CHAPTER 12Introduction to Binomial Trees Practice Questions Problem 12.8. A REF 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 12.9. OK 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 the value of a two-month European call option with a strike price 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: shares 1 option The value of the portfolio is either 48 or 53 4 in two months. If 48 53 4 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: 08 50 f where f is the value of the option. Since the portfolio must earn the risk-free rate of interest (08 50 f )e010212 384 i.e., f 223 The value of the option is therefore $2.23. This can also be calculated directly from equations (12.2) and (12.3). u 106 , d 096 so that e0102 12 096 p 05681 106 096 and f e 010212 05681 4 223 .) The value of the portfolio is either 85 or 75 5 in four months. 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). It is known that at the end of three months it will be either $45 or $35. For this value of the portfolio is therefore riskless. It is known that at the end of four months it will be either $75 or $85.3). The risk-free interest rate is 5% per annum with continuous compounding.2) and (12. u 10625 . We have constructed the portfolio so that it is +1 option and shares rather than 1 option and shares so that the initial investment .11. If 85 75 5 i. OK A stock price is currently $80. This can also be calculated directly from equations (12.. The current value of the portfolio is: 05 80 f where f is the value of the option.e. What is the value of a four-month European put option with a strike price of $80? Use no-arbitrage arguments. Consider a portfolio consisting of: shares 1 option (Note: The delta. of a put option is negative. 05 the value of the portfolio is certain to be 42.5.e. 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. 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).Problem 12. . The risk-free rate of interest with quarterly compounding is 8% per annum. Since the portfolio is riskless (05 80 f )e005412 425 i. f 180 The value of the option is therefore $1. Calculate the value of a three-month European put option on the stock with an exercise price of $40. d 09375 so that e005412 09375 p 06345 10625 09375 1 p 03655 and f e005412 03655 5 180 Problem 12. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same answers.80. Consider a portfolio consisting of: shares 1 option (Note: The delta. OK A stock price is currently $40.10. of a put option is negative. The risk-neutral probability of an up move.e.1. If: 35 5 45 i.) The value of the portfolio is either 35 5 or 45 .1.e. the value of the option is $2. 10 p 58 or: p 058 The expected value of the option in a risk-neutral world is: 0 058 5 042 210 This has a present value of 210 206 102 This is consistent with the no-arbitrage answer. The value of the call option is the lower number at each node in the figure. 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. p. 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 S12.5. OK A stock price is currently $50. Since the portfolio must earn the risk-free rate of interest (40 05 f ) 102 225 Hence f 206 i.e.06. is given by e005312 095 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.is positive. The value of the option is therefore 518 056892 e 005612 1635 This can also be calculated by working back through the tree as indicated in Figure S12.12. 05 the value of the portfolio is certain to be 22.. The risk-free interest rate is 5% per annum with continuous compounding.. The current value of the portfolio is 40 f where f is the value of the option. We must have 45 p 35(1 p ) 40 102 i. Over each of the next two three-month periods it is expected to go up by 6% or down by 5%.. Problem 12. . For this value of the portfolio is therefore riskless. The option should not be exercised at either node A or node B. OK but DO NOT FORGET TO TEST THE FIRST NODE FOR EXER. For the situation considered in Problem 12.Figure S12. Because this is greater than 2. 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.12 1376 50 51376 The value of the call plus the present value of the strike price is 1635 51e 0056 12 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. The value of the option is therefore (065 2 05689 04311 5875 043112 ) e 005612 1376 This can also be calculated by working back through the tree as indicated in Figure S12. At node C the payoff from immediate exercise is 51 475 35 .8664.13.1 Tree for Problem 12.12 Problem 12. the option should be exercised at this node.2. The value of the put plus the stock price is from Problem 12. If the put option were American.2. Figure S12.13 . 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 S12.12. We get a payoff of 51 5035 065 if the middle final node is reached and a payoff of 51 45125 5875 if the lowest final node is reached.2 Tree for Problem 12. and p when a binomial tree is constructed to value an option on a foreign currency. The risk-free interest rate is 10% per annum with continuous compounding. d . d 092 so that e010212 092 p 06050 108 092 and f e 010212 (06050 729 03950 529) 6393 Problem 12. f 6393 The value of the option is therefore $639. the domestic interest rate is 5% per annum. It is known that at the end of two months it will be either $23 or $27. The tree step size is one month. Calculate u .e. u 108 .3. This can also be calculated directly from equations (12.14.16.Problem 12. Consider a portfolio consisting of: shares 1 derivative The value of the portfolio is either 27 729 or 23 529 in two months. What is the value of a derivative that pays off ST2 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). In this case a e (005008)112 09975 u e012 112 10352 d 1 u 09660 p Problem 12. 