1. **Problem statement:** An investor holds 200,000 shares of ZP limited, current share price $S_0=60$. She wants to buy European put options with exercise price $K=50$, maturity $T=2$ years, risk-free rate $r=0.12$ per annum, volatility $\sigma=0.30$. We need to find: (i) The total cost of buying 200,000 put options. (ii) The change in wealth if after 1 year the share price is $S=30$. --- 2. **Calculate the price of one European put option using the Black-Scholes formula:** The Black-Scholes formula for a European put option is: $$P = Ke^{-rT}N(-d_2) - S_0 N(-d_1)$$ where $$d_1 = \frac{\ln(\frac{S_0}{K}) + (r + \frac{\sigma^2}{2})T}{\sigma \sqrt{T}}$$ $$d_2 = d_1 - \sigma \sqrt{T}$$ and $N(\cdot)$ is the cumulative distribution function of the standard normal distribution. --- 3. **Calculate $d_1$ and $d_2$:** $$\ln(\frac{60}{50}) = \ln(1.2) \approx 0.1823$$ $$d_1 = \frac{0.1823 + (0.12 + \frac{0.3^2}{2}) \times 2}{0.3 \times \sqrt{2}} = \frac{0.1823 + (0.12 + 0.045) \times 2}{0.3 \times 1.4142} = \frac{0.1823 + 0.33}{0.4243} = \frac{0.5123}{0.4243} \approx 1.2077$$ $$d_2 = 1.2077 - 0.3 \times 1.4142 = 1.2077 - 0.4243 = 0.7834$$ --- 4. **Find $N(-d_1)$ and $N(-d_2)$:** Using standard normal tables or a calculator: $$N(-d_1) = N(-1.2077) \approx 0.113$$ $$N(-d_2) = N(-0.7834) \approx 0.217$$ --- 5. **Calculate the put option price $P$:** $$P = 50 \times e^{-0.12 \times 2} \times 0.217 - 60 \times 0.113$$ Calculate discount factor: $$e^{-0.24} \approx 0.7866$$ So, $$P = 50 \times 0.7866 \times 0.217 - 60 \times 0.113 = 8.53 - 6.78 = 1.75$$ Price per put option is approximately 1.75. --- 6. **Calculate total cost for 200,000 put options:** $$\text{Total cost} = 200,000 \times 1.75 = 350,000$$ --- 7. **Calculate change in wealth if share price after 1 year is 30:** The investor holds 200,000 shares worth $30$ each, so value of shares: $$200,000 \times 30 = 6,000,000$$ The put option payoff at $S=30$ (exercise price $K=50$) is: $$\max(K - S, 0) = 50 - 30 = 20$$ Value of 200,000 put options at maturity: $$200,000 \times 20 = 4,000,000$$ Since the options mature in 2 years but the price is given at 1 year, the value at 1 year is not the full payoff but the intrinsic value plus time value. However, for simplicity, assuming the investor exercises or sells the options at 1 year at intrinsic value: Change in wealth from buying puts: $$\text{Gain from puts} - \text{Cost of puts} = 4,000,000 - 350,000 = 3,650,000$$ The shares have lost value from $12,000,000$ (200,000 x 60) to $6,000,000$, a loss of $6,000,000$. Net change in wealth: $$-6,000,000 + 3,650,000 = -2,350,000$$ So the investor's wealth decreases by 2,350,000 but the put options reduce the loss significantly. --- **Final answers:** (i) The investor pays approximately 350,000 for 200,000 put options. (ii) If the share price falls to 30 after 1 year, the investor's net wealth decreases by approximately 2,350,000 after accounting for the put options.