Deck 17: Single-Period Binomial Model

Question 1

Use the following data for a single-period binomial model to answer the questions that follow.
- The stock's price S is $50.After three months,it either goes up by the factor U = 1.16038286 or it goes down by the factor D = 0.85963276.
- Options mature after T =0 0.25 years.
- The continuously compounded risk-free interest rate r is 4 percent per year.

-Given the above data,the value of a call option with a strike price of $45 is:

A) $1.65
B) $3.45
C) $6.45
D) $7.08
E) None of these answers are correct.

Answer 1

$6.45

Question 2

Use the following data for a single-period binomial model to answer the questions that follow.
- The stock's price S is $50.After three months,it either goes up by the factor U = 1.16038286 or it goes down by the factor D = 0.85963276.
- Options mature after T =0 0.25 years.
- The continuously compounded risk-free interest rate r is 4 percent per year.

-Given the above data,consider an exotic option whose payoff at expiration is given by the square root of the stock price less the strike price (K = $6)if it has a positive value,zero otherwise,that is: max[ $\surd$ S(1)- 6,0].
The value of this exotic option is given by:

A) $0.55
B) $0.89
C) $1.08
D) $1.55
E) None of these answers are correct.

Answer 2

$1.08

Question 3

Use the following data for a single-period binomial model to answer the questions that follow.
- The stock's price S is $50.After three months,it either goes up by the factor U = 1.16038286 or it goes down by the factor D = 0.85963276.
- Options mature after T =0 0.25 years.
- The continuously compounded risk-free interest rate r is 4 percent per year.

-Given the above data,consider an exotic option whose payoff at expiration is given by the stock price S(1)squared less a strike price (K= $2,500)if it has a positive value,zero otherwise,that is: max[S(1)²- 2500,0].
Suppose a trader quotes a price of $450 for this option.Then you can make an immediate arbitrage profit of:

A) $21.08 by selling the traded call and buying the synthetic call involving buying 57.60 shares of stock and selling 2,451.28 units of the money market account
B) $226.50 by buying the traded call and selling the synthetic call involving selling 90.85 shares of stock and buying 3,866.21 units of the money market account
C) $50 by buying the traded call and selling the synthetic call involving selling 60.54 shares of stock and buying 2,321.21 units of the money market account
D) $337.76 by selling the traded call and buying the synthetic call involving buying 16.74 shares of stock and selling 641.35 units of the money market account
E) None of these answers are correct.

Answer 3

$21.08 by selling the traded call and buying the synthetic call involving buying 57.60 shares of stock and selling 2,451.28 units of the money market account

Question 4

USe the following data for a single-period binomial model to answer the questions that follow.
YBM's stock price S is $102 today.
- After six months,the stock price can either go up to $115.63212672,or go down to $93.52995844.
- Options mature after T = 6 months and have an exercise price of K =$105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-Given the above data,the pseudo-probability of an up movement and the discounted expected stock price using the pseudo-probabilities are given by:

A) 0.50 for the pseudo-probability and 100 for the stock price
B) 0.50 for the pseudo-probability and 102 for the stock price
C) 0.45 for the pseudo-probability and 102 for the stock price
D) 0.45 for the pseudo-probability and 105 for the stock price
E) None of these answers are correct.

Question 5

USe the following data for a single-period binomial model to answer the questions that follow.
YBM's stock price S is $102 today.
- After six months,the stock price can either go up to $115.63212672,or go down to $93.52995844.
- Options mature after T = 6 months and have an exercise price of K =$105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-Given the above data,the hedge ratio and the put option's value are given by:

A) -0.2523 for the hedge ratio and $2.35 for the put option's value
B) 0.2523 for the hedge ratio and $4.81 for the put option's value
C) 0.5190 for the hedge ratio and $5.59 for the put option's value
D) -0.5190 for the hedge ratio and $5.59 for the put option's value
E) None of these answers are correct.

