Final answer:
To calculate the value of a call option, the Black-Scholes model can be used. the value of a call option with an exercise price of $44 per share is $36.73 and the value of a call option with an exercise price of $69 per share is $29.96.
Step-by-step explanation:
To calculate the value of a call option, we use the Black-Scholes model. The formula to calculate the value of a call option is: Call Value = (Current Stock Price * N(d1)) - (Exercise Price * e^(-r*T) * N(d2))
Where:
- d1 = (ln(S/X) + (r + (σ^2)/2) * T) / (σ * sqrt(T))
- d2 = d1 - σ * sqrt(T)
- N(d1) and N(d2) are the cumulative probability distribution function of the standard normal distribution.
- S = Current Stock Price
- X = Exercise Price
- r = Risk-free Interest Rate
- T = Time to Expiration
- σ = Standard Deviation of the Stock Price Returns
In the first scenario, where the exercise price is $44 per share, and the current stock price is $71, let's assume a risk-free interest rate of 4% and a standard deviation of 0.4.
Plugging in the values into the Black-Scholes formula:
d1 = (ln(71/44) + (0.04 + (0.4^2)/2) * 1) / (0.4 * sqrt(1)) = 0.9654
d2 = 0.9654 - 0.4 * sqrt(1) = 0.5654
N(d1) = 0.8329
N(d2) = 0.7190
Call Value = (71 * 0.8329) - (44 * e^(-0.04 * 1) * 0.7190) = $36.7
Therefore, the value of the call option with an exercise price of $44 per share is $36.73.In the second scenario, where the exercise price is $69 per share, and the current stock price is $71, we use the same interest rate and standard deviation.
Plugging in the values into the Black-Scholes formula:
d1 = (ln(71/69) + (0.04 + (0.4^2)/2) * 1) / (0.4 * sqrt(1)) = 0.2534
d2 = 0.2534 - 0.4 * sqrt(1) = -0.1466
N(d1) = 0.5996
N(d2) = 0.4376
Call Value = (71 * 0.5996) - (69 * e^(-0.04 * 1) * 0.4376) = $29.96
Therefore, the value of the call option with an exercise price of $69 per share is $29.96.