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1、CHAPTER21Basic Nu me r ical ProceduresPractice QuestionsProblem 21.1.V.lich of the following can be estimated for an Alnerican option by constructing a single binonlial hee: delta, ga1runa, vega, tl1eta, rho?Delta, ga1mn a, and tl1eta can be detennined from a single binomial tree. Vega is determined
2、 by making a small change to tl1e vola tility and recomputing the option price using a new ttee. 灿 o i s calculated by making a stnall change to the interest rate and recomputing the option price using a new ttee.Problem 21.2.Calculate tl1e p ric e of a tlu ee-montl1Alner ic an put option on a non-d
3、ividend-paying stock 咘 en the stock price is $60, 小 e s 如 ke price is $60, tl1e risk-free intere st rate is 10%per a1mum, and tl1e volatility is 45% per arumm Use a binomial tre e with a time interval of one month.I11 tl1is case, 8iJ = 60 , K = 60 , r = 0 . 1 , a = 0.4 5 , T = 0.25 , and /1t = 0.083
4、3. Alsoll = ea 应 = e0.45严=1.1387-=-=d = 10.8782lla = e.-1:.t = eO.lxO.0833 = l. 0084a - dp = u-d= 0.49981- p = 0.5002TI1e output from De1ivaGem for tl1is exam ple is shown in the Figure S21.1. TI1e calculated price of tl1e option is $5.16.Gowth facto perstep, a= 1.0084 Probability of up move, p = 0.
5、4997Up step si ze, u = 1.1387 Downstepsize, d = 0.878277.800。84N odeT,me0.00000 08330.16670.2500Figwe S21.1:Tree for Problem21.2Problem 21.3.Explain how 小e control variate teclurique is itnplemented when a tree is used to value Atnerican options.TI1e conttol vari ate teclulique is implemented by1. V
6、aluing an American option using a binomial tree in the ustial way (众）2. Valuing tl1e European option witl1小e sam e parameters as the Atnerican option using小e same tree (= fi;:).3. Valuing tl1e European option using Black-Scholes-Merton（七 ）TI1eptice oftl1e Atnerican option is estimated as 众坛 丘Problem
7、 21.4.Calculate 小e price of a tune-month Atuerican call option on corn fi血 r es wl1en tl1e currentfi由11es price is 198 cents, the sttike price is 200 cents, 小e risk-丘ee interest rate is 8%per ammm , and tl1e volatility is 30% per ammm Use a binomial ttee witl1 a time interval of tluee months.It1 tl1
8、is case F 。;=198 , K = 200 , r = 0.08 , u=0.3, T =0.75, andt = 0.25 . AlsoU=e 03 应= 1.16181d = -=-= 0.8607ua = la- dp = 0.4626u-d1- p = 0.5373TI1e output from De1ivaGem for tl1is example is shown in the F屯ire S21.2. TI1e calculated price of tl1e option is 20.34 cents.Growh factor per st印 ，a = 1.0000
9、 Pr obabillty of up move, p = 0.4626Up step size, u = 1.1618 Dow,step size, d = 0.8607N ooe Time:0.00000.25000.50000.7500Figwe S21.2:Tree for Problem 21.4Problem 21.5.Consider an option tl1at pays o:fftl1e amount bywltich tl1e:final stock price exceeds the average stock price achieved during tl1e li
10、fe of the option. Can tltis be valued using the binontial hee approach? Explain your a 邯 werA binomial tree ca1111ot be used in the way desciibed in tllis chapter. Tilis is an example of what is known as a llistory-dependent option 兀 e payoff depends on the patl1 followed by tl1e stock price as well
11、 as its final value 兀 e option cannot be valu ed by star ting at the end of the hee and working backward since 小 e payoff at tl1e final branches is not known tmambiguously. Chapter 27 desciibes an extension oftl1e binomial hee approach tl1at can be used to handle options where tl1e payoff depends on
12、 tl1e average value of tl1e stock p1ice.Problem 21.6.F or a divide nd-paying stock, 小e tree for the stock price does not rec ombine; but the hee for 小 e stock price less the present va lue of 扣 tu.re di vidends does recombine. Explain this statement.Suppose a dividend eqtial to D is paid dming a cer
13、t ain time inte1v al. IfS is tl1e stock price at tl1e begimling of tl1e time inte1v al, it will be eitl1er 匈D or Sd - D at tl1e end of 小e time inte1v al. At the end of tl1e next time inteival, it will be one of（匈D)u ,(Su- D)d , (Sd - D)u and (Sd - D)d . Since（扣D) d does not eqt1al (Sci - D)u the tIe
14、e does not recombine. IfS is eqtial to the stock price less the present value of 和 血 e dividends, tllis problem is avoided.Pr oblem 21.7.Sl1owtl1at tl1e pro babi lities in a Cox, Ross, and 即 binstein binomial tIee are negative wl1en小e con dition in fooh1ote 8 holds.Witl1the ustial notationa-dp = u-d
15、u-a1- p = u-dIfa u , one oftl1e two probabilities is negative. Tilis happens whene(r 飞）心 e-(f 应ore归 ）丛 e(f应Tili s in hm1 happens when (q - r) /L汀 aor (r - q)-M 正 aHence negative probab 邮 es occur whenTili s i s tl1e con dition in fooh1ote 8.a and 1.5 montl1s we tl1en add tl1e pre s ent va lue of the
16、 dividend to tl1e stock ptice. TI1e result is tl1e ttee in Figme S21.6 TI1e ptice of the option calculated from tl1e ttee is 0.674. When 100 steps are used the ptice obtained is 0.690.Tree shows stock prices l ess PV of dividend at 0.125 years Growth factor per step, a =1.0025Probability of up move,
