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John Hull 習題解答[Numerical method, Other Derivatives]
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CHAPTEI{ 19 Basic Numerical Procedures for the Instructor Chapter 19 presents the standard numerical procedures used to value derivatives when analytic results are not avai1able. These involve binomialjtrinomial trees , Monte Carlo simulation, difference methods. Binomial trees are introduced in Chapter 11 , and Section 19.1 and 19.2 can be regarded as a review and more in-depth treatment of that material. When covering Section 19.1 , 1 usually go through in some detail the calculations for a number of nodes in an example such as the one in Figure 19.3. Once the basic tree building and roll back procedure has been covered it is fairly easy to explain how it can be extended to currencies , indices , futures , and stocks that pay dividends. Also the calculation of hedge statistics such as delta , gamma, and vega can be explained. The software DerivaGem is a convenient way of displaying trees in class as well as an important calculation tool for students. The binomial tree and Monte Carlo simulation approaches use risk-neutral valua- tion arguments. By contrast , the finite difference method solves the underlying differential equation directly. However , as explained in the book the explicit finite difference method is essentially the same as the trinomial tree method and the implicit finite difference method is essentially the same as a multinomial tree approach where the are branches ema- nating from each node. Binomial trees and finite difference methods are most appropriate for American options; Monte Carlo simulation is most appropriate for path-dependent options. Any of Problems 19.25 to 19.30 work well as assignment questions. QUESTIONS AND PROBLEMS Problem 19. 1. Which of the following can be estimated for an American option by constructing a single binomia1 tree: delta , gamma, theta , rho? Delta, gamma, and theta can be determined from a single binomial tree. Vega is determined by 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 prÎce using a new tree. Problem 19.2. Calculate the price of a American put option on a non-dividend-paying stock wlwn the stock príce is $60, the strike price is $60, the risk-free interest rate is 10% 237
Transcript of John Hull 習題解答[Numerical method, Other Derivatives]
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
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