Question 6

USe the following data for a single-period binomial model to answer the questions that follow.
YBM's stock price S is $102 today.
- After six months,the stock price can either go up to $115.63212672,or go down to $93.52995844.
- Options mature after T = 6 months and have an exercise price of K =$105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-Given the above data,the hedge ratio and the call option's value are given by:

A) 0.2523 for the hedge ratio and $4.1853 for the call option's value
B) 0.3810 for the hedge ratio and $5.5557 for the call option's value
C) 0.4810 for the hedge ratio and $5.1853 for the call option's value
D) 0.5810 for the hedge ratio and $6.2543 for the call option's value
E) None of these answers are correct.

Question 7

A necessary and sufficient condition to rule out arbitrage profits in the binomial tree is that the following must hold:

A) "up factor $\gt$ down factor $\gt$ dollar return" [U $\gt$ D $\gt$ (1 + R)]
B) U $\gt$ (1 +R) $\gt$ D
C) (1 + R)=U $\gt$ D
D) (1 + R)=(U + D)/2
E) None of these answers are correct.

Question 8

The following was NOT a major development in the history of option pricing:

A) the Black-Scholes-Merton model for pricing European options
B) the James-Jones model for dollar dividend adjustment
C) the Merton and Jarrow-Turnbull models for pricing derivatives with credit risk
D) Harrison-Kreps and Harrison-Pliska's martingale pricing methodology
E) Vasicek,Ho-Lee,and the Heath-Jarrow-Morton (HJM),and HJM Libor models

Question 9

Use the following data for a single-period binomial model to answer the questions that follow.
- The stock's price S is $50.After three months,it either goes up by the factor U = 1.16038286 or it goes down by the factor D = 0.85963276.
- Options mature after T =0 0.25 years.
- The continuously compounded risk-free interest rate r is 4 percent per year.

-Given the above data,consider an exotic option whose payoff at expiration is given by the stock price S(1)squared less a strike price (K= $2,500)if it has a positive value,zero otherwise,that is: max[S(1)²- 2500,0].
The value of this exotic option is given by:

A) $350.06
B) $428.92
C) $451.92
D) $676.50
E) None of these answers are correct.

Question 10

USe the following data for a single-period binomial model to answer the questions that follow.
YBM's stock price S is $102 today.
- After six months,the stock price can either go up to $115.63212672,or go down to $93.52995844.
- Options mature after T = 6 months and have an exercise price of K =$105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-Given the above data,suppose that a trader quotes a put price of $5.Then the arbitrage profit that you can make today by trading this call and related securities is:

A) $0
B) $0.33
C) $0.59
D) $1.54
E) None of these answers are correct.

Question 11

Which of the following statements is INCORRECT about the binomial option pricing model?

A) The binomial model had been used as a teaching tool at MIT and other places before its publication.
B) The binomial model was first printed in Sharpe's classic textbook Investments.
C) Cox,Ross,and Rubinstein;Rendleman and Bartter;and Jarrow and Rudd developed popular versions of the binomial model.
D) The binomial model is accurate because in the real world,stock price usually goes up or down with only two possible values.
E) Unlike the cost-of-carry models which can be priced with the help of the "no-arbitrage principle" alone,option-pricing models need an explicit assumption about the evolution of stock prices.

Question 12

Which of the following statements is INCORRECT about the binomial option pricing model?

A) The binomial model is popular because it is accessible and easy to understand.
B) The binomial model gives more accurate results than the Black-Scholes Merton model.
C) The binomial model is a useful teaching tool.
D) The binomial model is a versatile model that can price a variety of derivatives.
E) The binomial model illustrates the main tenets of martingale pricing,the key technique that lies at the heart of derivative pricing.

Question 13

The model that was the first true ancestor of modern option-pricing models was developed by:

A) Louis Bachelier
B) James Boness
C) Paul Samuelson
D) Case Sprenkle
E) Edward Thorp and Sheen Kassouf

Question 14

Which of the following statements is INCORRECT about the Troubled Asset Relief Program (TARP)of the US government during the financial crisis of 2007-09?

