Consider a call option with a strike price of $20, which expires in one year. The risk-free rate of interest is 5 percent. The underlying stock price is $30. Without arbitrage, which of the following is a possible price for the call option? (Round intermediate computations to two decimal places.)

Question

Quizplus · Accepted Answer

Using the Black-Scholes model, the call option's price can be calculated as: 
C = S*(N(d1)) - X*exp(-rT)*(N(d2)) 
where: 
S = 30 (underlying stock price) 
X = 20 (strike price) 
T = 1 (time to expiration) 
r = 0.05 (risk-free rate) 
d1 = [ln(S/X) + (r + σ^2/2)*T] / (σ*sqrt(T)) = [ln(30/20) + (0.05 + σ^2/2)*1] / (σ*sqrt(1)) 
d2 = d1 - σ*sqrt(T) = [ln(30/20) + (0.05 + σ^2/2)*1] / (σ*sqrt(1)) - σ*sqrt(1)

Using a trial and error method, we can find that when σ = 0.5, d1 = 0.8740 and d2 = 0.2276 
Plugging in all the values, we get: 
C = 30*(0.7991) - 20*exp(-0.05*1)*(0.5491) = $15.16

Therefore, the only possible price for the call option without arbitrage is $15.

Consider a Call Option with a Strike Price of $20