1、 CHAPTER 21 Basic Numerical Procedures Practice Questions Problem 21.1. Which of the following can be estimated for an American option by constructing a single binomial tree: delta, gamma, vega, theta, rho? Delta, gamma, and theta can be determined from a single binomial tree. Vega is determined by
2、making a small change to the volatility and recomputing the option price using a new tree. Rho is calculated by making a small change to the interest rate and recomputing the option price using a new tree. Problem 21.2. Calculate the price of a three-month American put option on a non-dividend-payin
3、g stock when the stock price is $60, the strike price is $60, the risk-free interest rate is 10% per annum, and the volatility is 45% per annum. Use a binomial tree with a time interval of one month. In this case, 0 60S , 60K , 01r , 045 , 025T , and 00833t . Also 0 45 0 08 330 1 0 08 331 138710 878
4、21 00840 49981 0 5002trtu e edua e eadpudp The output from DerivaGem for this example is shown in the Figure S21.1. The calculated price of the option is $5.16. Figure S21.1: Tree for Problem 21.2 G r o w t h f a c t o r p e r s t e p , a = 1 . 0 0 8 4P r o b a b i l i t y o f u p m o v e , p = 0 .
5、4 9 9 7U p s t e p s i z e , u = 1 . 1 3 8 7 8 8 . 5 9 3 2 8D o w n s t e p s i z e , d = 0 . 8 7 8 2 07 7 . 8 0 0 8 406 8 . 3 2 3 1 3 6 8 . 3 2 3 1 31 . 7 9 9 3 4 060 605 . 1 6 2 7 8 1 3 . 6 2 6 5 3 45 2 . 6 9 0 7 9 5 2 . 6 9 0 7 98 . 6 0 8 3 8 2 7 . 3 0 9 2 0 64 6 . 2 7 21 3 . 7 2 84 0 . 6 3 5 1 4
6、1 9 . 3 6 4 8 6N o d e T i m e : 0 . 0 0 0 0 0 . 0 8 3 3 0 . 1 6 6 7 0 . 2 5 0 0Problem 21.3. Explain how the control variate technique is implemented when a tree is used to value American options. The control variate technique is implemented by 1. Valuing an American option using a binomial tree in
7、 the usual way ()Af . 2. Valuing the European option with the same parameters as the American option using the same tree ()Ef . 3. Valuing the European option using Black-Scholes-Merton (=fBSM ). The price of the American option is estimated as fA + fBSM fE. Problem 21.4. Calculate the price of a ni
8、ne-month American call option on corn futures when the current futures price is 198 cents, the strike price is 200 cents, the risk-free interest rate is 8% per annum, and the volatility is 30% per annum. Use a binomial tree with a time interval of three months. In this case 0 198F , 200K , 008r , 03
9、 , 075T , and 025t . Also 0 3 0 25 1 161810 860710 46261 0 5373ueduaadpudp The output from DerivaGem for this example is shown in the Figure S21.2. The calculated price of the option is 20.34 cents. Figure S21.2: Tree for Problem 21.4 Problem 21.5. Consider an option that pays off the amount by whic
10、h the final stock price exceeds the average stock price achieved during the life of the option. Can this be valued using the binomial tree approach? Explain your answer. G r o w t h fa c t o r p e r s t e p , a = 1 . 0 0 0 0P r o b a b i l i t y o f u p m o v e , p = 0 . 4 6 2 6U p s t e p s i z e ,
