Given a risk-free rate of 2%, a volatility of 30% and a strike price of $100, what is the price of a call option on a stock currently priced at $105 with 6 months to expiration?

To solve this problem, we’ll need to use the Black-Scholes formula for pricing European call options. The formula is as follows:

C(S, K, T, r, σ) = SN(d₁) − Ke^− rTN(d₂)

Where:

- C is the price of the call option

- S is the current stock price

- K is the strike price of the call option

- T is the time to expiration (in years)

- r is the risk-free rate (annualized)

- σ is the volatility of the underlying stock (annualized)

- N(x) is the cumulative normal distribution function

- d₁ and d₂ are given by the following formulas:

$$d_1 = \frac{1}{\sigma \sqrt{T}} \left[\ln\left(\frac{S}{K}\right) + \left(r + \frac{\sigma^2}{2}\right)T \right]$$

$$d_2 = d_1 - \sigma \sqrt{T}$$

Now, let’s plug in the given values and compute the price of the call option:

- S = 105

- K = 100

- T = 0.5 (6 months)

- r = 0.02

- σ = 0.3

First, calculate d₁ and d₂:

$$d_1 = \frac{1}{0.3 \sqrt{0.5}} \left[\ln\left(\frac{105}{100}\right) + \left(0.02 + \frac{0.3^2}{2}\right)0.5 \right] \approx 0.6979$$

$$d_2 = 0.6979 - 0.3 \sqrt{0.5} \approx 0.4565$$

Next, compute the values of N(d₁) and N(d₂) using the cumulative normal distribution function:

N(d₁) = N(0.6979) ≈ 0.7580

N(d₂) = N(0.4565) ≈ 0.6760

Now we can find the price of the call option using the Black-Scholes formula:

C(105, 100, 0.5, 0.02, 0.3) = 105 × 0.7580 − 100e^{− 0.02(0.5)} × 0.6760 ≈ 16.6655

Hence, the price of the call option is approximately $16.67.