Let's carefully work through each part of the problem step-by-step. --- ### Given Data: | Parameter | Value | |--------------------------------|------------------------------| | Current stock price \( S_0 \) | \$55 | | Strike price \( K \) | \$60 | | Risk-free rate \( r \) (annual) | 4% (0.04) | | Volatility \( \sigma \) | 30% (0.30) | | Time to maturity \( T \) | 1 year | | Dividend payment at \( t=1 \) | \$1 | --- ## Part A: Find the price of the European put option using Black-Scholes formula ### Step 1: Adjust the stock price for dividends Since the dividend is paid at \( t=1 \) (at maturity), the stock price will drop by the dividend amount at that time. In the standard Black-Scholes model, dividends paid at maturity are often incorporated by adjusting the stock price or by subtracting the present value of the dividend from the stock price. But since the dividend is paid exactly at maturity, the stock price at \( T \) will effectively be: \[ S_T^{\text{effective}} = S_T - D \] where \( D = \$1 \). Alternatively, we can use the **dividend-adjusted stock price**: \[ S_0^{\text{adj}} = S_0 - D e^{-r T} = 55 - 1 \times e^{-0.04 \times 1} \] Calculate: \[ e^{-0.04} \approx 0.9608 \] So, \[ S_0^{\text{adj}} = 55 - 1 \times 0.9608 = 55 - 0.9608 = 54.0392 \] ### Step 2: Compute \( d_1 \) and \( d_2 \) The Black-Scholes formulas for options with dividends (assuming continuous dividend yield \( q=0 \), but here, dividends are discrete at maturity) are modified, but since the dividend is paid at maturity, we can approximate by subtracting its present value from the current stock price as above, and then treat the problem as a standard European option on the adjusted stock price. The formulas: \[ d_1 = \frac{\ln(S_0^{\text{adj}}/K) + (r + \frac{\sigma^2}{2}) T}{\sigma \sqrt{T}} \] \[ d_2 = d_1 - \sigma \sqrt{T} \] Calculate: \[ \ln(54.0392/60) = \ln(0.9007) \approx -0.1042 \] \[ r + \frac{\sigma^2}{2} = 0.04 + \frac{0.3^2}{2} = 0.04 + \frac{0.09}{2} = 0.04 + 0.045 = 0.085 \] Now, numerator for \( d_1 \): \[ -0.1042 + 0.085 \times 1 = -0.1042 + 0.085 = -0.0192 \] Denominator: \[ \sigma \sqrt{T} = 0.3 \times 1 = 0.3 \] Calculate: \[ d_1 = \frac{-0.0192}{0.3} \approx -0.064 \] Similarly, \[ d_2 = d_1 - 0.3 = -0.064 - 0.3 = -0.364 \] ### Step 3: Find \( N(d_1) \) and \( N(d_2) \) Using standard normal distribution: \[ N(d_1) \approx N(-0.064) \approx 0.474 \] (since \( N(0) = 0.5 \), and for small negative, approximately 0.474) \[ N(d_2) \approx N(-0.364) \approx 0.357 \] ### Step 4: Calculate the put option price The Black-Scholes formula for a European put: \[ P = K e^{-r T} N(-d_2) - S_0^{\text{adj}} N(-d_1) \] Calculate: \[ K e^{-r T} = 60 \times e^{-0.04} \approx 60 \times 0.9608 = 57.648 \] \[ N(-d_2) = 1 - N(d_2) = 1 - 0.357 = 0.643 \] \[ N(-d_1) = 1 - N(d_1) = 1 - 0.474 = 0.526 \] Finally, \[ P = 57.648 \times 0.643 - 54.0392 \times 0.526 \] Calculate: \[ 57.648 \times 0.643 \approx 37.095 \] \[ 54.0392 \times 0.526 \approx 28.461 \] So, \[ \boxed{ \text{Put Price} \approx 37.095 - 28.461 = \boxed{\$8.63} } \] **Answer for Part A:** **The approximate price of the European put option is \$8.63.** --- ## Part B: What is the price change if the stock price increases to \$6? ### New stock price: \[ S_0' = \$6 \] Repeat the calculations: ### Step 1: Adjusted stock price: \[ S_0^{\text{adj}}' = 6 - 1 \times e^{-0.04} = 6 - 0.9608 = 5.0392 \] ### Step 2: Compute \( d_1' \) and \( d_2' \) \[ \ln(5.0392/60) = \ln(0.083987) \approx -2.476 \] Calculate numerator: \[ -2.476 + 0.085 \times 1 = -2.476 + 0.085 = -2.391 \] Denominator remains: \[ \sigma \sqrt{T} = 0.3 \] Calculate: \[ d_1' = \frac{-2.391}{0.3} \approx -7.97 \] \[ d_2' = d_1' - 0.3 \approx -8.27 \] ### Step 3: Find \( N(d_1') \) and \( N(d_2') \) Since these are large negative numbers: \[ N(d_1') \approx 0 \] \[ N(d_2') \approx 0 \] Similarly, \[ N(-d_1') \approx 1 \] \[ N(-d_2') \approx 1 \] ### Step 4: Calculate new put price: \[ P' = 57.648 \times N(-d_2') - 5.0392 \times N(-d_1') \] \[ P' = 57.648 \times 1 - 5.0392 \times 1 = 57.648 - 5.0392 = \$52.61 \] ### **Price change:** \[ \Delta P = P' - P \approx 52.61 - 8.63 = \boxed{\$43.98} \] --- ## **Final answers:** **A) The price of the European put option is approximately \(\boxed{\$8.63}\).** **B) If the stock price increases to \$6, the option's price increases by approximately \(\boxed{\$43.98}\).** --- ## **Summary & Explanation:** - The initial put price reflects the likelihood of the stock falling below the strike, considering dividends and volatility. - When the stock price increases, the put option becomes less valuable. However, in this case, the increase from \$55 to \$6 has a huge impact because the option is deep in-the-money, leading to a significant increase in its value (from about \$8.63 to over \$52). **Note:** The large change highlights the sensitivity of options prices to underlying asset prices, especially when the option is deep in-the-money.