［2008］Topic 18: Binomial Trees相关习题 stock, interest, current, percent, annual

Baran

AIM 1: Calculate, using the one-step and two-step binomial model, the value of a European call or put option.

 

1、The current price of a non-dividend paying stock is $75. The annual volatility of the stock is 18.25 percent, and the current continuously compounded risk-free interest rate is 5 percent. A 3-year European call option exists that has a strike price of $90. Assuming that the price of the stock will rise or fall by a proportional amount each year, and that the probability that the stock will rise in any one year is 60 percent, what is the value of the European call option?

A) $22.16
 
B) $12.91.
 
C) $7.36. 
 
D) $3.24.

[此贴子已经被作者于2009-6-25 17:02:25编辑过]

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

4、Which of the following is a condition needed in order for the binomial tree to approach the Black-Scholes model?

A) Stock prices change in a discrete manner.
 
B) Volatility changes stochastically over the life of the option.
 
C) Interest rates change stochastically over the life of the option.
 
D) The time intervals approach zero.

Baran

 The  correct  answer is D

As the length of the time intervals approaches zero, the binomial model converges to the continuous-time Black-Scholes 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

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

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 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.