Question

a. A share of ARB stock sells for $80 and has a standard deviation of returns equal to 20% per year. The current risk-free rate is 9% 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 91-day European-style call option with an exercise price of $72. Use the modifyed model that assumes the dividend yield is paid continuously. Do not round intermediate calculations. Round your answer to the nearest cent.

b. What would be the price of a 91-day European-style put option on ARB stock having the same exercise price? Do not round intermediate calculations. Round your answer to the nearest cent.

c. 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. Do not round intermediate calculations. Round your answer to the nearest cent. The price would (increase/decrease) by $____

d. 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 35%, and (2) the risk-free rate decreases to 7%.

A increase in the volatility to 35% would (decrease/increase) the call's value. A decrease in the risk-free rate to 7% would (decrease/increase) the call's value.

Answer 1

ARB Stock-
Current Price S₀ = $80
Standard Deviation $\sigma$ = 20%
Risk free rate r = 9%
Time to expiry T = 91 days = 0.25 years (as explained in the question in dividend portion)
Exercise Price K = 72

Dividends :
1. $2, just before expiry - 0.25 years from now
2. $2 - 0.5 years from now

Note - 2nd dividend has no impact on the pricing of the option as it is paid beyond the expiry of option.

Quarterly compounded Dividend Yield = (Quarterly $ dividend/Current Stock Price)*(No. of quarters in an year) = (2/78)*4 = 0.1026 = 10.26%

Continuously Compounded Dividend Yield = n*(LN(1 + n-period compounded dividend yield/n))
where,
n = compounding frequency in an year ( n = 4 for quarterly compounding)
LN --> log base e function

Hence,
q = 4*(LN(1 + 10.26%/4))
q = 4*LN(1.025641)
q = 4*0.025318
q = 0.101271
q = 10.13%

Black-Scholes price of the European Options-
c = S₀e^-qT*N(d₁) - K*e^-rT*N(d₂)
p = K*e^-rT*N(-d₂) - S₀e^-qT*N(-d₁)

Where,
d₁ = (ln(S0/k) + (r - q + σ²/2)*T)/σ(√Δt)
d₂ = (ln(S0/k) + (r - q - σ²/2)*T)/σ(√Δt) = d₁ - σ(√Δt)
N is cumulative Normal Distribution with mean 0 and standard deviation 1.
S0 - current stock price
q - dividend yield
K - exercise price
r - risk free interest rate
T - time to expiry

using the formula given above, value of CALL and PUT option in excel can be calculated. Formulas in Excel are given below -

0.09 1 So 21 30 Time T 10.2 0.25 0.1013 9 di 10 d2 11 N(d1) 12 N(02) 13 N(-d1) 14 N(-d2) 15 C 16 p =(LN(B1/B5) + (B2-B6 +B3*B3*0.5) *B4)/(B3*SQRT(B4)) =B9-B3*SQRT(B4) =NORM.DIST(B9,0,1, TRUE) =NORM.DIST(B10,0,1,TRUE) =NORM.DIST(-B9,0,1,TRUE) =NORM.DIST(-810,0,1,TRUE) =B1*EXP(-86*B4) *B11 - B5*EXP(-B2*B4) *B12 =B5*EXP(-B2*B4) *B14-B1*EXP(-B6*B4) *B13 17

Values in the excel will look like below -

9% ДА 1 So 21 30 4 Time T 5 K 6 a 20% 0.25 10.13% 7 8 9 di 10 d2 11 N(D1) 12 N(D2) 13 N-d1) 14 N(-d2) 15 C 16 p 17 1.08 0.98 0.86 0.84 0.14 0.16 8.19 0.59

Part A:
Value of European CALL = $8.19

Part B:
Value of European PUT = $0.59

Part C:
No dividend payment means q = 0
Rest all formula values and the formula itself remain same.

Formulas in Excel will look like below -

0.09 0.2 0.25 72 1 So 21 30 4 Time T 5 K 69 7 8 9 di 10 d2 11 N(dl) 12 N(d2) 13 N(-d1) 14 N(-d2) =(LN(B1/B5) +(B2-B6 +B3*B3*0.5) *B4)/(B3*SQRT(B4)) =B9-B3*SQRT(B4) =NORM.DIST(B9,0,1,TRUE) =NORM.DIST(B10,0,1,TRUE) =NORM.DIST(-B9,0,1, TRUE) =NORM.DIST(-B10,0,1,TRUE) =B1*EXP(-B6*B4) *B11 - B5*EXP(-B2*B4) *B12 = B5*EXP(-B2*B4) *B14-B1*EXP(-B6*B4) *B13 15 C p 16 17

Values in excel will look like below -

В 9% 20% 0.25 0.00% 1 So 2 30 4 Time T 5 K 6q 71 8 9 di 10 d2 11 N(dl) 12 N(02) 13 N(-d1) 14 N(-02) 15 C 16 p 17 1.33 1.23 0.91 0.89 0.09 0.11 9.96 0.36

New value of the Call Option = 9.96
Increase in price = New CALL Price - Old CALL Price
Increase in Price = 9.96 - 8.19 = 1.77

Ans: CALL Option price would increase by $1.77 in case of no dividends.

Part D:

(1) Effect of change in volatility in Option Prices -
It is very simple to understand that Options provide protection against adverse movement in stock prices, which also means it is a protection against volatility on the stock prices. So, option's value will increase if volatility increases.and Options value will decrease if volatility decreases.

So, increase in volatility from 20% to 35% would increase the CALL option price from $8.19 to some higher value.

(2) Effect of change in interest rate in Option Prices -
Assuming all other factors remain constant, Increase in risk free interest will lead to increase in CALL Option price but decrease in PUT Option price. Decrease in risk free interest rate will lead to decrease in in CALL Option price but increase in PUT Option price.

Increase in interest rate provides interest advantage to the CALL Option holder. Suppose, if instead of CALL Option, option holder has to borrow money to buy stock immediately, his cost of borrowing would increase in case of increase in interest rate. Because CALL option is providing safety from this rise in interest rate, CALL option will become more valuable if interest rate rise. Converse is true for the PUT Option.

So, decrease in Risk Free Rate from 9% to 7% would decrease the CALL option price from $8.19 to some lower value.