CHAPTEI{19 BasicNumericalProcedures forthe Instructor Chapter 19 presents the standard numerical procedures used to value derivatives when analyticresultsarenotavai1able.Theseinvolvebinomialjtrinomialtrees, MonteCarlo simulation, differencemethods. Binomial trees are introduced in Chapter 11, and Section 19.1 and 19.2 can be regarded asareviewandmorein-depthtreatmentofthatmaterial.WhencoveringSection19.1, 1 usually gothrough insomedetailthecalculations foranumber of nodesin an example suchasthe onein Figure19.3.Oncethebasictreebuilding and rollback procedurehas beencovereditisfairlyeasytoexplainhowitcanbeextendedtocurrencies, indices, futures, andstocksthatpaydividends.Alsothecalculationof hedgestatisticssuchas delta, gamma, andvegacanbeexplained.The softwareDerivaGemisaconvenientway of displaying treesin classas wellasan important calculation toolforstudents. ThebinomialtreeandMonteCarlosimulationapproachesuserisk-neutralvalua-tion arguments.By contrast, the finitedifference method solves the underlying differential equation directly.However , as explained in the book the explicit finite difference method is essentially the same asthe trinomial tree method and the implicit finite difference method isessentially the same as amultinomial tree approach wherethe are branches ema-nating fromeachnode.Binomial treesand finitedifference methods are most appropriate forAmericanoptions;MonteCarlosimulationismostappropriateforpath-dependent options. Any of Problems19.25to19.30 workwellasassignment questions. QUESTIONSANDPROBLEMS Problem 19. 1. Whichof thefollowingcanbeestimatedforanAmericanoptionbyconstructinga singlebinomia1tree:delta, gamma, theta, rho? Delta,gamma, andthetacanbedeterminedfromasinglebinomialtree.Vegais determinedbymaking asmall changetothevolatilityandrecomputingtheoption price usinganewtree.Rhoiscalculatedbymakingasmallchangetotheinterestrateand recomputing the option prceusing anewtree. Problem 19.2. Calculatethepriceof aAmericanput optiononanon-dividend-paying stock wlwnthe stock prce is$60, the strike price is$60, the risk-free interest rate is10% 237 per annum, andthevolatility is45%per annum.Useabinomial treewithatime interval of one month. In thiscase, 80=60, K=60, r= 0.45, T=0.25, and 6.t=0.0833.Also u= eaVD. t= 1. 1387 d21:0.8782 u eTD.t= eO.lXO.0833= 1.0084 1- P =0.5002 The output fromDerivaGem forthis example is shown in the Figure 819. 1.The calculated price of the option is$5.16. Problem 19.3. Nodelim e: o 0000o 08330.16670.2500 Figure 819.1Tree forProblem 19.2 Explain how the control variate technique is implemented when atree is tovalue Americanoptions. 238 The controlvariate techniqueisimplemented by (a)valuinganAmerican optionusingabinomial treeinthe usual way(= f A)., (b)valuing the European option withthe same parameters astheAmerican option using the same tree(=fE). (c)valuing the European option using Black-8choles (= f BS).The price of the American option is Problem 19.4. prceof anne--monthAmercancalloptononcornfutureswhenthe currentfuturesprce s198cents, thestrke prce s200cents, the interestrate per annum, ands30%per annum.Useabinomaltreewithatme ntervalof three months. In thiscase K=200, r= 0.3, T=0.75, and f}. t=0.25.Also u=eO 3y'o'25=1.1618 d=1=08607 u 1 - p=0.5373 The output DerivaGem forthis example is shown in the Figure 819.2.The calculated price of the option is20.34cents. Problem 19.5. Consder anoptionthatpaysoff theamountby which stock prceexceeds theaveragestockprceachevedduringthelifeof theoption.Canthsbevaluedusing thebinomialtree approach?Explain your answer. Abinomialtreecannotbeusedinthewaydescribedinthischapter.Thisisan example of whatisknown asahistory-dependent option.The payoff depends on the path followed by the stock price as well as its final value.The option cannot be valued by starting atthe endof thetreeand workingbackward sincethepayoff at the finalbranches isnot knownunambiguously.Chapter26describesan extensionof thebinomial treeapproach thatcanbeusedto handle optionswheretheontheaveragevalueof the stock price. Problem 19.6. For advidend-paying stock, thetreeforthe stock price does notbut the tree forthe stock prce lessthe present value of futuredvdends does recombne."Explan this statement 239 Growthfactorperstep ,a=1 0000 P robabilityof upm ove ,p=0..4 626 u ps te ps iz e,U=11 618 00 W ns te ps iz e,d= 0..8607 NodeTime-o 0000o 2500o 5000o 7500 Figure 819.2TheeforProblem 19.4 Suppose adividend equal to Dispaid during acertain time interval.If Sisthe stock price at the beginning of the time interval, it willbe eitherSu - DorSd -- Dat the end of thetimeinterval.Attheendof thenexttimeinterval, itwillbeoneof(Su - (Su - D)d, (Sd - D)uand(Sd - D)d.Since(Su - D)ddoesnotequal(Sd ._- D)uthe treedoesnotrecombine.If Sisequaltothestockpricelessthepresentvalueof future dividends, this problem isavoided. Problem 19.7. Show thattheinaCox, Ross, and Rubinstein binomial tree are negative whenthecondition infootnote9 holds. With theusual notation p= u-d d or u , one of the twoprobabilities isnegative.This happens when e(T-q)At