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# HullFund8eCh12ProblemSolutions.pdf FINA2205

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Date Created: 05/31/15

CHAPTER 12 Introduction to Binomial Trees Practice Questions Problem 128 Consider the situation in which stock price movements during the life of a European option are governed by a twostep 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 A shares Because A changes during the life of the option this riskless portfolio must also change Problem 129 A stock price is currently 50 It is known that at the end of two months it will be either 53 or 48 The riskfree interest rate is 10 per annum with continuous compounding What is the value of a twomonth European call option with a strikeprice of 49 Use noarbitrage 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 A shares 1 option The value of the portfolio is either 48A or 53A 4 in two months If 48A 2 53A 4 ie A 08 the value of the portfolio is certain to be 384 For this value of A the portfolio is therefore riskless The current value of the portfolio is 08gtlt50 f where f is the value of the option Since the portfolio must earn the risk free rate of interest 08gtlt50 fe0 10gtlt212 384 ie f 223 The value of the option is therefore 223 This can also be calculated directly from equations 122 and 123 u 106 d 096 so that 010gtlt212 p 05681 106 096 and f M12 gtltO5681gtlt4 223 Problem 1210 A stock price is currently 80 It is known that at the end of four months it will be either 75 or 85 The riskfree interest rate is 5 per annum with continuous compounding What is the value of a fourmonth European put option with a strikeprice of 80 Use noarbitrage 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 A shares 1 option Note The delta A of a put option is negative We have constructed the portfolio so that it is 1 option and A shares rather than 1 option and A shares so that the initial investment is positive The value of the portfolio is either 85A or 75A 5 in four months If 85A 2 75A 5 ie A 05 the value of the portfolio is certain to be 425 For this value of A the portfolio is therefore riskless The current value of the portfolio is 05 x 80 f where f is the value of the option Since the portfolio is riskless 05 X 80 fe03905gtlt412 425 ie f 180 The value of the option is therefore 180 This can also be calculated directly from equations 122 and 123 u 10625 d 09375 so that 6005X412 06345 10625 09375 P 1 p 03655 and f W12 x03655gtlt5 2180 Problem 1211 A stock price is currently 40 It is known that at the end of three months it will be either 45 or 35 The riskfree rate of interest with quarterly compounding is 8 per annum Calculate the value of a threemonth European put option on the stock with an exercise price of 40 Verify that noarbitrage arguments and riskneutral 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 A shares 1 option Note The delta A of a put option is negative We have constructed the portfolio so that it is 1 option and A shares rather than 1 option and A shares so that the initial investment is positive The value of the portfolio is either 35A 5 or 45A If 35A 5 45A ie A 05 the value of the portfolio is certain to be 225 For this value of A the portfolio is therefore riskless The current value of the portfolio is 40A f where f is the value of the option Since the portfolio must earn the risk free rate of interest 40gtltO5 fgtlt102 225 Hence f 206 ie the value of the option is 206 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 45p 351 p 40gtlt102 ie 10p 2 58 or p 058 The eXpected value of the option in a risk neutral world is 0x058 5gtlt042 210 This has a present value of H 206 102 This is consistent with the no arbitrage answer Problem 1212 A stock price is currently 50 Over each of the next two threemonth periods it is expected to go up by 6 or down by 5 The riskfree interest rate is 5 per annum with continuous compounding What is the value of a sixmonth European call option with a strike price of 51 A tree describing the behavior of the stock price is shown in Figure 121 The risk neutral probability of an up move p is given by 005x312 p 05689 106 095 There is a payoff from the option of 5618 51 518 for the highest final node which corresponds to two up moves zero in all other cases The value of the option is therefore 518gtlt056892 gtlte3905X612 21635 This can also be calculated by working back through the tree as indicated in Figure 121 The value of the call option is the lower number at each node in the figure 5618 39 515 50 5035 1535 0 45125 0 Figure S121 Tree for Problem 1212 Problem 1213 For the situation considered in Problem 1212 what is the value of a sixmonth 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 8122 We get a payoff of 51 5035 065 if the middle final node is reached and a payoff of 51 45125 2 5875 if the lowest final node is reached The value of the option is therefore O65gtlt2gtltO5689gtltO43115875gtltO43112e 0 05X6 2 21376 This can also be calculated by working back through the tree as indicated in Figure S 122 The value of the put plus the stock price is from Problem 1212 137650 51376 The value of the call plus the present value of the strike price is 1635 51e o 05X612 51376 This verifies that put call parity holds To test whether it worth exercising the option early we compare the value calculated for the option at each node with the payoff from immediate exercise At node C the payoff from immediate exercise is 51 475 35 Because this is greater than 28664 the option should be exercised at this node The option should not be exercised at either node A or node B I 5515 0 5035 055 45125 5875 Figure 122 Tree for Problem 1213 Problem 1214 A stock price is currently 25 It is known that at the end of two months it will be either 23 or 27 The riskfree