18ÙSimple Implementing ofOption Oricing Models!W K1.In the EWMA model and the GARCH(1,1)model,the weights assigned to observa-tions decrease as the observations become older.2.The GARCH(1,1)model differs from the EWMA model in that some weight is alsoassigned to the.3.When the current volatility is above the long-term volatility,the GARCH(1,1)modelestimates a volatility term structure.4.Suppose thatλis0.8,the volatility estimated for a market variable for day n-1is2%per day,and during day n-1the market variable increased by3%.Then the estimate of the volatility for day n is.5.Suppose thatλ=0.9,and the estimate of the correlation between two variables Xand Y on day n-1is0.7.Supposeσx,n−1=2%,σy,n−1=3%,u x,n−1=0.4%,u y,n−1=2.5%.The covariance for day n wouled be.6.For an American option,the value at a node is the greater of,and thediscounted expected value if it is held for a further period of timeδt.7.Finite difference methods solve the underlying by covert it to a differenceequation.8.The explicit method is functionally the same as using a.9.involves dividing the distribution into ranges or intervals and sampling fromeach interval according to its probability.1218ÙSIMPLE IMPLEMENTING OF OPTION ORICING MODELS10.Currencies and futures contracts can,for the purposes of option evaluation,be con-sidered as assets providing known yields.In the case of a currency,the relevant yield is the;in the case of a futures contract,it is the.!üÀK(3z¢K o À Y¥ÀJ ( Y èW\K )ÒS)1.Which model is not used to produce estimates of volatilities()A.EWMAB.ARCHC.CRRD.GARCH2.In an EWMA model,the weights of the u i declines at rate as we move backthrough time.()C.λ2D.1−λA.λB.1λ3.When use variance targeting approach to estimate parameters in GARCH(1,1),thereare only parameters have to be estimated.()A.1B.2C.3D.44.The parameters of a GARCH(1,1)model are estimated asω=0.000004,α=0.05,β=0.92.What is the long-run average volatility?()A.0.00013B.0.013C.0.00025D.0.0255.The most recent estimate of the daily volatility of USD\GBP exchange rate is0.55%and the exchange rate at4p.m.yesterday was1.4950.The parameterλin the EWMA model is0.95.Suppose that the exchange rate at4p.m.today proves to be1.4850,the estimate of the daily volatility is()A.2.900B.2.874C.0.2874D.0.29006.Which of the following can be estimated for an American option by constructing asingle binomial tree?()A.DeltaB.VegaC.GammaD.Theta7.Which is not particularly useful when the holder has early exercise decisions in Amer-ican options?()A.Monte Carlo SimulationB.Binomial Frees modelC.Finite Difference MethodsD.Trinomial Trees model38.Which model can be used when the payoffdepends on the path followed by theunderlying variable S as well as when it depends only on thefinal value of S?()A.Binomial Trees modelB.Trinomial Trees modelC.Finite Difference MethodsD.Monte Carlo Simulation9.Consider a four-month American call option on index futures where the risk-freeinterest rate is9%per annum,and the volatility of the index is40%per annum.We divide the life of the option into four one-month periods for the purposes of con-structing the tree.Then the growth factor a equals:()A.1B.e9%×0.0833C.e9%D.e40%×t0.083310.In a binomial model for a dividend-paying stock,when the dollar amount dividend isknown,there are nodes on the tree at time iδt.()A.i+1B.iC.2iD.i2n!§äK(3 ( K )ÒS y”√”§ Ø K )ÒS y”×”)1.Black-Scholes model assume that the volatility of the underlying asset is not constant.()2.In the EWMA model,some weight is assigned to the long-run average variance rate.()3.When we build up models to forecast volatility,u is assumed to be zero.()4.In the ARCH(m)model,the older an observation,the less weight it is given.()5.The EWMA approach has the attractive feature that relatively little date need tobe stored.At any given time,we need to remember only the current estimate of the variance rate and the most recent observation on the value of the market variable.()6.For a stable GARCH(1,1)process,we requireα+β>1,then the GARCH(1,1)processis’mean reverting’rather than’meanfleeing’.()7.The EWMA model incorporates mean reversion,whereas the GARH(1,1)model dosenot.()8.For a series x i,the autocorrelation with a lag of k is the coefficient of correlationbetween x i and x i+k.