Let's solve the problem step by step: ### **Given Data** - **Dividend, \( D \) = \$2** (in 2 months) - **Current stock price, \( S_0 \) = \$64** - **Strike price, \( K \) = \$58** - **Time to expiration, \( T \) = 3 months = 0.25 years** - **Risk-free rate, \( r \) = 0.7% per month = 0.007 per month** - **Volatility, \( \sigma \) = 8% per month = 0.08 per month** - **European call option (but asked for American; for non-dividend stocks, American = European, but with dividends, early exercise may be optimal).** - **Dividend paid in 2 months (i.e., \( t_D = 2/12 = 1/6 \) years)** We are told to use the Black-Scholes model (which is for European options, but for American calls on dividend-paying stocks, the value is usually the same as the European value **unless early exercise is optimal**. But with a single dividend, we can use Black-Scholes with *adjusted spot price*.) --- ### **Step 1: Adjust Stock Price for Dividend** The ex-dividend date is in 2 months, before the option expires. In Black-Scholes for a discrete dividend, we **subtract the present value of the dividend from the spot price**: \[ PV(D) = D \cdot e^{-r t_D} \] Where \( t_D = 2 \) months \( = \frac{2}{12} = 0.1667 \) years, \( r = 0.007 \) per month \( = 0.007 \times 12 = 0.084 \) per year. But since we're working in months, let's keep all time in months: - \( r \) per month = 0.007 - \( t_D = 2 \) months \[ PV(D) = 2 \cdot e^{-0.007 \times 2} = 2 \cdot e^{-0.014} \] Calculate: \[ e^{-0.014} \approx 0.9861 \] \[ PV(D) = 2 \cdot 0.9861 \approx 1.9722 \] **Adjusted stock price:** \[ S_0^* = S_0 - PV(D) = 64 - 1.9722 = 62.0278 \] --- ### **Step 2: Black-Scholes Parameters** - \( S_0^* = 62.0278 \) - \( K = 58 \) - \( r = 0.007 \) per month \( = 0.021 \) for 3 months - \( \sigma = 0.08 \) per month - \( T = 3 \) months Convert \( T \) to years: \( T = 3/12 = 0.25 \) years. But since all rates are per month, it's easier to keep everything in months and \( T = 3 \). --- #### **Let’s use per-month Black-Scholes Formula:** \[ d_1 = \frac{\ln(\frac{S_0^*}{K}) + (r + \frac{1}{2}\sigma^2)T}{\sigma\sqrt{T}} \] \[ d_2 = d_1 - \sigma\sqrt{T} \] Where: - \( S_0^* = 62.0278 \) - \( K = 58 \) - \( r = 0.007 \) per month - \( \sigma = 0.08 \) per month - \( T = 3 \) months Calculate: \[ \ln(\frac{62.0278}{58}) = \ln(1.06939) = 0.06709 \] \[ rT = 0.007 \times 3 = 0.021 \] \[ \frac{1}{2} \sigma^2 T = 0.5 \times (0.08)^2 \times 3 = 0.5 \times 0.0064 \times 3 = 0.5 \times 0.0192 = 0.0096 \] \[ \sigma\sqrt{T} = 0.08 \times \sqrt{3} = 0.08 \times 1.73205 = 0.13856 \] So: \[ d_1 = \frac{0.06709 + 0.021 + 0.0096}{0.13856} = \frac{0.09769}{0.13856} = 0.705 \] \[ d_2 = 0.705 - 0.13856 = 0.566 \] --- ### **Step 3: Black-Scholes Formula for Call Option** \[ C = S_0^* N(d_1) - K e^{-r T} N(d_2) \] First, find \( N(d_1) \) and \( N(d_2) \): - \( d_1 = 0.705 \) - \( d_2 = 0.566 \) Using standard normal tables (or calculator): \[ N(0.705) \approx 0.7605 \] \[ N(0.566) \approx 0.7146 \] Calculate \( K e^{-rT} \): \[ e^{-rT} = e^{-0.007 \times 3} = e^{-0.021} \approx 0.9792 \] \[ K e^{-rT} = 58 \times 0.9792 = 56.7936 \] Now plug in the values: \[ C = 62.0278 \times 0.7605 - 56.7936 \times 0.7146 \] \[ 62.0278 \times 0.7605 = 47.176 \] \[ 56.7936 \times 0.7146 = 40.597 \] So, \[ C = 47.176 - 40.597 = 6.579 \] --- ### **Step 4: Conclusion** **The Black-Scholes value of the American call option is approximately**: \[ \boxed{6.58} \] (Rounded to 2 decimal places.) --- ## **Final Answer** \[ \boxed{6.58} \] *(This is the fair value of the call option, in dollars, using the Black-Scholes model for a European call. For an American call on a dividend-paying stock, this is a good approximation unless early exercise is optimal, but for a single dividend and these parameters, this is typically the upper bound.)*