Hull_OFOD9e_MultipleChoice_Questions_and_Answers_Ch21.doc

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 Answer: D To calculate vega it is necessary to increase volatility slightly and construct a new tree. The other three can be estimated from a single tree. 3. Which of the following is true for u in a Cox-Ross-Rubinstein binomial tree? A. It depends on the interest rate and the volatility B. It depends on the volatility but not the interest rate C. It depends on the interest rate but not the volatility D. It depends on neither the interest rate nor the volatility Answer: B u  e

t

. It therefore depends on volatility but not the interest rate.

4. How many different paths are there through a Cox-Ross-Rubinstein tree with four-steps? A. 5 B. 9 C. 12 D. 16 Answer: D

There are two choices for the path chosen at the initial node, two choices at the node reached at the end of the first time step, and so on. The total number of paths is 24 = 16. 5. When we move from assuming no dividends to assuming a constant dividend yield, which of the following is true for a Cox, Ross, Rubinstein tree? A. The parameters u and p change B. p changes but u does not C. u changes but p does not D. Neither p nor u changes Answer: B The formula for u does not change but the formula for p does change. 6. When the stock price is 20 and the present value of dividends is 2, which of the following is the recommended way of constructing a tree? A. Draw a tree for an initial stock price of 20 and subtract the present value of future dividends at each node B. Draw a tree for an initial stock price of 22 and subtract the present value of future dividends at each node C. Draw a tree with an initial stock price of 18 and add the present value of future dividends at each node D. Draw a tree with an initial stock price of 18 and add 2 at each node Answer: C We first subtract the present value of dividends from the initial stock price. We then draw the tree and then add back the present value of future dividends at each node 7. What is the recommended way of making interest rates a function of time in a Cox, Ross, Rubinstein tree? A. Make u a function of time B. Make p a function of time C. Make u and p a function of time D. Make the lengths of the time steps unequal Answer: B u does not depend on interest rates but p does. The impact of the interest rate being a function of time is therefore to make p a function of time. 8. What is the recommended way of making volatility a function of time in a Cox, Ross, Rubinstein tree? A. Make u a function of time B. Make p a function of time C. Make u and p a function of time D. Make the lengths of the time steps unequal

Answer: D To make volatility a function of time we make the lengths of time steps inversely proportional to variance. 9. A binomial tree prices an American option at $3.12 and the corresponding European option at $3.04. The Black-Scholes price of the European option is $2.98. What is the control variate price of the American option? A. $3.06 B. $3.18 C. $2.90 D. $3.08 Answer: A The control variate price is 3.12+(2.98-3.04) = 3.06. 10.The chapter discusses an alternative to the Cox, Ross, Rubinstein tree. In this alternative, which of the following are true: A. The relationship between u and d is: u=1/d B. The relationship between u and d is: u-1=1-d C. The probabilities on the tree are all 0.5 D. None of the above Answer: C Instead of using a degree of freedom to set u=1/d (as is done in CRR), the alternative method uses a degree of freedom to set both the up and down probabilities to 0.5. 11.Which of the following cannot be valued by Monte Carlo simulation A. European options B. American options C. Asian options (i.e., options on the average stock price) D. An option which provides a payoff of $100 if the stock price is greater than the strike price at maturity Answer: B American options cannot be valued in a simple way using Monte Carlo simulation because Monte Carlo simulation works forward from the beginning of the life of an option and we do not know whether the option should be exercised when a particular time is reached. 12.Which of the following is true? A. The implicit finite difference method relates prices at one node to three prices at nodes at a later time B. The implicit finite difference method relates prices at one node to three

prices at nodes at an earlier time C. The implicit finite difference method relates prices at one node to three prices at nodes at the same time D. None of the above Answer: B The implicit finite difference relates prices at time t to three prices at nodes at time t−t 13.Which of the following is true? A. The implicit finite difference method is equivalent to using a trinomial tree B. The explicit finite difference method is equivalent to using a trinomial tree C. Both methods are equivalent to using a trinomial tree D. Neither method is equivalent to using a trinomial tree Answer: B The explicit finite difference method is equivalent to using a trinomial tree. The implicit finite difference method is equivalent to using a multinomial tree. 14.The standard deviation of the values of an option calculated using 10,000 Monte Carlo trials is 4.5. The average of the values is 20. What is the standard error of this as an estimate of the option price? A. 4.5 B. 0.45 C. 0.045 D. 0.0045 Answer: A The standard deviation of 20 as an estimate of the price is the standard deviation of the calculated values divided by the square root of the number of trials. In this case the square root of the number of trials is 100 and so the standard error of the price estimate is 4.5/100 or 0.045. 15.The values of a stock price at the end of the second time step are $80, $100, $125. The corresponding values of an option are $0, $5, and $20 respectively. What is an estimate of gamma? A. 0.136 B. 0.146 C. 0.156 D. 0.166 Answer: C

The two delta estimates are (5−0)/(100-80)=0.25 and (20-5)/125-100)=0.6. The mid points of the ranges over which these are calculated are 90 and 112.5.The gamma estimate is therefore (0.6-0.25)/(112.5-90) = 0.156

16.What is the difference between valuing an American and a European option using a tree? A. The value of u is higher for American options B. The value of u is lower for American options C. The time steps for American options are not equal D. It is necessary to do two calculations at nodes where the option is in the money Answer: D When valuing American options it is necessary to calculate at each in-themoney node how much the option is worth if it is exercised (i.e., the intrinsic value at the node) and how much it is worth if it is not exercised (i.e., the roll back value) 17.A European option on a stock with a known dollar dividend is valued by setting the stock price variable equal to the stock price minus the present value of the dividend in the Black-Scholes-Merton formula. A second price can be obtained using the tree building procedure in the chapter. Which of the following is true when a very large number of time steps are used in the tree? A. The first price is higher than the second price B. The first price is lower than the second price C. The first price is sometimes higher and sometimes lower than the second price D. The two prices are almost exactly the same Answer: D The two prices are exactly the same because they are based on treating dividends in the same way. 18.Which of the following is possible in a modified Cox, Ross, Rubinstein binomial tree? A. The interest rate and volatility can both be functions of time B. The interest rate or the volatility can be a function of time, but not both C. The interest rate can be a function of time but the volatility cannot D. The interest rate and volatility must be constant Answer: A Both the t-period interest rate and the volatility can be made functions of time as described in the text.

19.Which of the following describes the way that the parameters in a binomial tree are chosen? A. The expected return during each time step is the risk-free rate B. The standard deviation of the return in each time step is, for small time steps, almost exactly equal to the volatility per annum times the square root of the length of the time step in years C. The tree recombines D. All of the above Answer: D A, B, and C are all true. 20.Which of the following can be valued without using a numerical procedure such as a binomial tree? A. American put options on a non-dividend paying stock B. American call options on a non-dividend paying tock C. American call options on a currency D. American put options on futures Answer: B The answer is B because an American call on a non-dividend paying stock is worth the same as the corresponding European call, which can be valued with BSM.