[2008]Topic 19: The Black-Scholes-Merton Model相关习题 option, following, assets

Baran

 

AIM 3: List, define and relate the assumptions underlying the Black-Scholes-Merton model.

1、Which of the following is NOT one of the assumptions of the Black-Scholes-Merton option-pricing model?

A) There are no cash flows over the term of the options. 

B) The volatility is known and remains constant over the term of the option.  

C) The yield curve for risk-free assets is fixed over the term of the option.  

D) There are no taxes and transactions costs are zero for options and arbitrage portfolios. 

Baran

The correct answer is C

 

The yield curve is assumed to be flat so that the risk-free rate of interest is known and constant over the term of the option. Having a fixed yield curve does not necessarily imply that the yield curve is flat.

Baran

 

2、Which of the following is NOT one of the assumptions of the Black-Scholes-Merton (BSM) option-pricing model?

A) Any dividends are paid at a continuously compounded rate.  

B) There are no taxes.  

C) There are no transaction costs.  

D) Options valued are European style. 

Baran

 

The correct answer is A

 

The BSM model assumes there are no cash flows on the underlying asset.

Baran

 

3、Which of the following is least likely one of the assumptions of the Black-Scholes-Merton option pricing model?

A) The risk-free rate of interest is known and does not change over the term of the option. 

B) There are no cash flows on the underlying asset.  

C) Changes in volatility are known and predictable.  

D) The options are European style.

Baran

 

The correct answer is C

 

The BSM model assumes that volatility is known and constant. The term predictable would allow for non-constant changes in volatility.

Baran

 

AIM 4: Compute the value of a European option using the Black-Scholes-Merton model on a non-dividend-paying stock.

1、The current price of a stock is $55. A put option with a $50 strike price that expires in 3 months is available. If N(d1) = 0.8133, N(d2) = 0.7779, the underlying stock exhibits an annual standard deviation of 25 percent, and current risk free rates are 3.25 percent, the Black-Scholes value of the put is closest to:

A) $1.25. 

B) $1.50. 

C) $0.75. 

D) $5.00.

Baran

 

The correct answer is C

 

This question is not as straight-forward as it looks because N(d1) and N(d2) are given instead of N(-d1) and N(-d2) which are needed to solve for the price of a put.

N(-d1) = 1 – 0.8133 = 0.1867; N(-d2) = 1 – 0.7779 = 0.2221

The Black-Scholes formula for pricing a put is P= Xe(-rT) N(-d2) – SoN(-d1)

P= ($50e(-0.0325*0.25) * 0.2221) - ($55 * 0.1867) = $0.75

Baran

 

2、Using the Black-Scholes model, compute the value of a European call option using the following imputs:

Underlying stock price: $100

Exercise price: $90

Risk-free interest rate: 5%

Volatility: 20%

Dividend yield: 0%

Time to expiration: one year

The Black-Scholes call option price is closest to:

A) $13.65. 

B) $16.71. 

C) $15.33. 

D) $17.99.

Baran

 

The correct answer is B

 

This value is obtained using the Black-Scholes model for call option without dividends:

So d1=(ln(100/90)+(0.05+0.202/2))/0.2√1=0.8768 and using the table, N(0.88)=0.8106. d2=0.8768-0.2√1=0.6768, so from the table,N(d2)=N(0.68)=0.7517.

So the call value is 100(0.8106)-90e(-0.05)(0.7517)=$16.71.