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Hull: Options, Futures, and other Derivatives, Ninth Edition
Chapter 21: Basic Numerical Procedures
Multiple Choice Test Bank: Questions with Answers
1. How many nodes are there at the end of a Cox-Ross-Rubinstein five-step
binomial tree?
A. 4
B. 5
C. 6
D. 7
Answer: C
The number of nodes at the end of one time step is 2; the number of
nodes at the end of two time steps is 3; and so on.
2. Which of the following cannot be estimated from a single binomial tree?
A. delta
B. gamma
C. theta
D. vega
A

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Hull: Options, Futures, and other Derivatives, Ninth EditionChapter 21: Basic Numerical ProceduresMultiple Choice est Ban!: uestions #ith $ns#ers
1.How many nodes are there at the end of a Cox-Ross-Rubinstein ve-step binomial tree?A.4.!C. #.$ Answer% C &he number of nodes at the end of one time step is '( the number of nodes at the end of two time steps is )( and so on.'.*hi+h of the followin, +annot be estimated from a sin,le binomial tree?A.delta.,ammaC.theta#.ve,a Answer% # &o +al+ulate ve,a it is ne+essary to in+rease volatility sli,htly and +onstru+t a new tree. &he other three +an be estimated from a sin,le tree. ).*hi+h of the followin, is true for u in a Cox-Ross-Rubinstein binomial tree?A.t depends on the interest rate and the volatility.t depends on the volatility but not the interest rateC.t depends on the interest rate but not the volatility#.t depends on neither the interest rate nor the volatility Answer%
.
t
eu
∆
=
σ
t therefore depends on volatility but not the interest rate.4.How many dierent paths are there throu,h a Cox-Ross-Rubinstein tree with four-steps?A.!./C.1'#.1 Answer% #
&here are two +hoi+es for the path +hosen at the initial node0 two +hoi+es at the node rea+hed at the end of the rst time step0 and so on. &he total number of paths is '
4
1 .!.*hen we move from assumin, no dividends to assumin, a +onstant dividend yield0 whi+h of the followin, is true for a Cox0 Ross0 Rubinstein tree?A.&he parameters u and p +han,e.p +han,es but u does notC.u +han,es but p does not#.2either p nor u +han,es Answer% &he formula for u does not +han,e but the formula for p does +han,e. .*hen the sto+3 pri+e is ' and the present value of dividends is '0 whi+h of the followin, is the re+ommended way of +onstru+tin, a tree?A.#raw a tree for an initial sto+3 pri+e of ' and subtra+t the present value of future dividends at ea+h node .#raw a tree for an initial sto+3 pri+e of '' and subtra+t the present value of future dividends at ea+h node C.#raw a tree with an initial sto+3 pri+e of 15 and add the present value of future dividends at ea+h node#.#raw a tree with an initial sto+3 pri+e of 15 and add ' at ea+h nodeAnswer% C*e rst subtra+t the present value of dividends from the initial sto+3 pri+e. *e then draw the tree and then add ba+3 the present value of future dividends at ea+h node$.*hat is the re+ommended way of ma3in, interest rates a fun+tion of time in a Cox0 Ross0 Rubinstein tree?A.6a3e u a fun+tion of time.6a3e p a fun+tion of timeC.6a3e u and p a fun+tion of time#.6a3e the len,ths of the time steps une7ualAnswer% u does not depend on interest rates but p does. &he impa+t of the interest rate bein, a fun+tion of time is therefore to ma3e p a fun+tion of time. 5.*hat is the re+ommended way of ma3in, volatility a fun+tion of time in a Cox0 Ross0 Rubinstein tree?A.6a3e u a fun+tion of time.6a3e p a fun+tion of timeC.6a3e u and p a fun+tion of time#.6a3e the len,ths of the time steps une7ual
Answer% # &o ma3e volatility a fun+tion of time we ma3e the len,ths of time steps inversely proportional to varian+e./.A binomial tree pri+es an Ameri+an option at 8).1' and the +orrespondin,9uropean option at 8).4. &he la+3-:+holes pri+e of the 9uropean option is8'./5. *hat is the +ontrol variate pri+e of the Ameri+an option?A.8). .8).15C.8'./#.8).5 Answer% A &he +ontrol variate pri+e is ).1';<'./5-).4= ). .1.&he +hapter dis+usses an alternative to the Cox0 Ross0 Rubinstein tree. n this alternative0 whi+h of the followin, are true%A.&he relationship between u and d is% u1>d.&he relationship between u and d is% u-11-dC.&he probabilities on the tree are all .!#.2one of the above Answer% C nstead of usin, a de,ree of freedom to set u1>d <as is done in CRR=0 the alternative method uses a de,ree of freedom to set both the up and down probabilities to .!.11.*hi+h of the followin, +annot be valued by 6onte Carlo simulationA.9uropean options.Ameri+an optionsC.Asian options <i.e.0 options on the avera,e sto+3 pri+e=#.An option whi+h provides a payo of 81 if the sto+3 pri+e is ,reater than the stri3e pri+e at maturity Answer% Ameri+an options +annot be valued in a simple way usin, 6onte Carlo simulation be+ause 6onte Carlo simulation wor3s forward from the be,innin, of the life of an option and we do not 3now whether the option should be exer+ised when a parti+ular time is rea+hed.1'.*hi+h of the followin, is true?A.&he impli+it nite dieren+e method relates pri+es at one node to threepri+es at nodes at a later time.&he impli+it nite dieren+e method relates pri+es at one node to three
pri+es at nodes at an earlier timeC.&he impli+it nite dieren+e method relates pri+es at one node to threepri+es at nodes at the same time #.2one of the aboveAnswer% &he impli+it nite dieren+e relates pri+es at time
t
to three pri+es at nodes attime
t
∆
t
1).*hi+h of the followin, is true?A.&he impli+it nite dieren+e method is e7uivalent to usin, a trinomial tree.&he expli+it nite dieren+e method is e7uivalent to usin, a trinomial treeC.oth methods are e7uivalent to usin, a trinomial tree#.2either method is e7uivalent to usin, a trinomial treeAnswer% &he expli+it nite dieren+e method is e7uivalent to usin, a trinomial tree. &he impli+it nite dieren+e method is e7uivalent to usin, a multinomial tree.14.&he standard deviation of the values of an option +al+ulated usin, 10 6onte Carlo trials is 4.!. &he avera,e of the values is '. *hat is the standard error of this as an estimate of the option pri+e? A.4.!..4!C..4!#..4!Answer% A &he standard deviation of ' as an estimate of the pri+e is the standard deviation of the +al+ulated values divided by the s7uare root of the number of trials. n this +ase the s7uare root of the number of trials is 1 and so the standard error of the pri+e estimate is 4.!>1 or .4!.1!.&he values of a sto+3 pri+e at the end of the se+ond time step are 850 810 81'!. &he +orrespondin, values of an option are 80 8!0 and 8' respe+tively.*hat is an estimate of ,amma? A..1) ..14 C..1! #..1 Answer% C

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