Question

6) Consider an option on a non-dividend paying stock when the stock price is $38, the exercise price is $40, the risk-free interest rate is 6% per annum, the volatility is 30% per annum, and the time to maturity is six months. Using Black-Scholes Model, calculating manually, a. What is the price of the option if it is a European call? b. What is the price of the option if it is a European put? c. Show that the put-call parity condition holds (or does not hold)?

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 =		38
t = time to expiry =		0.5
K = Strike price =		40
r = Risk free rate =		6.0%
q = Dividend Yield =		0.00%
σ = Std dev =		30%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(38/40)+(0.06-0+0.3^2/2)*0.5)/(0.3*0.5^(1/2))
d1 = 0.005688
d2 = d1-σ*t^(1/2)
d2 =0.005688-0.3*0.5^(1/2)
d2 = -0.206444
N(d1) = Cumulative standard normal dist. of d1
N(d1) =0.502269
N(d2) = Cumulative standard normal dist. of d2
N(d2) =0.418222
Value of call= 38*0.502269-0.418222*40*e^(-0.06*0.5)
Value of call= 2.85

b

As per Black Scholes Model
Value of put option = N(-d2)*K*e^(-r*t)-S*N(-d1)
Where
S = Current price =		38
t = time to expiry =		0.5
K = Strike price =		40
r = Risk free rate =		6.0%
q = Dividend Yield =		0.00%
σ = Std dev =		30%
d1 = (ln(S/K)+(r-q+σ^2/2)*t)/(σ*t^(1/2)
d1 = (ln(38/40)+(0.06-0+0.3^2/2)*0.5)/(0.3*0.5^(1/2))
d1 = 0.005688
d2 = d1-σ*t^(1/2)
d2 =0.005688-0.3*0.5^(1/2)
d2 = -0.206444
N(-d1) = Cumulative standard normal dist. of -d1
N(-d1) =0.497731
N(-d2) = Cumulative standard normal dist. of -d2
N(-d2) =0.581778
Value of put= 0.581778*40*e^(-0.06*0.5)-38*0.497731
Value of put= 3.67

c

As per put call parity

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

2.85+40*e^(-0.06*0.5)=38+3.67

2.85+38.817=38+3.67

41.67=41.67

Thus put call parity holds