50 the value of the portfolio is certain to be 621. For this value of the portfolio is therefore riskless. Suppose ST is the stock price at the end of two months. 09975 09660 04553 10352 09660 .2) and (12. Since the portfolio must earn the risk-free rate of interest (50 25 f )e 0102 12 621 i. The current value of the portfolio is: 50 25 f where f is the value of the derivative.e..3). the foreign interest rate is 8% per annum. A REF A stock price is currently $25. and the volatility is 12% per annum. If 27 729 23 529 i.15.. 025 )0.8805 p e ( 0.000 options (10 contracts). The risk-free rate is 4% per annum (continuously compounded) for all maturities and the dividend yield on the index is 2.3.18 0. What is the value of a four-month European call option with a strike price of $80 given by a two-step binomial tree. The initial delta is 9.1357 d 1 / u 0. The value of the option is $4. .01) which is almost exactly 0.8847 The tree is given in Figure S12. d. The risk-free rate is 3% per annum (continuously compounded) for all maturities.16 Problem 12.1303 d 1 / u 0.5 1.4898 1. The option is exercised at the lower node at the six-month point. Figure S12.480 given by a two-step binomial tree.1357 0. Calculate values for u. It is worth 78.04 0.67.500.302 / 12 0.5%.41.58/ (88.17.8805 The tree is shown in Figure S12.5 0.8805 0.3: Tree for Problem 12.8847 p e 0. Suppose a trader sells 1. What is the value a 12-month American put option with a strike price of 1.5 so that 500 shares should be purchased. u e 0.The volatility of a non-dividend-paying stock whose price is $78.30 0. is 30%.16 – 69.4977 1.1667 1. What position in the stock is necessary to hedge the trader’s position at the time of the trade? u e 0. A stock index is currently 1.4. d. and p when a two-month time step is used.1303 0. Calculate values for u. and p when a six-month time step is used.8847 0. Its volatility is 18%. 5a and that for valuing the put is in Figure S12.1503 d 1 / u 0.17 Problem 12.8694 The tree for valuing the call is in Figure S12.8694 0.Figure S12.18 The futures price of a commodity is $90.88.28 0.1503 0.19 Figure S12. 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 $93.8694 u 1 0.5b: Put . Figure S12. The volatility is 28% and the risk-free rate (all maturities) is 3% with continuous compounding. respectively. The values are 7.5a: Call Further Questions Problem 12.5b.25 1. u e 0.4: Tree for Problem 12.4651 1.94 and 10. Verify that no-arbitrage arguments and risk-neutral valuation arguments give the same . The risk-free rate of interest with continuous compounding is 12% per annum. Figure S12.20 In Problem 12. Problem 12.4273.76/ (179.1331 . If there is a down movement the trader should decrease the holding to zero. 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.4911.6b: Put Problem 12. respectively. 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 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.6a and that for valuing the put is in Figure S12.76 – 140) = 0. If there is an up movement the delta for the second period is 29.6a: Call Figure S12. (a) u e 0.64 – 123.25 0.5089 (d) The probability of a down movement is0. suppose that a trader sells 10. In both cases the strike price is $150. The trader should take a long position in 4. The percentage up movement is 13.8825) 0.21.75% (c) The probability of an up movement is (e 0.55) = 0.273 shares. The delta for the first period is 15/(158.The current price of a non-dividend-paying biotech stock is $140 with a volatility of 25%. The values are 7.8825. A stock price is currently $50.000 European call options. The risk-free rate is 4%. Calculate the value of a six-month European call option on the stock with an exercise price of $48.485 shares.8825) /(1.19.25 = 1.56 and 14.58.25 ) .7485.1331.31% (b) d = 1/u = 0.040. The percentage down movement is 11. It is known that at the end of six months it will be either $60 or $42. The tree for valuing the call is in Figure S12. 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. The trader should increase the holding to 7.6b. 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). Suppose that p is the probability of an upward stock price movement in a risk-neutral world.answers.96. Since the portfolio must earn the risk-free rate of interest (06667 50 f )e01205 28 i. What is the value of a six-month European put option with a strike price of $42? b. A tree describing the behavior of the stock price is shown in Figure S12. The riskneutral probability of an up move. For this value of the portfolio is therefore riskless. The second number at each node is . Problem 12. p .22.e. The current value of the portfolio is: 06667 50 f where f is the value of the option. 06667 the value of the portfolio is certain to be 28. 18 p 1109 or: p 06161 The expected value of the option in a risk-neutral world is: 12 06161 0 03839 73932 This has a present value of 73932e 006 696 Hence the above answer is consistent with risk-neutral valuation.7.e. Over each of the next two three-month periods it is expected to go up by 10% or down by 10%. a.. A stock price is currently $40. The risk-free interest rate is 12% per annum with continuous compounding. What is the value of a six-month American put option with a strike price of $42? a.7. We must have 60 p 42(1 p) 50 e006 i. f 696 The value of the option is therefore $6.. This can also be calculated by working back through the tree as shown in Figure S12. This can also be calculated using risk-neutral valuation.118. is given by e012312 090 p 06523 11 09 Calculating the expected payoff and discounting. If 42 60 12 i. Consider a