5%, respectively. Suppose a company has bought options on 1 million units of the foreign currency. What position in the foreign currency is initially necessary to hedge its risk?The tree is shown in Figure S13.13. The cost of an American option to buy one million units of the foreign currency is $18,100. The delta initially is (0.0346 ?0.0051)/(0.8261 – 0.7554) = 0.4176. The company should sell 417,600 units of the foreign currencyFigure S13.13: Tree for Problem 13.29。
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Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
4
The Waterfall
Asset Cash Flows
ห้องสมุดไป่ตู้
Senior Tranche
Mezzanine Tranche Equity Tranche
13
The Aftermath…
•A huge amount of new regulation (Basel II.5, Basel III, Dodd-Frank, etc). For example:
• Banks required to hold more equity capital with the definition of equity capital being tightened
93.3%
100%
100%
100%
Losses on Senior Tranche of ABS CDO
0%
28.2%
69.2%
100%
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
7
• In 2019 the bubble burst. Some borrowers could not afford their payments when the teaser rates ended. Others had negative equity and recognized that it was optimal for them to exercise their put options.
Copyright © John C. Hull 2014
2
Securitization
• Traditionally banks have funded loans with deposits
• Securitization is a way that loans can increase much faster than deposits
• Foreclosures increased supply and caused U.S. real estate prices to fall. Products, created from the mortgages, that were previously thought to be safe began to be viewed as risky
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
3
Asset Backed Security (Simplified)
Asset 1 Asset 2 Asset 3
Asset n Principal: $100 million
期权期货和其他衍生品-英文第9版Chap08-Securitization and the Credit Crisis of 2019
此处添加副标题内容
Chapter 8 Securitization and the Credit Crisis of 2019
Options, Futures, and Other Derivatives 9th Edition,
• A tranche with a certain rating cannot be equated with a bond with the same rating. For example, the BBB tranches used to create ABS CDOs were typically about 1% wide and had “all or nothing” loss distributions (quite different from BBB bond)
• There was a “flight to quality” and credit spreads increased to very high levels
• Many banks incurred huge losses
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
10
What Many Market Participants
Did Not Realize…
• Default correlation goes up in stressed market conditions
• Recovery rates are less in stressed market conditions
• Mortgage originators (Their prime interest was in in originating mortgages that could be securitized)
• Valuers (They were under pressure to provide high valuations so that the loan-tovalue ratios looked good)
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
12
Incentives
• The crisis highlighted what are referred to as agency costs
Copyright © John C. Hull 2014
14
谢谢观看
共同学习相互提高
11
Regulatory Arbitrage
• The regulatory capital banks were required to keep for the tranches created from mortgages was less than that for the mortgages themselves
• This is quite different from the loss distribution for a BBB bond from a BBB bond
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
• Features of the market: 100% mortgages, ARMs, teaser rates, NINJAs, liar loans, non-recourse borrowing
• Mortgages were packaged in financial products and sold to investors
• This, combined with very low interest rates, increased the demand for real estate and prices rose.
• To continue to attract first time buyers and keep prices increasing they relaxed lending standards further
Mezzanine Tranche (25%) BBB
Equity Tranche (10%)
Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
6
Losses to AAA Senior Tranche of ABS CDO (Table 8.1, page 189)
• Traders (They were focused on the next
end-oOfptyiones,aFurturbeso, annduOstheraDneridvatinveso9tthwEdiotiornr, ied about
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Options, Futures, and Other Derivatives 9th Edition,
Copyright © John C. Hull 2014
9
What happened...
• Banks found it profitable to invest in the AAA rated tranches because the promised return was significantly higher than the cost of funds and capital requirements were low
Losses on Subprime portfolios
10% 13% 17% 20%
Losses on Mezzanine Tranche of ABS
33.3%
53.3%
80.0%
100%
Losses on Equity Tranche of ABS CDO
100%
100%
100%
100%
Losses on Mezzanine Tranche of ABS CDO
• Banks required to satisfy liquidity ratios