Question

Consider an option on a non dividend paying stock when the stock price is $35, the exercise price is $31, the risk free interest rate is 4% per annum, the volatility is 20% per annum, and the time to maturity is four months.

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 an American call?

c. What is the price of the option if it is a European put?

d. Verify that put call parity holds

Answer 1

A) We will use Black Scholes Merton options valuation model

Value of European Call option = SN(d1) – e^-rt*XN(d2)

Where d1 = [ln(S/X) + (r + volatility^2/2)T] / volatility*T^0.5

And d2 = d1 - volatility*T^0.5

d1 = [ln(35/31) + (0.04 + 0.20^2/2)*4/12] / 0.20*(4/12)^0.5

= 1.224220933

Using Excel, the value of N(d1) is calculated as =Norm.dist(d1,0,1,True)

N(d1)= 0.889565553

d2 = d1 - volatility*T^0.5

d2 = 1.224220933 - 0.2*(4/12)^0.5

d2= 1.108750879

N(d2)= 0.866231167

Value of European Call option = 35*0.889565553 - e^-0.04*4/12 * 31*0.866231167 = 4.637294019

B) Since it is not a dividend paying stock, it's not worth exercising it early,

Hence value of American call option with same strike price = $4.637294019

C) Value of European Put option = e^-rt*XN(-d2) - SN(-d1)

Where d1 = [ln(S/X) + (r + volatility^2/2)T] / volatility*T^0.5

And d2 = d1 - volatility*T^0.5

d1 = [ln(35/31) + (0.04 + 0.20^2/2)*4/12] / 0.20*(4/12)^0.5

= 1.224220933

N(-d1)= 0.110434

d2 = 1.224220933 - 0.2*(4/12)^0.5

d2 = 1.108750879

N(-d2)= 0.133769

Value of European Put option = e^-0.04*4/12 * 31*0.133769 - 35*0.110434 = 0.226725

D) Put- Call parity :

Value of call option + Exercise price*e^-rt = Value of put option + stock price

4.637294019 + 31*0.986755162 = 0.226725 + 35

35.2267 = 35.2267

Since both are equal, put call parity does hold here.