interest rate is 10 per annum with continuous compounding Suppose ST is the stock price at the end of two months What is the value of a derivative that pays 0 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 A shares 1 derivative The value of the portfolio is either 27A 729 or 23A 529 in two months If 27A 729 23A 529 ie A 50 the value of the portfolio is certain to be 621 For this value of A the portfolio is therefore riskless The current value of the portfolio is 50x 25 f where f is the value of the derivative Since the portfolio must earn the risk free rate of interest 50X25 feo10x212 621 ie f 6393 The value of the option is therefore 6393 This can also be calculated directly from equations 122 and 123 u 108 d 092 so that 010X212 p 06050 108 092 and f e o 10X21206050gtlt729 03950gtlt529 6393 Problem 1215 Calculate M 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 a e005 008gtltl12 u 6mm 10352 d 1u 209660 p 09975 09660 2 04553 10352 09660 Problem 1216 The volatility of a nondividendpaying stock whose price is 78 is 3 0 The riskfree rate is 3 per annum continuously compounded for all maturities Calculate values for u d and p when a twomonth time step is used What is the value of a fourmonth European call option with a strike price of 80 given by a twostep binomial tree Suppose a trader sells 1000 options 10 contracts What position in the stock is necessary to hedge the trader s position at the time of the trade u 603OXVOJ667 d 1u 08847 O30gtlt212 p 6 08847 2 04898 11303 08847 The tree is given in Figure 123 The value of the option is 467 The initial delta is 958 8816 6901 which is almost exactly 05 so that 500 shares should be purchased 9965 1965 8816 958 7800 7800 467 000 6901 000 6105 000 Figure 123 Tree for Problem 1216 Problem 1217 A stock index is currently 1500 Its volatility is 18 The riskfree rate is 4 per annum continuously compounded for all maturities and the dividend yield on the index is 25 Calculate values for u d and p when a sixmonth time step is used What is the value a 2month American put option with a strike price of 1480 given by a twostep binomial tree u eOlsxm 11357 d 1u 08805 004 0025gtlto5 p 6 08805 2 04977 11357 08805 The tree is shown in Figure 124 The option is exercised at the lower node at the siX month point It is worth 7841 193484 000 170380 000 150000 150000 7841 000 132073 15927 118289 31711 Figure S124 Tree for Problem 1217 Problem 1218 The futures price of a commodity is 90 Use a threestep tree to value a a ninemonth American call option with strike price 93 and b a ninemonth American put option with strike price 93 The volatility is 28 and the riskfree rate all maturities is 3 with continuous compounding u 6028XV025 d lu 08694 u 1 08694 11503 08694 04651 The tree for valuing the call is in Figure S 125 a and that for valuing the put is in Figure S125b The values are 794 and 1088 respectively 13898 13898 4398 000 11908 11908 2808 000 10352 10352 10352 10352 1482 1052 418 000 9000 N 9000 9000 N 9000 794 488 1088 784 7824 7824 7824 7824 224 000 1888 1478 8802 8802 000 2498 5913 5913 000 3387 Figure S125a Call Figure S125b Put Further Questions Problem 1219 The current price of a nondividendpaying biotech stock is 140 with a volatility of 25 The riskfree rate is 4 For a threemonth 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 riskneutral world d What is the probability of a down movement in a riskneutral world Use a twostep tree to value a sixmonth European call option and a sixmonth European put option In both cases the strike price is 150 a u 60285 11331 The percentage up movement is 1331 b d 1u 08825 The percentage down movement is 1175 C The probability of an up movement is e03904X03925 8825 1 133 1 8825 05089 d The probability of a down movement is04911 The tree for valuing the call is in Figure S 126a and that for valuing the put is in Figure S126b The values are 756 and 1458 respectively 17976 17976 2976 000 15864 15864 1500 486 14000 14000 14000 14000 756 000 1458 1000 12355 12355 000 2496 10903 10903 000 4097 Figure S126a Call Figure S126b Put Problem 1220 In Problem 1219 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 threemonth 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 1515864 12355 04273 The trader should take a long position in 4273 shares If there is an up movement the delta for the second period is 2976 17976 140 07485 The trader should increase the holding to 7485 shares If there is a down movement the trader should decrease the holding to zero Problem 1221 A stock price is currently 50 It is known that at the end of six months it will be either 60 or 42 The riskfree rate of interest with continuous compounding is 12 per annum Calculate the value of a sixmonth European call option on the stock with an exercise price of 48 Verify that noarbitrage arguments and riskneutral 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 A shares 1 option The value of the portfolio is either 42A or 60A 12 in siX months If 42A 2 60A 12 ie A 06667 the value of the portfolio is certain to be 28 For this value of A the portfolio is therefore riskless The current value of the portfolio is 06667 x 50 f where f is the value of the option Since the portfolio must earn the risk free rate of interest 06667x50 fe03912X0395 28 ie f 696 The value of the option is therefore 696 This can also be calculated using risk neutral valuation Suppose that I is the probability of an upward stock price movement in a risk neutral world We must have 60 p 421 p 50Xe0 06 ie 18 p 1109 01 p 206161 The eXpected value of the option in a risk neutral world is 12gtltO6161OgtltO3839 73932 This has a present value of 73932e o 06 696 Hence the above answer is consistent with risk neutral valuation Problem 1222 A stock price is currently 40 Over each of the next two threemonth