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)?
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
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