Baran

AIM 2: Calculate, using a two-step binomial model, the value of an American call or put option.

 

1、The current price of Razor Manufacturing is $20. In each of the next two years you expect the stock price to either move up 20 percent or down 20 percent. The probability of an upward move is 0.65 and the probability of a downward move is 0.35. The risk-free rate is 5 percent. The value of a 2-year American put option with strike price of $24 is closest to:

A) $3.22.
 
B) $3.85.
 
C) $4.00.
 
D) $3.65. 

Baran

The correct answer is C

You need to use a two-step binomial model and consider the possibility of early exercise. First calculate the stock price tree. You have S0=20, so the first step results in either SU=20(1.2)=24 or SD=20(0.8)=16 at the end of year one. At the end of the second year the possible outcomes are SUU=24(1.2)=28.80, SUD= SDU=24(0.8)=19.20, or SDD=16(0.8)=12.80. The PV of the expected payoff in the up node is e-0.05[0.00(0.65)+4.80(0.35)]=$1.60. The payoff from early exercise in the up node is max{24-24, 0}=0. Since the PV of the expected payoff exceeds the payoff from early exercise, early exercise in the up node is not optimal. In the down node the PV of the expected payoff is e-0.05[4.80(0.65)+11.20(0.35)]=$6.70. The payoff from early exercise in the down node is max{24-16, 0} = $8.00. So early exercise is optimal in the down node. The value of the option can now be calculated as the PV of the expected payoffs at the end of the first year, or as e-0.05[1.60(0.65)+8.00(0.35)]=$3.65.

If the option is exercised early at the initial node it is worth $4 (=$24 - 20). This value is greater than $3.65, thus, the option should be exercised early at Node 0 and will be worth $4.

Baran

2、A stock that is currently trading at $30 can move up or down by 10 percent over a 6-month time period. The probability of the stock moving up in price in a 6-month period is 0.6074. The continuously compounded risk-free rate is 4.25 percent. The value of a 1-year American put option with an exercise price of $32.50 is closest to:

A) $3.42.
 
B) $5.50.
 
C) $2.49.
 
D) $2.75.

Baran

 

The correct answer is D

First calculate the probability of a down move as: pd = 1 – pu = 1 – 0.6074 = 0.3926

Next calculate the terminal values of the option at expiration for each node of the tree:

Suu = $30 × 1.10 × 1.10 = $36.30, Puu = $0

Sud = $30 × 1.10 × 0.90 = $29.70, Pud = $2.80

Sdu = $30 × 0.9 × 1.10 = $29.70, Pdu = $2.80

Sdd = $30 × 0.9 × 0.9 = $24.30, Puu = $8.20

Since this is an American option, we need to compare the discounted present value of the option to its intrinsic value after the end of the first 6-month period to see if the option is worth more dead than alive.

Baran

AIM 3: Discuss how the binomial model value converges as time periods are added.

 

1、As the binomial model of option prices is altered by increasing the number of periods:

A) the results stabilize at 30 periods. 
 
B) it eventually converges to the Black-Scholes-Merton option-pricing model. 
 
C) option values increase. 
 
D) volatility increases. 

Baran

The  correct  answer is B

As the option period is divided into more/shorter periods in the binomial option-pricing model, we approach the limiting case of continuous time and the binomial model results converge to those of the continuous-time Black-Scholes-Merton option pricing model.

Baran

2、Which of the following statements regarding the Black-Scholes-Merton option-pricing model is TRUE?

A) The Black-Scholes-Merton option-pricing model is the discrete time equivalent of the binomial option-pricing model.
 
B) As the number of periods in the binomial options-pricing model is increased toward infinity, it converges to the Black-Scholes-Merton option-pricing model. 
 
C) The Black-Scholes-Merton model is superior to the binomial option-pricing model in its ability to price options on assets with periodic cash flows. 
 
D) As the periods in the binomial option-pricing model are lengthened, it converges to the Black-Scholes-Merton option-pricing model.

Baran

The  correct  answer is B

As the option period is divided into more/shorter periods in the binomial option-pricing model, we approach the limiting case of continuous time and the binomial model results converge to those of the continuous-time Black-Scholes-Merton option pricing model.

Baran

3、The pricing results of the Black-Scholes-Merton model can be derived by:

A) lengthening the periods in the binomial model. 
 
B) taking the limit as the periods in the binomial model become shorter. 
 
C) using a regression model of prices on volatility. 
 
D) solving a system of simple mathematical equations.

Baran

The  correct  answer is B

As the option period is divided into more/shorter periods in the binomial option-pricing model, we approach the limiting case of continuous time and the binomial model results converge to those of the continuous-time Black-Scholes-Merton option pricing model.