HullFund8eCh12ProblemSolutions（精編版）

1、chapter 12introduction to binomial treespractice questionsproblem 12.8.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 ris

2、kless 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. becausechanges during the life of the option, this riskless portfolio must also change.problem 12.9.a stock price is currently $50. it is known that at the e

3、nd of two months it will be either $53 or $48. the risk-free interest rate is 10% per annum with continuous compounding. what is the value 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

4、the stock price is $53) or$0 (if the stock price is $48). consider a portfolio consisting of:shares1optionthe value of the portfolio is either48or534in two months. if48534i.e.,0 8the value of the portfolio is certain to be 38.4. for this value ofthe portfolio is therefore riskless. the current value

5、 of the portfolio is:0 850fwherefis the value of the option. since the portfolio must earn the risk-free rate of interesti.e.,(0 850f )e0 10 2 12f2 2338 4the value of the option is therefore $2.23.this can also be calculated directly from equations (12.2) and (12.3).u that1 06 ,d0 96soande0 10 2 120

6、 96p1 060 960 5681fe 0 10 2 120 568142 23problem 12.10.a stock price is currently $80. it is known that at the end of four months it will be either $75 or $85. the risk-free interest rate is 5% per annum with continuous compounding. what is the value of a four-month european put option with a strike

7、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

8、so that it is +1 option andshares rather than1 option andshares so that the initial investment is positive.)the value of the portfolio is either85or755in four months. if85755i.e.,0 5the value of the portfolio is certain to be 42.5. for this value ofthe portfolio is therefore riskless. the current va

9、lue of the portfolio is:0 580fwherefis the value of the option. since the portfolio is risklessi.e.,(0 580f )e0 05 4 12f1 8042 5the value of the option is therefore $1.80.this can also be calculated directly from equations (12.2) and (12.3).u1 0625,d0 9375so thate0 05 4 120 9375p1 06250 93750 63451p

10、0 3655andfe 0 05 4 120 365551 80problem 12.11.a stock price is currently $40. it is known that at the end of three 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 wit

11、h 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

12、: the delta, of a put option is negative. we have constructed the portfolio so that it is +1 option andshares rather than1 option andshares so that the initial investment is positive.)the value of the portfolio is either355or45. if:35545i.e.,0 5the value of the portfolio is certain to be 22.5. for t

13、his value ofthe portfolio is therefore riskless. the current value of the portfolio is40fwhere f is the value of the option. since the portfolio must earn the risk-free rate of interesthence(400 5ff )1 0222 52 06i.e., the value of the option is $2.06.this can also be calculated using risk-neutral va

14、luation. suppose thatpis the probability ofan upward stock price movement in a risk-neutral world. we must havei.e., or:45 p35(110 p pp)401 025 80 58the expected value of the option in a risk-neutral world is:00 5850 422 10this has a present value of2 102 061 02this is consistent with the no-arbitra

15、ge answer.problem 12.12.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?

16、a tree describing the behavior of the stock price is shown in figure s12.1. the risk-neutral probability of an up move, p, is given bye0 05 3 120 95p1 060 950 56892ethere is a payoff from the option of 56 18515 18for the highest final node (which corresponds to two up moves) zero in all other cases.

17、 the value of the option is therefore5 180 56890 05 6 121 635this can also be calculated by working back through the tree as indicated in figure s12.1. the value of the call option is the lower number at each node in the figure.problem 12.13.for the situation considered in problem 12.12, what is the

18、 value of a six-month european put option with a strike price of $51? verify that the european call and european put prices satisfy putcall 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 sh

19、own in figure s12.2. we get a payoff of5150 350 65if the middle final node is reached and a payoff of 5145 1255 875 if the lowest final node is reached. the value of the option is therefore(0 6520 56890 43115 8750 43112 )e0 05 6 121 376this can also be calculated by working back through the tree as

20、indicated in figure s12.2. the value of the put plus the stock price is from problem 12.121 3765051 376the value of the call plus the present value of the strike price is1 63551ethis verifies that putcall parity holds0 05 6 1251 376to test whether it worth exercising the option early we compare the

21、value calculated for the option at each node with the payoff from immediate exercise. at node c the payoff from immediate exercise is 5147 53 5. because this is greater than 2.8664, the option should be exercised at this node. the option should not be exercised at either node a or node b.problem 12.

22、14.a stock price is currently $25. it is known that at the end of two months it will be either $23or $27. the risk-free interest rate is 10% per annum with continuous compounding. supposestis the stock price at the end of two months. what is the value of a derivative that pays offst2at this time?at

23、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:shares1derivativethe value of the portfolio is either27729 or23529 in two months. if2772923529i.e.,50the value of the portfolio is cer

24、tain to be 621. for this value ofthe portfolio is therefore riskless. the current value of the portfolio is:5025fwherefis the value of the derivative. since the portfolio must earn the risk-free rate of interesti.e.,(5025ff )e0 10 2 12639 3621the value of the option is therefore $639.3.this can also

25、 be calculated directly from equations (12.2) and (12.3).u that1 08 ,d0 92soande0 10 2 120 92p1 080 920 60500 10 2 12fe(0 60507290 3950529)639 3problem 12.15.calculateu ,d , andpwhen a binomial tree is constructed to value an option on aforeign currency. the tree step size is one month, the domestic

26、 interest rate is 5% per annum, the foreign interest rate is 8% per annum, and the volatility is 12% per annum.in this caseae0 9975(0 05 0 08) 1 12ue0 12 1 121 0352d1u0 9660p0 99750 96600 45531 03520 9660problem 12.16.the volatility of a non-dividend-paying stock whose price is $78, is 30%. the risk

27、-freerate is 3% per annum (continuously compounded) for all maturities. calculate values for u, d, and p when 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 co

