Baran

The  correct  answer  is C

Given an up probability of 1.15, the down probability is simply the reciprocal of this number 1/1.15=0.87. Two down moves produce a stock price of 38 × 0.872 = 28.73 and a put value at the end of two periods of 6.27. An up and a down move, as well as two up moves leave the put option out of the money. The value of the put option is [0.322 × 6.27] / 1.062 = $0.57.

Baran

6、A stock is priced at 40 and the periodic risk-free rate of interest is 8 percent. What is the value of a two-period European call option with a strike price of 37 on a share of stock using a binomial model with an up factor of 1.20 and a (risk-neutral) up probability of 67 percent?

A) $20.60.
 
B) $3.57.
 
C) $9.25.
 
D) $9.07. 

Baran

The  correct  answer  is D

Two up moves produce a stock price of 40 × 1.44 = 57.60 and a call value at the end of two periods of 20.60. An up and a down move leave the stock price unchanged at 40 and produce a call value of 3. Two down moves result in the option being out of the money. The value of the call option is discounted back one year and then discounted back again to today. The calculations are as follows:

C+ = [20.6(0.67) + 3(0.33)] / 1.08 = 13.6962

C- = [3(0.67) + 0 (0.33)] / 1.08 = 1.8611

Call value today = [13.696(0.67) + 1.8611(0.33)] / 1.08 = 9.07

Baran

The  correct  answer  is D
The risk neutral probability of an up move is

 

Cuu = $40 × 1.05 × 1.05 = $44.10

Cud = $40 × 1.05 × 0.95 = $39.90

Cdu = $40 × 0.95 × 1.05 = $39.90

Cdd = $40 × 0.95 × 0.95 = $36.10

Since the strike price is at least $40 in all periods, we know that the option only has value if it follows an up, up path. In period 2, after following an up, up path, the option’s strike price is calculated as $40 + [(0 + 1) ′ 0.5] = $40.50. The intrinsic option value is $44.10 – $40.50 = $3.60.

The value of the option today is

Baran

5、A stock that currently trades at $40 can either move up or down by 5 percent each year. The continuously compounded risk-free rate is 4 percent. An over-the-counter European call option with 2 years until expiration is set up so that the strike price is determined by the formula $40 + [(years to expiration + 1) × 0.5] in periods when the stock price increases. In periods when the stock price declines, the strike price is $40. What is the value of this 2-year specialized OTC call option?

A) $3.12.
 
B) $3.27.
 
C) $2.56.
 
D) $2.74.

Baran

The  correct  answer  is D

 

In this case, u = 1.1, d = 0.9, r = 0.05, and the value of the option is $5 if the stock increases and 0 if the stock decreases. The risk-neutral probability of an up movement, p, can be calculated as:

 

 

The value of the call option is, therefore:

Baran

The  correct  answer  is D

One must first calculate the risk-neutral probability measure, π, which is [(e(rT) – d)/(u – d)]. In this case, r = 0.03, u = 1.1, d = 0.9, and T = 1, so π = [(e(0.03) –0.9)/(1.1 – 0.9)] = 0.6523. The value of the put at expiration if the stock price increases is 0, while it is $4 if the stock price decreases. The value of the put, therefore, is e(-0.03)(1 – 0.6523)($4) = $1.35.

Baran

4、A stock currently trades at $50. At the end of three months, the stock will either be $55 or $45. The continuously compounded risk-free rate of interest is 5 percent per year. The value of a 3-month European call option with a strike price of $50 is closest to:

A) $2.89.
 
B) $2.55.
 
C) $2.25.
 
D) $2.78. 

Baran

The  correct  answer  is C
 
First, we need to calculate the size of an upward movement in the asset’s price as eσ√t = e(0.1825)(1) = 1.20. The size of a downward movement in the stock’s price is 1/1.20 = 0.83.

Next, we project the various paths the stock’s price can follow over the 3 year period. The stock has 4 potential ending values:

Suuu = $75 × 1.2 × 1.2 × 1.2 = $129.60

Suud = Sduu = Sudu = $75 × 1.2 × 1.2 × 0.83 = $89.64

Sudd = Sdud Sddu = $75 × 1.2 × 0.83 × 0.83 = $62.00

Sddd = $75 × 0.83 × 0.83 × 0.83 = $42.89

The only point at which the option finishes in the money is after 3 upward moves, which as a probability of (0.60)(0.60)(0.60) = 0.216.

The value of the option today is therefore ($129.60 - $90) × 0.216 × e(-0.05)(3) = $7.36.

Baran

2、A stock that is currently trading at $50 and can either move to $55 or $45 over the next 6-month period. The continuously compounded risk-free rate is 2.25 percent. What is the risk-neutral probability of an up movement?

A) 0.6655.
 
B) 0.6565.
 
C) 0.5656.
 
D) 0.5566.