Question

A stock’s current price S is $100. Its return has a volatility of s = 25 percent per year. European call and put options trading on the stock have a strike price of K = $105 and mature after T = 0.5 years. The continuously compounded risk-free interest rate r is 5 percent per year. The Black-Scholes-Merton model gives the price of the European call as:

please provide explanation

Answer 1

The Black-Scholes-Merton model; European call option formula

C = SN (d1) - N (d2) Ke ^ (-rt)

Where

C = the price of the European call option =?

S = current stock price =$100

N = cumulative standard normal probability distribution

t = days until expiration = 0.5 years

Standard deviation, s = 25% per year = 0.25

K = option strike price = $105

r = risk free interest rate = 5% per year = 0.05

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ s^2 /2)* t}/s *√t

= {ln (100/105) + (0.05 + (0.25^2)/2) * 0.5} / 0.25 * √0.5

= 0.22981

d2 = d1 – s *√t = 0.22981 – 0.25 * √0.5 = 0.05303

Now putting the value in the above formula

C = $100 * N (0.22981) – N (0.05303)* $105 * e^ (-0.05*0.5)   [Note: use cumulative standard normal                   probability distribution table to calculate the value of N]

C = $100 * 0.59088 – 0.52115 * $105 * e^ (-0.05*0.5)  

= $8.67

Price of the call option is $8.67