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 C

The value at any given node in a binomial tree is the present value of the cash flows at the two possible states immediately to the right of the given node, discounted at the 1-period rate at the node under examination.

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

 

10、Which of the following statements concerning the calculation of value at a node in a binomial interest rate tree is most accurate? The value at each node is the:

A) present value of the two possible values from the next period.

B) sum of the present values of the two possible values from the next period.

C) average of the present values of the two possible values from the next period.

D) average of the future values of the two possible values from the next period.

Baran

 

The  correct answer is A

 

1、Calculate the payoffs on the call in percent for I++ and I+? (= I?+):

   I++ value = (0.0712 ? 0.0625) / 1.0712 = 0.00812173.

   I+? value = (0.0684 ? 0.0625) / 1.0684 = 0.00552228.

   Remember that the payoff on the call value is the present value of the interest rate difference based on the raterealized  at     t= 2 because the payment is received at t = 3.

2、Calculate the t = 1 values (the probabilities in an interest rate tree are 50%):

   At t = 1 the values are I+ = [0.5(0.00812173) + 0.5 (0.00552228)] / 1.0683 = 0.00638585.

   At t = 1 the values are I? = [0.5(0) + 0.5 (0.00552228)] / 1.0617 = 0.00260068.

3、Calculate the t = 0 value:

   At t = 0 the option value is [0.5(0.00638585) + 0.5(0.00260068)] / 1.06 = 0.00423893 0.00423893 × 100,000 = $423.89.

Baran

 

9、A two-period interest rate tree has the following expected one-period rates:

The price of a two-period European interest-rate call option on the one-period rate with a strike rate of 6.25% and a principal amount of $100,000 is closest to:

A) $423.89. 

B) $725.86.

C) $449.33.

D) $704.22. 

Baran

The  correct answer is C

There is a positive relationship between the volatility of the stock and the price of both puts and calls. A higher estimate of volatility implies that the prices of both puts and calls should be higher. 

 

Baran

All else being equal, the greater the dividend paid by a stock the:
A) lower the call price and the lower the put price.
 
B) lower the call price and the higher the put price.
 
C) higher the call price and the lower the put price. 
 
D) higher the call price and the higher the put price.

Baran

The  correct answer is B

When dividend payments occur during the life of the option, the price of the underlying stock is reduced (on the ex-dividend date). All else being equal, the lower price reduces the value of call options and increases the value of put options.