28、ntracts). what position in the stock is necessary to hedge the trader s position at the time of the trade?ue0.300.16671.1303d1/ u0.8847e0.30p2 / 120.88470.48981.13030.8847the tree is given in figure s12.3. the value of the option is $4.67. the initial delta is 9.58/(88.16 69.01) which is almost exac

29、tly 0.5 so that 500 shares should be purchased.99.6519.6588.169.5878.0078.004.670.0069.010.0061.050.00figure s12.3:problem 12.17.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 inde

30、x 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.ue0 .180.51.1357d1/ u0.8805e(0. 04p0.025 )0.50.88050.49771.13570.8805the tree is shown in figure s12.4. the

31、option is exercised at the lower node at the six-month point. it is worth 78.41.1703.600.001934.840.001500.001500.0078.410.001320.73159.271162.89317.11: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-m

32、onth american put option with strike price $93. the volatility is 28% and the risk-free rate (all maturities) is 3% with continuous compounding.ue0.280.251.1503d1 / u1u0.86940.86940.46511.15030.8694the tree for valuing the call is in figure s12.5a and that for valuing the put is in figure s12.5b. th

33、e values are 7.94 and 10.88, respectively.119.0826.08136.9843.98119.080.00136.980.00103.52103.5214.6210.5290.0090.007.944.8678.24068.020.0059.130.00103.52103.524.160.0090.0090.0010.887.8478.2478.2416.8814.7668.0224.9859.1333.87a: callb: putfurther questionsthe current price of a non-divi

34、dend-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 dow

35、n movement in 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) ue 0.250.25 = 1.1331. the percentage up movement is 13.31%(b) d = 1/u = 0.8825. the percentage down movement is 11.75%(c) t

36、he probability of an up movement is(e0.040.25).8825) /(1.1331.8825)0.5089(d) the probability of a down movement is0.4911.the tree for valuing the a and that for vab. the values are 7.56 and 14.58, respectively.179.7629.76158.6415.00140.00140.007.560.00123.550.00158.644.86109.030.00179.760.00140.0014

37、0.0014.5810.00123.5524.96109.0340.97a: callb: putin problem 12.19, 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 p

38、rice 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/(158.64123.55) = 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) = 0.74

39、85. the trader should increase the holding to 7,485 shares. if there is a down movement the trader should decrease the holding to zero.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

40、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 sameanswers.at the end of six months the value of the option will be either $12 (if the stock price is $6

41、0) or $0 (if the stock price is $42). consider a portfolio consisting of:sharesthe value of the portfolio is either421oroption6012in six months. if426012i.e.,0 6667the value of the portfolio is certain to be 28. for this value ofthe portfolio is therefore riskless. the current value of the portfolio

42、 is:0 666750fwherefis the value of the option. since the portfolio must earn the risk-free rate of interesti.e.,(0 6667500 12 0 5f )e28fthe value of the option is therefore $6.96.6 96this can also be calculated using risk-neutral valuation. suppose thatpis the probability ofan upward stock price mov

43、ement in a risk-neutral world. we must have0 06i.e., or:60 p42(118 p pp) 50e 11 090 6161the expected value of the option in a risk-neutral world is:120 616100 38397 3932this has a present value of7 3932 e0 066 96hence the above answer is consistent with risk-neutral valuation.problem 12.22.a stock p

44、rice 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 with a strike price of $42?b. what is the value of a si

45、x-month american put option with a strike price of $42?a. a tree describing the behavior of the stoc. the risk-neutral probability of an up move, p , is given bye0 12 3 120 90p1 10 90 6523calculating the expected payoff and discounting, we obtain the value of the option as2 420 65230 34779 60 34772

46、e0 12 6 122 118the value of the european option is 2.118. this can also be calculated by working back through t. 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 thetree. it is 2.537. this is greater t

47、han the value of the european option because it is optimal to exercise early at node c.40.0002.118a 2.53744.0000.8100.810bc 36.0004.7596.00048.4000.0000.00039.6002.4002.40032.4009.6009.600figuretree to evaluate european and american put options in pr. at each node, upper number is the stock price, t

48、he next number is the european put price, and the final number isthe american put price.using a“ tr-iand-error” approach, estimate how high the stripkreice has to be in problem12.17 for it to be optimal to exercise the option immediately.trial and error shows that immediate early exercise is optimal

49、 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 amountq . this increases the value of being at node c byqand the valueof being at node b by node a by0 3477 e0 03q0 3374 q . it therefore increases the value

50、 of being at(0 65230 3374q0 3477 q)e0 030 551qfor early exercise at node a we require 2 5370 551q2qorq1 196 . thiscorresponds to the strike price being greater than 43.196.problem 12.24.a stock price is currently $30. during each two-month period for the next four months it is expected to increase b

51、y 8% or reduce by 10%. the risk-free interest rate is 5%. use atwo-step tree to calculate the value of a derivative that pays off max(30s ) 02wheretstis the stock price in four months? if the derivative is american-style, should it beexercised early?this type of option is known as a power option. a

52、tree describing the behavior of the stock price is s. the risk-neutral probability of an up move, p , is given bye0 05 2 120 9p1 080 90 6020calculating the expected payoff and discounting, we obtain the value of the option as0 705620 60200 398032 490 3980 2 e0 05 4 125 394the value of the european o

53、ption is 5.394. this can also be calculated by working back through t. the second number at each node is the value of the european option.early exercise at node c would give 9.0 which is less than 13.2449. the option should therefore not be exercised early if it is american.30.0005.3940a32.4000.2785bc 27.00013.244934.922d0.00029.1600.7056e24.30032.49ftree to evaluate europe. at each node, upper number is the stock price and the next numberis the option price.consider a european call option on a non-dividend-paying stock where the