Suppose a stock is currently trading for $12,and in one period it will either increase to $15 or decrease to $8.If the one-period risk-free rate is 4%,what is the price of a European call option that expires in one period and has an exercise price of $7?
A)$4.68
B)$4.50
C)$5.27
D)$5.00
E)$7.00

Question

Quizplus · Accepted Answer

To calculate the price of the European call option, we need to find the probability-weighted average of the option's payoffs. 
The probability of the stock increasing to $15 is 0.5 (50%), and the probability of decreasing to $8 is also 0.5.

We can find the present value of both payoffs using the risk-free rate of 4%:

PV($15) = $15 / (1 + 0.04) = $14.42
PV($8) = $8 / (1 + 0.04) = $7.69

Using the Black-Scholes formula, the call option price is:

C = e^(-rT)[N(d1)S - N(d2)K]

where:
r = risk-free rate = 4%
T = time to expiration = 1 period
S = current stock price = $12
K = exercise price = $7
d1 = [ln(S/K) + (r + σ^2/2)T] / [σ√T]
d2 = d1 - σ√T
σ = volatility (not given)

However, since we are not given the volatility, we can use a shortcut known as the "probability method" to find the call option price:

C = e^(-rT)[(probability of stock price increasing) x (PV of stock price increase payoff) + (probability of stock price decreasing) x (PV of stock price decrease payoff) - PV of exercise price]

C = e^(-0.04 x 1)[(0.5) x ($14.42 - $7) + (0.5) x ($0 - $7) - $0]

C = $5.27

Therefore, the correct answer is choice C, $5.27.

Suppose a Stock Is Currently Trading for $12,and in One