Question

# Consider a six-month European call option on a non-dividend-paying stock. The stock price is \$30, the...

Consider a six-month European call option on a non-dividend-paying stock. The stock price is \$30, the strike price is \$29, and the continuously compounded risk-free interest rate is 6% per annum. The volatility of the stock price is 20% per annum. What is price of the call option according to the Black-Schole-Merton model? Please provide you answer in the unit of dollar, to the nearest cent, but without the dollar sign (for example, if your answer is \$1.02, write 1.02).

We can use The Black-Scholes Model call option formula

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

Where,

C = call value =?

S = current stock price =\$30

N = cumulative standard normal probability distribution

t = days until expiration = 6 months = 0.5 years

Standard deviation σ = 20% = 0.20

K = option exercise price = \$29

r = risk free interest rate = 6% = 0.06

Formula to calculate d1 and d2 are -

d1 = {ln (S/K) +(r+ σ^2 /2)* t}/σ *√t

= {ln (30/29) + (0.06 + (0.20^2)/2) * 0.5} / 0.20 * √0.5

= 0.52256

d2 = d1 – σ *√t = 0.52256 – 0.20 *√0.5 = 0.38114

Now putting the value in the above formula

C = 30 * N (0.52256) – N (0.38114)* 29 * e^ (-0.06*0.5)

= 30* 0.69936 – 0.64845*29 * 0.9704 = 2.73

Price of call option is \$ 2.73

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