A) TARP used nearly $700 billion to support banks with troubled assets.
B) The Treasury purchased preferred shares from banks and injected funds into these financial institutions.
C) The banks also issued warrants that allowed the government to buy the company's shares at a predetermined price over a ten-year period.
D) After the banks recovered,the US government sold these warrants back to the banks or to others at a "fair price" determined with the help of the Black-Scholes-Merton and binomial models.
E) Like most other government policies,TARP was a failure and cost the government significantly more than the amount that was originally estimated.

Question 15

Which of the following statements is INCORRECT?

A) The method of computing a stock's present value by its expected payoff using the actual probabilities is known as martingale pricing.
B) A martingale is a stochastic process X(t)whose time t value equals its expected value X(T)at some later date T.
C) Martingales are associated with "fair gambles" because what you have today is what you expect to have tomorrow.
D) In finance,martingales are used to price derivatives in a framework that allows no arbitrage opportunities.
E) The probabilities we use in martingale pricing are pseudo-probabilities and not the actual probabilities.

Question 16

Which of the following statements is INCORRECT?

A) The hedge ratio is the number of shares of the stock to hold for each written option to form the perfect hedge.
B) The hedge ratio lies between 0 and 1 in the case of a call option.
C) The hedge ratio lies between 0 and -1 in the case of a put option.
D) The hedge ratio may be defined as the ratio of the change in option price to change in stock price.
E) Pseudo-probabilities are useful for computing the hedge ratio.

Question 17

USe the following data for a single-period binomial model to answer the questions that follow.
YBM's stock price S is $102 today.
- After six months,the stock price can either go up to $115.63212672,or go down to $93.52995844.
- Options mature after T = 6 months and have an exercise price of K =$105.
- The continuously compounded risk-free interest rate r is 5 percent per year.

-Given the above data,suppose that a trader quotes a call price of $6.Then the arbitrage profit that you can make today by trading this call and related securities is:

A) $0
B) $0.25
C) $0.81
D) $1.22
E) None of these answers are correct.

Question 18

The following is NOT an assumption underlying the binomial option pricing model:

A) no market frictions
B) no credit risk
C) competitive and well-functioning markets
D) no interest rate uncertainty
E) stock prices follow a lognormal process

Question 19

Which of the following is an INCORRECT step in pricing an option by the no-arbitrage principle in a single-period binomial framework?

A) We consider a market where a stock,a money market account,and an option trade.
B) The stock can take one of two possible values at the expiration date and so does the option.
C) We create a portfolio of the stock and money market account to replicate an option's payoff.
D) We choose the portfolio of the stock and money market account to have the same value as the option today.Then,to prevent arbitrage,this "synthetic option" and the market-traded option must have the same payoffs on the option's expiration date.
E) A portfolio holding the market-quoted option and the synthetic option on opposite sides of the market are free from price risk.This demonstrates that while solving the pricing problem,we also learn how to hedge the exposure.

Question 20

Which of the following was NOT a key insight that helped Fischer Black,Myron Scholes,and Robert Merton formulate their 1973 option pricing model?

A) adjustment for known dollar dividends
B) the lognormal distribution for stock prices
C) the no-arbitrage principle
D) hedging an option with a stock and the creation of a "perfect hedge"
E) the focus on a stock's price return's volatility as opposed to measuring a risk-premium

Question 21

Which of the following statements about Robert Merton's "Trick" is INCORRECT?

A) Merton used to give the intuition of risk-neutral valuation to MIT students with an argument that he called The Trick.
B) The Trick involves considering two worlds: the real world (where investors could be risk-averse)and a pseudo-world (where investors would be risk-neutral).
C) The same securities: a bond (which may be viewed as a money market account),a stock,and an option trade in both worlds.
D) The probabilities of the stock going up and down are the same in both the real world and the pseudo-world.
E) One can easily move from the real world to the pseudo-world where things are tractable,price the option there by risk-neutral valuation,and bring that price back to the real world.