11、 u = 1 . 1 6 1 8 3 1 0 . 5 2 5 8D o w n s t e p s i z e , d = 0 . 8 6 0 7 1 1 0 . 5 2 5 82 6 7 . 2 7 26 7 . 2 7 2 0 42 3 0 . 0 4 3 2 2 3 0 . 0 4 3 23 7 . 6 7 7 7 1 3 0 . 0 4 3 1 8198 1982 0 . 3 3 7 0 8 1 3 . 6 2 1 91 7 0 . 4 2 0 2 1 7 0 . 4 2 0 26 . 1 7 6 3 1 4 01 4 6 . 6 8 201 2 6 . 2 5 0 40N o d e
12、 T i m e : 0 . 0 0 0 0 0 . 2 5 0 0 0 . 5 0 0 0 0 . 7 5 0 0 A binomial tree cannot be used in the way described in this chapter. This is an example of what is known as a history-dependent option. The payoff depends on the path followed by the stock price as well as its final value. The option cannot
13、be valued by starting at the end of the tree and working backward since the payoff at the final branches is not known unambiguously. Chapter 27 describes an extension of the binomial tree approach that can be used to handle options where the payoff depends on the average value of the stock price. Pr
14、oblem 21.6. “For a dividend-paying stock, the tree for the stock price does not recombine; but the tree for the stock price less the present value of future dividends does recombine.” Explain this statement. Suppose a dividend equal to D is paid during a certain time interval. If S is the stock pric
15、e at the beginning of the time interval, it will be either Su D or Sd D at the end of the time interval. At the end of the next time interval, it will be one of ()Su Du , ()Su Dd , ()Sd Du and ()Sd Dd . Since ()Su Dd does not equal ()Sd Du the tree does not recombine. If S is equal to the stock pric
16、e less the present value of future dividends, this problem is avoided. Problem 21.7. Show that the probabilities in a Cox, Ross, and Rubinstein binomial tree are negative when the condition in footnote 8 holds. With the usual notation 1adpuduapud If ad or au , one of the two probabilities is negativ
17、e. This happens when ()r q t tee or ()r q t tee This in turn happens when ()q r t or ()r q t Hence negative probabilities occur when ()| r q t | This is the condition in footnote 8. Problem 21.8. Use stratified sampling with 100 trials to improve the estimate of in Business Snapshot 21.1 and Table 2
18、1.1. In Table 21.1 cells A1, A2, A3,., A100 are random numbers between 0 and 1 defining how far to the right in the square the dart lands. Cells B1, B2, B3,.,B100 are random numbers between 0 and 1 defining how high up in the square the dart lands. For stratified sampling we could choose equally spa
19、ced values for the As and the Bs and consider every possible combination. To generate 100 samples we need ten equally spaced values for the As and the Bs so that there are 10 10 100 combinations. The equally spaced values should be 0.05, 0.15, 0.25,., 0.95. We could therefore set the As and Bs as fo
20、llows: A 1 A 2 A 3 A 1 0 0 0 5 A 1 1 A 1 2 A 1 3 A 2 0 0 1 5 A 9 1 A 9 2 A 9 3 A 1 0 0 0 9 5 and B 1 B 1 1 B 2 1 B 9 1 0 0 5 B 2 B 1 2 B 2 2 B 9 2 0 1 5 B 1 0 B 2 0 B 3 0 B 1 0 0 0 9 5 We get a value for equal to 3.2, which is closer to the true value than the value of 3.04 obtained with random samp