periods it is expected to go up by 10 or down by 10 The riskfree interest rate is 12 per annum with continuous compounding a What is the value of a sixmonth European put option with a strike price of 42 b What is the value of a sixmonth American put option with a strike price of 42 a A tree describing the behavior of the stock price is shown in Figure 8127 The risk neutral probability of an up move P is given by 012x312 p 06523 11 O9 Calculating the eXpected payoff and discounting we obtain the value of the option as 24gtlt2gtltO6523gtltO3477 96gtltO34772e 0 1zx612 2118 The value of the European option is 2118 This can also be calculated by working back through the tree as shown in Figure S 127 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 2537 This is greater than the value of the European option because it is optimal to exercise early at node C 8400 39 0000 0000 40000 39600 211 2400 2537 2400 36000 32400 4759 9600 6000 9600 Figure S127 Tree to evaluate European and American put options in Problem 1222 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 price Problem 1223 Using a trialanderror approach estimate how high the strike price has to be in Problem I 21 7 for it to be optimal to exercise the option immediately Trial and error shows that immediate early exercise is optimal when the strike price is above 432 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 03477e0 03q 03374q It therefore increases the value of being at node A by O6523gtlt 03374q 03477qe 03 0551q For early exercise at node A we require 2537 0551q lt 2 q or q gt 1196 This corresponds to the strike price being greater than 43196 Problem 1224 A stock price is currently 30 During each twomonth period for the next four months it is expected to increase by 8 or reduce by 10 The riskfree interest rate is 5 Use a two step tree to calculate the value of a derivative that pays 0quot max30 ST 02 where ST is the stock price in four months If the derivative is Americanstyle 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 8128 The risk neutral probability of an up move I is given by 005X212 e 09 p 06020 108 09 Calculating the expected payoff and discounting we obtain the value of the option as 07056gtlt2gtlt06020gtlt03980 3249gtlt039802e 03905gtlt412 5394 The value of the European option is 5394 This can also be calculated by working back through the tree as shown in Figure S 128 The second number at each node is the value of the European option Early exercise at node C would give 90 which is less than 132449 The option should therefore not be exercised early if it is American 32400 4922 D 0000 5394 07056 27000 132449 Figure S128 Tree to evaluate European power option in Problem 1224 At each node upper number is the stock price and the next number is the option price Problem 1225 Consider a European call option on a nondividendpaying stock where the stock price is 40 the strike price is 40 the riskfree rate is 4 per annam the volatility is 30 per annum and the time to maturity is six months a Calculate M d and P for a two step tree 9 Value the option using a two step tree 6 Verify that DerivaGem gives the same answer d Use DerivaGem to value the option with 5 50 100 and 500 time steps a In this case At 025 so that u 603945 211618 d l u 08607 and 004gtlt025 2 e 08607 204959 11618 08607 P b and C The value of the option using a two step tree as given by DerivaGem is shown in Figure 8129 to be 33739 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 1 With 5 50 100 and 500 time steps the value of the option is 39229 37394 37478 and 37545 respectively At each node Upper value Underlvina Asset Price nwer value Option Price Values in red are a result of early exercise Strike price 40 Discount factor per step 0 9900 Time step dt 02500 years 9125 days Growth factor per step a 10101 Probability of up move p 04959 Up step size u 11618 Down sten si7e d 0 860 Figure S129 Tree produced by DerivaGem to evaluate European option in Problem 1225 Problem 1226 Repeat Problem 1225 for an American put option on a futures contract The strike price and the futures price are 50 the riskfree rate is 10 the time to maturity is six months and the volatility is 40 per annum a In this case At 025 and u 2 604mm 212214 d 1 u 08187 and eO1gtlt025 1 04502 12214 08187 I9 b and C The value of the option using a two step tree is 48604 1 With 5 50 100 and 500 time steps the value of the option is 56858 53869 53981 and 5 4072 respectively Problem 1227 A stock index is currently 990 the riskfree rate is 5 and the dividend yield on the index is 2 Use a threestep tree to value an 18month American put option with a strike price of 1000 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 1210 The value of the option is 8751 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 23663 The gain made at the one year point is therefore 25390 23663 1727 151318 000 131363 000 114039 114039 3107 000 99000 99000 8751 6608 85944 85944 15275 14056 74610 25390 64771 35229 Figure 1210 Tree for Problem 1227 Problem 1228 Calculate the value of ninemonth American call option on a foreign currency using a threestep binomial tree The current exchange rate is 079 and the strike price is 080 both expressed as dollars per unit of the foreign currency The volatility of the exchange rate is 12 per annum The domestic and foreign riskfree 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 1211 The cost of an American option to buy one million units of the foreign currency is 18100 The delta initially is 00346 0005108261 07554 04176 The company should sell 417600 units of the foreign currency 09034 01034 08639 00639 08261 08261 00346 00261 07900 07900 00181 00115 07554 07554 00051 00000 07224 00000 06908 00000 Figure 1211 Tree for Problem 1228

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