Question

Problem 4.2 (15.30 in Hull) Consider an option on a non-dividend-paying stock when the stock price is $30, the exercise price is $29, the risk-free interest rate is 5%, the volatility is 25% per annum, and the time to maturity is 4 months. (a) what is the price of the option is it is a European call? (b) what is the price of the option if it is an American call? (c) what is the price of the option if it is European put? (d) verify that put-call parity holds

Answer 1

a

As per Black Scholes Model
Value of call option = (S)*N(d1)-N(d2)*K*e^(-r*t)
Where
S = Current price =		30
t = time to expiry =		0.33333
K = Strike price =		29
r = Risk free rate =		5.0%
q = Dividend Yield =		0%
σ = Std dev =		25%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(30/29)+(0.05-0+0.25^2/2)*0.33333)/(0.25*0.33333^(1/2))
d1 = 0.422516
d2 = d1-σ*t^(1/2)
d2 =0.422516-0.25*0.33333^(1/2)
d2 = 0.278179
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.663676
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.609563
Value of call= 30*0.663676-0.609563*29*e^(-0.05*0.33333)
Value of call= 2.53

b

As per Black Scholes Model
Value of put option = N(-d2)*K*e^(-r*t)-(S)*N(-d1)
Where
S = Current price =		30
t = time to expiry =		0.33333
K = Strike price =		29
r = Risk free rate =		5.0%
q = Dividend Yield =		0%
σ = Std dev =		25%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(30/29)+(0.05-0+0.25^2/2)*0.33333)/(0.25*0.33333^(1/2))
d1 = 0.422516
d2 = d1-σ*t^(1/2)
d2 =0.422516-0.25*0.33333^(1/2)
d2 = 0.278179
N(-d1) = Cumulative standard normal dist. of -d1
N(-d1) =0.336324
N(-d2) = Cumulative standard normal dist. of -d2
N(-d2) =0.390437
Value of put= 0.390437*29*e^(-0.05*0.33333)-30*0.336324
Value of put= 1.05

c

As per put call parity

Call price + PV of exercise price = Spot price + Put price

2.53+29*e^(-0.05*0.33333)=30+Put value

Put value = 1.05