Question

Consider the following questions on the pricing of options on the stock of ARB Inc.: a.[5 Points] A share of ARB stock sells for $75 and has a standard deviation of returns equal to 20 percent per year. The current risk-free rate is 9 percent and the stock pays two dividends: (1) a $2 dividend just prior to the option’s expiration day, which is 91 days from now (i.e., exactly one-quarter of a year), and (2) a $2 dividend 182 days from now (i.e., exactly one-half year). Calculate the Black-Scholes value for a European-style call option with an exercise price of $70. b.[5 Points] What would be the price of a 91-day European-style put option on ARB stock having the same exercise price? c.[5 Points] Calculate the change in the call option’s value that would occur if ARB’s management suddenly decided to suspend dividend payments and this action had no effect on the price of the company’s stock. d.[5 Points] Briefly describe (without calculations) how your answer in Part a would differ under the following separate circumstances: (1) the volatility of ARB stock increases to 30 percent, and (2) the risk-free rate decreases to 8 percent.

Answer 1

Answer:

Check the following calculations

Explanation:

Exercise Price = 70

Call Option = S* N(d1) - N(d2)*Ke-rt

Risk free Rate = 9% = .09

S = Current stock price = 75

K = Exercise price = 70

t = time to expiry = 91 days = .25

s = Volatility = .2

Dividend1 = 2

We will not include Dividend2 as it is occuring after the maturity of the option. It will not have any effect on stock option.

d1 = (ln (S/K) + (r + s2/2) * t) / s * t.5

d2 = d1 - s * t.5

d1 =( ln (75/70) + (.09 + .22/2) * .25 ) / .2 * .25.5

d1 = .965499

d2 = d1 - s * t.5 = .96492 - .2 *.25.5 = .865636

N(d1)= .8329 (It is Cummulative distribution function)

N(d2)= .8067 (It is Cummulative distribution function)

Present value of D1 =2 * e-.09*.25 = 1.9555

Call Option =((75 - 1.9555) * .8329) - (70 * .8067 * e-.09*.25) =(60.83876 - 55.2126) = 5.626

Call Option = 5.626

b Part:

Put Option = N(-d2)*Ke-rt -S* N(-d1)

N(-d2) = .5534

N(-d1) = .5138

Put Option = (.5534 * 70 * e-.09 * .25 ) - ((75 - 1.9555) * .5138)

Put Option = 37.8761 - 37.5302 = .3459

Put Option = .3459

c Part:

New call option when there are no dividends:

N(d1)= .8329 (It is Cummulative distribution function)

N(d2)= .8067 (It is Cummulative distribution function)

Call Option =((75 * .8329) - (70 * .8067 * e-.09*.25) =(62.4675 - 55.2126)

Call Option = 7.254

Change in Call value : 7.254 - 5.626 = 1.6288

D part:

If the volatility will increase the Call Price will also increase as both are directly related. We can measure it with Vega. Vega is the measurement of an option's sensitivity to changes in the volatility of the underlying asset

If the risk free rate decreases then the value of the call option will also decrease as these are also directly related. We can measure it with Rho.