17、p =0.4993 Up step size, u =1.0748Down step size, d =0.9 劝NodeTime:O.CXXJO0.08330.16670.2500Fi即 eS21.5:First Tree for Problem 21.12At each node:Uppervalue =Undelying Asset Price Lower value=Option PriceBolded values ae a result of exercisePobabi lit y of up move, p =0.4993Node Time:O,CXXX)0.08330.166
18、70.2500Figure S21.6:Final Tree for Problem 21.12Problem 21.13.A one-year Alnerican put option on a non-dividend-paying stockhas an exercise price of$18. TI1e current stock price is $20, 小e risk-freeinterest rate is 15%per ammm , and the volatility of the stock is 40% per anmun. Use tl1e DerivaGem so
19、ftware with four three-montl1 time steps to estimate tl1e value oftl1e option. Display tl1e tree and verify that the option prices at the:final and penultimate nodes are correct. Use DerivaGem to value tl1e European version of小e option. Use the conhol variate teclmique to iniprove your estimate oftl
20、1e price ofthe Atnerican option.111 tl1is case Sa= 20,K = 18 , r = 0.15, CJ= 0.40 , T = 1, and !it= 0.25. TI1e parameters for the ttee areU = e(I.J1;i = e o.4扫1.2214 d =1/u = 0.8187a = et = 1.0382p =a-d1.0382-0.8187=0.545u-d1.2214-0.8187TI1e廿ee produced by DerivaGem for the 心n e1ica11 option is show
21、n in Figure S21.7. TI1e estimated value oftl1e American option is $1.29At u ch node :Uppu valut Unde 由 ing Asset Pnce lowuvalut Optfon Price8oldt d values art a mult of exercin Growth factorper step, a L0382 Prob动 i;tyof up mov, p 0.5451Up ste p ,;ze , ,u 1.2214Down stop size, d 0.8187No d t Time:0.
22、00000.25000.50000.75001.0000Figure S21.7:Tree to evaluate凡 n erican option for Problem 21.13At u ch no de:Uppe,value: UnderlyingAsset Price Lowe,value: Option Price Boldedvalues are a . suh of exerOseG,owth fa cto, pe, ste p, a: L0382 P,obab 山 ty of up move, p: 0.5451 Up ste p 泣 e, u: 1.2214Down ste
23、p size, d: 0.8187Nod, Time0.00000. 25000.50000.75 00i ooooFigun S21.8:nee to evaluate European option in Problem 21.13As shown in Figure S21.8, the same tree can be used to value a European put option witl1 the same param eters 兀 e estimated value of the European option is $1.14 九 e option parameter
24、s are Sa=20, K =l 8, r = 0.15, a=0.40 and T=l斗111(20 / 1 8) + 0.15 + 0.402 /2= 0.83840.40斗 di - 0.40 = 0.4384N(-di) = 0.2009;N(屯）0.3306TI1e hue European put price is therefore18e-0u15 x 0.3306 - 20 x 0.2009 = 1.10Tilis can also be obtained from De1ivaGem. TI1e con廿ol variate estimate oftl1e American
25、 put price is therefore 1.29+1.10 -1.14 = $1.25.Problem 21.14A two-month.A!nerican put option on a stock index has an exercise price of 480. TI1e current level oft11e index is 484, tl1e risk- 丘ee interest rate is 10% per ammm, the dividend yield on 小 e index is 3% per ammm, and the volatility of the
26、 index is 25% per annum Divide the life of the option into four half-month periods and use tl1e binomial hee approach to estimate the value oftl1e option.I11tl1is caseSa = 484 , K = 480 , r = 0.10 , a=0.25 q=0.03, T =0.1667, an d!i.t = 0.04167ll = e(I 应 = e0.25,/6 可=l.0524 1d = .:. = 0.9502ua = e归 “
27、） 1.00292p =a - d1.0029-0.9502= 0.516u-d1.0524-0.9502TI1e 廿ee produced by DerivaGem is shown in the Figure S21.9 兀 e estimated price of the option is $14.93.At u ch node:Upper valu, Und e y; nc Am t Price Lowervalu, Opt ion PrictBolded va l u ts are a resu lt of exercise Growth fa ctor per ste p, a,
28、 1.0029 Probability of up mov, ,p 0.5159Up step sin , ,u 1.05 义Dow n stop sin , d 0.95020.00000.04170.0833。0.16(;7Node Time1250Figwe S21.9:飞 ee to evaluate option in Problem 21.14Problem 21.15How can the control variate approach to improve the estimate of tl1edelta ofan Alnerican option wl1en the bi
29、nomi al 廿 ee approach is used?First the delta of the Alne1ican option is estimated in tl1e usual way from the tree. Denote this by I!,. . TI1en小e delta of a European option which has the s ame parameters as tl1e Alnerican option is calculated in the same way using tl1e same 订 ee. D enote this by I!,
30、.; . Finally the tmeEuropean delta, /!,B. , i s calculated using tl1e fo rm ul as in Chapter 19 九 e con订ol va1iateestimate of delta is tl1en:丛A七ABProblem 21.16.沁 ppose that Monte Carlo simulation is being used to evaluate a European call option on a 11011- dividend-pa ng stock wl1en the vo la tility
31、 is stochastic. How could tl1e con 廿 ol variate and anti 小 etic varia ble technique be used to in 甲 rovenumerical efficiency? Explain why it isnecessary to calculate six values of the option in each si皿 tlation trial wi1en both the contiol va riate and the antithetic variable teclmiqi1e are used.I11 tl1is case a simulation reqi1ires two sets of samples from standardized normal distiibutions.TI1e first is to generate tl1e vo la tility movements. TI1e s econd is to g enera te the stock p1icem ovement