Baran

 

3、Using the Black-Scholes model compute the value of a European put option using the following inputs:

Underlying stock price: $90

Exercise price: $90

Risk-free interest rate: 5%

Volatility: 20%

Dividend yield: 0%

Time to expiration: one year

The Black-Scholes put option price is closest to:

A) $4.11. 

B) $5.89. 

C) $6.12.

D) $5.01.

Baran

 

The correct answer is D

 

It is found as follows:

Put Price = Xe-.rt * [1 - N(d2)] – S * [1 - N(d1)]

Where d1=(ln(90/90)+(.05+.04/2)1)/.2√1)=.35 and using the Normal Z table, 1-N(.35)=1-.6368=.3632.

d2=.35-.2√1=.15, and using the Normal Z table, 1-N(.15)=1-.5596=.4404.

So the put value = 90e(-.05*1)(.4404) - 90(.3632)=$5.01

Baran

 

4、Consider a 120-day call option at 28 on a stock selling at 30 with an annualized standard deviation of 0.20 when the continuously compounded risk-free rate is 7 percent. The value of the call is closest to: [round d1 and d2 rather than interpolate for N(.)]

CT = [ST x N(d1)] - [Xe-rTN(d2)]

where:

d1 = ln(ST / X) + [r + σ2/2]T / σ √T

d2 = d1 - σ √T

Figure 1: Cumulative Standard Normal Probability

0.03

0.04

0.05

0.06

0.6	0.7357	0.7389	0.7422	0.7454
0.7	0.7673	0.7704	0.7734	0.7764
0.8	0.7967	0.7995	0.8023	0.8051

A) $3.02.  

B) $3.12.  

C) $3.33. 

D) $3.07. 

Baran

 

The correct answer is D

 

T=120/365 = 0.328767
d₁ =0.8596 ≈ 0.86 N(d₁) = 0.8051
d₂ =0.7449 ≈ 0.74 N(d₂) = 0.7704
c = $3.07

Baran

 

5、 Consider a 145-day put option at 30 on a stock selling at 27 with an annualized standard deviation of 0.30 when the continuously compounded risk-free rate is 4 percent. The value of the put option is closest to: [round d1 and d2 rather than interpolate for N(.)].

P_T = [Xe^{-r (T)} × (1 - N(d₂))] - [S_T × (1 - N(d₁))]
where:
d₁ = [ln(S_t / X) + [r + σ²/2](T) ] / σ √(T-t)
d₂ = d₁ - σ √(T)

A) $3.64. 

B) $3.32.  

C) $3.97. 

D) $4.07.

Baran

 

The correct answer is A

 

T=145/365 = 0.39726

d₁ = [ln(27/30) + [.04 + .3²/2](.39726)] / (.3√.39726)

    = (-.10536052 + .0337671) / .18908569

    = -.07159342 / .18908569

    = -0.37863
d₁ = -0.37863 ≈ -0.38 N(d₁) = 1 -0.6480 = 0.3520

d₂ = -0.37863 - .3√.39726

    = -0.37863 - .18908569

    = -.56771569

    = -.56772
d₂ = -0.56772 ≈ -0.57 N(d₂) = 1 - 0.7157 = 0.2843

P_T = 30e^-.04(.39726) (1-.2843) – 27(1-.352)

    = (29.527056 × .7157) – 17.496

    = 21.1325 – 17.496
p = $3.64

Baran

 

AIM 6: Compute the value of a European option using the Black-Scholes-Merton model on a dividend-paying stock.

1、Dividends on a stock can be incorporated into the valuation model of an option on the stock by:

A) subtracting the present value of the dividend from the current stock price. 

B) subtracting the future value of the dividend from the current stock price.  

C) adding the future value of the dividend to the option value. 

D) adding the present value of the dividend to the current stock price. 

Baran

 

The correct answer is A

 

The option pricing formulas can be adjusted for dividends by subtracting the present value of the expected dividend(s) from the current asset price.

Baran

 

2、Compared to the value of a call option on a stock with no dividends, a call option on an identical stock expected to pay a dividend during the term of the option will have a:

A) lower value only if it is an American style option.  

B) lower value in all cases.  

C) higher value in all cases. 

D) higher value only if it is an American style option.

Baran

 

The correct answer is B

 

An expected dividend during the term of an option will decrease the value of a call option.