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