Question

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% per annum, the volatility is 25% per annum, and the time to maturity is four months.

1. What is the price of the option if it is a European call?
2. What is the price of the option if it is an American call?
3. What is the price of the option if it is a European put?
4. Verify that put-call parity holds. (Hint: Substitute the European put and call prices into the put call parity equation and demonstrate that the right hand side of the equation is equal to the left hand side of the equation.)

Answer 1

In the above question applying Black-Scholes Model

S_o = $30, K= $29, r= 5%, T = 4/12 , σ = 0.25

d₁ = ((r + σ²/2)*T + ln(S_o/K))/ σ √ T

d₁ = ((0.05+0.25²/2)*(4/12) + ln(30/29))/ (0.25*(4/12))

d₁ = 0.4225

d₂ = ((r + σ²/2)*T + ln(S_o/K))/ σ √ T

d₂ = ((0.05-0.25²/2)*(4/12) + ln(30/29))/ (0.25*(4/12))

d₂ = 0.2782

here Standard Normal denoted by N, the value can be find out from Cumulative Normal Distribution Table

N(0.4225) = 0.6637

N(0.2782) = 0.6096

N(-0.4225) = 0.3363

N(-0.2782) = 0.3904

a. The European Call Price formula is

C = S_o*N(d1) − Ke^-rTN(d₂)

C = 30*0.6637 - (29*e^-0.05*(4/12))*0.6096

C = 19.911 - 17.386

C = $2.524

b. The American Call Price will be similar to European Call Price which is $2.524

c. The price of European Put Option is

P = Ke^-rTN(-d₂) - S_o*N(-d1)

P= (29*e^-0.05*(4/12))*0.3904 - 30*0.3363

P = 11.1344 - 10.089

P = $1.045

d. The Put Call Parity formula is

P+S_o = C+Ke^-rT

1.045+30 = 2.524 + 29*e^-0.05*(4/12)

The Right hand side is equal to Left Hand Side , so Put Call Parity holds true.