［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 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.

Baran

The  correct  answer  is D

The risk-neutral probability, p, can be calculated as [e(rT)-d] / [u-d].  In this case, r = 0.0225, u = 1.1, d = 0.9, which makes p equal to [e[0.0225*(6/12)] - 0.9] / [1.1 - 0.9] = .5566

Baran

3、Calculate the value of a one-year put option today for a stock that currently trades at $40 and can either move to $44 or $36 at the end of a year. The continuously compounded risk-free rate is 3 percent and the put strike price is $40. The put option’s value is closest to:

A) $1.35.
 
B) $2.70.
 
C) $2.02.
 
D) $2.36.

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 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

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
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