21、ling in Table 21.1. Because samples are not random we cannot easily calculate a standard error of the estimate. Problem 21.9. Explain why the Monte Carlo simulation approach cannot easily be used for American-style derivatives. In Monte Carlo simulation sample values for the derivative security in a
22、 risk-neutral world are obtained by simulating paths for the underlying variables. On each simulation run, values for the underlying variables are first determined at time t , then at time 2t , then at time 3t , etc. At time ( 0 1 2 )i t i ? it is not possible to determine whether early exercise is
23、optimal since the range of paths which might occur after time it have not been investigated. In short, Monte Carlo simulation works by moving forward from time t to time T . Other numerical procedures which accommodate early exercise work by moving backwards from time T to time t . Problem 21.10. A
24、nine-month American put option on a non-dividend-paying stock has a strike price of $49. The stock price is $50, the risk-free rate is 5% per annum, and the volatility is 30% per annum. Use a three-step binomial tree to calculate the option price. In this case, 0 50S , 49K , 005r , 030 , 075T , and
25、025t . Also 0 30 0 250 05 0 251 161810 86071 01260 50431 0 4957trtu e edua e eadpudp The output from DerivaGem for this example is shown in the Figure S21.3. The calculated price of the option is $4.29. Using 100 steps the price obtained is $3.91 B o l d e d v a l u e s a r e a r e s u l t o f e x e
26、 r c i s eG r o w th f a c tor p e r s te p , a = 1 . 0 1 2 6P r o b a b i l i ty o f u p mo v e , p = 0 . 5 0 4 3U p s te p s i z e , u = 1 . 1 6 1 8 7 8 . 4 1 5 6 1D o w n s te p s i z e , d = 0 . 8 6 0 7 06 7 . 4 9 2 9 405 8 . 0 9 1 7 1 5 8 . 0 9 1 7 11 . 4 2 9 1 8 7 050 504 . 2 8 9 2 2 5 2 . 9 1
27、 9 6 84 3 . 0 3 5 4 4 3 . 0 3 5 47 . 3 0 8 2 1 4 5 . 9 6 4 6 0 13 7 . 0 4 0 9 11 1 . 9 5 9 0 93 1 . 8 8 1 4 11 7 . 1 1 8 5 9N o d e T i me : 0 . 0 0 0 0 0 . 2 5 0 0 0 . 5 0 0 0 0 . 7 5 0 0 Figure S21.3: Tree for Problem 21.10 Problem 21.11. Use a three-time-step tree to value a nine-month American c
28、all option on wheat futures. The current futures price is 400 cents, the strike price is 420 cents, the risk-free rate is 6%, and the volatility is 35% per annum. Estimate the delta of the option from your tree. In this case 0 400F , 420K , 006r , 035 , 075T , and 025t . Also 0 35 0 25 1 191210 8395
29、10 45641 0 5436ueduaadpudp The output from DerivaGem for this example is shown in the Figure S21.4. The calculated price of the option is 42.07 cents. Using 100 time steps the price obtained is 38.64. The options delta is calculated from the tree is ( 7 9 9 7 1 1 1 4 1 9 ) ( 4 7 6 4 9 8 3 3 5 7 8 3
30、) 0 4 8 7 When 100 steps are used the estimate of the options delta is 0.483. A t e a c h n o d e : U p p e r v a l u e = U n d e r l y i n g A s s e t P r i c e L o w e r v a l u e = O p ti o n P r i c eB o l d e d v a l u e s a r e a r e s u l t o f e x e r c i s eG r o w th f a c tor p e r s te p