()418ÙSIMPLE IMPLEMENTING OF OPTION ORICING MODELS9.When the current volatility is below the long-term volatility,it estimates an downward-sloping volatility term structure.()10.Suppose there is a big move in the market variable on day n-1,the estimate of thecurrent volatility moves upward.()11.Monte Carlo simulation is used primarily for derivatives where the payoffis depen-dent on the history of the underlying variable or where there are several underlying variables.()12.Binomial trees andfinite difference methods are not useful when the holder has earlyexercise decisions to make prior to maturity.()13.In binomial trees model,the derivatives can be value by discounting their expectedvalues at the risk-free interest rate.()14.Currencies can be considered as assets providing known yields,and the relevant yieldis domestic risk-free interest rate.()15.The tree does not recombine when the dividend yield is known.()16.The control variate technique used by binomial trees needs to calculate the Blank-Scholes price of the European option.()17.The CRR is the only way to construct binomial trees.()18.The advantage of Monte Carlo simulation is that it is computationally very time-saving.()19.In Monte Carlo simulation the uncertainty about the value of the derivative is in-versely proportional to the square root of the number of trials.()20.The implicitfinite difference has the advantage of being very robust.It always con-verges to the solution of the differential equation as s and t approach zero.()o!O K1.The volatility of a certain market variable is30%per annum.Calculate a99%confi-dence interval for the size of the percentage daily change in the variable.52.Assume that S&P500at close of trading yesterday was1040and the daily volatility ofthe index was estimated ai1%per day at that time.The parameters in a GARCH(1,1) model areω=0.000002,α=0.06,β=0.92.If the level of the index at close of trading today is1060,what is the new volatility estimate?3.Suppose that the current daily volatilities of asset A and asset B are1.6%and2.5%,respectively.The prices of the assets at close of trading yesterday were$20and$40and the estimate of the coefficient of correlation between the returns on the two assets made at that time was0.25.The parameterλused in the EWMA model is0.95.a.Calculate the current estimate of the covariance between the assets.b.On the assumption that the prices of the assets at close of trading today are$20.5and$40.5,update the correlation estimate.4.A three-month American call option on a stock has a strike a strike price of$20.Thestock price is$20,the risk-free rate is3%per annum,and the volatility is25%per annum.A dividend of$2is expected e a three-step binomial tree to calculate the option price.5.A two-month American put option on a stock index has an exercise price of480.Thecurrent level of the index is484,the risk-free interest rate is10%per annum,the dividend yield on the index is3%per annum,and the volatility of the index is25% per annum.Divide the life of the option into four half-month periods and use the tree approach to estimate the value of the option.6.Provide formulas that can be used for obtaining three random samples from standardnormal distributions when the correlation between sample i and sample j isρi,j.Ê!¶c)º1.maximum likelihood method2.volatility term structure3.mean reversion4.GARCH(1,1)5.antithetic variable technique618ÙSIMPLE IMPLEMENTING OF OPTION ORICING MODELS6.stratified sampling7.finite difference8.hopscotch method8!{ K1.What is the difference between the exponentially weighted moving average modeland the GARCH(1,1)model for updating volatilities?2.A company uses the GARCH(1,1)model for updating volatility.The three parame-ters areω,αandβ.Describe the impact of making a small increase in each of the parameters while keeping the othersfixed.3.Which of the following can be estimated for an American option by constructing asingle binomial tree:delta,gamma,vega,theta,rho?4.Explain how the control variate technique is implemented when a tree is used tovalue American options.5.Explain why the Monte Carlo simulation approach cannot easily be used for American-style derivatives.。