portfolio consisting of: shares 1 option The value of the portfolio is either 42 or 60 12 in six months.. we obtain the value of the option as [24 2 06523 03477 96 03477 2 ]e 0126 12 2118 The value of the European option is 2.e. This can be also shown to be true algebraically. Use a two2 step tree to calculate the value of a derivative that pays off max[(30 ST ) 0] where ST is the stock price in four months? If the derivative is American-style. b. During each two-month period for the next four months it is expected to increase by 8% or reduce by 10%.537.23. Trial and error shows that immediate early exercise is optimal when the strike price is above 43. upper number is the stock price. This is greater than the value of the European option because it is optimal to exercise early at node C. The value of the American option is shown as the third number at each node on the tree. Problem 12. It is 2. the next number is the European put price. Figure S12.196.17 for it to be optimal to exercise the option immediately.7 Tree to evaluate European and American put options in Problem 12. should it be exercised . A stock price is currently $30.24. At each node. This corresponds to the strike price being greater than 43. Suppose the strike price increases by a relatively small amount q .2.22. The risk-free interest rate is 5%. It therefore increases the value of being at node A by (06523 03374q 03477 q )e 003 0551q For early exercise at node A we require 2537 0551q 2 q or q 1196 . and the final number is the American put price Problem 12.the value of the European option. estimate how high the strike price has to be in Problem 12. This increases the value of being at node C by q and the value of being at node B by 03477e 003q 03374q . Using a “trial-and-error” approach. 25. Early exercise at node C would give 9. is given by e0052 12 09 p 06020 108 09 Calculating the expected payoff and discounting.8. and 500 time steps.early? This type of option is known as a power option. A tree describing the behavior of the stock price is shown in Figure S12. The option should therefore not be exercised early if it is American. p . 100. This can also be calculated by working back through the tree as shown in Figure S12.24.2449. Use DerivaGem to value the option with 5.8. . Calculate u . a. Verify that DerivaGem gives the same answer d. the risk-free rate is 4% per annum. we obtain the value of the option as [07056 2 06020 03980 3249 039802 ]e 0054 12 5394 The value of the European option is 5.394. the strike price is $40. c. and the time to maturity is six months. d . The second number at each node is the value of the European option. and p for a two step tree b. Value the option using a two step tree. The risk-neutral probability of an up move. upper number is the stock price and the next number is the option price Problem 12. At each node. the volatility is 30% per annum. Figure S12.0 which is less than 13.8 Tree to evaluate European power option in Problem 12. Consider a European call option on a non-dividend-paying stock where the stock price is $40. 50. d 1 u 08607 . and the . the risk-free rate is 10%. Figure S12. and select Binomial European as the Option Type.26. 3.9229.7545. Repeat Problem 12.9 Tree produced by DerivaGem to evaluate European option in Problem 12.25 for an American put option on a futures contract.7478.3739.9 to be 3. respectively.(a) In this case t 025 so that u e030 025 11618 . and e004025 08607 p 04959 11618 08607 (b) and (c) The value of the option using a two-step tree as given by DerivaGem is shown in Figure S12.7394. and 3. The strike price and the futures price are $50. (d) With 5. 100. the time to maturity is six months. and 500 time steps the value of the option is 3. To use DerivaGem choose the first worksheet. 50. select Equity as the underlying type. 3.25 Problem 12. After carrying out the calculations select Display Tree. and 500 time steps the value of the option is 5. 100. The cost of an American option to buy one million units .4072. and the dividend yield on the index is 2%. Use a three-step tree to value an 18-month American put option with a strike price of 1. If early exercise were not possible the value at this node would be 236.79 and the strike price is 0. How much does the option holder gain by being able to exercise early? When is the gain made? The tree is shown in Figure S12. (a) In this case t 025 and u e040 025 12214 .6858.000 when the volatility is 20% per annum.27 A stock index is currently 990.63.8604. The gain made at the one year point is therefore 253. Problem 12. What position in the foreign currency is initially necessary to hedge its risk? The tree is shown in Figure S12. and e01025 08187 p 04502 12214 08187 (b) and (c) The value of the option using a two-step tree is 4. Figure 12. The current exchange rate is 0. and 5.10.3869. It is optimal to exercise at the lowest node at time one year. d 1 u 08187 . The volatility of the exchange rate is 12% per annum.10 Tree for Problem 12.volatility is 40% per annum.11. Suppose a company has bought options on 1 million units of the foreign currency.27.63 = 17. The value of the option is 87. (d) With 5.27 Problem 12. respectively.80 (both expressed as dollars per unit of the foreign currency).28 Calculate the value of nine-month American call option on a foreign currency using a three-step binomial tree.90 – 236. 50. 5.51. 5. respectively. the risk-free rate is 5%. The domestic and foreign risk-free rates are 2% and 5%.3981. 4176.of the foreign currency is $18.100.7554) = 0.11: Tree for Problem 12.28 . The company should sell 417.0346 −0.600 units of the foreign currency Figure S12.0051)/(0.8261 – 0. The delta initially is (0.