31、 , a = 1 . 0 0 0 0P r o b a b i l i ty o f u p mo v e , p = 0 . 4 5 6 4U p s te p s i z e , u = 1 . 1 9 1 2 6 7 6 . 1 8 3 5D o w n s te p s i z e , d = 0 . 8 3 9 5 2 5 6 . 1 8 3 55 6 7 . 6 2 71 4 7 . 6 2 74 7 6 . 4 9 8 5 4 7 6 . 4 9 8 57 9 . 9 7 1 5 6 . 4 9 8 4 9400 4004 2 . 0 6 7 6 7 2 5 . 3 9 9 8
32、53 3 5 . 7 8 2 8 3 3 5 . 7 8 2 81 1 . 4 1 8 9 4 02 8 1 . 8 7 5 202 3 6 . 6 2 2 10N o d e T i me : 0 . 0 0 0 0 0 . 2 5 0 0 0 . 5 0 0 0 0 . 7 5 0 0 Figure S21.4: Tree for Problem 21.11 Problem 21.12. A three-month American call option on a stock has a strike price of $20. The stock price is $20, the r
33、isk-free rate is 3% per annum, and the volatility is 25% per annum. A dividend of $2 is expected in 1.5 months. Use a three-step binomial tree to calculate the option price. In this case the present value of the dividend is 0 03 0 1252 1 9925e . We first build a tree for 0 2 0 1 9 9 2 5 1 8 0 0 7 5S
34、 , 20K , 0 3r , 025 , and 025T with 0 08333t . This gives Figure S21.5. For nodes between times0 and 1.5 months we then add the present value of the dividend to the stock price. The result is the tree in Figure S21.6. The price of the option calculated from the tree is 0.674. When 100 steps are used
35、 the price obtained is 0.690. T r e e s h o w s s toc k p r i c e s l e s s P V o f d i v i d e n d a t 0 . 1 2 5 y e a r sG r o w th f a c tor p e r s te p , a = 1 . 0 0 2 5P r o b a b i l i ty o f u p mo v e , p = 0 . 4 9 9 3U p s te p s i z e , u = 1 . 0 7 4 8 2 2 . 3 6 0 4 7D o w n s te p s i z
36、e , d = 0 . 9 3 0 42 0 . 8 0 3 61 9 . 3 5 5 1 2 1 9 . 3 5 5 1 21 8 . 0 0 7 5 1 8 . 0 0 7 51 6 . 7 5 3 7 1 1 6 . 7 5 3 7 11 5 . 5 8 7 2 11 4 . 5 0 1 9 3N o d e T i me : 0 . 0 0 0 0 0 . 0 8 3 3 0 . 1 6 6 7 0 . 2 5 0 0 Figure S21.5: First Tree for Problem 21.12 A t e a c h n o d e : U p p e r v a l u e
37、 = U n d e r l y i n g A s s e t P r i c e L o w e r v a l u e = O p ti o n P r i c eB o l d e d v a l u e s a r e a r e s u l t o f e x e r c i s eP r o b a b i l i ty o f u p mo v e , p = 0 . 4 9 9 32 2 . 3 6 0 4 52 . 3 6 0 4 5 32 0 . 8 0 3 5 81 . 1 7 5 6 1 42 1 . 3 5 2 6 1 1 9 . 3 5 5 1 11 . 3 5
38、2 6 0 9 020 1 8 . 0 0 7 4 90 . 6 7 3 6 6 2 01 8 . 7 5 1 2 1 6 . 7 5 3 6 90 01 5 . 5 8 7 201 4 . 5 0 1 9 20N o d e T i me : 0 . 0 0 0 0 0 . 0 8 3 3 0 . 1 6 6 7 0 . 2 5 0 0 Figure S21.6: Final Tree for Problem 21.12 Problem 21.13. A one-year American put option on a non-dividend-paying stock has an ex
39、ercise price of $18. The current stock price is $20, the risk-free interest rate is 15% per annum, and the volatility of the stock is 40% per annum. Use the DerivaGem software with four three-month time steps to estimate the value of the option. Display the tree and verify that the option prices at
40、the final and penultimate nodes are correct. Use DerivaGem to value the European version of the option. Use the control variate technique to improve your estimate of the price of the American option. In this case 0 20S , 18K , 015r , 040 , 1T , and 025t . The parameters for the tree are 0 4 0 2 5 1
41、2 2 1 41 0 8 1 8 71 0 3 8 21 0 3 8 2 0 8 1 8 7 0 5 4 51 2 2 1 4 0 8 1 8 7trtu e eduaeadpud The tree produced by DerivaGem for the American option is shown in Figure S21.7. The estimated value of the American option is $1.29. A t e a c h n o d e : U p p e r v a l u e = U n d e r l y i n g A s s e t P
42、 r i c e L o w e r v a l u e = O p ti o n P r i c eB o l d e d v a l u e s a r e a r e s u l t o f e x e r c i s eG r o w th f a c tor p e r s te p , a = 1 . 0 3 8 2 4 4 . 5 1 0 8 2P r o b a b i l i ty o f u p mo v e , p = 0 . 5 4 5 1 0U p s te p s i z e , u = 1 . 2 2 1 4 3 6 . 4 4 2 3 8D o w n s te
43、 p s i z e , d = 0 . 8 1 8 7 02 9 . 8 3 6 4 9 2 9 . 8 3 6 4 90 02 4 . 4 2 8 0 6 2 4 . 4 2 8 0 60 . 3 8 6 5 0 2 020 20 201 . 2 8 7 8 6 1 0 . 8 8 2 0 3 4 01 6 . 3 7 4 6 2 1 6 . 3 7 4 6 22 . 4 7 5 9 5 4 2 . 0 1 2 8 8 61 3 . 4 0 6 4 1 3 . 4 0 6 44 . 5 9 3 5 9 9 4 . 5 9 3 5 9 91 0 . 9 7 6 2 37 . 0 2 3 7
44、6 78 . 9 8 6 5 7 99 . 0 1 3 4 2 1N o d e T i me : 0 . 0 0 0 0 0 . 2 5 0 0 0 . 5 0 0 0 0 . 7 5 0 0 1 . 0 0 0 0 Figure S21.7: Tree to evaluate American option for Problem 21.13 A t e a c h n o d e : U p p e r v a l u e = U n d e r l y i n g A s s e t P r i c e L o w e r v a l u e = O p ti o n P r i c
45、eB o l d e d v a l u e s a r e a r e s u l t o f e x e r c i s eG r o w th f a c tor p e r s te p , a = 1 . 0 3 8 2 4 4 . 5 1 0 8 2P r o b a b i l i ty o f u p mo v e , p = 0 . 5 4 5 1 0U p s te p s i z e , u = 1 . 2 2 1 4 3 6 . 4 4 2 3 8D o w n s te p s i z e , d = 0 . 8 1 8 7 02 9 . 8 3 6 4 9 2 9
46、. 8 3 6 4 90 02 4 . 4 2 8 0 6 2 4 . 4 2 8 0 60 . 3 8 6 5 0 2 020 20 201 . 1 4 3 9 7 3 0 . 8 8 2 0 3 4 01 6 . 3 7 4 6 2 1 6 . 3 7 4 6 22 . 1 4 7 5 8 7 2 . 0 1 2 8 8 61 3 . 4 0 6 4 1 3 . 4 0 6 43 . 8 4 4 2 3 3 4 . 5 9 3 5 9 91 0 . 9 7 6 2 36 . 3 6 1 2 6 78 . 9 8 6 5 7 99 . 0 1 3 4 2 1N o d e T i me :
47、0 . 0 0 0 0 0 . 2 5 0 0 0 . 5 0 0 0 0 . 7 5 0 0 1 . 0 0 0 0 Figure S21.8: Tree to evaluate European option in Problem 21.13 As shown in Figure S21.8, the same tree can be used to value a European put option with the same parameters. The estimated value of the European option is $1.14. The option par
48、ameters are 0 20S , 18K , 015r , 040 and 1T 2121l n ( 2 0 1 8 ) 0 1 5 0 4 0 2 0 8 3 8 40 4 00 4 0 0 4 3 8 4ddd 12( ) 0 2 0 0 9 ( ) 0 3 3 0 6N d N d The true European put price is therefore 0 1 51 8 0 3 3 0 6 2 0 0 2 0 0 9 1 1 0e This can also be obtained from DerivaGem. The control variate estimate
49、of the American put price is therefore 1.29+1.10 1.14 = $1.25. Problem 21.14 A two-month American put option on a stock index has an exercise price of 480. The current level of the index is 484, the risk-free interest rate is 10% per annum, the dividend yield on the index is 3% per annum, and the vo
50、latility of the index is 25% per annum. Divide the life of the option into four half-month periods and use the binomial tree approach to estimate the value of the option. In this case 0 484S , 480K , 010r , 025 003q , 01667T , and 0 04167t 0 25 0 04 16 7()1 0 5 2 410 9 5 0 21 0 0 2 9 21 0 0 2 9 0 9
